IMPROVEMENT OF THE BUCKLING CAPACITY OF
SLENDER TIMBER COLUMNS
by
Daniel Edwin Sass
B.S., Purdue University, 1975
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado at Denver
in partial fulfillment
of the requirements for the degree of
Master of Science
Civil Engineering
1994
This thesis for the Master of Science degree by
Daniel Edwin Sass
has been approved for the
Department of Civil Engineering
by
Sass, Daniel Edwin (M.S., Civil Engineering)
Improvement of the Buckling Capacity of Slender Timber Columns
Thesis directed by Associate Professor Judith J. Stalnaker
ABSTRACT
This document presents a simple method to improve the buckling capacity of
timber columns. The theoretical development is presented for a simply supported
column with a reinforced section. This method is then used to evaluate the improved
buckling capacity of selected timber columns. The buckling capacities predicted by the
method presented are compared to Finite Element model predictions for selected
cases. Finally, design curves and a modified National Design Specification buckling
equation are presented to be used by the engineer.
This abstract accurately represents the content of the candidates thesis. I recommend
its publication.
Signed
Judith J/St&lnaker
In loving memory of my father
Edward John Sass
ACKNOWLEDGMENTS
I would like to thank Truswal Systems of Colorado Springs, Colorado, and
their chief engineer, Scott Welker, who originated the concept of using the truss plates
to reinforce timber columns to improve their buckling capacity. Truswal provided
continued assistance throughout the conceptual development phase of this document.
I would also like to thank Dr. Judith Stalnaker, Associate Professor in the
Department of Civil Engineering, University of Colorado at Denver, who has provided
continuing guidance and support throughout the three years I have been at UCD.
Dr. Stalnaker is a gifted instructor who has always had time for her students. Her
timely professional engineering support throughout all phases of this project were
indispensable. It has been a great privilege to have been one of Dr. Stalnakers
students.
VI
CONTENTS
Chapter
1 Introduction...................................................1
2 Theory.........................................................4
2.1 Development of Methodology for Buckling Capacity..............5
2.2 Comparison to Euler Column Buckling............................9
2.3 Column Buckling Capacity with Reinforcement and without
Reinforcement Eccentricity....................................11
2.4 Column Buckling Capacity with Reinforcement and with
Reinforcement Eccentricity....................................17
2.5 Effects of Reinforcement Length on Column Buckling Capacity...26
2.6 Effects of Timber Modulus of Elasticity on Column Buckling
Capacity......................................................39
2.7 Chapter Summary...............................................47
3 Finite Element Analysis.......................................48
3.1 Validation of Plate Stiflhess.................................48
3.2 Validation of Column Buckling Capacity........................51
3.3 Detailed Column Buckling Analysis.............................52
4 Practical Considerations......................................56
4.1 Limitations...................................................56
4.2 National Design Specification Buckling........................57
4.3 NDS Modified Buckling Capacity................................59
vii
Chapter
5 Recommendations and Conclusions..............................63
Appendix
A. Column Buckling Factor Notes.................................64
B. Column with Steel Reinforcement 2x4x72.....................72
C. Column with Steel Reinforcement 2x4x96.....................82
D. Column with Steel Reinforcement 2x4x 120...................92
E. Column with Aluminum Reinforcement 2x4x72.................102
F. Column with Aluminum Reinforcement 2x4x96.................112
G. Column with Aluminum Reinforcement 2x4x120................122
H. Column with Steel Reinforcement 2x6x72....................132
I. Column with Steel Reinforcement 2x6x96....................142
J. Column with Steel Reinforcement 2x6x120...................152
K. Column with Aluminum Reinforcement 2x6x72.................162
L. Column with Aluminum Reinforcement 2x6x96.................172
M. Column with Aluminum Reinforcement 2x6x 120...............182
References.........................................................192
viii
FIGURES
Figure
1.1 Typical Timber Truss Configuration............................1
1.2 Typical Truss Plate Connection................................2
2.1 Column Property Definition....................................5
2.2 Column Plate Reinforcement...................................11
2.3 Column Box Plate Reinforcement...............................15
2.4 Column Plate Reinforcement with Eccentricity.................18
2.5 Buckling Capacity of a 6 Foot 2x4 1650fl ,4E Reinforced
Column......................................................27
2.6 Buckling Capacity of an 8 Foot 2x4 1650fl ,4E Reinforced
Column..................................................... 28
2.7 Buckling Capacity of a 10 Foot 2x4 1650fl ,4E Reinforced
Column......................................................29
2.8 Buckling Capacity of a 6 Foot 2x6 1650fl .4E Reinforced
Column......................................................30
2.9 Buckling Capacity of an 8 Foot 2x6 1650fl ,4E Reinforced
Column......................................................31
