STABILITY ANALYSIS OF PIPE RACKS FOR
INDUSTRIAL FACILITIES
By
David A. Nelson
B.S., Walla Walla University, 2008
A thesis submitted to
University of Colorado Denver
in partial fulfillment
of the requirements for the degree of
Master of Science, Civil Engineering
2012
This thesis for the Master of Science
degree by
David A. Nelson
has been approved
by
Fredrick Rutz
Kevin Rens
Rui Liu
Nelson, David A. (M.S., Civil Engineering)
Stability Analysis of Pipe Racks for Industrial Facilities
Thesis directed by Professor Fredrick Rutz
ABSTRACT
Pipe rack structures are used extensively throughout industrial facilities
worldwide. While stability analysis is required in pipe rack design per the AISC
Specification for Structural Steel Buildings (AISC 36010), the most compelling
reason for uniform application of stability analysis is more fundamental. Improper
application of stability analysis methods could lead to unconservative results and
potential instability in the structure jeopardizing the safety of not only the pipe rack
structure but the entire industrial facility.
The direct analysis method, effective length method and first order method are
methods of stability analysis that are specified by AISC 36010. Pipe rack structures
typically require moment frames in the transverse direction creating intrinsic
susceptibility to second order effects. This tendency for large second order effects
demands careful attention in stability analysis. Proper application as well as clear a
understanding of the limitations of each method is crucial for accurate pipe rack
design.
A comparison of the three AISC 36010 methods of stability analysis was
completed for a representative pipe rack structure using the 3D structural analysis
program STAAD.Pro V8i. For the model chosen, all three methods of stability
analysis met AISC 36010 requirements.
For typical pipe rack structures, all three methods of stability analysis are
acceptable as long as limitations are met and the methods are applied correctly. The
first order method typically provided conservative results while the effective length
method was determined to underestimate the moment demand in beams or
connections that resist column rotation. The direct analysis method was found to be a
powerful analysis tool as it requires no additional calculations to calculate additional
notional loads, calculate effective length factors or verify AISC 36010 limitations.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Fredrick Rutz
ACKNOWLEDGEMENT
I would like to thank first and foremost Dr. Fredrick Rutz for the support and
guidance in completion of this thesis. I would also like to thank Dr. Rens and Dr. Li
for participating on my graduate advisory committee. Lastly, I would like to thank
various work associates for their help with either editing or discussion of the topic.
TABLE OF CONTENTS
LIST OF FIGURES..................................................................ix
LIST OF TABLES...................................................................xii
Chapter
1. Introduction................................................................1
1.1 Stability Analysis of Steel Structures......................................1
1.2 Pipe Racks in Industrial Facilities.........................................3
2. Problem Statement...........................................................6
2.1 Introduction................................................................6
2.2 Significance of Research....................................................8
2.3 Research Obj ective.........................................................8
3. Literature Revi ew.........................................................10
3.1 Introduction...............................................................10
3.2 Pipe Rack Loading..........................................................10
3.2.1 Load Definitions......................................................10
3.2.2 Dead Loads............................................................14
3.2.3 Live Loads............................................................15
3.2.4 Thermal and Self Straining Loads......................................16
3.2.5 Snow Load and Rain Loads..............................................16
VI
.17
.21
.22
.25
.29
.29
.29
.34
.35
.37
.41
.43
.46
.47
.52
.58
.61
.63
.68
.68
.72
Wind Loads
Seismic Loads...........................
Load Combinations.......................
Column Failure and Euler Buckling............
Stability Analysis...........................
AISC Specification Requirements.........
Second Order Effects....................
Flexural, Shear and Axial Deformation...
Geometric Imperfections.................
Residual Stresses and Reduction in Stiffness
AISC Methods of Stability Analysis...........
Rigorous Second Order Elastic Analysis..
Approximate Second Order Elastic Analysis
Direct Analysis Method..................
Effective Length Method.................
First Order Method......................
Research Plan................................
Member Design................................
Pipe Rack Analysis...........................
Generalized Pipe Rack........................
Pipe Rack Loading............................
VII
6.3 Pipe Rack Load Combinations.............................................83
6.4 Strength and Serviceability Checks......................................86
6.5 Base Support Conditions.................................................87
6.6 Effective Length F actor................................................88
6.7 Notional Load Development for First Order Method........................91
6.8 STAAD Benchmark Validation..............................................93
7. Comparison of Results...................................................96
8. Conclusions............................................................Ill
References....................................................................116
Appendix A STAAD Input Pinned Base Analysis Effective Length Method.......123
Appendix B STAAD Input Pinned Base Analysis Direct Analysis Method........132
Appendix C STAAD Input Pinned Base Analysis First Order Method............141
viii
LIST OF FIGURES
Figure
1 1 Typical FourLevel Pipe Rack Consisting of Eight Transverse Frames
Connection by Longitudinal Struts.......................................4
2 1 Typical Elevation View of Pipe Rack......................................7
2 2 Section View Showing Moment Resisting Frame..............................7
3 1 Load vs. Deflection Yielding of Perfect Column........................26
32 Visual Definition of Critical Buckling Load Pcr.........................27
33 Load vs. Deflection Euler Buckling....................................28
34 Second Order P8 and PA moments (Adapted from Ziemian, 2010)........30
35 Comparison of First Order Analysis to Second Order Analysis (Adapted from
Gerschwindner, 2009)..........................................................31
36 Quebec Bridge Prior to Collapse (Canada, 1919)..........................32
37 Quebec Bridge after Failure (Canada, 1919)..............................33
38 Deformation from Flexure, Shear and Axial (Adapted from Gerschwindner,
2009)..................................................................35
39 Load vs. Deflection Real Column Behavior with Initial Imperfections..37
310 Residual Stress Patterns in Hot Rolled Wide Flange Shapes...............38
IX
311 Influence of Residual Stress on Average StressStrain Curve (Salmon and
Johnson, 2008)...........................................................39
312 Idealized Residual Stresses for Wide Flange Shape Members Lehigh Pattern
(Adapted from Ziemian, 2010).............................................40
313 Load vs. Deflection Comparison of Analysis Types (Adapted from White
and Hajjar, 1991)........................................................42
314 Visual Representation of IncrementalIterative Solution Procedure (Adapted
from Ziemian, 2010)......................................................45
315 Reduced Modulus Relationship (Powell, 2010)..............................52
316 Alignment Chart Sidesway Inhibited (Braced Frame) (Adapted from AISC
36010).........................................................................54
317 Alignment Chart Sidesway Uninhibited (Moment Frame)
(Adapted from AISC 36010)...............................................55
51 Simple Cantilever Design Example.........................................64
5 2 Simple Cantilever Design Example Results.................................66
6 1 Isometric View of Typical Pipe Rack Structure Used for Analysis..........70
62 Section View of Moment Frame in Typical Pipe Rack.......................71
63 Section View of Moment Frame Operating Dead Load......................74
64 Section View of Moment Frame Pipe Anchor Load.........................77
x
65 Section View of Moment Frame Wind Load..............................81
66 Section View of Moment Frame Operating Seismic......................83
67 Effective Length Factor K Pinned Base...............................90
68 Effective Length Factor K Fixed Base................................91
69 AISC Benchmark Problems (Adapted from AISC 36010).......................94
XI
LIST OF TABLES
Table
31 Force coefficient, Cf for open structures trussed towers (Adapted from ASCE
705)...................................................................18
32 Cf force coefficient (Adapted from ASCE 705) ............................20
33 Comparison of direct analysis method and equivalent length method (Adapted
from Nair, 2009)........................................................57
61 Velocity pressure for cable tray and structural members...................78
62 Velocity pressure for pipe................................................79
63 Resultant design wind force from pipe.....................................80
64 Lateral seismic forces operating........................................82
65 Lateral seismic forces empty............................................82
6 6 Benchmark solutions.......................................................95
7 1 Ratio A2/A1 effective length method pinned base.........................97
72 Ratio A2/Ai direct analysis method pinned base..........................99
73 Maximum demand to capacity ratio pinned base...........................100
74 Maximum demand forces pinned base......................................101
75 Ratio A2/Ai effective length method fixed base.........................103
xii
76 Ratio A2/A1 direct analysis method fixed base...........................104
77 Maximum demand to capacity ratio fixed base.............................105
78 Maximum demand forces fixed base........................................105
79 Ratio A2/Ai effective length method pinned base serviceability limits...107
710 Ratio A2/Ai direct analysis method pinned base serviceability limits....108
711 Maximum demand to capacity ratio pinned base serviceability limits......109
xiii
1.
Introduction
1.1 Stability Analysis of Steel Structures
The engineering knowledge base continues to grow and expand. This growth
creates ongoing challenges as designs demand adaptation in response to new
information and technology. Although the value of stability analysis has long been
recognized, implementation in design has historically been difficult as calculations
were performed primarily by hand. Various methods were created to simplify the
analysis and allow the engineer to partially include the effects of stability via hand
calculations. However, with the development of powerful analysis software, rigorous
methods to account for stability effects were developed. While stability analysis
calculations can still be done by hand, most engineers now have access to software
that will complete a rigorous stability analysis. The majority of the methods
presented here assume that software analysis is utilized.
Stability analysis is a broad term that covers many aspects of the design
process. According to the 2010 AISC Specification for Structural Steel Buildings
(AISC 36010) stability analysis shall consider the influence of second order effects
(PA and P8 effects), flexural, shear and axial deformations, geometric
imperfections, and member stiffness reduction due to residual stresses.
1
Both the 2005 and 2010 AISC Specification for Structural Steel Buildings
recognize at least three methods for stability analysis: (AISC 36005 and AISC 360
10)
1. FirstOrder Analysi s Method
2. Effective Length Method
3. Direct Analysis Method
Other methods for analysis may be used as long as all elements addressed in the
prescribed methods are considered.
Stability analysis is required for all steel structures according to AISC 36010.
The application of methods for stability analysis in design of structures varies greatly
from firm to firm and from engineer to engineer. A crucial principle for engineers in
the process of design is the inclusion of stability analysis in design. If stability
analysis is not performed or a method of analysis is incorrectly applied, the ability of
the structure to support the required load is potentially jeopardized. The analysis of
nearly all complex structures is completed using advanced analysis software capable
of performing various methods of analysis. Therefore omitting stability analysis in
the design of structures creates unnecessary risk and is unjustified.
2
1.2 Pipe Racks in Industrial Facilities
Pipe racks are structures used in various types of plants to support pipes and
cable trays. Although pipe racks are considered nonbuilding structures, they should
still be designed with the effects of stability analysis considered.
Pipe racks are typically long, narrow structures that carry pipe in the
longitudinal direction. Figure 11 shows a typical pipe rack used in an industrial
facility. Pipe routing, maintenance access, and access corridors typically require that
the transverse frames are momentresisting frames. The moment frames resist
gravity loads as well as lateral loads from either pipe loads or wind and seismic loads.
The transverse frames are typically connected using longitudinal struts with one bay
typically braced. Any longitudinal loads are transferred to the longitudinal struts and
carried to the braced bay. (Drake and Walter, 2010)
3
Figure 11 Typical FourLevel Pipe Rack Consisting of Eight Transverse Frames Connection by
Longitudinal Struts
Pipe racks are essential for the operation of industrial facilities but because
pipe racks are considered nonbuilding structures, code referenced documents will
usually not cover the design and analysis of the structure. The lack of industry
standards for pipe rack design leads to each individual firm or organization adopting
its own standards, many without clear understanding of the concepts and design of
pipe rack structures. (Bendapodi, 2010) Process Industry Practices Structural Design
Criteria (PIP STC01015) has tried to develop a uniform standard for design but it
should be noted that this is not considered a code document.
4
The lack of code referenced documents can lead to confusion in the design of
pipe racks. The concept of stability analysis should not be ignored based the lack on
code referenced documents AISC 36010 should still be used as reference for stability
analysis and design.
5
2.
Problem Statement
2.1 Introduction
Industrial facilities typically have pipes and utilities running throughout the
plant which require large and lengthy pipe racks. Pipe racks not only are used for
carrying pipes and cable trays, but many times defines access corridors or roadways.
It is relatively easy to add a braced bay in the longitudinal direction of a pipe rack
because pipes and utilities run parallel to access roads. It is much more difficult to
add bracing to the pipe rack in the transverse direction because of the potential for
interference with pipes, utilities, corridors and access roads. Therefore moment
connections in the transverse direction of the pipe rack are typically used. Figure 21
shows an elevation view of a length of pipe rack. Figure 22 shows a section view of
the same pipe rack showing the moment resisting frame.
Pipe racks are a good example of structures that can be subject to large second
order effects. The current AISC 36010 defines three methods for stability analysis:
1. First Order Analysis Method
2. Effective Length Method
3. Direct Analysis Method
Limitations restrict practical application for certain methods.
6
Figure 21 Typical Elevation View of Pipe Rack
Figure 22 Section View Showing Moment Resisting Frame
7
2.2 Significance of Research
If stability analysis is not performed or a method is incorrectly applied, this
could jeopardize the ability of the structure to support the required loads.
Most of the current literature on pipe racks discusses the application of loads
and has suggestions on design and layout of pipe racks, while little applicable
information is available on comparing the three methods of stability analysis for pipe
racks. Currently the design engineer must research each method of stability analysis
and decide which method to apply for analysis. After the analysis is completed, the
engineer must then verify that the pipe rack meets all the requirements of the applied
analysis method. If the requirements of AISC 36010 methods are not met for the
structure, then the engineer must completely reanalyze the structure using a new
method of stability analysis which will meet the requirements. Comparing the
various types of stability analysis will not only show the engineer which method will
provide the most accurate analysis based on method limitations, but will also show
why stability analysis is crucial.
2.3 Research Objective
The main purpose of this thesis will be to analyze various types of pipe rack
structures, compare the results from stability analyses, and describe both positive and
negative aspects of each method of stability analysis as it applies specifically to pipe
8
rack structures. The paper will also look at some of the various issues with applying
each of the methods.
Some engineers are accustomed to braced frames structures, which are not
susceptible to large second order effects, therefore those designers can tend to neglect
or incorrectly apply methods of stability analysis. This thesis will not only show the
importance of stability analysis, but also provide suggestions on practical
implementation of each method. This could potentially save time in analysis and
design because the process of selecting the appropriate stability analysis method will
no longer be based on trial and error but rather on educated considerations that can
easily be verified after analysis.
9
3.
Literature Review
3.1 Introduction
This section will focus on review of the available literature on the subject of
both pipe rack loading as well as stability analysis. Literature on the general theory
of stability analysis will be reviewed. The main focus of this literature review will be
on the three methods prescribed by AISC 36010. Layout and loading guidelines for
pipe racks will also be reviewed as this has a major influence on stability.
3.2 Pipe Rack Loading
3.2.1 Load Definitions
Pipe racks are unique structures that have unique loading when compared to
typical buildings and structure. Pipe racks design is not covered under Minimum
Design Loads for Buildings and Other Structures (ASCE 705) or International
Building Code (IBC 2009) however the design philosophies should remain the same
as that for all structures. Most company design criteria and Process Industry Practices
(PIP) documents will list ASCE 705 or IBC as the basis for load definition and load
combinations. There are several primary loads which should be considered in the
design of pipe racks in addition to loads defined by ASCE 705 or IBC 2009. ASCE
705 primary load cases are as follows:
10
Ak = load or load effect arising from extraordinary event A
D = dead load
D; = weight of ice
E = earthquake load
F = load due to fluids with well defined pressures and maximum heights
Fa = flood load
H = load due to lateral earth pressure, ground water pressure, or pressure of
bulk materials
L = live load
Lr= roof live load
R = rain load
S = snow load
T = selfstraining force
W = wind load
W; = windonice determined in accordance with ASCE 705 Chapter 10
According to AISC 36010, regardless of the method of analysis,
consideration of notional loads is required. The notional loads may be required in all
load combinations if certain requirements of the stability analysis are not satisfied.
11
The magnitude of notional load will vary based on the method used. Therefore the
additional primary load cases per AISC 36010 are as follows:
N = notional load per AISC, applied in the direction that provides the
greatest destabilizing effect
PIP STC01015 states that pipe racks shall be designed to resist the minimum
loads defined in ASCE 705 as well as the additional loads described therein. PIP
STC01015 breaks down the dead load into various categories that are not defined in
ASCE 705. In addition, various loads from plant operation are defined and required
for consideration in design.
PIP STC01015 breaks down the ASCE 705 Dead Load (D) by dividing the
dead load into the subcategories listed below.
Ds = Structure dead load is the weight of materials forming the structure
(not the empty weight of process equipment, vessels, tanks, piping nor
cable trays), foundation, soil above the foundation resisting uplift, and
all permanently attached appurtenances (e.g., lighting, instrumentation,
HVAC, sprinkler and deluge systems, fireproofing, and insulation,
etc...).
Df = Erection dead load is the fabricated weight of process equipment or
vessels.
12
De = Empty dead load is the empty weight of process equipment, vessels,
tanks, piping, and cable trays.
D0 = Operating dead load is the empty weight of process equipment,
vessels, tanks, piping and cable trays plus the maximum weight of
contents (fluid load) during normal operation.
Dt = Test dead load is the empty weight of process equipment, vessels,
tanks, and/or piping plus the weight of the test medium contained in
the system.
PIP STC01015 also provides additional primary load cases from the effects of
thermal loads caused from operational temperatures in the pipes.
T = Selfstraining thermal forces caused by restrained expansion of
horizontal vessels, heat exchangers, and structural members in pipe
racks or in structures. This is essentially the same load case as defined
in ASCE 705.
Af = Pipe anchor and guide forces.
Ff = Pipe rack friction forces cause by the sliding of pipes or friction forces
cause by the sliding of horizontal vessels or heat exchanges on their
supports, in response to thermal expansion.
13
Seismic loads are also discussed in PIP STC01015. Seismic events can occur
either when the plant is in operation or during shutdown when the pipes are empty.
Therefore two seismic load cases are defined as follows:
E0 = Earthquake load considering the unfactored operating dead load and
the applicable portion of the unfactored structure dead load.
Ee = Earthquake load considering the unfactored empty dead load and the
applicable portion of the unfactored structure dead load.
3 .2.2 Dead Loads
Further information on the dead loads specifically for pipe racks is defined in
PIP STC01015. The operating dead load for piping on a pipe rack shall be 40 psf
uniformly distributed over each pipe level. The 40 psf load is equivalent to 8 inch
diameter, schedule 40 pipes, full of water, at 15 inch spacing. For pipes larger than 8
inch, the actual load of pipe and contents shall be calculated and applied as a
concentrated load.
The empty dead load (De) is defined for checking uplift and minimum load
conditions. Empty dead load (De) is approximately 60% of the operating dead load
(D0) which is equivalent to 24 psf uniformly distributed over each pipe rack level.
This is an acceptable approximation unless calculations indicate a different
percentage should be used. (PIP STC01015)
14
Pipe racks for industrial applications are usually designed with consideration
for potential future expansion. Therefore, additional space or an additional level
should be provided and the rack should be uniformly loaded across the entire width to
account for pipes that may be placed there in the future.
Cable trays are often supported on pipe racks and typically occupy a level
within the rack specifically designated for cable tray. The operating dead load (D0)
for cable tray levels on pipe racks shall be 20 psf for a single level of cable tray and
40 psf for a double level. These uniform loads are based on estimates of full cable
tray over the area of load application. (PIP STC01015)
The degree of usage for cable trays can vary greatly. The empty dead load
(De) should be considered on a case by case basis. Engineering judgment should be
used in defining the cable tray loading, because empty dead load (De) is defined for
checking uplift and minimum load conditions.
3 .2.3 Live Loads
Live load should be applied to pipe racks as needed. Pipe racks typically have
very few platform or catwalks. When platforms are required for access to valves or
equipment located on the pipe rack structure, the platform and supporting structure
should be designed in accordance with ASCE 705 Live Loads.
15
3.2.4 Thermal and Self Straining Loads
Temperature effects on structural steel members should be included in design.
PIP STC01015 introduces two additional self straining loads. These additional loads
are caused by the operation effects on the pipes. The operational temperatures of
pipes need to be considered in design.
Support conditions of pipes vary greatly and need to be considered in design.
A pipe may be supported to resist gravity only, or may have varying degrees of
restraint from guided in a single direction to fully anchored supports. Pipe stress
analysis can be completed for all the pipes located in the pipe rack. This stress
analysis takes into account the support type and location for each support and
provides individual design forces for each pipe at that specific location. These
resultant pipe loads can be used for design. However, application of loads in this
manner does not include additional loads for futures expansion. Therefore, a uniform
load at each level of the rack is typically applied in lieu of actual pipe forces. Local
support condition should also be verified where large anchor forces are present.
3.2.5 Snow Load and Rain Loads
Snow loading should be considered in the design of pipe racks. Pipe racks
typically do not have roofs or solid surfaces that large amounts of snow can collect
on, therefore the engineer may reduce the snow load by a percentage using
engineering judgment based on percentage of solid area and operational temperatures
16
of pipes. Based on the reduced area for snow to accumulate, snow load combinations
will usually not control the design of pipe racks. (Drake, Walter, 2010)
Rain loads are intended for roofs where rain can accumulate. Because pipe
racks typically have no solid surfaces where rain can collect, rain load usually does
not need to be considered in design of pipe racks. (Drake, Walter, 2010)
3 2 6 Wind Loads
ASCE 705 provides very little, if any guidance for application of wind load
for pipe racks. The most appropriate application would be to assume the pipe rack is
an open structure and design the structure assuming a design philosophy similar to
that of a trussed tower. See Table 31 below for Cf, force coefficient. This method
requires the engineer to calculate the ratio of solid area to gross area of one tower face
for the segment under consideration. This may become very tedious for pipe rack
structures because each face can have varying ratios of solids to gross areas.
17
Table 31 Force coefficient, Cf for open structures trussed towers (Adapted from ASCE 705)
Tower Cross Section cf
Square 4.0e25.9e+4.0
Triangle 3.4e24.7e+3.4
Notes:
1. For all wind directions considered, the area Af consistent with the specified force
coefficients shall be the solid area of a tower face projected on the plane of that face for
the tower segment under consideration.
2. The specified force coefficients are for towers with structural angles or similar flat
sided members.
3. For towers containing rounded member, it is acceptable to multiply the specified force
coefficients by the following factor when determining wind forces on such members:
0.51e2+5.7, but not > 1.0
4. Wind forces shall be applied in the directions resulting in maximum member forces and
reactions. For towers with square crosssections, wind forces shall be multiplied by the
following factor when the wind is directed along a tower diagonal: l+0.75e, but not >1.2
5. Wind forces on tower appurtenances such as ladders, conduits, lights, elevators, etc.,
shall be calculated using appropriate force coefficients for these elements.
6. Loads due to ice accretion as described in Section 11 shall be accounted for.
7. Notation:
e: ratio of solid area to gross area of one tower face for the segment under consideration.
The method generally used for pipe rack wind load application comes from
Wind Loads for Petrochemical and Other Industrial Facilities (ASCE, 2011). This
report provides an approach for wind loading based on current practices, internal
company standards, published documents and the work of related organizations.
18
Design wind force is defined as: (ASCE 705 Eqn 5.1)
F = qz*G* Cf*A
With:
qz = Velocity pressure determined from ASCE 705 Section 6.5.10
G = Gust effect factor determined from ASCE 705 Section 6.5.8
Cf is defined as the force coefficient and varies based on the shape and
direction of wind. Structural members can have force coefficients between 1.5 and 2.
Cf can be taken as 1.8 for all structural members or equal to 2 at and below the first
level and 1.6 above the first level. No shielding shall be considered. Cf for pipes
should be 0.7 as a minimum. Cf for cable should be taken as 2.0. (ASCE, 2011)
These values of Cf are developed based on the Table 32 below. Cable tray are
considered square in shape with h/D = 25 corresponding to Cf = 2.0. Pipe are round
in shape with h/D = 25 and a moderately smooth surface corresponding to Cf = 0.7.
19
Table 32 Cf force coefficient (Adapted from ASCE 705)
CrossSection Type of Surface h/D
1 7 25
Square (wind normal to face) All 1.3 1.4 2
Square (wind along diagonal) All 1 1.1 1.5
Hexagonal or octagonal All 1 1.2 1.4
Round (D^/qz > 2.5) Moderately smooth 0.5 0.6 0.7
Rough (D'/D = 0.02) 0.7 0.8 0.9
Very rough (D'/D = 0.08) 0.8 1 1.2
Round (D^/qz < 2.5) All 0.7 0.8 1.2
Notes:
1. The design wind force shall be calculated based on the area of the structure projected on a plane
normal to the wind direction. The force shall be assumed to act parallel to the wind direction.