2.10 Buckling Capacity of a 10 Foot 2x6 1650f1,4E Reinforced
Column.............................................................32
2.11 A % for a 6 Foot 2x4 1650fl ,4E Reinforced Column...........33
2.12 A% for an 8 Foot 2x4 1650fl ,4E Reinforced Column...........34
2.13 A% for a 10 Foot 2x4 1650f1.4E Reinforced Column............35
2.14 A% for a 6 Foot 2x6 1650fl .4E Reinforced Column............36
2.15 A% for an 8 Foot 2x6 1650fl ,4E Reinforced Column...........37
LX
Figure
2.16 A% for a 10 Foot 2x6 1650f1.4E Reinforced Column............38
2.17 A% for a 6 Foot 2x4 3300f2.6E Reinforced Column.............41
2.18 A% for an 8 Foot 2x4 3300f2.6E Reinforced Column............42
2.19 A% for a 10 Foot 2x4 3300f2.6E Reinforced Column............43
2.20 A % for a 6 Foot 2x6 3300f2.6E Reinforced Column............44
2.21 A% for an 8 Foot 2x6 3300f2.6E Reinforced Column............45
2.22 A% for a 10 Foot 2x6 3300f2.6E Reinforced Column............46
3.1 Typical Reinforcing Plate Configuration..................... 49
3.2 Reduced Analytical Model.....................................50
3.3 Beam Finite Element Model....................................51
3.4 3D Finite Element Model.....................................52
3.5 Plane Stress Finite Element Configurations...................53
3.6 Internal Compressive Stress for Reinforced Plates with and
without Prongs...............................................54
4.1 Column Buckling Factor for a 6 Foot Timber Column
Modulus of 1.4E6 psi.........................................62
B. 1 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement.........73
B.2 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement.........74
B.3 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement.........75
Figure
B.4 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........76
B.5 Column Buckling Factor for a 6 Foot 2x4 Column with a 1 8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........77
B.6 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........78
B.7 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........79
B.8 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........80
B. 9 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........81
C. 1 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........83
C.2 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........84
C.3 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........85
C.4 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........86
C.5 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........87
C.6 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........88
C.7 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement...............89
XI
Figure
C.8 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement...............90
C. 9 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........91
D. 1 Column Buckling Factor for a 10 Foot 2x4 Column with a 1.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........93
D.2 Column Buckling Factor for a 10 Foot 2x4 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........94
D.3 Column Buckling Factor for a 10 Foot 2x4 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........95
D.4 Column Buckling Factor for a 10 Foot 2x4 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........96
D.5 Column Buckling Factor for a 10 Foot 2x4 Column with a 1.8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........97
D.6 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........98
D.7 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........99
D.8 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........100
D. 9 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........101
E. 1 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............103
E.2 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............104
XU
Figure
E.3 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............105
E.4 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............106
E.5 Column Buckling Factor for a 6 Foot 2x4 Column with a 1.8E6 psi
Modulus of Elasticity Aluminum Reinforcement..............107
E.6 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............108
E.7 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............109
E.8 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............110
E. 9 Column Buckling Factor for a 6 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............Ill
F. 1 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.0E6 psi
Modulus of Elasticity Aluminum Reinforcement.................113
F.2 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............114
F.3 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............115
F.4 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............116
F.5 Column Buckling Factor for an 8 Foot 2x4 Column with a 1.8E6 psi
Modulus of Elasticity Aluminum Reinforcement..............117
F.6 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............118
xm
Figure
F.7 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............119
F.8 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement...............120
F. 9 Column Buckling Factor for an 8 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement...............121
G. 1 Column Buckling Factor for a 10 Foot 2x4 Column with a 1 0E6 psi
Modulus of Elasticity Aluminum Reinforcement...............123
G.2 Column Buckling Factor for a 10 Foot 2x4 Column with a 1 2E6 psi