2. Linear interpolation is permitted for h/D vales other than shown.
3. Notation:
D: Diameter of circular crosssection and least horizontal dimension of square, hexagonal or
octagonal crosssection at elevation under consideration in feet
D': Depth of protruding elements such as ribs and spoilers, in feet
h: Height of structure, in feet
qz: Velocity pressure evaluated at height z above ground, in pounds per square foot
The tributary area (A) for pipes is based on the diameter of the largest pipe
(D) plus 10% of the width of the pipe rack (W), then multiplied by the length of the
pipes (L) (usually the spacing of the bent frames). The tributary area for pipes is the
projected area of the pipes based on wind in the direction perpendicular to the length
of pipe. Wind load parallel to pipe is typically not considered in design since there is
typically very little projected area of pipe for applying wind pressure. (ASCE, 2011)
20
A = L(D+0.1W)
The tributary area takes into account the effects of shielding on the leeward
pipes or cable tray. The 10% of width of pipe rack is added to account for the drag of
pipe or cable tray behind the first windward pipe. It is based on the assumption that
wind will strike at an angle horizontal with a slope of 1 to 10 and that the largest pipe
is on the windward side. (ASCE, 2011)
The tributary area for structural steel members and other attachments should
be based on the projected area of the object perpendicular to the direction of the wind.
Because the structural members are typically spaced at greater distances than pipes,
no shielding effects should be considered on structural members and the full wind
pressures should be applied to each structural member.
The gust effect factor G, and the velocity pressure qz, should be determined
based on ASCE 705 sections referenced above.
3 .2.7 Seismic Loads
Pipe racks are typically considered nonbuilding structures, therefore seismic
design should be carried out in accordance with ASCE 705, Chapter 15. A few
slight variations from ASCE 705 are recommended. The operating earthquake load
E0 is developed based on the operating dead load as part of the effective seismic
weight. The empty earthquake load Ee is developed based on the empty dead load as
part of the effective seismic weight. (Drake and Walter, 2010)
21
The operating earthquake load and the empty earthquake load are discussed in
more detail in the load combinations for pipe racks. Primary loads, E0 and Ee are
developed and used in separate load combinations to envelope the seismic design of
the pipe rack.
ASCE Guidelines for Seismic Evaluation and Design of Petrochemical
Facilities (1997) also provides further guidance and information on seismic design of
pipe racks. The ASCE guideline is however based on the 1994 Ciniform Building
Code (UBC) which has been superseded in most states by ASCE 705 or ASCE 710.
Therefore the ASCE guideline should be considered as a reference document and not
a design guideline.
3 .2.8 Load Combinations
Based on the inclusion of additional primary load cases as specified by PIP
STC01015, additional load combinations need to be considered. PIP STC01015
specifies load combinations to be used for pipe rack design. Both LRFD and ASD
load combinations are specified. LRDF load combinations will be the focus of this
section as AISC LRFD will be used for analysis and design. ASD load combinations
should be considered when checking serviceability limits on pipe racks.
Because additional primary load cases are included in the design and ASCE 7
05 does not govern the design of pipe racks because they are typically considered
nonbuilding structures, PIP STC01015 load combinations should be used. In
22
practice, PIP STC01015 load combinations and ASCE 7050 load combinations are
very similar and a combination of the specified load combinations can be used.
ASCE 705 primary load cases must be redefined with the additional subcategories of
loads defined by the general primary load cases. Example: Dead load as defined by
ASCE 705 needs to be broken down into additional primary load cases such as the
dead load of the structure, the dead load of the empty pipe, etc...
PIP STC01015 LRFD load combinations specified for pipe racks are listed
below:
1. 1.4(Ds+D0+FffT+Af)
2. 1 2(DS+D0+ Af)+(1,6W or 1,0Eo)
3. 0.9(Ds+De)+1.6W
4. a) 0.9(Ds+Do)+1.2Affl.0Eo
b) 0.9(Ds+De)+1.0Ee
5. 1.4(DS+Dt)
6. 1.2(DS+Dt)+1.6WP
ASCE 705 LRFD load combinations are listed below:
1. 1.4(D+F)
2. 1.2(D+F+T)+1.6(L+H)+0.5(Lr or S or R)
3. 1.2D+1.6(Lr or S or R)+(L or 0.8W)
4. 1.2D+1.6W+L+0.5(Lr or S or R)
5. 1.2D+1.0E+L+0.2S
23
6. 0.9D+1.6W+1.6H
7. 0.9D+1.0E+1.6H
When comparing the two sets of load combinations, there are some
similarities. Certain primary loads such as live load, live roof load, snow load and
rain load do not typically apply or control the design of the pipe racks, therefore most
load combinations with these primary load cases will not control the design.
Therefore ASCE 705 load combination 2 and 3 will not be considered in design.
Taking into account the subcategories of primary load cases used in PIP STC01015,
ASCE 705 load combinations can be compared directly and a comprehensive list of
all load combinations can be developed.
Below is listed the combined load combinations to be used in this research for
design of pipe racks referenced from PIP STC01015. Reference of specific load
combination number from ASCE 705 is also included if applicable.
1. 1.4(DS+D0+Ff+T+Af) ASCE 1
2. 1.2(DS+D0+Af)+( 1.6W or 1,0Eo) ASCE 4 and 5
3. 0.9(Ds+De)+1.6W ASCE 6
4. a) 0.9(Ds+Do)+1.2Af+1.0Eo ASCE 7
b) 0.9(Ds+De)+1.0Ee
5. 1.4(DS+Dt)
6. 1.2(DS+Dt)+1.6WP
24
It can be seen in the above load combinations that ASCE 705 load
combinations 1, 4, 5, 6 and 7 are covered by the PIP STC01015 load combinations.
Slight changes such as the inclusion of Af are added to load combinations per the
direction of PIP STC01015. Additional load combinations to cover test load
conditions, partial wind, Wp, during test and seismic on the empty condition are
covered by PIP. Engineering judgment should be used to determine if any additional
load combinations should be considered in design.
ASD load combinations from both PIP STC01015 and ASCE705 are
combined in a similar fashion to come up with a combined list of load combinations
used for design.
3.3 Column Failure and Euler Buckling
An ideal column is considered to be perfectly straight with the load applied
directly through the centroid of the cross section. Theoretically the load on an ideal
column can increase until the limit state occurs by yielding or rupture. Figure 31
shows a graph of axial load P vs lateral deflection y. The axial load is increased
until yielding occurs with no lateral deflection.
25
p
Yield
A*Fy
(>
1  y
Figure 31 Load vs. Deflection Yielding of Perfect Column
For slender columns, this yielding is never reached. The axial load is
increased to a point of critical loading where the column is on the verge of becoming
unstable. The critical load is determined as the point where, if a small lateral load (F)
were applied at the midspan of the column, the column would remain in the
deflected position even after the lateral load was removed. Any additional load will
cause further lateral displacement. This is shown in Figure 32
26
Figure 32 Visual Definition of Critical Buckling Load Pcr
This critical load for slender columns is based on Euler buckling. Euler
buckling load is the theoretical maximum load that an ideal pin ended column can
support without buckling. (Euler, 1744) It is stated as:
Pcr = Euler Buckling Load or Critical Buckling Load
L = Length of Column
E = Modulus of Elasticity
I = Moment of Inertia of Column
Bifurcation is the point when the column is in a state of neutral equilibrium as
the critical buckling load is applied to the column. At the point of bifurcation, the
column is on the verge of buckling. Instead of the graph shown in Figure 31, the
27
graph now is shown in Figure 33. The load is increased to the critical load where the
column becomes unstable and buckling can occur. (Hibbeler, 2005)
P
Buckling
y
Figure 33 Load vs. Deflection Euler Buckling
Eulers formula for critical load was derived based on the assumption of an
ideal column. However, ideal columns do not exist. The load is never applied
directly through the centroid and the column is never perfectly straight. The
existence of load eccentricities, out of plumb members, member geometric
imperfections, material flaws, residual stresses, therefore second order effects become
the basis for stability analysis. Based on the discussion above, most real columns will
never suddenly buckle but will slowly bend due to the eccentricities and out of
straightness. (Hibbeler, 2005)
28
3.4 Stability Analysis
3.4.1 AISC Specification Requirements
AISC 36010 Specification for Structural Steel Building states in section Cl.
Stability shall be provided for the structure as a whole and for
each of its elements. The effects of all of the following on the
stability of the structure and its elements shall be considered: (1)
flexural, shear and axial member deformations, and all other
deformations that contribute to displacements of the structure;
(2) secondorder effects (both PA and P8 effects); (3) geometric
imperfections; (4) stiffness reductions due to inelasticity; and (5)
uncertainty in stiffness and strength. All loaddependant effects
shall be calculated at a level of loading corresponding to LRFD
load combinations of 1.6 times ASD load combinations.
3.4.2 Second Order Effects
Second order effects are a means to account for the increase in forces based on
the deformed shape of the member or frame. Second order effects can be further
broken down to P8 and PA effects. PA effects are the effects of loads acting on the
displaced location of joints or nodes in a structure. P8 effects are the effects of loads
acting on the deflected shape of a member between joints or nodes. Figure 34b
shows the effects of P8 and Figure 34a shows the effects of PA. (AISC 36010)
(Ziemian, 2010)
29
(a) Sway Permitted
(a) Sway Restrained
Figure 34 Second Order P6 and PA Moments (Ziemian, 2010)
Second order effects are typically the driving factor for stability analysis. All
the other requirements for stability design are implemented based on the effect that
they will have on the second order effects. The importance of second order effects
becomes readily apparent when comparing an example frame using first order elastic
analysis and a second order elastic analysis as shown in Figure 35. Lateral
displacements and as a result, member forces, can be drastically underestimated if a
first order analysis is performed.
30
Second Order Analysis Results
0 4 8 12 16 20
Lateral Displacement (in.)
Figure 35 Comparison of First Order Analysis to Second Order Analysis
(Adapted from Gerschwindner, 2009)
Second order analysis can be carried out per AISC 36010 methods of either
rigorous second order analysis or the approximate second order analysis, both of
which are discussed in more detail in further sections.
The importance of second order effects on overall stability of structure can be
seen in various structural failures over the years. For example, in 1907, the Quebec
Bridge collapsed during construction due to failure of the compression chords of the
31
truss. In the weeks previous to the collapse, deflections in the chords was noticed and
reported but nothing was done and work continued until the eventual collapse of the
bridge. While many factors led to the failure of the bridge, one of the errors made in
the design of the bridge was in the design of the compression chords. The
compression chords were fabricated slightly curved for aesthetic reasons. However,
because this member curvature complicated the calculations, the members were
designed as straight members. Therefore, the actual second order effects were
increased and the buckling capacity was reduced when compared to the straight
member as designed. (Delatte, 2009) As can be seen from Figure 34 and 35, the
second order effects can increase both the displacement and moments to failure much
before Euler buckling load is ever reached. Figure 36 shows the Quebec Bridge
prior to collapse and Figure 37 shows the result of the failure.
Figure 36 Quebec Bridge Prior to Collapse (Canada, 1919)
32
Figure 37 Quebec Bridge after Failure (Canada, 1919)
Some general points concerning second order analysis made by Ziemian
(2010) are as follows:
1. Second order behavior can affect all components and internal member
forces within a structure.
2. Second order moments do not necessarily have the same distribution as
the first order moments and therefore the first order moments cannot be
simply amplified. LeMessurier (1977) and Kanchanalai and Lu (1979)
however give several practical applications where amplification of first
order moments to achieve an approximation of second order moments is
applicable.
33
3. All structures will experience both P8 and PA effects but the magnitude
of the effects will vary greatly between structures.
4. Linear superposition of effects cannot be used with second order analysis;
the response is nonlinear.
3.4.3 Flexural, Shear and Axial Deformation
Second order effects are required in stability analysis as explained in the
previous section. Accurate deformation must be calculated because second order
effects are based on deformation to determine the amplified moments and forces.
The AISC 36010 requires that flexural, shear and axial deformations be
considered in the design. Although flexural deformations will usually be the largest
contributor to overall structural deformation, axial and shear deformations should not
be ignored. Figure 38 shows a frame and the calculated deformations from flexural,
shear and axial. If shear and axial deformations were ignored in the design and
analysis, the amplifications of moments and forces from second order effects could be
underestimated and the structure could become unstable.
34
Elastic Analysis
Aflex+Aaxial = 2.42 in.
Aflex+Aaxial+Ashear = 2.94 in.
Figure 38 Deformation from Flexure, Shear and Axial (Adapted from Geschwindner,
2009)
3.4.4 Geometric Imperfections
Geometric imperfections refer to the outofstraightness, outofplumbness,
material imperfections and fabrication imperfections. The maximum allowable
geometric imperfections are set by AISC Code of Standard Practice for Steel Building
and Bridges. The main geometric imperfections of concern in stability analysis are
member outofstraightness and frame outofplumbness. Member outofstraightness
is limited to L/1000, where L is the member length between brace or frame points.
35
W36X230
Frame outofplumbness is limited to H/500, where H is the story height. (AISC 360
10) (AISC 30310)
Geometric imperfections cause eccentricities for axial loads in the structure
and members. These eccentricities need to be accounted for in stability analysis
because they can cause destabilizing effects and increased moments. Eccentricities
also increase the second order effects in the analysis of the structure. Maximum
tolerances specified by AISC Code of Standard Practice for Steel Buildings and
Bridges should be assumed for analysis unless actual imperfection values are known.
(AISC 36010)
Columns are never perfectly straight and contain either member outof
straightness or general outofplumbness, therefore an initial eccentricity of the axial
load will be experienced. Figure 39 shows how the real column will behave with an
initial imperfection (y0). Nominal column capacity (Pn) will be reached well before
Euler Buckling load (Pcr). Second order effects are included in this graph of axial
load vs. deformation. Based on the presence of the initial displacement, moment will
develop and the column will typically yield based on flexure and compression while
the theoretical Euler buckling will never be reached.
36
p
Figure 39 Load vs. Deflection Real Column Behavior with Initial Imperfections
3.4.5 Residual Stresses and Reduction in Stiffness
Residual stresses are internal stresses contained in a structural steel member.
There are several sources of residual stresses: (Salmon and Johnson, 2008)
1. Uneven cooling after hot rolling of the structural member.
2. Cold bending or cambering during fabrication.
3. Punching holes or cutting during fabrication.
4. Welding.
Uneven cooling and welding typically produce the largest residual stresses in
a member. Local welding for connections does produce residual stresses but the
presence of these stresses tend to be localized and are not considered in overall
column or beam design strength. Residual stress from uneven cooling happens when
the rolled shape is cooled at room temperature from the rolling temperatures. Certain
37
areas of the member will cool more rapidly than others. For example, the flange tips
of a wide flange shape are surrounded by air on three sides and cool more rapidly
than the material at the junction of the flange and web. As the flange tips cool, they
can contract freely because the other regions have yet to develop axial stiffness.
When the slower cooling sections begin to cool and contract, the axial stiffness from
the cooled regions restrains the contraction thus creating compression on the faster
cooling section and tension in the slower cooling sections. Figure 310 shows the
residual stresses typically seen in hot rolled wide flange shapes from uneven cooling.
(Vinnakota, 2006) (Huber and Beedle, 1954) (Yang et al., 1952)
Light Deep Section
Heavy Shallow Section
Figure 310 Residual Stress Patterns in Hot Rolled Wide Flange Shapes
Welding of built up sections produces residual stresses as a result of the
localized heating applied during the welding. A built up wide flange shape will have
38
compression on the flange tips and middle of the web and tension around the junction
of the web and flange. (Vinnakota, 2006)
The presence of residual stresses results in a nonlinear behavior of the stress
strain curve. The average yield stress of the section is reduced by the amount of
residual stress in the member. Therefore the section will start to yield before the
stress reaches the theoretical yield stress of a member with no residual stress. Linear
elastic behavior is experienced to the point of theoretical yield stress (Fy) minus the
residual stress (see Figure 311). After this point, nonlinear behavior is experienced
and plasticity begins to spread through the section. (Salmon and Johnson, 2008)
Average Compressive Strain Ã‚Â£
Figure 311 Influence of Residual Stress on Average StressStrain Curve (Salmon and
Johnson, 2008)
39
Residual stresses need to be considered in stability analysis because of the
effect of general softening of the structure from the spread of plasticity through the
cross section causing reduced stiffness. The reduced stiffness increases deflections
and therefore increases the second order effects on the structure. (AISC 36010)
Beam and column design strength is calculated based on empirical equations
which take into account the residual stress which is assumed to follow a Lehigh
pattern which is a linear variation across the flanges and uniform tension in the web.
The AISC 36010 strength equations were developed and calibrated based on
research from Kanchanalai (1977) and ASCE Task Committee (1997). Figure 312
shows the idealized residual stresses for typical wide flange sections which follows
the Lehigh pattern. The residual stresses are assumed to be 0.3Fy in wide flange
shapes. (AISC 36010) (Ziemian, 2010) (Deierlien and White, 1998)
fr = 0.3Fy
+ fr
Figure 312 Idealized Residual Stresses for Wide Flange Shape Members Lehigh
Pattern (Adapted from Ziemian, 2010)
40
3.5 AISC Methods of Stability Analysis
As discussed before, AISC 36010 states that any method that considers the
influence of secondorder effects, flexural, shear and axial deformation, geometric
imperfections, and member stiffness reduction due to residual stresses on the stability
of the structure and its elements is permitted.
Various types of methods of have been developed and AISC 36010 detailed
the requirements for a few of these methods. Each method listed, does in some way,
address all the various requirements specified by AISC 36010. All methods listed in
AISC 36010, excluding the first order analysis method, require a second order
analysis. Table 22 from AISC 36010 provides a summary of requirements and
limitations of each of the methods. AISC 36010 allows two types of second order
analysis; Approximate Second Order Analysis and Rigorous Second Order Analysis.
Both methods of second order analysis either accurately account for or
approximate geometric nonlinear behavior. In reality, geometric nonlinear behavior
is only one of the nonlinear types of behavior that should be considered in design.
Material nonlinear behavior (inelastic analysis) caused by reduction in stiffness
should also be considered in design. Figure 313 shows the results from various types
of analyses.
41
Figure 313 Load vs. Deflection Comparison of Analysis Types (Adapted from White and
Hajjar, 1991)
While software is available that performs a true second order inelastic
analysis, it is very computationally expensive, and therefore other measures must be
considered in analysis and design. AISC 36010 strength equations are typically
based on the results from a second order elastic analysis. The AISC LRFD general
approach for strength and stability can be represented by the following equation:
Ey;*Qi < ORn
42
The left hand side of the formula represents the effects of factored loads on a
structural member, connection and the right side represents the design resistance or
design strength of the specified element with:
Qi = internal forces created by applied load
Rn = Nominal member or connection strength
Yi = factor to account for variability in load (load factor)
= factor to account for variability in resistance (strength reduction factor)
Geometric nonlinear behavior can be accounted for using a second order
elastic analysis (left side of the equation). These load effects are then compared to
resistance based on material and geometric inelasticity (right side of the equation).
(Yura el al., 1996) As Figure 313 shows, a direct comparison of load effects from
second order elastic analysis and member resistance is not compatible because the
inelastic material deflections are not considered in an elastic second order analysis.
Therefore AISC 36010 design equations should be calibrated for the results of an
elastic second order analysis or the effects of material inelasticity must be accounted
for in the elastic second order analysis. (Ziemian, 2010) This can be accomplished
through various methods discussed in more detail in further sections.
3.5.1 Rigorous Second Order Elastic Analysis
To fully capture the second order effects as described in previous sections,
nonlinear geometric behavior should be accurately calculated. With the constant
43
increase of computational capabilities, rigorous second order analysis is becoming
much more common in design practice. It should be noted that the AISC 36010
definition of rigorous second order analysis is typically not meant to represent a true
nonlinear second order analysis but will still produce results that accurately calculate
the second order effects. Many methods can be used for analysis but the general form
is usually expressed as: (Ziemian, 2010)
{dF}{dR} = K{dA}
With:
(dF} = Vector of incremental applied nodal forces
{dR} =Vector of unbalanced nodal forces, difference between current internal
forces and applied loads
K = Stiffness matrix
{dA} = Vector of incremental nodal displacements and rotations
Most solutions use an iterative approach for solving for second order effects.
The unbalanced forces are calculated based on the deformed geometry at the end of
each iteration and used as the basis for the next iteration. Iterations can be performed
until the unbalanced force vector is determined to be negligible. Figure 314 shows a
method commonly referred to as the NewtonRaphson incremental iterative
solution.
44
O
CO
Figure 314 Visual Representation of Incremental Iterative Solution Procedure (Adapted from
Ziemian, 2010)
Additional methods based on the NewtonRaphson incrementaliterative
solution have been developed based on the limitation imposed by the use of this
solution. McQuire et al. (2000) and Chen and Lui (1991) have provided general
overview of additional methods.
A variation of these methods that can be used by STAAD.Pro V8i is referred
to as the Lagrangian procedure. This procedure revises the stiffness matrix, K, to
take the following form:
[K] = [Ke] + [Kg]
45
With Ke being the standard linear elastic stiffness matrix and Kg being the geometric
stiffness matrix. This method recognizes that as the structure deforms the stiffness
associated with the member forces is changed at each increment of the solutions.
This leads to more accurate results and the ability to use this type of solution for
dynamic analyses because the method accounts for change in the natural period due
to second order effects and the stiffening of the structure. (Galambos, 1998) This
procedure is given in more detail by Crisfield (1991), Yang and Quo (1994) and
Bathe (1996). STAAD Technical Manual (2007) also gives details of how this
method is applied for analysis.
3.5.2 Approximate Second Order Elastic Analysis
In lieu of a rigorous second order elastic analysis which is iterative and can
demand extensive computational effort, approximate second order analysis methods
have been developed over the years to potentially simplify analysis. Rutenberg
(1981, 1982), White et al (2007a,b) and LeMessurier (1976, 1977) have each
developed approximate methods of analysis to either simplify analysis in computer
applications or simplify hand calculations. (Ziemian, 2010)
The AISC 36010 method of approximate second order analysis uses
amplification factors B1 and B2 to account for P8 and PA effects, respectively. See
AISC 36010 Appendix 8 for the procedure.
46
Because this approximate second order analysis is typically used for hand
calculations of frame analysis, no additional discussion will ensue based on the
assumption that the engineer has access to software capable of a rigorous second
order analysis.
3.5.3 Direct Analysis Method
Introduced in AISC 36005, the direct analysis method represents a
fundamentally new alternative to traditional stability analysis methods. (Griffis and
White, 2010) The most significant development addressed in this method is that
column strength can be based on the unbraced length of the member therefore
eliminating the need to calculate the effective length of the member (K may be taken
as 1 for all members). (Ziemian, 2010)
While all the requirements for AISC stability analysis are covered with this
method, slight variations in each requirement are allowed and discussed in further
detail. One major advantage of the direct analysis method is that it has been
developed and verified for application to all types of structural systems and therefore
has no limitations for use. (Maleck and White, 2003)
Accurate second order analysis is the cornerstone of the direct analysis
method. As previously discussed, two types of second order analysis are allowed by
AISC 36010, rigorous second order analysis and approximate second order analysis.
As with all AISC 36010 methods that use second order analysis results, the direct
47
analysis method is built around the assumption of LRFD loads and therefore if ASD
loads are to be used in analysis, they must be multiplied by 1.6 before second order
analysis is completed because of the nonlinearity of second order effects. (AISC 360
10) (Nair, 2009)
Flexural, shear and axial deformations need to be considered in analysis. As
previously discuss, accurate deformations are needed for calculation of second order
effects. However, in discussion of shear and axial deformation AISC 36010
explicitly uses the word consider instead of include. This allows the engineer to
ignore certain deformations based on the type of structural system. For example, the
shear deformations could feasibly be neglected in a low rise moment frame and
produce results with an error of less than 3%. High rise moment frame systems on
the other hand, could produce much higher errors if shear deformations were to be
neglected. (AISC 36010) Most modem analysis software is capable of calculating
accurate flexure, shear and axial deformations, and very little effort by the engineer is
required to achieve the most accurate results available by the software.
The direct analysis method is calibrated on the assumption that geometric
imperfections are equal to the maximum material, fabrication and erection tolerances
permitted by the Code of Standard Practice for Steel Buildings and Bridges (AISC
30310). Geometric imperfection may be accounted for by two methods; direct
modeling of imperfections or notional loads. (AISC 36010) Direct modeling of
imperfections can become quite tedious because as a minimum, four models must be
48
developed, each with the deflection in one of the four principle directions with the
additional member out of straightness modeled corresponding to the worst case
direction. Notional loads are defined as horizontal forces added to the structure to
account for the effects of geometric imperfections. (Ericksen, 2011) For the direct
analysis method the magnitude of notional loads applied to the structure is 0.2% of
the total factored gravity load at each story.
N; = 0.002aYi
With:
a = 1.0 (LRFD); 1.6 (ASD)
N; = Notional lateral load applied at level i, kips
Y; = Gravity load applied at level i from the LRFD or ASD load
combinations as applicable, kips
It can be see that 0.2% of gravity loads is appropriately selected as 1/500
which also corresponds to the maximum outofplumbness for columns from AISC
30310. Analysis will show that either applying a notional load of 0.2% or directly
modeling outofplumbness will produce similar results. (Malek and White, 1998)
Member outofplumbness is accounted for by notional loads, but member outof
straightness still needs to be considered in design. AISC 36010 has developed the
column strength equations based on maximum outofstraightness tolerances.