Modulus of Elasticity Aluminum Reinforcement...............124
G.3 Column Buckling Factor for a 10 Foot 2x4 Column with a 1 4E6 psi
Modulus of Elasticity Aluminum Reinforcement...............125
G.4 Column Buckling Factor for a 10 Foot 2x4 Column with a 1 6E6 psi
Modulus of Elasticity Aluminum Reinforcement...............126
G.5 Column Buckling Factor for a 10 Foot 2x4 Column with a 1 8E6 psi
Modulus of Elasticity Aluminum Reinforcement...............127
G.6 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement...............128
G.7 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement...............129
G.8 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement...............130
G. 9 Column Buckling Factor for a 10 Foot 2x4 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement...............131
H. 1 Column Buckling Factor for a 6 Foot 2x6 Column with a 1 0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement.........133
XIV
Figure
H.2 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........134
H.3 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........135
H.4 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........136
H.5 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........137
H.6 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........138
H.7 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........139
H.8 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........140
H. 9 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........141
I. 1 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........143
1.2 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........144
1.3 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........145
1.4 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........146
1.5 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........147
XV
Figure
1.6 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........148
1.7 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........149
1.8 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........150
1.9 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........151
J. 1 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........153
J.2 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........154
J.3 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........155
J.4 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........156
J.5 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........157
J.6 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........158
J.7 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........159
J.8 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........160
J.9 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Steel and Wood Reinforcement........161
XVI
Figure
K. 1 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............163
K.2 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............164
K.3 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............165
K.4 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............166
K.5 Column Buckling Factor for a 6 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Aluminum Reinforcement..............167
K.6 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............168
K.7 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............169
K.8 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............170
K. 9 Column Buckling Factor for a 6 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............171
L. 1 Column Buckling Factor for an 8 Foot 2x6 Column with a 1 0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............173
L.2 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............174
L.3 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............175
L.4 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............176
xvn
Figure
L.5 Column Buckling Factor for an 8 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Aluminum Reinforcement..............177
L.6 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............178
L.7 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............179
L.8 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............180
L. 9 Column Buckling Factor for an 8 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............181
M. 1 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............183
M.2 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............184
M.3 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............185
M.4 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.6E6 psi
Modulus of Elasticity Aluminum Reinforcement..............186
M. 5 Column Buckling Factor for a 10 Foot 2x6 Column with a 1.8E6 psi
Modulus of Elasticity Aluminum Reinforcement..............187
M.6 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.0E6 psi
Modulus of Elasticity Aluminum Reinforcement..............188
M.7 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.2E6 psi
Modulus of Elasticity Aluminum Reinforcement..............189
M.8 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.4E6 psi
Modulus of Elasticity Aluminum Reinforcement..............190
XVUl
Figure
M.9 Column Buckling Factor for a 10 Foot 2x6 Column with a 2.6E6 psi
Modulus of Elasticity Aluminum Reinforcement...............191
XIX
TABLES
Tables
2.1 Selected Wood Material Properties..............................10
2.2 Section Properties of Selected Standard Dressed Sawn Lumber....10
2.3 Comparison of Euler Buckling Load with Eqn 2.11................10
2.4 Reinforcement Material Properties..............................12
2.5 Effective Section Inertia for 1650fl,4E MSR Rated Wood
with Plate Reinforcement......................................13
2.6 Column Capacity with Plate Reinforcement.......................14
2.7 Effective Section Inertia for 1650fl ,4E MSR Rated Wood
with Box Plate Reinforcement..................................16
2.8 Column Capacity with Box Plate Reinforcement...................17
2.9 Column Capacity with Plate Reinforcement Eccentricity..........18
2.10 Column Capacity with Box Plate Reinforcement Eccentricity......21
2.11 Percentage Decrease in Capacity Due to Eccentricity............23
3.1 Comparison of Theoretical and FEM Buckling Capacities..........52
3.2 Maximum Internal Compressive Stress............................55
4.1 Comparison of NDS Column Capacity with Euler Buckling Load.... 59
A. 1 Selected Wood Material Properties..............................65
A.2 Section Properties of Selected Standard Dressed Sawn Lumber...65
A. 3 Effective Section Inertia for a Nominal Column Modulus of
1.0E6 psi.....................................................66
A.4 Effective Section Inertia for a Nominal Column Modulus of
1.2E6 psi.....................................................66
XX
Tables
A. 5 Effective Section Inertia for a Nominal Column Modulus of
1.4E6 psi.......................................................67
A. 6 Effective Section Inertia for a Nominal Column Modulus of
1.6E6 psi.......................................................67
A. 7 Effective Section Inertia for a Nominal Column Modulus of
1.8E6 psi.......................................................68
A. 8 Effective Section Inertia for a Nominal Column Modulus of
2.0E6 psi.......................................................68
A. 9 Effective Section Inertia for a Nominal Column Modulus of
2.2E6 psi.......................................................69
A. 10 Effective Section Inertia for a Nominal Column Modulus of
2.4E6 psi.......................................................69
A. 11 Effective Section Inertia for a Nominal Column Modulus of
2.6E6 psi.......................................................70
XXI
1.
Introduction
Wood has long been a major material used in the construction of a variety of
structures. These structures included in the past castles and fortifications; today we
choose wood as a building material for churches, businesses, and homes. The design
process over this span of time has allowed the architect, designer, and engineer to
assemble structures which utilize the capacity of this material to the maximum extent
possible. The factors controlling the design of wood structures include material
properties, environmental conditions, connections, geometric effects, and economics
to name a few.
In particular, the design of many wood structures today include component
truss systems built in a factory setting. These components are transported to the
building site and assembled into the final structure (Figure 1.1).
l
In the past, the light truss frame connections used nails or screws as the
fastening mechanism. Today, trusses are fastened using perforated metal plates.
These plates are easy to apply and produce connections which are superior to those
using nails or screws. These plates are inexpensive to produce and come in a variety
of thicknesses and lengths (Figure 1.2).
Figure 1.2 Typical Truss Plate Connection
Each truss is subjected to a variety of loading conditions which produce the
design conditions for the members of the frame. These design conditions define the
required compressive or tensile loads the truss members are required to sustain.
The compression members of these truss systems are normally long and
slender. The design of these members is therefore normally controlled by buckling
rather than the compression capacity of the member. The design engineer must add
lateral support to these members to obtain the maximum compressive strength of the
2
element. The addition of the bracing material not only adds weight to the structure,
but also adds significant material and manpower costs to the structure.
Increasing the moment of inertia over part of the length of the column should
increase its buckling capacity. Timoshenko (Ref. 1, Sect. 2.14) discussed this type of
problem. He examined a long slender column with a continuously varying area
moment of inertia and demonstrated that a substantial increase in the buckling capacity
was possible.
This thesis will utilize Timoshenkos technique to develop the theory required
to predict the increase in buckling capacity for a simply supported slender column with
truss plates as reinforcement. The theoretical model will be validated through
numerical analysis and a simplified analysis method will be presented to be used by the
design engineer in conjunction with the National Design Specification (NDS1
requirements (Ref. 2).
This thesis will show that a member of varying cross sectional properties is a
reasonable alternative to the addition of lateral support to meet current buckling
requirements.
3
2. Theory
This chapter will present the development of the methodology to predict the
buckling capacity of a slender column with reinforcement. In this development, the
column is assumed to be simply supported and axially loaded without any eccentricity.
The materials are assumed to be homogeneous and isotropic.