(Ziemian, 2010) (White et al., 2006)
49
For the direct analysis method, notional loads should be applied in
combination with all gravity and lateral load combinations to create the worst effects.
AISC 36010 does however, allow notional loads to be applied only to gravity load
combinations as long as the ratio of second order to first order drifts does not exceed
1.5 using the unreduced elastic stiffness or 1.7 if the reduced elastic stiffness is used
in analysis. The errors seen by this simplification are relatively small as long as the
ratio of drifts remains below the specified limits. (AISC 36010)
Reduction in stiffness of the structure is caused by partial yielding of
members. This yielding is further accentuated by residual stresses. The direct
analysis method specifies reduced stiffnesses of El* and EA* with:
El* = 0.8xbEI
EA* = 0.8EA
The reduced stiffness factor of 0.8 is applied for two reasons. The first and
most readily apparent is the reduction in stiffness in intermediate and stocky
members, namely columns, due to inelastic softening of members before they reach
their design strength. The second reason relates to slender members that are governed
by elastic stability. 0.8 is roughly equivalent to the product of 0= 0.9 and the factor
0.877. These factors are used in development of the AISC column curve (AISC 360
10 Eqn. E33) which is modified by the above factors for slender elements to account
for member outofstraightness. (Ziemian, 2010) (AISC 36010)
50
The Tb value is an adjustment factor to account for additional reduction in
stiffness in cases where high axial stresses are present which can reduce the bending
stiffness of the member. (AISC 36010)
Tb = 1.0 when aPr/Py< 0.5
Tb = 4(aPr/Py)[l( aPr/Py)] when aPr/Pv ^ 0.5
where:
a = 1.0 (LRFD); 1.6 (ASD)
Pr = Required axial compressive strength using LRFD or ASD load
combinations.
Py = Axial Yield strength (Ag*Fy)
AISC 36010 does allow Tb = 1.0 for all cases if the notional load is increased
by 0. l%Yi. This additional notional load is meant to increase the lateral deformation
to envelope the effects caused by reduction in stiffness in high axial loaded members.
However, Powell notes that this method does not appear to be logical because
notional loads are meant to account for initial out of plumbness and not for reduction
in stiffness (Powell, 2010). Figure 315 shows a graphical representation of the effect
of Tb on the reduction in stiffness.
51
Er/E
Modulus for EA
Figure 315 Reduced Modulus Relationship (Powell, 2010)
3.5.4 Effective Length Method
In recent years, the traditional method for stability analysis has been the
effective length method. In general, the effective length method calculates the
nominal column buckling resistance using an effective length (KL) and the load
effects are calculated based on either a rigorous or approximate second order analysis.
(Ziemian, 2010)
As with the direct analysis method, the effective length is built around
determining accurate second order effects. This can be done by either a rigorous
second order analysis or by an approximate second order analysis. (AISC 36010)
Both methods of second order analysis are based on the assumption that flexural,
shear and axial deformations are considered in calculations of second order effects.
52
Prior to AISC 36005, there were few limitations on applications of the
effective length method. Several studies by Deierlein et al. (2002), Maleck and White
(2003), and SurovekMaleck and White (2004a and 2004b) have shown that use of
the effective length method could produce significantly unconservative results in
certain types of framing systems. Therefore AISC 36005 imposed additional
requirements and limitations.
1. Notional loads need to be included in gravity only load combinations to
account for member outofplumbness.
2. The ratio of second order drift to first order drift or B2 is limited to 1.5.
Geometric imperfections are covered by applications of notional loads or
direct modeling of imperfections for analysis. Notional loads are applied in the same
manner as described with the direct analysis method with: (AISC 36010)
N; = 0.002aYi
However, based on the limitation of the ratio of second order drift to first
order drift or B2, by definition, notional loads need only be applied to gravity load
combinations.
One of the main components of the effective length method is the calculation
of the effective length factor K. The most common method for determining K is
through the use of the alignment charts found in the commentary for Appendix 7 in
AISC 36010 which can be seen in Figures 316 and 317.
53
Ga K Gb
00 50.0 = r1.0 00 F 50.0
10.0 ^ ^ 10.0
5.0 = 4.0 0.9 = 5.0 4.0
3.0 3.0
2.0 2.0
0.8
1.0 0.9 1.0 0.9
0.8 0.7 0.6 0.7 0.8 0.7 0.6
0.5 0.4  0.5 1 0.4
0.3 0.6 0.3
0.2 0.2
0.1 0.1
0.0 0.5 0.0
Figure 316 Alignment Chart Sidesway Inhibited (Braced Frame) (AISC 36010)
54
Ga k Gb
00 00 F 20 0 00
100.0 = 100 100.0
50.0= =50.0
30.0 ^ 5.0 30.0
20.0 4.0 20.0
10.0 3.0 10.0
8.0 8.0
7.0 7.0
6.0 6.0
5.0 5.0
4.0 2.0 4.0
3.0 3.0
2.0 2.0
1.5 
1.0 1.0
0.0 1.0 0.0
Figure 317 Alignment Chart Sidesway Uninhibited (Moment Frame) (AISC 36010)
The alignment charts were developed based on the following assumptions:
1. Behavior is purely elastic.
2. All members have constant cross section.
3. All j oints are rigid.
4. For columns in frames with sidesway inhibited, rotations at opposite ends
of the restraining beams are equal in magnitude and opposite in direction,
producing single curvature bending.
55
5. For columns in frames with sidesway uninhibited, rotation at opposite
ends of the restraining beams are equal in magnitude and direction,
producing reverse curvature bending.
6. The stiffness of parameter La/(P/EI) of all columns is equal.
7. Joint restraint is distributed to the column above and below the joint in
proportion to EI/L for the two columns.
8. All columns buckle simultaneously
9. No significant axial compression force exists in the girders.
The assumptions listed above, seldom if ever are seen in a real structure and
therefore additional methods of determining K have been developed.
Geschwindner(2002) and ASCE Task Committee (1997) have provided an overview
of the various methods for determining accurate values of K. In addition to methods
listed in AISC 36010, Yura (1971) and LeMessurier (1995) presented various
approaches for calculation of the effect length factor K. Folse and Nowak (1995) also
presented examples that included the effect of leaning columns.
The table below gives a comparison of the equivalent length method and the
direct analysis method and describes how each method addresses the stability analysis
requirements of AISC 36010.
56
Table 33 Comparison of Direct Analysis Method and Equivalent Length Method (Adapted from
Nair 2009)
Comparison of Basic Stability Requirements with Specific Provisions
Basic Requirement in Section 1 of This Model Specification Provisions in Direct Analysis Method (DAM) Provisions in Effective Length Method (ELM)
(1) Consider secondorder effects (both PA and P5) 2.1(1) Consider secondorder effects (both PA and P5)** Consider second order effects (both P A and P5)**
(2) Consider all deformations 2.1(2) Consider all deformations Consider all deformations
(3) Consider geometric imperfections which include jointposition imperfections* and member imperfections Effects of joint position imperfections* on structural response 2.2a. Direct modeling or 2.2b. Notional loads Apply notional loads
Effects of member imperfections on structural response Included in the stiffness reduction specified in 2.3 All these effects are considered by using KL from a sidesway buckling analysis in the member strength check. Note that the only difference between DAM and ELM is that: DAM uses reduced stiffness in the analysis; KL=L in the member strength check ELM uses full stiffness in the analysis; KL from sidesway buckling analysis in the member strength check for frame members
Effects of member imperfections on member strength Included in member strength formulas, with KL=L
(4) Consider stiffness reduction due to inelasticity which affects structure response and member strength Effects of stiffness reduction on structural response Included in the stiffness reduction specified in 2.3
Effects of stiffness reduction on member strength Included in member strength formulas, with KL=L
(5) Consider uncertainty in strength and stiffness which affects structure response and member strength Effects of stiffness/strength uncertainty on structural response Included in the stiffness reduction specified in 2.3
Effects of stiffness/strength uncertainty on member strength Included in member strength formulas, with KL=L
* In typical building structures, the "jointposition imperfections" are the column outofplumbness. ** Secondorder effects may be considered either by rigorous secondorder analysis or by amplifications of the results of firstorder analysis (using the B1 and B2 amplifiers in the AISC Specification).
57
3.5.5 First Order Method
The first order analysis method is a simplified method derived from the direct
analysis method. (Kuchenbecker et al., 2004) The main benefit of this method is that
only a first order analysis is required. In addition, because it is based on the direct
analysis method, K can be set at 1 for all cases. (AISC 36010)
The main simplification in this method comes from the assumption that the
ratio of second order drift to first order drift or B2, is assumed to be equal tol.5.
From this assumption, equivalent notional lateral loads can be back calculated which
simulate the effects of second order effects as well as reduction in stiffness due to
partial yielding which are both accounted for using the direct analysis method.
(Ziemian, 2010)
The simplification and assumptions which are made in development of the
first order method lead to limitations on use of this method. The ratio of second order
drift to first order drift or B2 is assumed to be 1.5, which sets the maximum allowed
value for application of this method. The stiffness reduction of 0.8 was also assumed
in calculations of additional notion loads and therefore, based on the previous
discussion of the reduction in stiffness, aPr/Py < 0.5 must be maintained for the 0.8
reduction in stiffness to remain true.
The notional load Ni for the first order analysis method is defined as:
(Kuchenbecker et al., 2004)
58
N1 =
Bn
1 0.2B9
v 2 y
Y; >
B.
\
1 0.2Bn
v 2 y
0.002Y
If the above assumptions of B2=1.5 and ib = 1.0 are made and substituted into
the equation above, it can be simplified to the form seen in AISC 36010
Ni = 2.1(A/L)Y; > 0.0042Y;
With:
N; = Notional load applied at level i, kips
a = 1.0 (LRFD); 1.6 (ASD)
Y; = Gravity load applied at level i from the LRFD or ASD load
combinations as applicable, kips
A = First order interstory drift
L = Height of story
Unlike the effective length and direct analysis methods, the notional load is
required to be added to all load combinations regardless of gravity only or lateral load
combinations.
While this method has been simplified to eliminate the need for a second
order analysis and any calculation of K, the limitations and verification can hinder the
use of this method. The simplifications made in the development of this method
make it a very useful tool when frame analysis is done by hand because a second
order analysis is not required. Most pipe rack structures are designed using modem
59
analysis software which is capable of performing a second order analysis, therefore
the first order method will likely see limited use for pipe rack applications.
60
4.
Research Plan
A literature review was first conducted to gather and review the available
information pertaining to the design and engineering of pipe rack structures for use in
industrial facilities. The industry is constantly evolving, and the most current
literature discussing the design and engineering of pipe racks was targeted for review.
Next, a literature review was conducted to gather and review the available
information pertaining to AISC stability analysis. Literature that described the
various methods used by both AISC 36005 and AISC 36010 were the focus of the
review. The direct analysis method literature was of particular interest as it was a
relatively new development in regards to stability analysis.
A general plan for the research that was conducted is presented here and is
described as follows:
1. Use Benchmark Problems from AISC 36010 to test the second order analysis
capabilities of STAAD and report on the validity of the STAAD approach.
2. Describe in detail a typical pipe rack to be used for comparison of the
methods.
3. Develop general loads and load combinations for use in the analysis models.
61
4. Develop a general STAAD.Pro V8i model that can be used for analysis of the
Equivalent Length Method, Direct Analysis Method and First Order Method
with input from [2] and [3],
5. Complete a first order analysis of the pipe rack structure developed in [4] for
use in calculation of the A2/A1 ratio as well as for use in the First Order
Method and discuss the results and validity of the method based on AISC
limitations
6. Optimize strength only design of test pipe rack structure developed in [4]
using Equivalent Length Method and determine validity of method for current
structure based on AISC limitations
7. Optimize the strength only design of the test pipe rack structure developed in
[4] using the Direct Analysis Method and compare the results to the
Equivalent Length Method.
8. Use the models developed in [6] and [7] and vary member sizes and base
fixity based on the serviceability limits and compare the results.
9. Compare the results of [5 to 8],
62
5.
Member Design
The available strength of members should be calculated in accordance with
the provisions of AISC 36010 Chapters D, E, F, G, H, I, J and K. These chapters
should be used regardless of the method of stability analysis chosen. Column design
can become a major point of focus during stability analysis based on several factors.
The effective length factor for column design can become complex for even simple
structures when using the effective length method of stability analysis. Columns
typically experience combined flexural, shear and axial load and the interaction
between each stress must be investigated.
The two methods of stability analysis; effective length and direct analysis, as
expected, produce varying load effects. A design example is shown in Figure 51
showing a simple cantilever W10X60 column bent about the strong axis with an axial
load P and the horizontal load H = 10%P. The column is assumed to be supported
out of plane resulting in only strong axis buckling. Stability analyses using both the
equivalent length method and the direct analysis method were performed. A linear
elastic analysis was performed as well to establish a first order baseline for
comparison. STAAD.Pro V8i was used for both methods of stability analysis while
hand calculations were done to compute the linear elastic forces.
63
p
H=0.01P
LO
o
CO
X
o
Ã‚Â§
Figure 51 Simple Cantilever Design Example
The cantilever column was then checked against the code specified available
strength. The column in this example will experience both axial and flexural loads,
therefore AISC 36010 Chapter H will be used to determine the available strength.
A simple cantilever column was chosen to simplify the selection of the effect
length factor K. The theoretical value of K for a fixed base cantilever column is 2.
Therefore the effective length of the column used for the determining the available
strength using the effective length method will be 30 ft.
64
Figure 52 shows a graph of axial load vs. moment for the simple cantilever
example. AISC 36010 equations Hlla and Hllb are included on the graph for
design purposes. When comparing methods of analysis, with the linear elastic
method as a baseline, the differences are readily apparent. As expected, the linear
elastic method produces linear results of axial load vs. moment. Based on the theory
of stability analysis, the linear elastic method is expected to produce results that
overestimate the axial resistance and underestimate the moment demand. When
comparing the results of both the effective length and direct analysis methods, the
resistance based on AISC 36010 equations Hlla and Hllb must be adjusted based
on the effective length factor K. This will affect the axial resistance of the column
section. The direct analysis method assumes that K = 1 and the effective length
method assumes that K = 2 for a fixed base cantilever column. This changes the
anchor point interaction equations.
65
Figure 52 Simple Cantilever Design Example Results
One of the main differences in results between the effective length method
and the direct analysis method is the moment demand. The axial load resistance is
comparable between the two methods but the moment demands can differ
significantly. The column capacity is adjusted between the methods based on the
effective length factor which calibrates the axial capacity. The difference in moment
demand is due to the reduction in stiffness used in the direct analysis which increases
deformations which in turn increases eccentricities and therefore moment demand is
increased.
66
Based on the results of this design example, several observations can be made:
(AISC 36010)
1. Accurate calculations of the effective length factor, K, are critical to
achieving accurate results using the effective length method.
2. The moment demand is underestimated when using the effective length
method. This can significantly affect the design loads for beams and
connections which provide rotational resistance for the column.
67
6 Pipe Rack Analysis
6.1 Generalized Pipe Rack
A typical pipe rack will be developed and used for comparison purposes for
this thesis. The typical pipe rack was chosen and modeled based on idealized
conditions. A width of 15 feet was chosen to allow oneway traffic along the pipe
rack corridor. The height of the first level of the pipe rack was set at 20 feet to
provide sufficient height clearance along the access corridor.
The overall length of the pipe rack was set at 100 feet. Longer pipe rack
sections are typically broken into shorter segments (100 to 200 feet) with each shorter
segment separated by expansion joints to allow thermal expansion or contraction
between segments. One of the lengthwise central bays of each segment is typically
braced in the longitudinal direction. This allows the length of the pipe rack to expand
and contract about a central braced bay and reduces thermally induced loads cause
from restraint of thermal movement. If each end of the segment were to consist of a
braced frame, the length of the pipe rack would essentially be locked in place and
higher thermally induced loads would be seen.
Moment frames are typically spaced at 1520 feet. This spacing is typically
chosen based on the maximum allowable spans for the pipes or cable trays being
supported. This spacing can vary based on the estimated size and allowable
deflection limits of the pipe being supported
68
Longitudinal struts are usually offset from the beams used to support the
pipes. Levels of the pipe rack are assumed to be fully loaded with pipe, and when the
pipes need to exit the rack to the side to connect to equipment, a flat turn cannot be
used as this would clash with the other pipes on the same level. The pipe is typically
routed to turn either up or down and then out of the rack at the level of the
longitudinal struts where the pipe can be supported on the longitudinal struts before
exiting the rack.
To allow room for pipes to enter and exit the pipe rack, a spacing of 5 feet
between levels is typically used. If the pipe rack carries larger pipes, additional room
may be required between levels. This spacing should be determine with the help of
the piping engineer on the project. Spacing between pipe rack levels of 5 feet will be
used in this thesis.
Figure 61 shows an isometric view of the typical pipe rack that will be used
for analysis and comparison of stability analysis methods. While this is not
representative of all pipe racks, it will still provide a useful basis for comparison
purposes in pipe rack structures.
69
Figure 61 Isometric View of Typical Pipe Rack Used for Analysis
To simplify the design and analysis, a typical moment frame will be selected
and isolated for analysis and design. Figure 62 shows an elevation view of a typical
moment frame.
70
Figure 62 Section View of Moment Frame in Typical Pipe Rack
Outofplane supports were added at the locations of longitudinal struts which
will restrain any movement in and out of the page (see Figure 61). Longitudinal
struts all tie into the braced bay, therefore relatively small deflections will be
experienced in the weak axis of the columns and the restraint of any movement in this
direction is a reasonable assumption.
Based on initial calculations that compare the results of the isolated moment
frame and the entire pipe rack segment, relatively small differences were seen.
Because the braced bay supports any longitudinal loading, relatively very little weak
axis column moment or longitudinal deflection occurs that would affect the design of
71
the columns or beams that are part of the moment frame. Ratios of demand to
capacity showed errors of less than 5% on member design when using the single
frame compared to the full pipe rack structure. Therefore, analysis of a single
moment frame will be used to simplify calculations. The focus of this thesis will
mainly be on the analysis of the moment frame; the braced frame will not be
considered in analysis and design.
In actual design, engineering judgment should be used to determine if the
analysis of a single frame simplification can be made. In many cases, pipe racks are
not symmetric and loading can vary from frame to frame and the overall structure
should be analyzed as a whole to determine the load effects. Bracing systems should
also be designed to resist the longitudinal loads of the entire segment and therefore
modeling of the entire pipe rack segment may be required to determine load path and
design loads for struts and braces.
6.2 Pipe Rack Loading
The pipe rack used for analysis and comparison of methods will have
consistent loading between methods to limit the number of variables. In general,
loads in the longitudinal direction will not be considered in design and comparison
because the focus of analysis will be on the behavior and performance of the moment
frame. Therefore, wind and seismic loading, will only be considered in the transverse
directions.
72
In practice, engineering judgment should be used in determining all applicable
loads. The following discussion is meant to define loads only for analysis and
comparison of the stability analysis and therefore certain simplifications are made to
facilitate analysis but still provide results that are typical of pipe racks.
All loads are developed based on the assumption that the pipe rack is
considered an Occupancy Category III. (PIP STC01015) This will affect the
importance factor used for development of wind and seismic loading.
The primary load cases, which were defined previously, were chosen
according to PIP STC01015. Additional primary load cases may be required based
on the requirements of AISC 36010 for notional loads. Loads are developed based
in input from ASCE 705. While ASCE 710 is available, PIP STC01015 has not
been updated to reflect the changes made by ASCE 710.
The first primary load case defined was the dead load of the structure (Ds).
This is considered as the selfweight of the steel. For design, an additional 10% of
the selfweight was added to account for fabrication tolerances and additional
materials used for connections. No other loads were assumed at this time for the dead
load of the structure.
The operating dead load as discussed previously is typically applied at 40 psf,
which assumes a fully loaded level with 8 inch pipes full of water spaced at 15
inches. The representative pipe rack will use 40 psf applied over the entire tributary
73
area of the pipe rack. The spacing of the moment frames was chosen as 20 feet,
therefore the uniform load applied at each level of the pipe rack is 800 pounds per
foot. This load can be seen in Figure 63.
80( ' .000 lb /ft 1 '
SOI ' .000 lb /ft 1
SOI .000 lb /ft 1
SOI 1 .00 1 lb /ft '
Figure 63 Section View of Moment Frame Operating Dead Load
The empty dead load of the pipe can be taken as 60% of the operating dead
load unless further information is known. This calculates to 480 pounds per foot.
The empty dead load of the pipe is applied in a similar fashion as seen in Figure 63.
Test dead load is the weight of the pipe plus the weight of the test medium.
This type of loading will typically control when the majority of the pipes in the pipe
74
rack are filled with gas or steam during operation. Hydrotesting is typically done to
test the piping prior to startup. Pipes were assumed to be full of water for the
operating load, therefore for this analysis, the test dead load is equivalent to the
operating dead load and the load is exactly as seen in Figure 63.
The erection dead load can account for any additional loads or reduction in
loads due to erection activities. This is typically used for any equipment and is based
on the fabricated weight. For piping, the erection dead load and the empty dead load
are typically the same. Therefore, the erection dead load case is not defined at this
time and if required in load combinations, the empty dead load can be used in place
of the erection dead load.
Pipe anchor and pipe friction loads are typically based on actual loading
conditions of pipes located in the pipe rack. However, without final pipe loading, an
estimate of pipe loads must be made. Friction forces can be estimated based on the
coefficient of friction between the pipe shoe and support beam. This coefficient of
friction is usually assumed to be 0.4. Application of 40% of the operating dead load
tends to be extremely conservative. Friction loads are cause by expansion and
contraction of pipes. Based on the expansion and contraction of pipes, friction loads
are typically seen in the longitudinal direction of the pipe rack. Because it is highly
unlikely that all pipes will expand and contract simultaneously and some pipes may
contract while others expand, a more realistic value of 10% of operating dead load
will be applied in the longitudinal direction. Friction loads are applied to the pipe
75
rack because the location of the loading could cause additional second order effects in
the beams in the moment frames.
While friction loads and anchor loads are typically only seen in the
longitudinal direction of the pipe rack, cases where anchor loads are seen in the
transverse direction could happen. Therefore apply 5% of the operating dead load as
a conservative estimate for the representative pipe rack. Figure 64 shows the
application of pipe anchor loads. In practice, local members should be checked for
pipe anchor loads and friction loads as the individual member design may be
controlled by high anchor loads.
76
Figure 64 Section View of Moment Frame Pipe Anchor Load
Selfstraining thermal loads will not be considered in design. A design AT of
0 degrees Fahrenheit will be applied to all members. Thermal loading can cause
problems if members are restrained from expansion or contraction. As the 2D
moment frame has very little resistance for thermal movements, thermal loads will
not be considered in design of the representative model. Additional thermal
considerations could be considered but are outside the scope of this thesis.
77
Live load is typically only applicable to platforms or walkways required for
access, therefore live load will not be applied based on the assumption of no access
platforms or walkways on the simplified pipe rack.
Similarly, snow load typically will not control the design and therefore will
not be considered in the simplified analysis model. (PIP STC01015)
Wind load is applied consistent with ASCE 705 principles. For comparison
purposes, a 3 second gust wind velocity was assumed to be 100 miles per hour which
should cover a larger majority of sites. Additional information given by ASCE Wind
Loads for Petrochemical Facilities is included in wind load development. The
following velocity pressures at specified heights was developed according to ASCE
705. Table 61 shows velocity pressures for cable tray and structural members and
Table 62 shows velocity pressures for pipes.
Table 61 Velocity pressures for cable tray and structural members
Cable Tray and
Structural Members
Height q7
(ft) (psf)
015 21.270
20 22.522
25 23.523
30 24.524
40 26.025
78
Table 62 Velocity pressures for pipe
Pipe
Height q7
(ft) (psf)
015 23.77
20 25.17
25 26.29
30 27.41
40 29.09
The design wind force for structural members varies based on the projected
area perpendicular to the wind direction and therefore will vary based on member
size. In the case where the wind is parallel to the strong axis, the flange width defines
the projected area. Structural shapes are assumed to have an average coefficient of
drag (Cf) of 2.0 for all wind directions. Pipes were assumed to have a coefficient of
drag (Cf) of 0.8 for wind perpendicular to pipe.
The design wind force for pipe is calculated based on the assumption of the
largest pipe being 8 inches. 10% of the pipe rack width is added to the pipe added to
the largest pipe diameter and multiplied by the bay spacing to determine a tributary
area. (ASCE, 2011) To simplify the design, the total force from the pipes on each
level is evenly divided and applied to each joint. This load could also be applied as a
uniform load across the entire length of the beam. Table 63 shows the resulting joint
loads resulting from wind loading on pipes.