4
2.1 Development of Methodology for Buckling Capacity
Finding the buckling capacity of the column with reinforcement shown in
Figure 2.1 requires solving the equation
E I 42 = Py
dx2 y
(2.1)
i
P
Figure 2.1 Column Property Definition
5
The solution to this equation is for 0 < x < lx
y = ^IjCOS^jX + Bx sinÂ£,x
(2.2)
for /, < x < lx +12
y = A2 cosk2x + B2 sin&2x
for /, +12 < x < /
y = A2coskxx + B3s\nkxx
where Arx2 = and A:,2 = ^
Vi
E2I2
The boundary conditions are
y = 0 at x = 0 0 = ^cos^jO + i^sin^O
_y = 0 at x = / 0=^(3 coskj + B3 sin kxl
The compatibility requirements are
at = A yv^ = yih+)
Ax cos kxlx + Blsinkllx = A2cosk2lx + B2 sink2lx
at x = lx
dy dy
) ^(h *)
Axkxsmkxlx + Bxkx coskxlx =
A2k2 sink2lx + B2k2 cosk2lx
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
6
_ yUi^z)+
at x = /j + l2
A2 cosk2 (/, +/2) + f?2 sin k2 (/, +/2) =
j43 cos^ (/, +/2) + B3sinkl(ll +/2)
at x = /, +l2 =
^*(1, Hi) ^(',h2)+
~A2k2 sink2 (/] +/2) + B2k2 cosk2 (Ji )
A^^sink^ +/2) + cosÂ£,(/, +/2)
The system of resulting equations may be written as
(2.9)
(2.10)
7
o ,
cs
cq ^
m m
dq ^ oq
cs
cn
+ +
cn ^
^ CO
O
S O
**e
c
+ +
w <4*
1( CO
S 8
^H
CO
(N
.3
r
O
u
^Â£S
+
f 3 s
s
~r
+
tN
A*
tN
V
T"* +
~ ^ + ~
id "Ss fl
g .3 ^
? ^
* 8
a
f
Jd w
Co
y
55
*
.3
CO co
8 H
w I8
8
The non trivial solution of this homogeneous system of equations is found by
solving the corresponding characteristic equation for the minimum value of P which is
defined as the critical buckling load.
It is impractical to solve the characteristic equation in a closed form, therefore
the buckling load will be calculated using numerical methods available in the
commercially available PC software package Mathcad by MathSoft (Ref. 3).
2.2 Comparison to Euler Column Buckling
The first obvious requirement of this solution is to produce the predicted Euler
buckling load for a column without reinforcement. The Euler buckling load is given
by
P. =
ir2 El
l2
(2.12)
For this comparison, the material and geometric properties presented in Tables
2.1 and 2.2 will be used. These properties are representative of the materials used in
wood construction. For a complete list of these properties, refer to the National
Design Specification for Wood Construction (Ref. 2).
Table 2.3 shows the comparison of the results obtained using equation 2 .12
and 2 .11 with Â£,/, = E2I2. Table 2.3 serves as validation of the derivation of the
methodology for equation 2.11
9
Table 2.1 Selected Wood Material Properties (*)
Commercial Grade Allowable Bending Stress Fb (psi) Allowable Tension parallel to grain Ft (psi) Allowable Compression parallel to grain Fc (psi) Modulus of Elasticity E (psi)
900f1.0E 900 350 1050 1,000,000
1200f1.2E 1200 600 1400 1,200,000
1650f1.4E 1650 1020 1700 1,400,000
1800f1.6E 1800 1175 1750 1,600,000
2100f1.8E 2100 1575 1875 1,800,000
2400f2.0E 2400 1925 1975 2,000,000
2700f2.2E 2700 2150 2100 2,200,000
3000f2.4E 3000 2400 2200 2,400,000
3300f2.6E 3300 2650 2325 2,600,000
* Note: Obtained from the National Design Specification for Wood (Ref. 2).