79
Table 63 Resultant design wind force from pipe
Height F F/2
(ft) (lbs) (lbs)
015 701 350
20 742 371
25 775 387
30 808 404
40 857 429
Figure 65 shows the application of wind load for the typical pipe rack. The
resultant load from pipe wind load is evenly distributed at the end joints. The design
wind force shown for structural members is based on a column size of W10X33 with
a flange width of 8 inches. This design wind force will vary based on column size but
will provide a basis for application of wind load.
80
Figure 65 Section View of Moment Frame Wind Load
Seismic loading is based on specific site information. The ASCE 705
Equivalent Lateral Force Procedure will be used for seismic loading. Because this is
a representative model, estimates on seismic loading will be made to simulate the
general seismic load effects. The pipe rack is assumed be in located in site class C or
below. An R value of 3 is chosen to simplify detailing requirements. The assumed
seismic response coefficient, Cs, will be 0.15. While the natural period of the
structure will depend on member size and base support conditions, the natural period
81
will be assumed to be greater than 0.5 seconds which is used to determine the vertical
distribution.
Based on the above assumptions, seismic forces can be developed and applied
to the structure. The effective seismic mass is based on previously discussed dead
loads, both operating and empty dead loads. Table 64 shows the applied operating
seismic load applied at each level of the pipe rack while Table 65 shows the applied
empty seismic load. The lateral force at each level will be evenly divided and applied
to each joint. See Figure 66 for operating seismic loading. Empty seismic loading is
similar to loading shown in Figure 66
Table 64 Lateral seismic forces operating
Fx (lbs) Fx/2 (lbs)
Level 1 950 475
Level 2 1450 725
Level 3 2100 1050
Level 4 2850 1425
Table 65 Lateral seismic forces empty
Fx (lbs) Fx/2 (lbs)
Level 1 600 300
Level 2 900 450
Level 3 1250 625
Level 4 1700 850
82
Figure 66 Section View of Moment Frame Lateral Seismic Load Operating Seismic
6.3 Pipe Rack Load Combinations
Load combinations used for the representative model were developed from the
PIP STC01015 load combinations as discussed in previous sections. As the model
was simplified to isolate a moment resisting frame and study the results from
essentially a 2D analysis, loading in the transverse direction was the focus for
83
developing load combinations. Because lateral loads are typically reversible, this can
create hundreds of load combinations if all load directions are considered. Transverse
only loading simplified the loading and resulted in fewer load combinations. In the
cases where notional loads are required, the notional load was considered to act only
in the direction causing the worst effect on stability which is typically in the same
direction as the lateral load.
Two sets of load combinations were developed for analysis and comparison
purposes. The first set applied notional loads to only the gravity only load
combinations. This set of load combinations is used for analysis using both the
effective length and direct analysis methods. The first set of load combinations can
only be used for the direct analysis when the ratio of second order to first order drift
is less than 1.7 if using reduced stiffness or 1.5 using unreduced stiffness per AISC
36010.
The second set of load combinations applies the notional loads as additive in
all load combinations to create the worst effect on stability. The second set of load
combinations was developed for analysis using the direct analysis method where the
ratio of second order to first order drift is greater than 1.7 if using reduced stiffness or
1.5 using unreduced stiffness. The second set of load combinations can also be used
for the first order method where the notional loads are additive for all load
combinations. Although the magnitude of notional loads varies between the direct
analysis and first order method, the notional loads can be adjusted in the primary load
84
combinations and the same load combinations can be used. Based on the limitation
for use of the effective length method, the second set of load combinations with
notional load applied in all load combinations is not required to be used in the
effective length method.
A few observations on application of methods can be made by investigating
the load combinations and requirements. First, in cases where the ratio of second
order drift to first order drift is greater than 1.7 if using reduced stiffness or 1.5 using
unreduced stiffness, the direct analysis method using the set of load combinations
with additive notional loads in all load combinations is the suggested method of
AISC 36010. Next, all three methods can be used for analysis when the ratio of
second order drift to first order drift is less than 1.7 if using reduced stiffness or 1.5
using unreduced stiffness. If this is the case, the direct analysis and effective length
method do not require load combinations where the notional loads are additive in all
cases, while the first order method requires additive notional loads in all load
combinations.
Strength and serviceability will both be of interest in analysis and design,
therefore both LRFD and ASD load combinations will be developed. LRFD load
combinations will be used for member strength checks, while the serviceability
checks will be made using the ASD load combinations. The load combinations used
in analysis can be seen in Appendix 1 through 3.
85
6.4 Strength and Serviceability Checks
Strength and serviceability check are made using the capabilities of
STAAD.Pro V8i. Strength checks are based on AISC 36005. While AISC 36010
has been released, STAAD.Pro V8i has yet to include the specification in design
capabilities. Very few changes have been made in member capacity calculations and
therefore the AISC 36005 can be used for determining member capacity. The ratio
of demand to capacity is a point of comparison between the various methods of
stability analysis.
Serviceability checks are made using the calculated deflections from
STAAD.Pro V8i. Unfactored loads combinations are used to calculated service
deflections. Various limits on serviceability can be set based on specific project
requirements.
AISC 36010 states that both geometric imperfections and reduction in
stiffness are not required in determining serviceability checks. Therefore, notional
loads are not needed in service load combinations. Reduction in stiffness is also not
included in calculations of deformations used for serviceability checks.
It should be noted that when using the direct analysis method in STAAD.Pro
V8i, the reported deformations are calculated based on the reduced stiffness. AISC
does not require the reduced stiffness to be used in serviceability checks, therefore the
models should be analyzed using the unreduced stiffness for serviceability checks or
86
the reported deformations should be adjusted to account for the inclusion of reduced
stiffness. Although not exact, reported deformations based on reduced stiffness could
be multiplied by 0.8 to provide relatively accurate estimates of the actual
deformations to be used in serviceability checks. This was based on several test
models that were analyzed using both methods and the results were compared and
found to be reasonable.
6.5 Base Support Conditions
Column base support conditions are affected by various factors. True fixed
base columns, in actual conditions, can be very hard to achieve. Foundation types
and anchor bolt layout and design can significantly affect the rotational resistance of
the column base. Fixed base moment frames typically can see savings in member
size but additional considerations in foundation and anchor bolt design could offset
the savings in member sizing. Fixed base moment frames will also typically see a
reduction in deformations due to the additional moment capacity generated by the
base fixity. Pinned base moment frames on the other hand will typically require
heavier members and experience potentially larger deformations compared to similar
fixed base moment frames. Base support conditions can have a significant effect on
overall frame behavior, therefore both fixed and pinned conditions were analyzed.
Base support conditions also become important in the calculation of the
effective length factor (K) used in the effective length method. The alignment charts
87

PAGE 1
STABILITY ANALYSIS O F PIPE RACKS FOR INDUSTRIAL FACILITIE S By David A. Nelson B.S., Walla Walla University, 2008 A thesis submitted to University of Colorado Denver in partial fulfillment of the requirements for the degree of Master of Science, Civil Engineering 2012
PAGE 2
This thesis for the Master of Science degree by David A. Nelson has been approved by Fredrick Rutz Kevin Rens Rui Liu
PAGE 3
Nelson, David A. (M.S., Civil Engineering) Stability Analysis of Pipe Racks for Indus trial Facilities Thesis directed by Professor Fredrick Rutz ABSTRACT Pipe rack structures are used extensively throughout industrial facilities worldwide. While stability analysis is required in pipe rack design per the AISC Specification for Structural Steel Buildings ( AISC 36010) the most compelling reason for uniform application of stability analysis is more fundamental. Improper application of stability analysis methods could lead to unconservative results and potential instability in the structur e jeopardizing the safety of not only the pipe rack structure but the entire industrial facility. The direct analysis method, effective length method and first order method are methods of stability analysis that are specified by AISC 36010. Pipe rack s tructures typically require moment frames in the t ransverse direction creating intrinsic susceptibility to second order effect s This tendency for large second order effects demands careful attention in stability analysis. Proper application as well as clear a understanding of the limitations of each method is crucial for accurate pipe rack design
PAGE 4
A comparison of the three AISC 36010 methods of stability analysis was completed for a representative pipe rack structure using the 3D structural analysis program STAAD.Pro V8i. For the model chosen, all three methods of stability analysis met AISC 360 10 requirements. For typical pipe rack structures, all three methods of stability analysis are acceptable as long as limitations are met and the methods are applied correctly. The first order method typically provided conservative results while the effective length method was determined to underestimate the moment de mand in beams or connections that resist column rotation. The direct analysis method was f ound to be a powerful analysis tool as it requires no a dditional calculations to calculate additional notional loads calculate effective length factors or verify AISC 36010 limitations This abstract accurately represents the content of the candidates thesis. I recommend its publication. Fredrick Rutz
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ACKNOWLEDGEMENT I would like to thank first and foremost Dr. Fredrick Rutz for the support and guidance in completion of this thesis. I would also like to thank Dr. Rens and Dr. Li for p articipating on my graduate advisory committee. Lastly I would like to thank various work associates for t heir help with either editing or discussion of the topic.
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vi TABLE OF CONTENTS LIST OF FIGURES ........................................................................................................... ix LIST OF TABLES ............................................................................................................ xii Chapter 1. Introduction ..............................................................................................................1 1.1 Stability Analysis of Steel Structures ......................................................................1 1.2 Pipe Racks in Industrial Facilities ............................................................................3 2. Problem Statement ...................................................................................................6 2.1 Introduction ..............................................................................................................6 2.2 Significance of Research ..........................................................................................8 2.3 Research Objective ..................................................................................................8 3. Literature Review ...................................................................................................10 3.1 Introduction ............................................................................................................10 3.2 Pipe Rack Loading .................................................................................................10 3.2.1 Load Definitions ............................................................................................10 3.2.2 Dead Loads ....................................................................................................14 3.2.3 Live Load s .....................................................................................................15 3.2.4 Thermal and Self Straining Loads .................................................................16 3.2.5 Snow Load and Rain Loads ...........................................................................16
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vii 3.2.6 Wind Loads ....................................................................................................17 3.2.7 Seismic Loads ................................................................................................21 3.2.8 Load Combinations ........................................................................................22 3.3 Column Failure and Euler Buckling ......................................................................25 3.4 Stability Analysis ...................................................................................................29 3.4.1 AISC Specification Requirements .................................................................29 3.4.2 Second Order Effects .....................................................................................29 3.4.3 Flexural, Shear and Axial Deformation .........................................................34 3.4.4 Geometric Imperfections ...............................................................................35 3.4.5 Residual Stresses and Reduction in Stiffness ................................................37 3.5 AISC Methods of Stabi lity Analysis ......................................................................41 3.5.1 Rigorous Second Order Elastic Analysis .......................................................43 3.5.2 Approximate Second Order Elastic Analysis ................................................46 3.5.3 Direct Analysis Method .................................................................................47 3.5.4 Effective Length Method ...............................................................................52 3.5.5 First Order Method ........................................................................................58 4. Research Plan .........................................................................................................61 5. Member Design ......................................................................................................63 6. Pipe Rack Analysis ................................................................................................68 6.1 Generalized Pipe Rack ...........................................................................................68 6.2 Pipe Rack Loading .................................................................................................72
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viii 6.3 Pipe Rack Load Combinations ...............................................................................83 6.4 Strength and Serviceability Checks .......................................................................86 6.5 Base Support Conditions ........................................................................................87 6.6 Effective Length Factor .........................................................................................88 6.7 Notional Load Development for First Order Method ............................................91 6.8 STAAD Benchmark Validation .............................................................................93 7. Comparison of Results ...........................................................................................96 8. Conclusions ..........................................................................................................111 Refer ences ........................................................................................................................116 Appendix A STAAD Input Pinned Base Analysis Effective Length Method ...........123 Appendix B STAAD Input Pinned Base Analysis Direct Analysis Method ..............132 Appendix C STAAD Input Pinned Base Analysis First Order Method .....................141
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ix LIST OF FIGURES Figure 11 Typical Four Level Pipe Rack Consisting of Eight T ransverse Frames Connection by Longitudinal Struts ....................................................................4 21 Typical Elevation View of Pipe Rack ................................................................7 22 Section View Showing Moment Resisting Frame .............................................7 31 Load vs. Deflection Yielding of Perfect Column .........................................26 32 Visual Definition of Critical Buckling Load Pcr ..............................................27 33 Load vs. Deflection Euler Buckling ..............................................................28 34 Second Order P moments ( Adapted from Ziemian, 2010) .............30 35 Comparison of First Order Analysis to Second Order Analysis ( Adapted from Gerschwindner, 2009) ......................................................................................31 36 Quebec Bridge Prior to Collaps e (Canada, 1919) ............................................32 37 Quebec Bridge after Failure (Canada, 1919) ...................................................33 38 Deformation from Flexure, Shear and Axial ( Adapted from Gerschwindner, 2009) ................................................................................................................35 39 Load vs. Deflection Real Column Behavior with Initial Imperfections .......37 310 Residual Stress Patt erns in Hot Rolled Wide Flange Shapes ..........................38
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x 311 Influence of Residual Stress on Average Stress Strain Curve (Salmon and Johnson, 2008) .................................................................................................39 312 Idealized Residual Stresses for Wide Flange Shape Members Lehigh Pattern ( Adapted from Ziemian, 2010) ........................................................................40 313 Loa d vs. Deflection Comparison of Analysis Types ( Adapted from White and Hajjar, 1991) .............................................................................................42 314 Visual Representation of Incremental Iterative Solution Procedure ( Adapted from Ziemian, 2010) ........................................................................................45 315 Reduced Modulus Relationship (Powell, 2010) ..............................................52 316 Alignme nt Chart Sidesway Inhibited (Braced Frame) ( Adapted from AISC 36010) .............................................................................................................54 317 Alignment Chart Sidesway Uninhibited (Moment Frame) ( Adapted from AISC 36010) ..........................................................................55 51 Simple Cantilever Design Example .................................................................64 52 Simple Cantilever Design Example Results ....................................................66 61 Isometric View of Typical Pipe Rack Structure Used for Analysis ................70 62 Section View of Moment Frame in Typical Pipe Rack ...................................71 63 Section View of Moment Frame Operating Dead Load ...............................74 64 Section View of Moment Frame Pipe Anchor Load ....................................77
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xi 65 Section View of Moment Frame Wind Load ................................................81 66 Section View of Moment Frame Operating Seismic ....................................83 67 Effective Length Factor K Pinned Base ........................................................90 68 Effective Length Factor K Fixed Base ..........................................................91 69 AISC Benchmark Problems ( Adapted from AISC 36010) ............................94
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xii LIST OF TABLES Table 31 Force coefficient, Cf for open structures trussed towers ( Adapted from ASCE 705) .................................................................................................................18 32 Cf force coefficient ( Adapted from ASCE 7 05) ...........................................20 33 Comparison of direct analysis method and equivalent length method ( Adapted from Nair, 2009) ..............................................................................................57 61 Velocity pressure for cable tray and structural members .................................78 62 Velocity pressure for pipe ................................................................................79 63 Resultant design wind force from pipe ............................................................80 64 Lateral seismic forces operating ...................................................................82 65 Lateral seismic forces empty ........................................................................82 66 Benchmark solutions ........................................................................................95 71 21 effective length method pinned base .........................................97 72 21 direct analysis method pinned base ...........................................99 73 Maximum demand to capacity ratio pinned base .......................................100 74 Maximum demand forces pinned base .......................................................101 75 21 effective length method fixe d base ..........................................103
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xiii 76 21 direct analysis method fixed base ............................................104 77 Maximum demand to capacity ratio fixed base ..........................................105 78 Maximum demand forces fixed base ..........................................................105 79 21 effective length method pinned base serviceability lim its .......107 710 21 direct analysis method pinne d base serviceability limits .........108 711 Maximum demand to capacity ratio pinned base serviceability limits .......109
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1 1. Introduction 1.1 Stability Analysis of Steel Structures The engineering knowledge base continues to grow and expand. This growth creates on going challenges as designs demand adaptation in response to new information and technology Although the value of stability analysis has long been recognized implementation in design has historically been difficult as calculations were performed primarily by hand. Various methods were created to simplify the analysis and allow the engineer to partially include the effects of stability via hand calculations. However, w ith the development of powerful analysi s software, rigorous methods to account for stability effects were developed. While stability analysis calculations can still be done by hand most engineers now have access to software that will complete a rigorous stability analysis. The majority of th e methods presented here assume that software analysis is utilized Stability analysis is a broad term that covers many aspects of the design process. According to the 2010 AISC Specification for Structural Steel Buildings (AISC 36010) stability analys is shall consider the influence of second order effects (P imperfections, and member stiffness reduction due to residual stress es
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2 Both the 2005 and 2010 AISC Specification for Struc tur al Steel Buildings recognize at least three methods for stability analysis : (AISC 36005 and AISC 36010) 1. First Order Analysis Method 2. Effective Length Method 3. Direct Analysis Method Other methods for anal ysis may be used as long as all elements addressed i n the prescribed methods are considered Stability analysis is required for all st eel structures according to AISC 360 10. The application of methods for stability analysis in design of structures varies greatly from firm to firm and from engineer to en gineer. A crucial principle for engineers in the process of design is the inclusion of stability analysis in design. If stability analysis is not performed or a method of analysis is incorrectly applied, the ability of the structure to support the requir ed load is potentially jeopardized The analysis of nearly all complex structures is completed using advanced analysis software capable of performing various methods of analysis. T herefore omitting stability analysis in the design of structures creates u nnecessary r isk and is unjustified
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3 1.2 Pipe Racks in Industrial Facilities Pipe racks are structures used in various types of plants to support pipes and cable trays. Although pipe racks are considered nonbuilding st ructures, they should still be designed w ith the effects of stability analysis considered. Pipe racks are typically long, narrow structures that carry pipe in the longitudinal direction. Figure 1 1 shows a typical pipe rack used in an industrial facility. Pipe routing, maintenance access, an d access corridors typically require that the transverse frames are moment resisting frames. The moment frames resist gravity loads as well as lateral loads from either pipe loads or wind and seismic loads. The transverse frames are typically connected using longitudinal struts with one bay typically braced. Any longitudinal loads are transferred to the longitudinal struts and car ried to the braced bay. (Drake and Walter, 2010)
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4 Figure 1 1 Typical Four Level Pipe Rack Consisting of Eight Transverse Frames Co nnection by Longitudinal Struts Pipe racks are essential for the operation of industrial facilities but because pipe racks are considered non building structures, code referenced documents will usually not cover the design and analysis of the str ucture. The lack of industry standards for pipe rack design leads to each individual firm or organization adopting its own standards many without clear understanding of the concepts and design of pipe rack structures. (Bendapodi, 2010) Process Industry Practices Structural Design Criteria ( PIP STC01015) has tried to develop a uniform standard for design but it should be noted that this is not considered a code document.
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5 The lack of code referenced documents can lead to confusion in the design of pipe ra cks. The concept of stability analysis should not be ignored based the lack on code referenced documents AISC 36010 should still b e used as reference for stability analysis and design
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6 2. Problem Statement 2.1 Introduction In dustrial facilities typically have pipes and utilities running throughout the plant which require large and lengthy pipe racks Pipe racks not only are used for carrying pipes and cable trays, but many times define s access corridors or roadways. I t is rel atively easy to add a braced bay in the longitudinal direction of a pipe rack because pipes and utilities run parallel to access roads It is much more difficult to add bracing to the pipe rack in the transverse direction because of the potential for inte rference with pipes, utilities, corridors and access roads Therefore moment connections in the transverse direction of the pipe rack are typically used Figure 2 1 shows an elevation view of a length of pipe rack. Figure 2 2 shows a section view of the same pipe rack showing the moment resisting frame. Pipe racks are a good example of structures that can be subject to large second order effects. The current AISC 36010 defines three methods for stability analysis: 1. First Order Analysis Method 2. Effectiv e Length Method 3. Direct Analysis Method L imitations restrict practical application for certain methods.
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7 Figure 2 1 Typical Elevation View of Pipe Rack Figure 2 2 Section View Showing Moment Resisti ng Frame
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8 2.2 Significance of Research If stability analysis is not performed or a method is incorrectly applied, this could jeopardize the ability of the structure to support the required loads. Most of the current literature on pipe racks discusses the application of loads and has suggestions on design and layout of pipe racks, while little applicable information is available on comparing the three methods of stability analysis for pipe racks. Currently the design engineer must research each method of stability analysis and decide which method to apply for analysis After the analysis is completed, the engineer must then verify that the pipe rack meets all the requirements of the applied analysis method. If the requirements of AISC 36010 methods are not met for the structure, then the engineer must completely reanalyze the structure using a new method of stability analysis which will meet the requirements Comparing the various types of stability analysis will not only show the engi neer which method wil l provide the most accurate analysis based on method limitations but will also show why stability analysis is crucial. 2.3 Research Objective The main purpose of this thesis will be to analyze various types of pipe rack structures, compare the results from st ability analyses, and describe both positive and negative aspects of each method of stability analysis as it applies specifically to pipe
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9 rack structures The paper will also look at some of the various issues with applying each of the methods. Some eng ineers are accustomed to braced frames structures, which are not susceptible to large second order effects therefore those designers can tend to neglect or incorrectly apply methods of stability analysis This thesis will not only show the importance of stability analy sis, but also provide suggestions on practical implementation of each method. This could potentially save time in analysis and design because the process of selecting the appropriate st ability analysis method will no longer be based on tria l and error but rather on educated considerations that can easily be verified after analysis.
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10 3. Literature Review 3.1 Introduction This section will focus on review of the available literature on the subject of both pipe rack loading as well as stabil ity analysis. Literature on the general theory of stability analysis will be reviewed. The main focus of this literature review will be on the three methods prescribed by AISC 36010. Layout and loading guidelines for pipe racks will also be reviewed as this has a major influence on stability 3.2 Pipe Rack Loadin g 3.2.1 Load Definitions Pipe racks are unique structures that have unique loading when compared to typical buildings and structure. Pipe racks design is not covered under Minimum Design Loads for Buil dings and Other Structures ( ASCE 7 05) or International Building Code ( IBC 2009) however the design philosophies should remain the same as that for all structures Most company design criteria and Process Industry Practices (PIP) documents will list ASCE 705 or IBC as the basis for load definition and loa d co mbinations. There are several primary load s which should be considered in the design of pipe racks in addition to loads defined by ASCE 705 or IBC 2009. ASCE 705 primary load cases are as follows:
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11 Ak = load or load effect arising from extraordinary event A D = dead load Di = weight of ice E = earthquake load F = load due to fluids with well defined pressures and maximum heights Fa = flood load H = load due to lateral earth pressure, ground water pr essure, or pressure of bulk materials L = live load Lr = roof live load R = rain load S = snow load T = self straining force W = wind load Wi = wind onice determined in accordance with ASCE 7 05 Chapter 10 According to AISC 36010, regardless of the method of analysis, consideration of notional loads is required. The notional loads may be required in all load combinations if certain requirements of the stability analysis are not satisfied.
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12 The magnitude of notional load will vary based on the method used. Therefore the additional primary load cases per AISC 36010 are as follows: N = notional load per AISC, applied in the direction that provides the greatest destabilizing effect PIP STC01015 states that pipe racks shall be designed to resist the minimum loads defined in ASCE 705 as well as the additional loads described therein PIP STC01015 breaks down the dead load into various categories that are not defined in ASCE 7 05. In addition, various loads from plant operation are defined and required for c onsideration in design. PIP STC01015 breaks down the ASCE 705 Dead Load (D) by dividing the dead load into the subcategories listed below. Ds = Structure dead load is the weight of materials forming the structure (not the empty weight of process equipme nt, vessels, tanks, piping nor cable trays), foundation, soil above the foundation resisting uplift, and all permanently attached appurtenances (e.g., lighting, instrumentation, HVAC, sprinkler and deluge systems, f ireproofing, and insulation, etc ). Df = Erection dead load is the fabricated weight of process equipment or vessels.
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13 De = Empty dead load is the empty weight of process equipment, vessels, tanks, piping, and cable trays. Do = Operating dead load is the empty weight of pr o cess equipment, vessels, tanks, piping and cable trays plus the maximum weight of contents (fluid load) during normal operation. Dt = Test dead load is the empty weight of process equipment, vessels, tanks, and/or piping plus the weight of the test medium contained in the system. PIP STC01015 also provides additional primary load cases from the effects of thermal loads caused from operational temperatures in the pipes. T = Self straining thermal forces caused by restrained expansion of horizontal vessels, heat exchangers, and st ructural members in pipe racks or in structures. This is essentially the same load case as defined in ASCE 7 05. Af = Pipe anchor and guide forces. Ff = Pipe rack friction forces cause by the sliding of pipes or friction forces cause by the sliding of hor izontal vessels or heat exchanges on their supports, in response to thermal expansion.
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14 Seismic loads are also discussed in PIP STC01015. S eismic event s can occur either when the plant is in operation or during shutdown when the pipes are empty. T herefore two seismic load cases are defined as follows : Eo = Earthquake load considering the unfactored operating dead load and the applicable portion of the unfactored structure dead load. Ee = Earthquake load considering the unfactored empty dead load and the ap plicable portion of the unfactored structure dead load. 3.2.2 Dead Loads Further information on the dead loads specifica lly for pipe racks is defined in PIP STC01015. The operating dead load for piping on a pipe rack shall be 40 psf uniformly distributed over e ach pipe level. The 40 psf load is equivalent to 8 inch diameter, schedule 40 pipes, full of water, at 15 inch spa cing. For pipes larger than 8 inch, the actual load of pipe and contents shall be calculated and applied as a concentrated load. The emp ty dead load (De) is defined for checking uplift and minimum load conditions. Empty dead load ( De) is approximately 60% of the operating dead load (Do) which is equivalent to 24 psf uniformly distributed over each pipe rack level. This is an acceptable approximation unless calculations indicate a different percentage should be used. (PIP STC01015)
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15 Pipe racks for industrial applications are usually designed with consideration for potential future expansion. Therefore, additional space or an additional lev el should be provided and the rack should be uniformly loaded across the entire width to acco unt for pipes that may be placed there in the future. Cable trays are often supported on pipe racks and typically occupy a level within the rack specifically desig nated for cable tray The operating dead load (Do) for cable tray levels on pipe racks shall be 20 psf for a single level of cable tray and 40 psf for a double level. These uniform loads are based on estimates of full cable tray over the area of load app lication. (PIP STC01015) The degree of usage for cable tray s can vary greatly. T he empty dead load (De) should be considered on a case by case basis. E ngineering judgment should be used in defining the cable tray loading because empty dead load ( De) is defined for checking upl ift and minimum load conditions. 3.2.3 Live Loads Live load should be applied to pipe racks as needed. Pipe racks typically have very few platform or catwalks. When platforms are required for access to valves or equipment located on t he pipe rack structure, the platform and supporting structure should be designed in accordance with ASCE 705 Live Loads.
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16 3.2.4 Thermal and Self Straining Loads Temperature effects on structural steel members should be included in design. PIP STC01015 introduces two additional self straining loads. These additional loads are caused by the operation effects on the pipes. The operational temperatures of pipes need to be considered in design. Support conditions of pipes vary greatly and need to be considered in design. A pipe may be supported to resist gravity only, or may have varying degrees of restraint from guided in a single dire ction to fully anchored supports Pipe stress analysis can be completed for all the pipes located in the pipe rack. This stre ss analysis takes into account the support type and location for each support and provides individual design forces for each pipe at that specific location These resultant pipe loads can be used for design. However, application of loads in this manner does not include additional loads for futures expansion. Therefore, a uniform load at each level of the rack is typically applied in lieu of actual pipe forces. Local support condition should also be verified where large anchor forces are present. 3.2.5 Snow L oad and Rain Loads Snow loading should be considered in the design of pipe racks. P ipe racks typically do not have roofs or solid surfaces that large amounts of snow can collect on, therefore the engineer may reduce the snow load by a percentage using eng ineering judgment based on percentage of solid area and operational temperatures
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17 of pipes. Based on the reduced area for snow to accumulate, snow load combinations will usually not control the design of pipe racks. (Drake, Walter, 2010) Rain loads are int ended for roofs where rain can accumulate. Because pipe racks typically have no solid surfaces where rain can collect, rain load usually does not need to be considered in design of pipe racks. (Drake, Walter, 2010) 3.2.6 Wind Loads ASCE 7 05 provides very littl e, if any guidance for application of wind load for pipe racks. The most appropriate application would be to assume the pipe rack is an open structure and design the structure assuming a design philosophy similar to that of a trussed tower. See Table 3 1 below for Cf, force coefficient. This method requires the engineer to calculate the ratio of solid area to gross area of one tower face for the segm ent under consideration. This may become very tedious for pipe rack structures because each face can have varying ratios of solids to gross areas.
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18 Table 3 1 Force coefficient, Cf for open structures trussed towers ( Adapted from ASCE 7 05) Tower Cross Section Cf Square 4.0 2Triangle 2Notes: 1. For all wind directions considered, the area Af consistent with the specified force coefficients shall be the solid area of a tower face projected on the plane of that face for the tower segment under consideration. 2. The specified force coefficients are for towers with structural angles or similar flat sided members. 3. For towers containing rounded member, it is acceptable to multiply the specified force coefficients by the following f actor when determining wind forces on such members: 2 4. Wind forces shall be applied in the directions resulting in maxim um member forces and reactions. For towers with square cross sections, wind forces shall be multiplied by the 5. Wind forces on tower appurtenances such as ladders, conduits, lights, elevators, etc., shall be calculated using appropriate force coefficients for these elements. 6. Loads due to ice accretion as described in Section 11 shall be accounted for. 7. Notation: a to gross area of one tower face for the segment under consideration. The method generally used for pipe rack wind load application comes from Wind Loads for Petrochemical and Other Industrial Fac ilities (ASCE, 2011). This report provides an approach for wind loading based on current practices, internal company standards, published documents and the work of related organizations.
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19 Design wind force is defined as: (ASCE 705 Eqn 5.1) F = qz*G* Cf *A With: qz = Velocity pressure determined from ASCE 7 05 Section 6.5.10 G = Gust effect factor determined from ASCE 7 05 Section 6.5.8 Cf is defined as the force coefficient and varies based on the shape and direction of wind. Structural members can have force coefficients between 1.5 and 2. Cf can be taken as 1.8 for all structural members or equal to 2 at and below the first level and 1.6 above the first level. No shielding shall be considered. Cf for pipes should be 0.7 as a minimum. Cf for cable should be taken as 2.0. (ASCE, 2011) These values of Cf are developed based on the Table 3 2 below Cable tray are considered square in shape with h/D = 25 corresponding to Cf = 2.0. Pipe are round in shape with h/D = 25 and a moderately smooth surface corresponding to Cf = 0.7.
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20 Table 3 2 Cf force coefficient ( Adapted from ASCE 7 05) Cross Section Type of Surface h/D 1 7 25 Square (wind normal to face) All 1.3 1.4 2 Square (wind along diagonal) All 1 1.1 1.5 Hexagonal or octagonal All 1 1.2 1.4 Round (D Moderately smooth 0.5 0.6 0 .7 Rough (D`/D = 0.02) 0.7 0.8 0.9 Very rough (D`/D = 0.08) 0.8 1 1.2 Round (D All 0.7 0.8 1.2 Notes: 1. The design wind force shall be calculated based on the area of the structure projected on a plane normal to the wind direction. The force shall be assumed to act parallel to the wind direction. 2. Linear interpolation is permitted for h/D vales other than shown. 3. Notation: D: Diameter of circular cross section and least horizontal dimension of square, hexagonal or octagonal cross section at elevation under consideration in feet D`: Depth of protruding elements such as ribs and spoilers, in feet h: Height of structure, in feet q z : Velocity pressure evaluated at height z above gro und, in pounds per square foot The tributary area (A) for pipes is based on the diameter of the largest pipe (D) plus 10% of the width of the pipe rack (W), then multipl ied by the length of the pipes ( L ) (usually the spacing of the bent frames). The tr ibutary area for pipes is the projected area of the pipes based on wind in the direction perpendicular to the length of pipe. Wind load parallel to pipe is typically not considered in design since there is typically very little projected area of pipe for applying wind pressure (ASCE, 2011)
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21 The tributary area takes into account the effects of shielding on the leeward pipes or cable tray. The 10% of width of pipe rack is added to account for the drag of pipe or cable tray behind the first windward pipe. It is based on the assumption that wind will strike at an angle horizontal with a slope of 1 to 10 and that the largest pipe is on the windward side. (ASCE, 2011) The tributary area for structural steel members and other attachments should be based on the projected area of the object perpendicular to the direction of the wind. Because the structural members are typically spaced at greater distances than pipes, no shielding effects should be considered on structural members and the full wind pressures should be applied to each structural member. The gust effect factor G and the velocity pressure qz, should be determined based on ASCE 705 sections referenced above. 3.2.7 Seismic Loads Pipe racks are typically considered non building structures, therefore seismic design should be carried out in accordance with ASCE 705, Chapter 15. A few slight variations from ASCE 7 05 are recommended. The operating earthquake load Eo is developed based on the operating dead load as part of the effective seis mic weight. The empty earthquake load Ee is developed based on the empty dead load as part of the effective seismic weight. (Drake and Walter, 2010)
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22 The operating earthquake load and the empty earthquake load are discussed in more detail in the load comb inations for pipe racks. Primary loads, Eo and Ee are developed and used in separate load combinations to envelope the seismic design of the pipe rack. ASCE Guidelines for Seismic Evaluation and Design of Petrochemical Facilities (1997) als o provides furt her guidance and information on seismic design of pipe racks. The ASCE guideline is however based on the 1994 Uniform Building Code (UBC) which has been superseded in most st ates by ASCE 7 05 or ASCE 710. Therefore the ASCE guideline should be considere d as a reference document and not a design guideline 3.2.8 Load Combinations Based on the inclusion of additional primary load cases as specified by PIP STC01015, additional load combinations need to be considered. PIP STC01015 specifies load combinations to be used for pipe rack design. Both LRFD and ASD load combinations are specified. LRDF load combinations will be the focus of this section as AISC LRFD will be used for analysis and design. ASD load combinations should be considered when checking servic eability limits on pipe racks. Because additional primary load cases are included in the design and ASCE 705 does not govern the design of pipe racks because they are typically considered nonbuilding structures, PIP STC01015 load combinations should be used. In
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23 practice, PIP STC01015 load combinations and ASCE 7050 load combinations are very similar and a combination of the specified load combinations can be used. ASCE 7 05 primary load cases must be redefined with the additional subcategories of loa ds defined by the general primary load cases. Example: Dead load as defined by ASCE 7 05 needs to be broken down into additional primary load cases such as the dead load of the structure, the dead load of the empty pipe, etc PIP STC01015 LRFD load com binations specified for pipe racks are listed below: 1. 1.4( Dso Fff) 2. 1.2( Dso Afo) 3. 0.9( Dse 4. a) 0.9( Dsofo b) 0.9( Dsee 5. 1.4( Dst) 6. 1.2( Dstp ASCE 7 05 LRFD load combinations are listed below: 1. 2. r or S or R) 3. r W ) 4. r or S or R) 5.
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24 6. 7. When comparing the two sets of load combinations, there are some similarities. C ertain primary loads such as live load, live roof load, snow load and rain load do not typically apply or control the design of the pipe racks, therefore most load combinations with these primary load cases will not control the design. Therefore ASCE 7 05 load combination 2 and 3 will not be considered in design. Taking into account the subcategories of primary load cases used in PIP STC01015, ASCE 7 05 load combinations can be compared directly and a comprehensive list of all load combinations can be developed. Below is listed the combined load combinations to be used in this research for design of pipe racks referenced from PIP STC01015. Reference of specific load combination number from ASCE 7 05 is also included if applicable. 1. 1.4(Dsoff) ASCE 1 2. 1.2( Dso Afor 1.0Eo) ASCE 4 and 5 3. 0.9(Dse ASCE 6 4. a) 0.9( Dsofo ASCE 7 b) 0.9( Dsee 5. 1.4( Dst) 6. 1.2( Dstp
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25 It can be seen in the above load combinations that ASCE 705 load combinations 1, 4, 5, 6 and 7 are covered by the PIP S TC01015 load combinations. Slight changes such as the inclusion of Af are added to load combinations per the direction of PIP STC01015. Additional load combinations to cover test load conditions partial wind, Wp, during test and seismic on the empty condition are covered by PIP. Engineering judgment should be used to determine if any additional load combinations should be considered in design. ASD load combinations from both PIP STC01015 and ASCE705 are combined in a similar fashion to come up with a c ombined list of load combinations used for design. 3.3 Column Failure and Euler Buckling An ideal column is considered to be perfectly straight with the load applied directly through the centroid of the cross section. Theoretically the load on an ideal column can increase until the limit state occurs by yielding or rupture. Figure 31 shows a graph of axial load P vs lateral deflection y. The axial load is increased until yielding occurs with no lateral deflection
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26 Figu re 3 1 Load vs. Deflection Yielding of Perfect Column For slender columns, this yielding is never reached. The axial load is increased to a point of critical loading where the column is on the verge of becoming unstable. The critical load is determine d as the point where if a small lateral load (F) were applied at the mid span of the column, the column would remain in the deflected position even after the later al load was removed. Any additional load will cause further lateral displacement. This is shown in Figure 32
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27 Figure 3 2 Visual Definition of Critical Buckling Load Pcr This critical load for slender columns is based on Euler buckling. Euler buckling load is the theoretical maximum load that an ideal pin ended c olumn can support without buckling. (Euler, 1744) It is stated as: P cr 2E I L2 Pcr = Euler Buckling Load or Critical Buckling Load L = Length of Column E = Modulus of Elasticity I = Moment of Inertia of Column Bifurcation is the point when the column is in a state of neutral equilibrium as the critical buckling load is applied to the column. At the point of bifurcation, the column is on the verge of buckling. Instead of the graph shown in Figure 31, the
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28 graph now is shown in Figure 33. The load is increased to the critical load where the column becomes unstable and buckling can occur. (Hibbeler, 2005) Figure 3 3 Load vs. Deflection Euler Buckling Euler s formula for critical load was derived based on the assumption of an ideal column. However, ideal columns do not exist T he load is never applied directly through the centroid and the column is never perfectly straight. The existence of load eccentricities, out of plumb members, member geometric imperfe ctions material flaws residual stresses, therefore second order effects become the basis for stability analysis. Based on the discussion above, most real columns will never suddenly buckle but will slowly bend due to the eccentricities and out of straig htness. (Hibbeler, 2005)
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29 3.4 Stability Analysis 3.4.1 AISC Specification Requirements AISC 36010 Specification fo r Structural Steel Building states in section C1. Stability shall be provided for the structure as a whole and for each of its elements. The effects of all of the following on the stability of the structure and its elements shall be considered: (1) flexural, shear and axial member deformations, and all other deformations that contribute to displacements of the structure; (2) second order effects (both P and P imperfections; (4) stiffness reductions due to inelasticity; and (5) uncertainty in stiffness and strength. All loaddependant effects shall be calculated at a level of loading corresponding to LRFD load combinations of 1.6 times ASD load combinations. 3.4.2 Second Order Effects Second order effects are a means to account for the increase in forces based on the deformed shape of the member or frame. Second order effects can be further broken down to P effects. P effects are the effects of loads acting on the displaced location of joints or nodes in a structure. P actin g on the deflected shape of a member between joints or nodes. Figure 34b shows the effects of P and Figure 3 4a shows the effects of P (AISC 36010) (Ziemian, 2010)
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30 Figure 3 4 Second Order P M oments (Ziemian, 2010) Second order effects are typically the driving factor for stability analysis. All the other requireme nts for stability design are implemented based on the effect that they will have on the second order effects. The importance of second order effects becomes readily apparent when comparing an example frame using first order elastic analysis and a second order elastic analysis as shown in Figure 3 5. Lateral displacements and as a result, member forces, can be dr astically underestimated if a first order analysis is performed.
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31 Figure 3 5 Comparison of First Order Analysis to Second Order Analysis ( Adapted from Gerschwindner, 2009) Second order analysi s can be carried out per AISC 36010 methods of either rigorous second order analysis or the approximate second order analysis, both of which are discussed in more detail in further sections. The importance of second order effects on overall stability of structure can be seen in various structural failures over the years. For example, i n 1907, the Quebec Bridge collapsed during construction due to failure of the compression c hord s of the
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32 truss. In the weeks previous to the collapse, deflection s in the c hords was notice d and reported but nothing was done and work continued until the eventual collapse of the bridge While many factors led to the failur e of the bridge, one of the e rrors made in the design of the bridge was in the design of the compression chords. The compression chords were fabricated slightly curved for aesthetic reasons. However, because this member curvature complicated the calculations, the members were design ed as straight members. Therefore, the actual second order effects were increased and the buckling capacity was reduced when compared to the straight member as designed (Delatte, 2009) As can be seen from Figure 3 4 and 35, the second order effects can increase both the displacement and moments to failure much before Euler buckling load is ever reached. Figure 3 6 shows the Quebec Bridge prior to collapse and Figure 37 shows the result of the failure. Figure 3 6 Quebec Bridge Prior to C ollapse (Canada, 1919)
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33 Figure 3 7 Quebec Bridge after F ailure (Canada, 1919) Some general points concerning second order analysis made by Ziemian (2010) are as follows: 1. Second order behavior can affect all components and internal member forces within a structure. 2. Se cond order moments do not necessarily have the same distribution as the first order moments and therefore the first order moments cannot be simply amplified. LeMessurier (1977) and Kanchanalai and Lu (19 79) however give several practical applications wher e amplification of first order moments to achieve an approximation of second order moments is applicable.
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34 3. All structures will experience both Peffects but the magnitude of the effects will vary greatly between structures. 4. Linear superposition of effects cannot be used with second order analysis; the response is nonlinear. 3.4.3 Flexural, Shear and Axial Deformation Second order effects are required in stability analysis as explained in the previous section. A ccurate deformation must be calculated because second order effects are based on deformation to determine the amplified moments and forces. The AISC 36010 requires that flexural, shear and axial deformations be considered in the design. Although flexural deformations will usually be the larg est contributor to overall structural deformation, axial and shear deformations should not be ignored. Figure 38 shows a frame and the calculated deformations from flexural, shear and axial. If shear and axial deformations were ignored in the design and analysis, the amplifications of moments and forces from second order effects could be underestimated and the structure could become unstable.
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35 Figure 3 8 Deformation from Flexure, Shear and A xial ( Adapted from Geschwindner, 2009) 3.4.4 Geometric Imperfections Geometric imperfections refer to the out of straightness, out of plumbness material imperfections and fabrication imperfections. The maximum allowable geometric imperfections are set by AISC Code of Standard Practice for Ste el Building and Bridges. The main geometric imperfections of concern in stability analysis are member out of straightness and frame out of plumbness. Member out of straightness is limited to L/1000, where L is the member length between brace or frame poi nts.
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36 Frame out of plumbness is limited to H/500, where H is the story height. (AISC 36010) (AISC 30310) Geometric imperfections cause eccentricities for axial loads in the structure and members These eccentricities need to be accounted for in stabili ty analysis because they can cause destabilizing effects and increased moments Eccentricities also increase the second order effects in the analysis of the structure. Maximum tolerances specified by AISC Code of Standard Practice for Steel Buildings and Bridges should be assumed for analysis unless actual imperfection values are known. (AISC 36010) C olumns are never perfectly straight and contain either member out of straightness or general out of plumbness, therefore an initial eccentricity of the axial load will be experienced. Figure 3 9 shows how the real column will behave with an initial imperfection ( yo). Nominal column capacity (Pn) will be reached well before Euler Buckling load (Pcr). Second order effects are included in this graph of ax ial load vs deformation. Based on the presence of the initial displacement, moment will develop and the column will typically yield based on flexure and compression while the theoretical Euler b uckling will never be reached.
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37 Figure 3 9 Load vs. Deflecti on Real Column Behavior with Initial I mperfection s 3.4.5 Residual Stresses and Reduction in Stiffness Residual stresses are internal stresses contained in a structural steel member. There are several sources of residual stres ses: (Salmon and Johnson, 2008) 1. Uneven cooling after hot rolling of the structural member. 2. Cold bending or cambering during fabrication. 3. Punching holes or cutting during fabrication. 4. Welding. Uneven cooling and welding typically produce the largest res idual stresses in a member. Local welding for connections does produce residual stresses but the presence of these stresses tend to be localized and are not considered in overall column or beam design strength. Residual stress from uneven cooling happens when the rolled shape is cooled at room temperature from the rolling temperatures. Certain
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38 areas of the member will cool more rapidly than others. For example, the flange tips of a wide flange shape are surrounded by air on three sides and cool more rap idly than the material at the junction of the flange and web. As the flange tips cool, they can contract freely because the other regions have yet to develop axial stiffness. When the slower cooling sections begin to cool and contract, the axial stiffnes s from the cooled regions restrains the contraction thus creating compression on the faster cooling section and tension in the slower cooling sections. Figure 3 10 shows the residual stresses typically seen in hot rolled wide flange shapes from uneven cooling. (Vinnakota, 2006) (Huber and Beedle, 1954) (Yang et al. 1952) Figure 3 10 Residual Stress Patterns in Hot Rolled Wide Flange Shapes Welding of built up sections produces residual stresses as a result of the localized heating applied during the welding. A built up wide flange shape will have
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39 compression on the flange tips and middle of the web and tension around the junction of the web and flange. (Vinnakota, 2006) The presence of residual stress es results in a nonli near behavior of the stress strain curve. The average yield stress of the section is reduced by the amount of residual stress in the member. Therefore the section will start to yield before the stress reaches the theoretical yield stress of a member with no residual stress. Linear elastic behavior is experienced to the point of theoretical yield stress (Fy) minus the residual stress (see Figure 3 11). After this point, nonlinear behavior is experienced and plasticity begins to spread through the section. (Salmon and Johnson, 2008) Figure 3 11 Influence of Residual Stress on Average Stress Strain C urve (Salmon and Johnson, 2008)
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40 Residual stresses need to be considered in stability analysis because of the effect of general softening of the structure from the spread of plasticity through the cross section causing reduced stiffness. The reduced stiffness increases deflections and therefore increases the second order effects on the structure. (AISC 36010) Beam and column des ign strength is calculated based on empirical equations which take into account the residual stress which is assumed to follow a Lehigh pattern which is a linear variation across the flanges and uniform tension in the web. The AISC 36010 strength equations were developed and calibrated based on research from Kanchanalai (19 77) and ASCE Task Committee (1997). Figure 312 shows the idealized residual stresses for typical wide flange sections which follows the Lehigh pattern The residual stresses are assum ed to be 0.3Fy in wide flange shapes. (AISC 36010) (Ziemian, 2010) (Deierlien and White, 1998) Figure 3 12 Idealized Residual Stresses for Wide Flange Shape M embers Lehigh Pattern ( Adapted from Ziemian, 2010)
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41 3.5 AISC Method s of Stability Analys is As discussed before, AISC 36010 states that any method that considers the influence of secondorder effects, flexural, shear and axial deformation, geometric imperfections, and me mber stiffness reduction due to residual stresses on the stability of the structure and its elements is permitted. Various types of methods of have been developed and AISC 36010 detail ed the requirements for a few of these methods. Each method listed, does in some way, address all the various requirements specified by AISC 36010. All methods listed in AISC 36010, excluding the f irst order a nalysis m ethod, require a second order analysis Table 2 2 from AISC 36010 provides a summary of requirements and limitations of each of the methods. A ISC 36010 allows two types of second order analysis; Approxi mate Second Order Analysis and Rigorous Second Order Analysis. Both methods of second order analysis either accurately account for or approximate geometric nonlinear behavior. In reality, geometric nonl inear behavior is only one of the nonlinear types of behavior that should be considered in design. Material nonlinear behavior (inelastic analysis) caused by reduction in stiffness should also be considered in design. Figure 313 shows the resul ts from v arious types of analyses.
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42 Figure 3 13 Load vs. Deflection Comparison of Analysis Types ( Adapted from White and Hajjar, 1991) While software is available that perform s a true second order inelastic analysis, it is very computationally expensive, and therefore other measures must be considered in analysis and design. AISC 36010 strength equations are typically based on the results from a second order elastic analysis. The AISC LRFD general approach for strength and sta bility can be represented by the following equation: i*Qi n
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43 The left hand side of the formula represents the effects of factored loads on a structural member, connection and the right side represents the design resistance or design strength of the specified element with: Qi = internal forces created b y applied load Rn = Nominal member or connection strength i = factor to account for variability in load (load factor) (strength reduction factor) Geometric nonlinear behavior can be accounted for using a second order elastic analysis (left side of the equation). These load effects are then compared to resistance based on material and geometric inelasti city (right side of the equation). (Yura el al. 1996) As Figure 3 13 shows, a direct comparison of loa d effects from second order elastic analysis and member resistance is not compatible because the inelastic material deflections are not considered in an elastic second order analysis. Therefore AISC 36010 design equations should be calibrated for the res ults of an elastic second order analysis or the effects of material inelasticity must be account ed for in the elastic second order analysis. (Ziemian, 2010) This can be accomplished through various methods discussed in more detail in further sections. 3.5.1 R igorous Second Order Elastic Analysis To fully capture the second order effects as described in previous sections, nonlinear geometric behavior should be accurately calculated. With the constant
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44 increase of computational capabilities, rigorous second order analysis is becoming much more common in design practice. It should be noted that the AISC 36010 definition of rigorous second order analysis is typically not meant to represent a true nonlinear second order analysis but will still produce results that accurately calculate the second order effects. Many methods can be used for analysis but the general form is usually expressed as: (Ziemian, 2010) {dF}{dR} = K{d With: {dF } = Vector of i ncremental applied nodal forces {dR } = Vector of unbalanced nodal forces, difference between current internal forces and applied loads K = Stiffness matrix {d = Vector of incremental nod al displacements and rotations Most solutions use an iterative approach for solving for second order effects. The unbalanced forces are calculated based on the deformed geometry at the end of each iteration and used as the basis for the next iteration. Iterations can be performed until the unbalanced force vector is determined to be negligible. Figure 3 14 shows a method commonly referred to as the NewtonRaphson incremental iterative solution.
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45 Figure 3 14 Visual Representation of Incremental Iterative S olu tion P rocedure ( Adapted from Ziemian, 2010) Additional methods based on the NewtonRaphson incremental iterative solution have been developed based on the limitation imposed by the use of this solution. McQuire et al. (2000) and Chen and Lui (1991) have provided general overview of additional methods A variation of these methods that can be used by STAAD.Pro V8 i is referred to as the Lagrangian procedure. This procedure revises the stiffness matrix, K, to take the following form:
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46 With Ke being the standard linear elastic stiffness matrix and Kg being the geometric stiffness matrix. This method recognizes that as the structure deforms the stiffness associated with the member forces is changed at each increment of the solutions. This l eads to more accurate results and the ability to use this type of solution for dynamic analyses because the method accounts for change in the natural period due to second order effects and the stif fening of the structure. (Galambos, 1998) This procedure i s given in more detail by Crisfield (1991), Yang and Quo (1994) and Bathe (1996). STAAD Technical Manual (2007) also gives details of how this method is applied for analysis. 3.5.2 Approximate Second Order Elastic Analysis In lieu of a rigorous second order el astic analysis which is iterative and can demand extensive computational effort, approximate second order analysis methods have been developed over the years to potentially simplify analysis. Rutenberg (1981, 1982), White et al (2007a,b) a nd LeMessurier ( 1976, 1977) have each developed approximate methods of analysis to either simplify analysis in computer applications or simplify hand calculations. (Ziemian, 2010) The AISC 36010 method of approximate second order analysis uses amplification factors B1 a nd B2 to account for P See AISC 36010 Appendix 8 for the procedure.
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47 Because this approximate second order analysis is typically used for hand calculations of frame analysis, no additional discussion will ensue based on the ass umption that the engineer has access to software capable of a rigorous second order analysis. 3.5.3 Direct Analysis Method Introduced in AISC 36005, the direct analysis method represents a fundamental ly new alternative to traditional stability analysis methods. (Griffis and White, 2010) The most significant development addressed in this method is that column strength can be based on the unbraced length of the member therefore eliminating the need to calculate the effective length of the member (K may be take n a s 1 for all members). (Ziemian, 2010) While all the requirements for AISC stability analysis are covered with this method, slight variations in each requirement are allow ed and discussed in further detail. One major advantage of the direct analysis metho d is that it has been developed and verified for application to all types of structur al system s and therefore has no limitations for use. (Maleck and White, 2003) Accurate second order analysis is the cornerstone of the direct analysis method. As previously discussed, two types of second order analysis are allowed by AISC 36010, rigorous second order analysis and approximate second order analysis. As with all AISC 36010 methods that use second order analysis results the direct
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48 analysis method is built around the assumption of LRFD loads and therefore if ASD loads are to be used in analysis, they must be multiplied by 1.6 before second order analysis is completed because of the nonlinearity of second order effects (AISC 36010) (Nair, 2009) Flexural, shear and axial deformations need to be considered in analysis. As previously discuss, accurate deformations are needed for calculation of second order effects. However, in discussion of shear and axial deformation AISC 36010 explicitly uses the word c onsider instead of include. This allows the engineer to ignore certain deformations based on the type of structural system. For example, the shear deformations could feasibly be neglected in a low rise moment frame and produce results with an error of less than 3%. High rise moment frame systems on the other hand, could produce much higher errors if shear deformations were to be neglected. (AISC 36010) Most modern analysis software is capable of calculating accurate flexure, shear and axial deformat ions, and very little effort by the engineer is required to achieve the most accurate results available by the software. The direct analysis method is calibrated on the assumption that geometric imperfections are equal to the maximum material, fabrication and erection tolerances permitted by the Code of Standard Practice for Steel Buildings and Bridges ( AISC 30310) Geometric imperfection may be accounted for by two methods; direct modeling of imperfections or notional loads. (AISC 36010) Direct modelin g of imperfections can become quit e tedious because as a minimum four models must be
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49 developed, each with the deflection in one of the four principle directions with the additional member out of straightness modeled corresponding to the worst case directi on. Notional loads are defined as horizontal forces added to the structure to account for the effects of geometric imperfections. (Ericksen, 2011) For the direct analysis method the magnitude of notional loads applied to the structure is 0.2% of the total factored gravity load at each story. Ni i With: = 1.0 (LRFD); 1.6 (ASD) Ni = Notional lateral load applied at level i, kips Yi = Gravity load applied at level i from the LRFD or ASD load combinations as applicable, kips It can be see that 0.2% of gravity loads is appropriately select ed as 1/500 which also corresponds to the maximum out of plumbness for columns from AISC 30310. Analysis will show that either applying a notional load of 0.2% or directly modeling out of plumbness will produce similar results (Malek and White, 1998) M e mber out of plumbness is accounted for by notional loads, but member out of straightness still needs to be considered in design. AISC 36010 has developed the column strength equations based on maximum out of straightness tolerances. (Ziemian, 2010) (Whi te et al., 2006)
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50 For the direct analysis method, notional loads should be applied in combination with all gravity and lateral load combinations to create the worst effects. AISC 36010 does however, allow notional loads to be applied only to gravity load combinations as long a s the ratio of second order to first order drifts does not exceed 1.5 using the unreduced elastic stiffness or 1.7 if the reduced elastic stiffness is used in analysis. The errors seen by this simplification are relatively small as l ong as the ratio of drifts remains below the specified limits. (AISC 36010) Reduction in stiffness of the structure is cause d by partial yielding of memb ers This yielding is further accentuated by residual stress es The direct analysis method specifie s reduced stiffness es of EI* and EA* with: EI* = bEI EA* = 0.8EA The reduced stiffness factor of 0.8 is applied for two reasons. The first and most readily apparent is the reduction in stiffness in intermediate and stocky members, namely columns, due to inelastic softening of members before they re ach their design strength. The second reason relates to slender members that are governed by elastic stability. 0.8 is roughly equivalent to the product of = 0.9 and the factor 0.877. These factors are used in development of the AISC column curve (AISC 36010 Eqn. E33) which is modified by the above factors for slender elements to account for member out of straightness. (Ziemian, 2010) (AISC 36010)
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51 b value is an adjustment factor to account for additional reduction in stiffness in cases where hi gh axial stresses are present which can reduce the bending stiffness of the member. (AISC 36010) b Pr/Py b r/Py)[1 ( r/Pyr/Py 0.5 where: = 1.0 (LRFD); 1.6 (ASD) Pr = Required axial compressive strength using LRFD or ASD load combinations. Py = Axial Yield strength (Ag*Fy) AISC 360b = 1.0 for all cases if the notional load is increased by 0.1%Yi. This additional notional load is meant to increase the lateral deformation to envelope the effects caus ed by reduct ion in stiffness in high axial loaded members. However, Powell notes that this method does not appear to be logical because notional loads are meant to account for initial out of plumbness and not for reduction in stiffness (Powell, 2010) Fi gure 3 15 shows a graphical representation of the effect of b on the reduction in stiffness.
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52 Figure 3 15 Reduced Modulus Relationship (Powell, 2010) 3.5.4 Effective Length Method In recent years, the traditional method for stability analysis has been the effective length method. In general, the effective length method calculates the nominal column buckling resistance using an effective length ( KL) and the load effects are calculated based on either a rigorous or approximate second order analysis. (Ziemian, 2010) As with the direct anal ysis method, the effective length is built around determining accurate second order effects. This can be done by either a rigorous second order analysis or by an approximate second order analysis. (AISC 36010) Both methods of second order analysis are based on the assumption that flexural, shear and axial deformations are considered in calculations of second order effects.
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53 Prior to AISC 36005, there were few limitations on applications of the effective length method. Several studies by Deierlein et al. (2002), Maleck and White (2003), and Surovek Maleck and White (2004a and 2004b) have shown that use of the effective length method could produce significantly unconservative results in certain types of framing systems. Therefore AISC 360 05 imposed additional requirements and limitations. 1. Notional loads need to be included in gravity only load combinations to account for member out of plumbness. 2. The ratio of second order drift to first order drift or B2 is limited to 1.5. Geometric imperfections are covered by applications of notional loads or direct modeling of imperfections for analysis. Notional loads are applied in the same manner as described with the direct analysis method with: (AISC 360 10) Ni i However, based on the limitation of the ratio of second order drift to first order drift or B2, by definition, notional loads need only be applied to gravity load combinations. One of the main components of the effective length method is the calculation of the effective length factor K. The most common method for determining K is through the use of the alignment charts found in the commentar y for Appendix 7 in AISC 36010 which can be seen in Figures 316 and 317.
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54 Figure 3 1 6 Alignment Chart Sidesway I nhibited (Braced F rame ) (AISC 360 10)
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55 Figure 3 1 7 Alignment Chart Sidesway Uninhibited (Moment F rame) (AISC 360 10) The alignment charts were developed based on the followi ng assumptions: 1. Behavior is purely elastic. 2. All members have constant cross section. 3. All joints are rigid. 4. For columns in frames with sidesway inhibited, rotations at opposite ends of the restraining beams are equal in magnitude and opposite in direction, producing single curvature bending.
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56 5. For columns in frames with sidesway uninhibited, rotation at opposite ends of the restraining beams are equal in magnitude and direction, producing reverse curvature bending. 6. The stiffness of parameter L (P/EI) of all columns is equal. 7. Joint restraint is distributed to the column above and below the joint in proportion to EI/L for the two columns. 8. All columns buckle simultaneously 9. No significant axial compression force exists in the girders. The assumptions listed above, seldom if ever are seen in a real structure and therefore additional methods of determining K have been developed. Geschwindner(2002) and ASCE Task Committee (1997) have provided a n overview of the various methods for determi ning accurate values of K. In addition to methods listed in AISC 36010, Yura ( 1971) and LeMessurier (19 95) presented various approaches for calculation of the effect length factor K. Folse and Nowak (1995) also presented examples that included the effect of leaning columns The table below gives a comparison of the equivalent length method and the direct analysis method and describes how each method addresses the stability analysis requirements of AISC 36010.
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57 Table 3 3 Comparison of Direct Analysis Me thod and Equivalent Length Method ( Adapted from Nair 2009) Comparison of Basic Stability Requirements with Specific Provisions Basic Requirement in Section 1 of This Model Specification Provisions in Direct Analysis Method (DAM) Provisions in Effective Length Method (ELM) (1) Consider second order effects (both P and P secondorder effects (both P and P Consider second order effects (both P (2) Consider all deformations deformations Consider al l deformations (3) Consider geometric imperfections which include joint position member imperfections Effects of joint position structural response Notional loads Apply notional loads Effects of member imperfections on structural response Included in the stiffness reduction All these effects are considered by using KL from a sidesway buckling analysis in the member strength only difference betw een DAM and ELM is that: Effects of member imperfections on member strength Included in member strength formulas, with KL=L (4) Consider stiffness reduction due to inelasticity which affects structure response and member strength Effects of stiffness reduction on structural response Included in the stiffness reduction Effects of stiffness reduction on member strength Included in member strength formulas, with KL=L DAM uses reduced s tiffness in the analysis; KL=L in the member strength check (5) Consider uncertainty in strength and stiffness which affects structure response and member strength Effects of stiffness/strength uncertainty on structural response Included in the stiff ness reduction ELM uses full stiffness in the analysis; KL from sidesway buckling analysis in the member strength check for frame members Effects of stiffness/strength uncertainty on member strength Included in member strength formulas, with KL=L the column out of order effects may be considered either by rigorous second order analysis or by amplifications of the results of first o rder analysis (using the B1 and B2 amplifiers in the AISC
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58 3.5.5 First Order Method The first order analysis method is a simplified method derived from the direct analysis method. (Kuchenbecker et al., 2004) The main benefit of this method is that only a first order analysis is required. In addition, because it is based on the direct analysis method, K can be set at 1 for all cases. (AISC 360 10) The main simplification in this method comes from the assumption that the ratio of second o rder drift to first order drift or B2, is assumed to be equal to 1.5. From this assumption, equivalent notional lateral loads can be back calculated which simulate the effects of second order effects as well as reduction in stiffness due to partial yielding whi ch are both accounted for using the direct analysis method. (Ziemian, 2010) T he simplification and assumptions which are made in development of the first order method lead to limitations on use of this method. The ratio of second order drift to first orde r drift or B2 is a ssumed to be 1.5, which sets the maximum allowed value for application of this method. The stiffness reduction of 0.8 was also assumed in calculations of additional notion loads and therefore, based on the previous discussion of the redur/Py reduction in stiffness to remain true. The notional load Ni for the first order analysis method is defined as: (Kuchenbecker et al., 2004)
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59 N i B 2 1 0.2 B 2 L Y i B 2 1 0.2 B 2 0.002 Y i If the above assumptions of B2 b = 1.0 are made and substituted into the equation above, it can be simplified to the form seen in AISC 36010 Ni = 2.1( /L)Yi 0.0042Yi With: Ni = Notional load applied at level i, kips = 1.0 (LRFD); 1.6 (ASD) Yi = Gravity load applied at level i from the LRFD or ASD load combinations as applicable, kips = First order interstory drift L = Height of story Unlike the effective length and direct analysis methods, the notional load is required to be added to all load combinations regardless of gravity only or lateral load combinations. While this method has been simplified to eliminate the need for a second order analysis and any calculation of K, the limitations and verification can hinder the use of this method. The simplifications made i n the development of this method make it a very useful tool when frame analysis is done by hand because a second order analysis i s not required. M ost pipe rack structures are designed using modern
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60 analysis software which is capable of performing a second order analysis, therefore the first order method will likely see limited use for pipe rack applications.
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61 4. Research Plan A literature review was first conducted to gather and review the available information pertaining to the design and e ngineering of pipe rack structures for use in industrial facilities. T he industry is constantly evolving, and the most current literature discussing the design and engineering of pipe racks was targeted for review. Next, a literature review was conducted to gather and review the available information pertaining to AISC stability analysis. Literature that described the various methods used by both AISC 36005 and AISC 36010 were the focus of the review. The direct analysis method literature was of parti cular interest as it was a relatively new development in regards to stability analysis. A general plan for the research t hat was conducted is presented here and is described as follows: 1. Use Benchmark Problems from AISC 36010 to test the second order ana lysis capabilities of STAAD and report on the validity of the STAAD approach. 2. Describe in detail a typical pipe rack to be used for comparison of the methods. 3. Develop general loads and l oad combinations for use in the analysis models.
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62 4. Develop a general S TAAD.Pro V8i model that can be used for analysis of the Equivalent Length Method, Direct Analysis Method and First Order Method with input from [2] and [3]. 5. Complete a first order analysis of the pipe rack structure developed in [4] for use in calculatio n of the 21 ratio as well as for use in the F irst O rder M ethod and discuss the results and validity of the method based on AISC limitations 6. Optimize strength only design of test pipe rack structure developed in [4] using Equivalent Length Method and det ermine validity of method for current structure based on AISC limitations 7. Optimize the strength only design of the test pipe rack structure developed in [4] using the Direct Analysis Method and compare the results to the Equivalent Length Method. 8. Use the m odels developed in [ 6 ] and [ 7 ] and vary member sizes and base fixity based on the serviceability limits and compare the results. 9. Compare the results of [5 to 8].
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63 5. Member Design The available strength of members should be calculated in accordance with t he provisions of AISC 36010 Chapters D, E, F, G, H, I, J and K. These chapters should be used regardless of the method of stability analysis chosen. Column design can become a major point of focus during stability analysis based on several factors. The effective length factor for column design can become complex for even simple structures when using the effective length method of stability analysis. Columns typically experience combined flexural, shear and axial load and the interaction between each st ress must be investigated. The two methods of stability analysis ; effective length and direct analysis, as expected produce varying load effects A design example is shown in Figure 51 showing a simple cantilever W10X60 column bent about the strong ax is with an axial load P and the horizontal load H = 10%P. The column is assumed to be support ed out of plane resulting in only strong axis buckling. Stability analys e s using both the equivalent length method and the direct analysis method were performed. A linear elastic analysis was performed as well to establish a first order baseline for comparison. STAAD.Pro V8i was used for both methods of stability analysis while hand calculations were done to compute the linear elastic forces
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64 Figure 5 1 Simple Cantilever Design Example The cantilever column was then checked against the code spe cified available strength. T he column in this example will experience both axial and flexural loads, therefore AISC 36010 Chapter H will be used to determine the available strength. A simple cantilever column was chosen to simplify the selection of the effect length factor K. The theoretical value of K for a fixed base cantilever column is 2. Therefore the effective length of the column used for the determining the available strength using the effective length method will be 30 ft.
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65 Figure 5 2 shows a graph of axial load vs. moment for the simple cantilever example. AISC 36010 equations H11a and H1 1b are included on the graph for de sign purposes. When comparing methods of analysis, with the linear elastic method as a baseline, the differences are readily apparent. As expected, the linear elastic method produces linear results of axial load vs. moment Based on the theory of stabil ity analysis, the linear elastic method is expect ed to produce results that overestimate the axial resistance and underestimate the moment demand. When comparing the results of both the effective length and direct analysis methods, the resistance based on AISC 36010 equations H11a and H1 1b must be adjusted based on the effective length factor K. This will affect the axial resistance of the column section. The direct analysis method assumes that K = 1 and the effective length method assumes that K = 2 for a fixed base cantilever column. This changes the anchor point interaction equations.
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66 0 100 200 300 400 500 600 700 800 900 0 50 100 150 200 250 300 M (ftkips) P (kips) Direct Analysis Method (Demand) Effective Length Method (Demand) Linear Elastic Method (Demand) cPn K=1 cPn KL AISC Eqn. H11a (Resistance) AISC Eqn. H11b (Resistance) Figure 5 2 Simple Cantilever Design Example Results One of the main differences in results between the effective length method and the direct analysis method is the moment demand. The axial load resistance is comparable between the two methods but the moment demands can differ significantly. The column capacity is adjusted between the methods based on the effective length factor which calibrates the axial capacity The difference in moment demand is due to the reduction in stiffness used in the direct analysis which increases deformations which in turn increases eccentricities and therefore moment demand is increased.
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67 Based on the results of this design examp le, several observations can be made : (AISC 36010) 1. Accurate calculations of the effective length factor, K, are critical to achieving accurate results using the effective length method. 2. The moment demand is underestimated when using the effective length m ethod. This can significantly affect the design loads for beams and connections which provide rotational resistance for the column.
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68 6. Pipe Rack Analysis 6.1 Generalized Pipe Rack A typical pipe rack will be developed and used for comparison purposes for this thesis. The typical pipe rack was chosen and modeled based on idealized conditions. A width of 15 feet was chosen to allow one way traffic along the pipe rack corridor. The height of the first level of the pipe rack was set at 20 feet to provi de sufficient height clearance along the access corridor. The overall length of the pipe rack was set at 100 feet. Longer pipe rack sections are typically broken into shorter segments (100 to 200 feet) with each shorter segment separated by expansion jo ints to allow thermal expansion or contraction between segments One of the lengthwise central bays of each segment is typically braced in the longitudinal direction. This allows the length of the pipe rack to expand and contract about a central braced bay and reduces thermally induced loads cause from restraint of thermal movement If each end of the segment were to consist of a braced frame, the length of the pipe rack would essentially be locked in place and higher thermally induced loads would be seen. Moment frames are typically spaced at 15 20 feet. This spacing is typically chosen based on the maximum allowable spans for the pipes or cable tr ays being supported. This spacing can vary based on the estimated size and allowable deflection limits of the pipe being supported
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69 Longitudinal struts are usually offset from the beams used to support the pipes. Levels of the pipe rack are assumed to be fully loaded with pipe, and when the pipes need to exit the rack to the side to connect to equipment, a flat turn cannot be used as this would clash with the other pipes on the same level. The pipe is typically routed to turn either up or down and then out of the rack at the level of the longitudinal struts where the pipe can be supported on the longitudinal struts before exiting the rack. To allow room for pipes to enter and exit the pipe rack, a spacing of 5 feet between levels is typically used If the pipe rack carries larger pipes, additional room may be requi red between levels. This spacing should be determine with the help of the piping engineer on the project. Spacing between pipe rack levels of 5 feet will be used in this thesis. Figure 6 1 show s an isometric view of the typical pipe rack that will be used for analysis and comparison of stability analysis methods. While this is not representative of all pipe racks, it will still provide a useful basis for comparison purposes in pipe rack structures.
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70 Figure 6 1 Isometric View of Typical Pipe Rack Used for Analysis To simplify the design and analysis, a typical moment frame will be selected and isolated for analysis and design. Figure 62 shows an elevation view of a typical moment frame.
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71 Figure 6 2 Section View of Moment Frame in Typical Pipe Rack Out of plane s upports were added at the locations of longitudinal struts which will restrain any movement in and out of the page (see Figure 61 ) L ongitudinal struts all tie into the braced bay, therefore relatively small deflections will be experienced in the weak axis of the columns and the re straint of any movement in this direction is a reasonable assumption. Based on initial calculations that compare the results of the isolated moment frame and the entire pipe rack segment, relatively small differences were seen. Because the braced bay su pports any longitudinal loading, relatively very little weak axis column moment or longitudinal deflection occurs that would affect the design of
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72 the columns or beams that are part of the moment frame. Ratios of demand to capacity showed errors of less than 5% on member design when using the single frame compared to the full pipe rack structure Therefore, analysis of a single moment frame will be used to simplify calculations. T he focus of this thesis will mainly be on t he analysis of the moment frame; t h e braced frame will not be considered in analysis and design. In actual design, engineering judgment should be used to determine if the analysis of a single frame simplification can be made. In many cases, pipe racks are not symmetric and loading can va ry from frame to frame and the overall structure should be analyzed as a whole to determine the load effects. Bracing systems should also be designed to resist the longitudinal loads of the entire segment and therefore modeling of the entire pipe rack seg ment may be required to determine load path and design loads for struts and braces 6.2 Pipe Rack Load ing The pipe rack used for analysis and comparison of methods will have consistent loading between methods to limit the number of variables. In general, load s in the longitudinal direction will not be considered in design and comparison because the focus of analysis will be on the behavior and performance of the moment frame. Therefore, wind and seismic loading will only be considered in the transverse directions.
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73 In practice, engineering judgment should be used in determining all applicable loads. The following discussion is meant to define loads only for analysis and comparison of the stability analysis and therefore certain simplifications are made to f acilitate analysis but still provide results that are typical of pipe racks. All loads are developed based on the assumption that the pipe rack is considered an Occupancy Category III. (PIP STC01015) This will affect the importance factor used for development of wind and seismic loading. The primary load cases, which were defined previously, were chosen according to PIP STC01015. Additional primary load cases may be required based on the requirements of AISC 36010 for notional loads. Loads are developed based in input from ASCE 705. While ASCE 710 is available, PIP STC01015 has not been updated to reflect the changes made by ASCE 7 10. The first primary load case defined was the dead load of the structure ( Ds). This is considered as the self weight o f the steel. For design, an additional 10% of the self weight was added to account for fabrication tolerances and additional materials used for connections. No other loads were assumed at this time for the dead load of the structure. The operating dead l oad as discussed previously is typically applied at 40 psf, which assumes a fully loaded level with 8 inch pipes full of water spaced at 15 inches. The representative pipe rack will use 40 psf applied over the entire tributary
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74 area of the pipe rack. The spacing of the moment frames was chosen as 20 feet, therefore the uniform load applied at each level of the pipe rack is 800 pounds per foot This load can be see n in Figure 6 3. Figure 6 3 Section View of Moment Frame Operating Dead Load The empty dead load of the pipe can be taken as 60% of the operating dead load unless further information is known. This calculates to 480 pounds per foot. The empty dead load of the pipe is applied in a similar fashion as see n in Figure 6 3. Test dead load is the w eight of the pipe plus the weight of the test medium. This type of loading will typically control when the majority of the pipes in the pipe
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75 rack are filled with gas or steam during operation. Hydrotesting is typically done to test the piping prior to startup. Pipes were assumed to be full of water for the operating load, therefore for this analysis, the test dead load is equivalent to the operating dead load and the load is exactly as seen in Figure 6 3. The erection dead load can account for any addi tional loads or reduction in loads due to erection activities. This is typically used for any equipment and is based on the fabricated weight. For piping, the erection dead load and the empty dead load are typically the same. Therefore, the erection dea d load case is not defined at this time and if required in load combinations, the empty dead load can be used in place of the erection dead load. Pipe anchor and pipe friction loads are typically based on actual loading conditions of pi pes located in the p ipe rack. However, without final pipe loading, an estimate of pipe loads must be made. Friction forces can be estimated based on the coefficient of friction between the pipe shoe and support beam. This coefficient of friction is usually assumed to be 0.4. Application of 40% of the operating dead load tends to be extremely conservative. Friction loads are cause by expansion and contraction of pipes. Based on the expansion and contraction of pipes, friction loads are typically seen in the longitudinal direction of the pipe rack. Because it is highly unlikely that all pipe s will expand and contract simultaneously and some pipes may contract while others expand, a more realistic value of 10% of operating dead load will be applied in the longitudinal direction Friction loads are applied to the pipe
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76 rack because the location of the loading could cause additional second order effects in the beams in the moment frames. While friction loads and anchor loads are typically only seen in the longitudinal directio n of the pipe rack, cases where anchor loads are seen in the transverse direction could happen. Therefore apply 5% of the operating dead load as a conservative estimate for the representative pipe rack Figure 6 4 shows the application of pipe anchor load s. In practice, local members should be checked for pipe anchor loads and friction loads as the individual member design may be controlled by high anchor loads.
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77 Figure 6 4 Section View of Moment Frame Pipe Anchor Load Self straining thermal loads will not be considered in design. A design T of 0 degrees Fahrenheit will be applied to all members T hermal loading can cause problems if members are restrained from expansion or contraction. As the 2D moment frame has very little resistance for thermal movements, thermal loads will not be considered in design of the representative model. Additional thermal considerations could be considered but are outside the scope of this thesis.
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78 Live load is typically only applicable to platforms or walkways required for acces s, therefore live load will not be applied based on the assumption of no access platforms or walkways on the simplified pipe rack. Similarly, snow load typically will not control the design and therefore will not be considered in the simplified analysis model. (PIP STC01015) Wind l oad is applied consistent with ASCE 705 principles. For comparison purposes, a 3 second gust wind velocity was assumed to be 100 miles per hour which should cover a larger majority of sites Additional information given by ASCE Wind Loads for Petrochemical Facilities is included in wind load development. The following velocity pressure s at specified heights was developed according to ASCE 705. Table 6 1 shows velocity pressures for cable tray and structural members and Table 6 2 shows velocity pressure s for pipes. Table 6 1 Velocity pressures for cable tray and structural m embers Cable Tray and Structural Members Height q z (ft) (psf) 0 15 20 25 30 40
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79 Table 6 2 Velocity pressures for p ipe Pipe Height q z ( ft) (psf) 0 15 20 25 30 40 The design wind force for structural members varies based on the projected area perpendicular to the wind direction and therefore will vary based on member size In the case where the wind is parallel to the strong axis, the flange width defines the projected area. Structural shapes are assumed to have an average coefficient of drag (Cf) of 2.0 for all wind directions. Pipes were assumed to have a coefficient of drag (Cf) of 0.8 for wind per pendicular to pipe. The design wind force for pipe is calculated based on the assumption of the largest pipe being 8 inches. 10% of the pipe rack width is added to the pipe added to the largest pipe diameter and multiplied by the bay spacing to determine a tributary area. (ASCE, 2011) To simplify the design the total force from the pipes on each level is evenly divided and applied to each joint This load could also be applied as a uniform load across the entire length of the beam. Table 63 shows the resulting joint loads resulting from wind loading on pipes.
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80 Table 6 3 Resultant design wind force from pipe Height F F/2 (ft) (lbs) (lbs) 0 15 701 350 20 742 371 25 775 387 30 808 404 40 857 429 Figure 6 5 shows the application of wind load for the typical pipe rack. The resultant load from pipe wind load is evenly distributed at the end joints. The design wind force shown for structural members is based on a column size of W10X33 with a flange width of 8 inches. This design wind force will vary based on column size but will provide a basis for application of wind load.
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81 Figure 6 5 Section View of Moment Frame Wind Load Seismic loading is based on specific site information. The ASCE 705 Equivalent Lateral Force Procedure will be used for seismic loading. Because this is a representative model, estimates on seismic loading will be made to simulate the general seismic load effects. The pipe rack is assumed be in located in site class C or below. An R value of 3 is chosen to simplify detai ling requirements. The assumed seismic response coefficient, Cs, will be 0.15. While the natural period of the structure will depend on member size and base support conditions, the natural period
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82 will be assumed to be greater than 0.5 seconds which is us ed to determine the vertical distribution Based on the above assumptions, seismic forces can be developed and applied to the structure. The effective seismic mass is based on previously discussed dead loads, both operating and empty dead loads. Table 6 4 shows the applied operating seismic load applied at each level of the pipe rack while Table 6 5 shows the applied empty seismic load The lateral force at each level will be evenly divided and applied to each joint. See Figure 66 for operating seismi c loading Empty seismic loading is similar to loading shown in Figure 6 6 Table 6 4 Lateral seismic f orces o perating F x (lbs) F x /2 (lbs) Level 1 950 475 Level 2 1450 725 Level 3 2100 1050 Level 4 2850 1425 Table 6 5 L ateral seismic forces e mp ty F x (lbs) F x /2 (lbs) Level 1 600 300 Level 2 900 450 Level 3 1250 625 Level 4 1700 850
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83 Figure 6 6 Section View of Moment Frame Lateral Seismic Load Operating Seismic 6.3 Pipe Rack Load Combinations Load combinations used for the representative m odel were developed from the PIP STC01015 load combinations as discussed in previous sections. As the model was simplified to isolate a moment resisting frame and study the results from essentially a 2 D analysis, loading in the transverse direction was t he focus for
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84 developing load combinations. Because lateral loads are typically reversible, this can create hundreds of load combinations if all load directions are considered. Transverse only loading simplified the loading and resulted in fewer load combinations. In the cases where notional loads are required, the notional load was considered to act only in the direction causing the worst effect on stability which is typically in the same direction as the lateral load. Two sets of load combinations were developed for analysis and comparison purposes. The first set applied notional loads to only the gravity only load combinations. This set of load combinations is used for analysis using both the effective length and direct analysis methods. The first se t of load combinations can only be used for the direct analysis when the ratio of second order to first order drift is less than 1.7 if using reduced stiffness or 1.5 using unreduced stiffness per AISC 36010. The second set of load combinations applies the notional loads as additive in all load combinations to create the worst effect on stability. The second set of load combinations was developed for analysis using the direct analysis method where the ratio of second order to first order drift is great er than 1.7 if using reduced stiffness or 1.5 using unreduced stiffness. The second set of load combinations can also be used for the first order method where the notional loads are additive for all load combinations. Although the magnitude of notional l oad s varies between the direct analysis and first order method, the notional loads can be adjusted in the primary load
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85 combinations and the same load combinations can be used. Based on the limitation for use of the effective length method, the second set of load combinations with notional load applied in all load combinations is not required to be used in the effective length method. A few observations on application of methods can be made by investigating the load combinations and requirements. First, in cases where the ratio of second order drift to first order drift is greater than 1.7 if using reduced stiffness or 1.5 using unreduced stiffness, the direct analysis method using the set of load combinations with additive notional loads in all load comb inations is the suggested method of AISC 36010. Next, all three methods can be used for analysis when the ratio of second order drift to first order drift is less than 1.7 if using reduced stiffness or 1.5 using unreduced stiffness. If this is the case t he dire ct analysis and effective length method do not require load combinations where the notional loads are additive in all cases, while the first order method requires additive notional loads in all load combinations. Strength and serviceability will both be of interest in analysis and design, therefore both LRFD and ASD load combinations will be developed. LRFD load combinations will be used for member strength checks, while the serviceability checks will be made using the ASD load combinations. T he load combinations used in analysi s can be seen in Appendix 1 through 3.
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86 6.4 Strength and Serviceability Checks Strength and serviceability check are made using the capabilities of STAAD.Pro V8i. Strength checks are based on AISC 36005. While AISC 36010 has been released, STAAD.Pro V8i has yet to include the specification in design capabilities. Very few changes have been made in member capacity calculations and therefore the AISC 36005 can be used for determining member capacity. The ratio of demand t o capacity is a point of comparison between the various methods of stability analysis. Serviceability checks are made using the calculated deflections from STAAD.Pro V8i. Unfactored loads combinations are used to calculated service deflections. Various l imits on serviceability can be set based on specific project requirements. AISC 36010 states that both geometric imperfections and reduction in stiffness are not required in determining serviceability checks. Therefore, notional loads are not needed in service load combinations. Reduction in stiffness is also not included in calculations of deformations used for serviceability checks. It should be noted that when using the direct analysis method in STAAD.Pro V8i, the reported deformations are calculated based on the reduced stiffness. AISC does not require the reduced stiffness to be used in serviceability checks, therefore the models should be analyzed using the unreduced stiffness for serviceability checks or
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87 the reported deformations should be adjust ed to account for the inclusion of reduced stiffness. Although not exact, reported deformations based on reduced stiffness could be multiplied by 0.8 to provide relatively accurate estimate s of the actual deformations to be used in serviceability checks. This was based on several test models that were analyzed using both methods and the results were compared and found to be reasonable. 6.5 Base Support Conditions Column base support conditions are affected by various factors. True fixed base column s in ac tual conditions, can be very hard to achieve. Foundation types and anchor bolt layout and design can significantly affect the rotational resistance of the column base. Fixed base moment frames typically can see saving s in member size but additional consi derations in foundation and anchor bolt design could offset the savings in member sizing Fixed base moment frames will also typically see a reduction in deformations due to the additional moment capacity generated by the base fixity. Pinned base moment frames on the other hand will typically require heavier members and experience potentially larger deformations compared to similar fixed base moment frames. B ase support conditions can have a significant effect on overall frame behavior, therefore both fi xed and pinned conditions were analyzed. Base support conditions also become important in the calculation of the effective length factor (K) used in the effective length method. The alignment charts
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88 from AISC 36010 are based on the value G which is a f unction of the rotational resistance provided by end constrains. F or pinned conditions, G is theoretically infinity. However, unless the connection is design as a true frictionfree pin, G should be assumed to be 10 for use in practical design. Unless t rue pin connections are used, t his recognizes that pin connections have some moment resistance which affects the effective length factor. On the other hand, rigid connections have a theoretical G value of 0, but for practical design, a value of 1 should be used. (AISC 36010) This again recognizes that rigid or moment connections are not completely rigid and can have a slight rotation before full rigidity is reached. Based on AISC 36010 recommendation, G is assumed to equal 10 for the pin column support condition and 1 for the fixed column support condition. Additional modification in the calculation of the effective length factor, K, will be discussed in further sections. 6.6 Effective Length Factor Many papers and books have extensive discussions on the c alculation of the effective length factor K. The previous discussion of this topic in the Literature Review section provide d additional sources and information for calculation of K. As the representative pipe rack structure used for analysis is a relativ ely straightforward moment resisting frame, the standard AISC 36010 determination of K can be used. However, AISC 36010 does suggest a few adjustments when using the alignment charts. The adjustments are recommended based on the fact that the alignment charts
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89 are developed based on the previously discussed assumptions. Adjustments made based on the column end conditions was discussed in the previous section. Additional adjustments must be made based girder end connections, significant axial loads in g irders, column inelasticity, and connection flexibility. (AISC 36010) For pinned base support conditions the following adjustments were used: G = 10 for pinned base column condition. EI/L for girders was multiplied by 2/3 to account for fixed end girders EI for columns was reduced to 0.8EI to account for column inelasticity. Small axial loads were assumed in the girders, therefore no adjustments were made based on the axial load present in the girders. K values were calculated using the alignment chart s and the equations for G from AISC 36010. For pinned base support conditions the following values were determined. See Figure 6 7 for K values of each column section. K values shown are based on W 10X49 columns and W 10X33 girders. For models where the member sizes were changed, K values were recalculated based on actual members.
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90 Figure 6 7 Effective Length Factor K Pinned Base For the fixed base column condition, all the same adjustments were made except for the adjustment for column end condition. For the fixed base condition, G was set equal to 1. See Figure 6 8 for K values for each column section. K values shown are based on W 10X33 columns and W 10X26 girders. For models where the member sizes were changed, K values wer e recalculated based o n actual members.
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91 Figure 6 8 Effective Length Factor K Fixed Base 6.7 Notional Load Development for First Order Method Notional loads must be developed for use of the first order method. Notional loads used in the first order method account for second order effects, geometric imperfections and reduction in stiffness. Notional load were developed based on the following AISC 36010 equation as defined previously:
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92 Ni = 2.1( /L)Yi 0.0042Yi Notional loads based on the above equation must be developed based on the drift ratio at strength load levels. Initial target drifts may be calculated and therefore be used in the calculation of notional loads. These notional loads are then applied and the strength level drifts are verified to be less than the targeted drifts. The closer the strength level drift is to the target drift used in calculation of notional loads, the more accurate the results will be with this method. Several iterations in the development of notional loads were completed to optimize and accurately apply the first order method. For the pinned base support condition, a target drift ratio /L was set at h/ 65. Using the drift ratio of h/ 65, the notional load can be calculated to be 0.0323Yi or 3.23% of the gravity load at each level. It shoul d be noted that this is significantly higher than the 0.002Yi required for the effective length and direct analysis methods. Analysis based on the application of the previously calculated notional loads resulted in a strength level drift ratio of approxim ately h/ 67, therefore notional loads are correct. Additional calculations could be completed to determine notional loads which would provide more accurate results but the accuracy of the above target drift ratio is sufficient for comparison purposes. Similarly, t he target drift ratio for the fixed base condition was set at h/175. This drift ratio calculates to 0.012Yi or 1.2% of the gravity load at each level. Analysis based on the application of the previously calculated notional loads resulted
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93 in a str ength level drift ratio of approximately h/182. Again, additional calculations could be completed which provide more accurate results. However, the initial target drift ratio is sufficient for comparison purposes. Notional load used in the first order method are additive in all load combinations. Load combinations were developed to apply these notional loads in all load combinations to create the most destabilizing effect. See Appendix 3 for STAAD input file for the representative model using the firs t order method. 6.8 STAAD Benchmark Validation Some analysis software packages are capable of performing a rigorous second order analysis which can be used in any method that requires the inclusion of second order effects such as the direct analysis method or equivalent length method. AISC 36005 and 36010 both give two benchmark problems to determine if the analysis procedure meets the requirements of a rigorous second order analysis. The benchmark problems can be seen in Figure 69.
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94 Figure 6 9 AISC Benchmark Problems (AISC 360 10) STAAD.Pro V8i was used for modeling and analysis of pipe racks. To verify second order analysis capabilities in STAAD.Pro V8i, both benchmark problems from AISC 36010 were run with the results listed in Table 66. AISC 360 10 specifies that moment corresponding to all axial load cases should agree within 3% and deflections within 5%. When comparing the solutions, it can be seen that the STAAD results have less than 0.5% variance from the AISC solutions. (AISC 36010) Most of these small differences can probably be explained by rounding errors or precision of reported solutions. Therefore, STAAD.Pro V8i can be assumed to be correctly carr ying out a rigorous second order analysis and therefore can be used to
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95 analyze pipe racks using both the direct analysis method and equivalent length method. Table 6 6 Benchmark solutions Benchmark Problem 1 Axial Load,P (kip) 0 150 300 450 mid (in) AISC Solution 0.202 0.23 0.269 0.322 STAAD Solution 0.201 0.23 0.268 0.322 Mmid (kip*in) AISC Solution 235 270 316 380 STAAD Solution 235.2 269.7 315.7 380.1 Benchmark Problem 2 Axial Load,P (kip) 0 100 150 200 mid (in) AISC Solution 0.907 1.34 1.77 2.6 STAAD Solution 0. 905 1.339 1.765 2.594 Mmid (kip*in) AISC Solution 336 470 601 856 STAAD Solution 336 470 601 855
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96 7. Comparison of Results Both pinned base and fixed base support condition models were developed for analysis and comparison of the three methods of stability analysis. See Appendix 1 for STAAD.Pro V8i input file for pipe rack analysis using the effective length method. Appendix 2 and 3 contain similar inputs for the direct analysis and first order method respectively. The first model was ana lyzed with a pinned base column. The member sizes were chosen without regard to serviceability and picked only to satisfy the load demand. F irst order method, effective length method and direct analysis method were all applied to the model and the results compiled. A first order linear elastic analysis was completed to provide a benchmark for comparison and calculation of the ratio of second order drift to first order drift. Table 7 1 shows the ratio of second order to first order drift 21) based on the comparison of the benchmark linear elastic analysis to the effective length method analysis. It should be noted that these maximum deflections are based on LRFD load combinations. See Appendix 1 and 2 for details of LRFD load combina tions. The maximum 21 ratio is calculated as 1.15.
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97 Table 7 1 21 e ffective length method pinned base LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Effective Length Method Maximum Deflection (inch) 21 201 202 203 204 205 206 207 208 209 210 211 212 5 213 214 215 216 217 218 219 220 221 222 223 224 2 1 = A few other obs ervations can be made from the results seen in Table 7 1. The limitation of 21 for use of the first order method set by AISC 36010 is 1.5. Therefore, for the representative pinned base pipe rack, the first order method is a valid method for stability analysis. Also, AISC 36010 sets limitations for use of notional loads. Because the maximum 21 is less than 1.5, notional load only need
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98 be applied to the gravity only load combinations for use in the effective length method. Table 7 2 shows the ratio of second order to first order drift 21) based on the comparis on of the benchmark linear elastic analysis to the direct analysis method analysis. As expected, the ratio 21 is slightly higher based on the reduction in stiffness. The benchmark first order linear elastic analysis for this comparison included a reduced stiffness used in analysis. The increase in the ratio 21 seen in Table 7 2 shows that the reduction in stiffness can a mplify the second order effects. The maximum ratio 21 is 1.21. Because the ratio 21 is less than 1.7 (reduced stiffness is used to calculate drift), notional load need only be applied in the gravity only load combinations. (AISC 36010)
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99 Table 7 2 21 direct analysis method pinned base LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Reduced Stiffness Effective Length Method Maximum Deflection (inch) 21 201 202 3 203 204 205 206 207 208 209 210 211 212 213 2 14 215 216 217 218 7 219 220 221 222 223 224 2 1 = Both Table 71 and 72 show the importance of consideration of stability analysis in design for pinned base conditions. For the representative pinned base model, stability analysis can amplify the deformation by up to 21% for this s pecific model. Deformation may not always be the focus of analysis and design but when
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100 checking serviceability limits, stability analysis can increase deformation s significantly when compared to an elastic first order analysis. The first order method w as performed on the same model but as the method name implies, only a first order analysis is done and therefore the ratio 21 cannot be directly calculated based on the drifts alone. However, based on the results of the previous two analyses, the ratio 21 will be well below the 1.5 limitation set be AISC 36010. Therefore the first order method is a valid type of stabi lity analysis for the representative pinned base pipe rack. Demand to capacity for members should also be used when comparing the types of stability analysis methods. Maximum demand to capacity ratio for both column and beam design is shown in Table 73. Table 7 3 Maximum demand to capacity ratio pinned base Column (10X49) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method Beam (W10X33) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method The linear elastic analysis was included as a benchmark for comparison. The linear elastic analysis can be seen to underestimate the demand to capacity ratios of
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101 members, sometimes significantly. When comparing the direct analysis method and the first order method, it can be seen that the demand to capacity ratio is slightly higher when using the first order method. This is to be expected since the first order method is a simplification of the direct analysis built on conservative assumptions which will envelope the design. The effective length method has slightly higher ratios for column design and s lightly lower for beam design. As discussed in previous sections for the effective length method, the column strength equations are adjusted using K to account for reduction in stiffness but the moment can be underestimated for beams and connections whic h resist column rotation. The actual demand forces are listed in Table 7 4. Table 7 4 Maximum demand forces pinned base Column (W12X53) Maximum Forces Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method St r ong Axis Moment Axial Load (kip) Beam (W12X40) Maximum Forces Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method St r ong Axis Mome nt Axial Load (kip) Based on Table 7 3 and 74 g ood correlation can be seen between the methods The demand to capacity ratios for each method show results that are
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102 expected based on the theory us ed to develop each method. The member forces have slight variation between methods based on the slight differences required in analysis in the methods All results show similar relationships between each method. It should be noted that varying geometry could have a significant effect on the ratio 21 which could limit the use of either the first order method or effective length method. In general for the pinned base support condition, columns have relatively low axial demand when compared to the compre ssion failure load Py. Large moments are developed in both the columns and beams and therefore the majority of the member capacity is used to resist the moment demand Similar tables to those seen above were also developed based on the fixed base support c ondition which can be seen on the following pages
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103 Table 7 5 Ratio 21 effective length method fixed base LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Effective Length Method Maximum Deflection (inch) 21 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 2 1 = When comparing the fixed base ratio 21 for the effective length method, it can be seen that the maximum value is 1.05. While this is slightly less than for the pinned base support condition model, it still shows the significance of stability analysis in design. Table 76 below shows the same ratio 21 for the direct analysis with both the linear elastic and direct analysis using the reduced stiffness in
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104 calculation of deformations As discussed previously, it is expected that the ratio 21 is slight higher based on the reduced stiffness. Both tables do show however that the representative model with fixed base satisfies all the requirements for stability analysis by any of the three methods. Table 7 6 21 direct anal ysis method fixed base LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Reduced Stiffness Effective Length Method Maximum Deflection (inch) 21 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 4 216 217 218 219 220 221 222 223 224 2 1 =
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105 Table 7 7 Maximum demand to capacity ratio fixed base Column (W10X33) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method Beam (W10X26) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method Table 7 8 Maximum demand forces fixed b ase Column (W10X33) Maximum Forces Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method St r ong Axis Moment Axial Load (kip) Beam (W10X26) Maximum Forces Linear Elastic A nalysis First Order Method Effective Length Method Direct Analysis Method St r ong Axis Moment Axial Load (kip) In general the first order method tends to be inherently more conservative based on the analysis assumptions which increase demand to account for the second order effects and reduction in stiffness The effective length method on the other hand, tends to underestimate the demand while the capacity is adjusted to account for
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106 the underest imation in demand by using the K factor. The direct analysis method will typically provide the most accurate results. When comparing the two support condition models, several observations can be made. The representative fixed base model tends to have s lightly lower second order effects compared to the pinned based model. The fixed base model also tends to have lower deformations even when smaller member sizes are used. When demand to capacity is the only consideration in design, the deformations can e asily become relatively significant and exceed standard serviceability limits especially in the case of pinned base support conditions. Serviceability limits were considered for additional analysis models. The representative pin based model was the focu s based on large drift ratios when strength was the only consideration. A target serviceability limit was set at H /200. Members were re sized based on this target and the results can be seen in the following tables.
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107 Table 7 9 21 effective length method pinned base serviceability limits LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Effective Length Method Maximum Deflection (inch) 21 201 202 203 204 205 206 207 208 209 210 211 212 213 214 2 215 216 217 218 219 220 221 222 223 224 2 1 =
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108 Table 7 10 21 direct analysis method pinned base serviceability limits LRFD Load Combination Number Linear Elastic Analysis Maximum Deflection (inch) Reduced Stiffness Effective Length Method Maximum Deflection (in ch) 21 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 1 224 2 1 = Comparing the results of the pinned base (Tables 7 1 and 72) with the pinned base with serviceability limits imposed (Tables 7 9 and 710) shows an overall reduction in second order effects when serviceability controls the design. The overall structural stiffness is increased to limit deflections due to serviceability and therefore second order effects are minimized since the loads are acting on a less deformed shape.
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109 Demand to capacity ratios show similar results when compared to previous results. Table 7 11 shows demand to capacity ratios when serviceability limits are imposed on the design. The demand to capacity ratios are much lower based on the additional stiffness require d to meet serviceability limits. Table 7 11 Maximu m demand to capacity ratio pinned base serviceability l imits Column (W 12X65 ) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method 418 452 483 447 Beam (W 12X45 ) Maximum Demand to Capacity Ratio Linear Elastic Analysis First Order Method Effective Length Method Direct Analysis Method 515 555 547 As seen in the results, both from the representative models and the cantilever column example the effective length method tends to underestimate the moment demand in both the column and any beam or connection that resists column rotation. This fact should be considered in design if the effective length method is used for stability analysis. Th e results also show the importance of stability analysis. Linear elastic analysis is not sufficient especially when designing moment frames. The results from above showed that the demand to capacity ratio could be underestimated by approximately 1020% f or the worst case when conducting a linear elastic analysis.
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110 Varying the stiffness or ge ometry could easily produce greater errors in analysis if sta bility analysis is not considered.
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111 8. Conclusions For the representative pipe rack model both pinned and fixed base conditions, the first order, effective length, and direct analysis methods were all found to be valid methods of stability analysis according to AISC 36010. When the ratios 21 and Pr/Py are below the limits specified by AISC 360 10, all methods gave comparable results. Several observations on each method can be made based on the analysis and results. The direct analysis method provides the most accurate results since reduction in stiffness is considered in analysis There are also many benefits in application of this method with the most significant benefit being that the effective length factor K can be set to 1 for all cases. This can significantly simplify calculati ons required to perform analysis especially for moment frames. Additionally, the direct analysis method has no limitations for use. Without limitations, no validation after analysis is required and therefore less time is required for this method. Howeve r, the best results from use of this method typically come from utilization of modern analysis software capable of performing rigorous second order analysis. Although an approximate second order analysis can be done by hand, this can become quite tedious and time consuming when other methods could produce similar results with less work.
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112 The first order analysis is a mathematically simplified analysis method. The greatest benefit as the name implies, is that the method only requires a first order analysis As with the direct analysis method, K can also be set at 1 for all cases. Based on the above simplifications, analysis can be significantly simplified. The simplified analysis tends to be the most beneficial for hand calculations of frames. The simpl ifications made in development of the method do however impose limitations on use. While the analysis portion may be less intensive, the post analysis validation required can sometimes outweigh the benefits if the simplified analysis. Additionally, sever al iterations in analysis may be required to achieve the most accurate results since notional loads are calculated based on target drift limits. Application of notional load in all load combinations can also create significantly more load combinations whi ch need to be considered in design. Simplifications in analysis also tend to create the most conservative results when compared to other methods of stability analysis. The effective length method is probably the most well known method for stability analys is. However as the name implies, the effective length factor K must be calculated. Based on the discussions in previous sections, this can become very complex even for relatively simple structures. The accuracy of the effective length method is critically linked to accurate calculation of the effective length factor. Various methods for determining K have been developed but the most widely known is still the a lignment charts When using the alignment charts from AISC 36010 the
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113 engineer should be aware of the assumptions made in the development of the charts and the limitations for use. AISC 36010 has set limitations for use of this method. Verification of applicability of effective length method after analysis could limit the use of this method. Se cond order effects should be considered in design which can be done using either a rigorous or approximate analysis. Results from this method show that the moment demand for beams and connections that resist column rotation can be underestimated. The und erestimation of moments in elements resisting column rotation is due to no reduction in stiffness considered in analysis. The engineer should be aware of the underestimation of moment demand when designing beams and connections with results found using the effective length method. When comparing the pros and cons of each method, some observations can be made. If modern software analysis is available, the direct analysis method will most likely require the least amount of work to achieve the most accur ate results. Although the effective length method is the most well known, engineers should be aware of limitations and the tendency for underestimation of moments in results. The first order method can be a power ful method if certain limitations are met and slightly conservative results are acceptable. However, with modern analysis software, the first order method will likely see limited applicab ility unless calculations are completed by hand. Stability analysis as it relates specifically to pipe rack structures provides several points of interest. Base support conditions have a relatively large influence on
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114 results of stability analysis P inned base pipe racks could have large deformations which tend to produce larger second order effects. If fixed base support conditions can be developed, member strength can be further utilized before serviceability limits are reached. However, fixed base support conditions in many cases can be difficult to achieve If each method is applied appropriately to pipe rack structures, based on the results above, good correlation can be seen between the first order, effective length and direct analysis methods. However, AISC 36010 limitation must be verified for use of the first order and effective length methods on a case by case basis. For the representative models used above, all three methods met AISC 36010 requirements and were found acceptable for stability analysis. Pipe racks moment frames tend to support relatively large lateral loads through flexural res istance. Because the majority of the member capacity is used to resist the lateral loads, little capacity is left to resist axial loads, therefore the member is sized based primarily on the flexural demand. Based on the lower axial demands, the additiona l reduction in stiffness used in the direct analysis method due to large axial demand typically is not required. The representative pipe rack chosen for analysis was a moderately simple frame and calculation of the effective length factor was relatively st raightforward. However, actual pipe racks may require more complex structure s which can greatly complicate the calculation of the effective length factor. Based on the possibility for
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115 complex effective length factor calculations, the direct analysis and first order method could be much less calculation intensive and provide more accurate results. Based on the above results and observations, I recommend the direct analysis as the first choice in stability analysis for pipe racks. While both the effectiv e length and first order method provide relatively accurate results as long the ir respective requirements are met t he direct analysis provides the most accurate results and has no limitations for use. The direct analysis method can also be the simplest me thod to apply if modern software analysis is utilized as no front end calculations or post analysis verification are required
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116 REFERENCES American Institute of Steel Construction (AISC) (2005) Specification for s tructural s teel buildings (ANSI/AISC 360 05). American Institute of Steel Construction, Inc. Chicago. American Institute of Steel Construction (AISC) (2010) Specification for structural steel buildings (ANSI/AISC 36010). American Institute of Steel Construction, Inc. Chicago. American Institute of Steel Construction (AISC) (2010) Code of standard practice for steel buildings and bridges ( AISC 30310) . American Institute of Stee l Construction, Inc. Chicago. ASCE Task Committee on Effective Length (1997) Effective length and notional load approaches for assessing frame stability: implications for American steel design. American Society of Civil Engineers, New York American Society of Civil Engineers (ASCE). (2006) Minimum Design Loads for Buildings and Other Structures ( AS CE 7 05). American Society of Civil Engineers, Reston, VA. American Society of Civil Engineers (ASCE). (2010) Minimum design loads for buildings and other s tructures ( ASCE 7 10). American Society of Civil Engineers, Reston, VA.
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117 American Society of Civil Engineers (ASCE). (2011) Wind load for petrochemical and other industrial f acilities. American Society of Civil Engineers, Reston, VA. American Society of Civil Engineers (ASCE). (1997) Guideline for seismic evaluation and design of petrochemical f acil ities . American Society of Civil Engineers, Reston, VA. Bathe, K. J. (1995). Finite element procedures. Prentice, Upper Saddle River, NJ. Hall. Bendapudi, K. V. (2010). Structural design of steel pipe support s tructures . Structure Magazine 17 (2) 68. Bendapudi, K. V. (2010). Discussion on structural design of steel pipe support s tru ctures. Structure Magazine, 17 (11), 3840. Canada Department of Railw ays and Canals. (1919). The Quebec brid ge over the St. Lawrence River near the city of Quebec o n the l ine of the Canadian National Railways ( Report No. 1). Canada Department of Railw ays and Canals, Ottawa Canada Che n, W. F., and Lui, E. M. (1991). Stability design of s teel f rames . CRC Press, Boca Raton, FL. Crisfield, M. A. (1991). Nonlinear fin ite element analysis of solids and structures . Wiley, New York.
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118 Deierlein, G. G., Hajjar, J. F., Yura, J. A., White, D. W. and Baker, W. F. (2002) Proposed new p rovisions for f rame s tability using s econdorder a nalysis. Proceedings 2002 Annual Technic al Session, Seattle April. Structural Stability Research Council, Rolla, MO. Delatte, N. J. (2009). Beyond f ailure : forensic case studies for civil engineers . ASCE Press Reston, VA, 5170. Drake, R .M., & Walter, R.J. (2010). Design of structural steel pipe racks. AISC Engineering Journal, 47 (4) 241252. Euler, L. (1744). Methodus i nveniendi l ineas curvas. Bousquet, Lausanne Ericksen, J. R. (2011). A how t o a pproach to notional l oads . AISC Modern Steel Construction, 51 (1) 4445 Folse, M. D., and Nowak, P. S. (1995). Steel rigid frames with leaning columns 1993 LRFD example. AISC Engineering Journal, 32 (4). 125131. Galambos T. V. ed (1998). Guide to stability design criteria for metal s tructures, 5th Edition. John Wiley & Sons, Inc., Hoboken, NJ. Geschwindner, L. F. (2002) A practical l ook at f rame a nalysis stability and leaning columns. AISC Engineering Journal, 39 (4). 167181 Geschwindner, L. F. (2009) Design for s tability using 2005 AISC specification. seminar notes, presente d Sept 18, 2009, Sponsored by AISC, Chicago, IL. Huber, A. W., and Beedle, L. S. (1954). Residual s tress and compressive strength of steel. Welding Journal December. 589614.
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119 Hibbeler, R. C. (2005). Mechanics of materials 6th Edition. Pearson Prentice Hall, New Jersey. Kanchanalai, T. (1977) The Design and Behavior of Beam Columns in Unbraced Steel Frames . AISI Project Number 189, Report Number 2, Civil Engineering/Structures Research Laboratories, University of Texas, Austin, TX. Kanchanalai, T. a nd Lu, L.W. (1979) Analysis and de sign of framed columns under minor axis bending . AISC Engineering Journal, 16 (2). 2941. Kuchenbecker, G. H., White, D. W. and SurovekMalek, A. E. (2004). Simplified design of building f rames using f irst order analy sis and K = 1. Proceedings of the Annual Technical Session and Meetings Lo ng Beach, CA, March 2427, 2004. Structural Stability R esearch Council, Rolla, MO. 119138. LeMessurier, W J. (1976). A practical m ethod of s econd order a nalysis part 1 pin j oi nted s ystems . AISC Engineering Journal, 13 (4). 8996 LeMessurier, W J. (1977). A practical m ethod of s econd order a nalysis part 2 r igid f rames . AISC Engineering Journal, 14 (2). 4967 LeMessurier, W. J. (1995) Simplified K f actors for s tiffness c ontrolled designs , r estructuring: America and beyond. Proceedings of ASCE Structures Congress XIII Boston, MA April 25. American Society of Civil Engineers, New York. 1797 1812.
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120 Maleck, A.E. and White, D.W. (1998). Effects of i mperfections on s teel f raming s ystems . Proceedings 1998 Annual Technical Sessions Atlanta, September. Structural Stability Research Council, Rolla, MO. Maleck, A. E. and White, D. W. (2003). Direct a nalysis a pproach for the a ssessment of f rame stability: verification studi es. Proceedings Annual Technical Sessions Structural Stability Research Council, Rolla, MO. 18 McGuire, W., Gallegher, R. H. and Ziemian, R. D. (2000). Matrix s tructural a nalysis, 2nd Ed ition. John Wiley & Sons Inc., Hoboken, NJ. Nair, S. R. (2009). A m odel s pecification for s tability d esign by direct analysis. AISC Engineering Journal, 46 (1). 2938 Pr ocess Industry Practices (PIP) (2007). Structural design c riteria ( PIP STC01015). Construction Industry Institute, P rocess Industry Practices, Austi n, TX. Powell, G. H. (2010) Modeling for s tructural a nalysis: behavior and b asics . Computers and Structures, Inc. Berkeley, CA. Rutenberg, A. (1981) A direct pdelta a nalysis using s tand plane f rame c omputer programs. ASCE Journal Engineering Mechani cal Division, 86 (EM1). 6178. Rutenberg, A. (1982). Simplified pdelta ana lysis for asymmetric structures. ASCE Journal Engineering Mechanical Division, 108 ( ST9 ). 19952013.
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121 Salmon, C. G., and Johnson, J. E. (2008). Steel s tructures: design and b ehavi or, e mphasizing l oad and r esistance design 5th Ed ition. Harper Collins, New York. STAAD.Pro V8i (2007) Technical Referen ce Manual. Bentley Systems, Inc., Yorba Linda, CA. SurovekMaleck, A. E., and White, D. W. (2004a) Alternative a pproaches for e las tic a nalysis and design of s teel f rames. I: overview . ASCE Journal of Structural Engineering, 130 (8). 11861196. SurovekMaleck, A.E., and White, D.W. (2004b) Alternative a pproaches for e lastic a nalysis and design of s teel f rames. II: verification s tudi es . ASCE Journal of Structural Engineering, 130 (8). 11971205. Trakov, J. (1986). Quebec bridge: a disaster in the m aking . Invention & Technology, American Heritage, 1 (4). 1017 Vinnakota, S. (2006) Steel s tructures: b ehavior and LRFD, 1st Ed ition. McGraw Hill, New York White, D. W., and Hajjar, J. F. (1991). Application of s econd o rder e lastic a nalysis in LRFD: r esearch to practice. AISC Engineering Journal, 28 (4). 133148 White, D. W., Surovek, A. E., Alemdar, B. N., Chang, C., Kim, Y. D., a nd Kuchenbecker, G. H. (2006). Stability a nalysis and design of s teel building f rames usin g the AISC 2005 s pecification. International Journal of Steel Structures 6. 7191
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122 White, D. W., Surovek, A. E. and Kin, S C. (2007a). Direct a nalysis and design using a mplified f irst o rder a nalysis. part 1 c ombined braced and gravity f raming s ystems . AISC Engineering Journal, 44 (4) 305322. White, D. W., Surovek, A. E. and Chang, C J. (2007b). D irect a nalysis and design using a mplified f irst o rder a nalysis part 2 m oment f rames and general f raming s ystems . AISC Engineering Journal, 44 (4). 323340. Yang, C. H., Beedle, L. S., and Johnson, B. G. (1952). Residual s tress and the yield strength of steel beams. Welding Journal, April 1952. 589614. Yang, C. H., and Quo, S. R. (1994) Theory and a nalysis of nonlinear f ramed s tructures . Prentice Hall, Upper Saddle River, NJ. Yura, J. A. ( 1971). The e ffective l ength of c olumns in unbraced f rames . AISC Engineering Journal, 8 (2) 3742. Yura, J. A., Kanc hanalai, T., and Chotichanathawenwon g, S. (1996). Verification of steel beam column desig n based on the AISC LRFD method. Proc eeding 5th International Colloquium Stability Metal Structures Lehigh University, Bethlehem, PA, Structural Stability Research Council, Rolla, MO. 2130. Ziemianm, R. A. ed (2010). Guide to s tability d esign c riteria for m etal s tructures, 6th Ed ition . John Wiley & Sons, Inc., Hoboken, NJ.
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123 Appendix A STAAD Input Pinned Base Analysis Effective Length Method STAAD SPACE S TART JOB INFORMATION ENGINEER DATE 1 25 2012 ENGINEER NAME DAN END JOB INFORMATION JOINT COORDINATES 1 0 0 0; 2 0 20 0; 3 0 25 0; 4 0 30 0; 5 0 35 0; 6 15 0 0; 7 15 20 0; 8 15 25 0; 9 15 30 0; 10 15 35 0; MEMBER INCIDENCES 1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 2 7; 6 3 8; 7 4 9; 8 5 10; 9 6 7; 10 7 8; 11 8 9; 12 9 10; DEFINE MATERIAL START ISOTROPIC STEEL e+006 006 END DEFINE MATERIAL MEMBER PROPERTY AMERICAN 1 TO 4 9 TO 12 TABLE ST W1 0X49 5 TO 8 TABLE ST W1 0X33 CONSTANTS MATERIAL STEEL ALL
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124 1 6 PINNED DEFINE WIND LOAD TYPE 1 LOAD 1 DEAD LOAD (DS) STEEL AND FIREPROOFING SELFWEIGHT Y LOAD 2 OPERATING DEAD LOAD (DO) MEMBER LOAD LOAD 3 EMPTY DEAD LOAD (DE) MEMBER LOAD LOAD 4 TEST DEAD LOAD (DT) MEMBER LOAD LOAD 5 WIND LOAD X DIRECTION (WX)
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125 PROJECTED AREA JOINT LOAD LOAD 7 SEISMIC LOAD X DIRECTION (EX) JOINT LOAD 3 LOAD 14 SEISMIC LOAD Y DIRECTION (EY) SELFWEIGHT Y MEMBER LOAD LOAD 9 PIPE FRICTION LOAD (FF) MEMBER LOAD LOAD 10 PIPE ANCHOR LOAD (AF) MEMBER LOAD LOAD 11 THERMAL EXPA NSION FORCE (T)
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126 1 TO 12 TEMP 0 LOAD 15 NOTIONAL LOAD (N) MEMBER LOAD (DS) STEEL AND FIREPROOFING SELFWEIGHT x MEMBER LOAD MEMBER LOAD modal calculation requested LOAD 101 DS+DO+FF+T+AF REPEAT LOAD LOAD 102 DS+DO FF+T AF REPEAT LOAD LOAD 103 DS+DO+FF T+AF REPEAT LOAD LOAD 104 DS+DO FF T AF REPEAT LOAD
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127 LOAD 105 DS+DO+AF+WX REPEAT LOAD LOAD 106 DS+DO AF+WX REPEAT LOAD LOAD 107 DS+DO+AF WX REPEAT LOAD LOAD 108 DS+DO AF WX REPEAT LOAD REPEAT LOAD LOAD 110 DS+DO REPEAT LOAD LOAD 111 DS+DO+AF REPEAT LOAD LOAD 112 DS+DO AF REPEAT LOAD LOAD 113 DS+DE+WX REPEAT LOAD LOAD 114 DS+DE WX REPEAT LOAD
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128 REPEAT LOAD REPEAT LOAD REPEAT LOAD 7 AF REPEAT LOAD REPEAT LOAD REPEAT LOAD LOAD 121 DS+DT+WPX REPE AT LOAD LOAD 122 DS+DT WPX REPEAT LOAD
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129 REPEAT LOAD FF+T AF) REPEAT LOAD T+AF) REPEAT LOAD 4 FF T AF) REPEAT LOAD REPEAT LOAD REPEAT LOAD REPEAT LOAD AF) REPEAT LOAD REPEAT LOAD REPEAT LOAD LOA REPEAT LOAD AF)
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130 REPEAT LOAD REPEAT LOAD REPEAT LOAD EY REPEAT LOAD EY REPEAT LOAD EX EY REPEAT LOAD LO EX EY REPEAT LOAD EEY REPEAT LOAD 6 14 EEXEEY REPEAT LOAD REPEAT LOAD
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131 LOAD 222 1 NX) REPEAT LOAD REPEAT LOAD 6WX) REPEAT LOAD PDELTA KG ANALYSIS LOAD LIST 201 TO 224 PARAMETER 1 FYLD 7200 ALL KY 1 ALL CHECK CODE ALL FINISH
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132 Appendix B STAAD Input Pinned Base Analysis Direct Analysis Method STAAD SPACE START JOB INFORMATION ENGINEER DATE 1 25 2012 ENGINEER NAME DAN END JOB INFORMATION T KIP JOINT COORDINATES 1 0 0 0; 2 0 20 0; 3 0 25 0; 4 0 30 0; 5 0 35 0; 6 15 0 0; 7 15 20 0; 8 15 25 0; 9 15 30 0; 10 15 35 0; MEMBER INCIDENCES 1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 2 7; 6 3 8; 7 4 9; 8 5 10; 9 6 7; 10 7 8; 11 8 9; 12 9 10; DEFINE MATERIAL START ISOTROPIC STEEL 006 END DEFINE MATERIAL MEMBER PROPERTY AMERICAN 1 TO 4 9 TO 12 TABLE ST W1 0X49 5 TO 8 TABLE ST W1 0X33 CONSTANTS MATERIAL STEEL ALL
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133 1 6 PINNED DEFINE DIRECT ANALYSIS FYLD 7200 ALL FLEX 1 ALL AXIAL ALL END DEFINE WIND LOAD TYPE 1 LOAD 1 DEAD LOAD (DS) STEEL AND FIREPROOFING SELFWEIGHT Y LOAD 2 OPERATING DEAD LOAD (DO) MEMBER LOAD LOAD 3 EMPTY DEAD LOAD (DE) MEMBER LOAD LOAD 4 TEST DEAD LOAD (DT) MEMBER LOAD
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134 LOAD 5 WIND LOAD X DI RECTION (WX) JOINT LOAD 71 LOAD 7 SEISMIC LOAD X DIRECTION (EX) JOINT LOAD LOAD 14 SEISMIC LOAD Y DIRECTION (EY) SELFWEIGHT Y MEMBER LOAD LOAD 9 PIPE FRICTION LOAD (FF) MEMBER LOAD LOAD 10 PIPE ANCHOR LOAD (AF)
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135 MEMBER LOAD LOAD 11 THERMAL EXPANSION FORCE (T) 1 T O 12 TEMP 0 LOAD 15 NOTIONAL LOAD (N) MEMBER LOAD SELFWEIGHT x MEMBER LOAD MEMBER LOAD modal calculation requested L+S LOAD 101 DS+DO+FF+T+AF REPEAT LOAD LOAD 102 DS+DO FF+T AF REPEAT LOAD
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136 LOAD 103 DS+DO+FF T+AF REPEAT LOAD LOAD 104 DS+DO FF T AF REPEAT LOAD LOAD 105 DS+DO+AF+WX REPEAT LOAD LOAD 106 DS+DO AF+WX REPEAT LOAD LOAD 107 DS+DO+AF WX REPEAT LOAD LOAD 108 DS+DO AF WX REPEAT LOAD REPEAT LOAD LOAD 110 DS+DO REPEAT LOAD LOAD 111 DS+DO+AF REPEAT LOAD LOAD 112 DS+DO AF REPEAT LOAD LOAD 113 DS+DE+WX
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137 REPEAT LOAD LOAD 114 DS+DE WX REPEAT LOAD REPEAT LOAD REPEAT LOAD REPEAT LOAD AF REPEAT LOAD 14 REPEAT LOAD REPEAT LOAD 1 0LOAD 121 DS+DT+WPX REPEAT LOAD LOAD 122 DS+DT WPX REPEAT LOAD
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138 REPEAT LOAD FF+T AF) REPEAT LOAD T+AF) REPEAT LOAD FF T AF) REPEAT LOAD REPEAT LOAD REPEAT LOAD LOA REPEAT LOAD AF) REPEAT LOAD REPEAT LOAD
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139 REPEAT LOAD REPEAT LOAD 2(DS+DO AF) REPEAT LOAD REPEAT LOAD LOAD REPEAT LOAD EY REPEAT LOAD 0 EY REPEAT LOAD EX EY REPEAT LOAD EX EY REPEAT LOAD EEY REPEAT LOAD EEXEEY REPEAT LOAD
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140 REPEAT LOAD NX) REPEAT LOAD REPEAT LOAD REPEAT LOAD 014 LOAD LIST 201 TO 224 PARAMETER 1 FYLD 7200 ALL KY 1 ALL CHECK CODE ALL FINISH
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141 Appendix C STAAD Input Pinned Base Analysis First Order Method STAAD SPACE START JOB INFORMATION ENGINEER D ATE 1 25 2012 ENGINEER NAME DAN END JOB INFORMATION JOINT COORDINATES 1 0 0 0; 2 0 20 0; 3 0 25 0; 4 0 30 0; 5 0 35 0; 6 15 0 0; 7 15 20 0; 8 15 25 0; 9 15 30 0; 10 15 35 0; MEMBER INCIDENCES 1 1 2; 2 2 3; 3 3 4; 4 4 5; 5 2 7; 6 3 8; 7 4 9; 8 5 10; 9 6 7; 10 7 8; 11 8 9; 12 9 10; DEFINE MATERIAL START ISOTROPIC STEEL AL 006 END DEFINE MATERIAL MEMBER PROPERTY AMERICAN 1 TO 4 9 TO 12 TABLE ST W1 0X49 5 TO 8 TABLE ST W1 0X33 CONSTANTS MATERIAL STEEL ALL
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142 1 6 PIN NED DEFINE WIND LOAD TYPE 1 CASES LOAD 1 DEAD LOAD (DS) STEEL AND FIREPROOFING SELFWEIGHT Y LOAD 2 OPERATING DEAD LOAD (DO) MEMBER LOAD LOAD 3 EMPTY DEAD LOAD (DE) MEMBER LOAD LOAD 4 TEST DEAD LOAD (DT) MEM BER LOAD LOAD 5 WIND LOAD X DIRECTION (WX)
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143 OPEN JOINT LOAD LOAD 7 SEISMIC LOAD X DIRECTION (EX) JOINT LOAD LOAD 14 SEISMIC LOAD Y DIRECTION (EY) SELFWEIGHT Y MEMBER LOAD LOAD 9 PIPE FRICTION LOAD (FF) MEMBER LOAD LOAD 10 PIPE ANCHOR LOAD (AF) MEMBER LOAD LOAD 11 THERMAL EXPANSION FORCE (T)
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144 1 TO 12 TEMP 0 LOAD 15 NOTIONAL LOAD (N) MEMBER LOAD AND FIREPROOFING SELFWEIGHT x MEMBER LOAD MEMBER LOAD modal calculation requested COMBINATIONS LOAD COMB 101 DS+DO+FF+T+AF LOAD COMB 102 DS+DO FF+T AF LOAD COMB 103 DS+DO+FF T+AF LOAD COMB 104 DS+DO FF T AF
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145 LOAD COM B 105 DS+DO+AF+WX LOAD COMB 106 DS+DO AF+WX LOAD COMB 107 DS+DO+AF WX LOAD COMB 108 DS+DO AF WX LOAD COMB 110 DS+DO LOAD COMB 111 DS+DO+AF LOAD COMB 112 DS+DO AF LOAD COMB 113 DS+DE+WX LOAD COMB 114 DS+DE WX LOAD COM AF
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146 LOAD COMB 121 DS+DT+WPX LOAD COMB 122 DS+DT WPX ATIONS (WITH NOTIONAL) FF+T AF N X) T+AF+NX) FF T AF NX) NX) 1 1
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147 AF NX) AF LOAD AF+NX) AF+NX) AF L NX) AF NX) NX)
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148 EY EY EX EY EX EY EY NX) EY 14 EX EY NX) EX EY EEY NX) EEX EEY DT) NX)
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149 NX) PERFORM ANALYSIS LOAD LIST 201 TO 236 PARAMETER 1 FYLD 7200 ALL L KY 1 ALL CHECK CODE ALL FINISH