Table 2.2 Section Properties of Selected Standard Dressed Sawn Lumber (*)
Nominal Size Dressed Size (in) Cross Sectional Area (in2) XX axis Moment of Inertia Lx (in4) YY axis Moment of Inertia Iw (in4)
1x4 3/4 x 31/2 2.625 2.680 0.123
2x4 11/2 x 31/2 5.250 5.359 0.984
2x6 11/2 x 51/2 8.250 20.80 1.547
2x8 11/2 x 71/4 10.88 47.63 2.039
* Note: Obtained from the National Design Specification for Wood (Ref. 2).
Table 2.3 Comparison of Euler buckling load with Eqn 2.11
Column Length (ft) 2x4 Buckling Load (lb) 2x6 Buckling Load (lb)
Euler Eqn 2.11 Euler Eqn 2.11
6 2623 2623 4123 4123
8 1475 1475 2319 2319
10 944 944 1484 1484
10
2.3 Column Buckling Capacity with Reinforcement and without
Reinforcement Eccentricity
The previous section provided evidence that equation 2.11 does predict the
Euler buckling load accurately for unreinforced compression members. This section
will present the possible increase in capacity to be achieved using common building
materials as reinforcement.
Many of the homes and businesses constructed of wood use trusses built and
assembled offsite in the factory. These trusses are assembled using perforated steel
plates at the joints. These steel plates can also make an excellent reinforcement for the
wood columns for increasing the buckling capacity. For comparison purposes,
aluminum and wood reinforcement is also examined. The plates will be centered along
the width and length of the column as shown in Figure 2.2. The material and
geometric properties for this configuration are presented in Table 2.4.
1 J ^ 1 ^
* 11 * * 2* * 13
Figure 2.2 Column Plate Reinforcement
11
Table 2.4 Reinforcement Material Properties
Materials
Material Modulus of Elasticity (psi) Effective Cross Sectional Area (%)
Steel 29,000,000 53%
Aluminum 10,200,000 53%
Wood 1,400,000 100%
Section Properties
Material Thickness (in) Width (in) Cross Sectional Area A (in2) Effective Cross Sectional Area Aeff (in2)
Steel (20 Gauge) 0.036 3.200 0.1152 0.0611
Steel (18 Gauge) 0.048 3.200 0.1536 0.0814
Steel (16 Gauge) 0.060 3.000 0.1800 0.0954
Aluminum (20 Gauge) 0.036 3.200 0.1152 0.0611
Aluminum (18 Gauge) 0.048 3.200 0.1536 0.0814
Aluminum (16 Gauge) 0.060 3.000 0.1800 0.0954
Wood (1x4) 0.750 3.500 2.625 2.625
The transformed area moment of inertia about the centriodal axis for the
composite reinforced section is given by
I
I mod +2
+/<#*)
^ wood
(2.13)
The local moment of inertia of the reinforcing element (Ieffp]ate) is of secondary
importance and may be neglected for the steel and aluminum plates. In the case of the
12
wood plate, the local moment of inertia of the plate is significant. The moment of
inertia of the transformed section for a MSR 1650f1.4E timber column is shown in
Table 2.5 for various composite sections.
Table 2.5 Effective Section Inertia for 1650f1.4E MSR Rated Wood
with Plate Reinforcement
2x4 Section
Reinforcing Material Iwood (in4) Iplale (in4) Ieff (in4) Ieff* (in4) lefl^Iwood (%)
Steel (20 Gauge) 0.984 1.24E5 2.407 2.407 244.60%
Steel (18 Gauge) 0.984 2.95E5 2.881 2.882 292.79%
Steel (16 Gauge) 0.984 5.40E5 3.207 3.209 325.93%
Aluminum (20 Gauge) 0.984 1.24E5 1.484 1.485 150.86%
Aluminum (18 Gauge) 0.984 2.95E5 1.651 1.652 167.81%
Aluminum (16 Gauge) 0.984 5.40E5 1.766 1.767 179.47%
Wood (1x4) 0.984 0.123 7.629 7.875 800.30%
2x6 Section
Reinforcing Material I\VDod (in4) Iplate (in4) Ieff (in4) Ieff* (in4) Iefl/Iwood (%)
Steel (20 Gauge) 1.547 1.24E5 2.970 2.970 191.97%
Steel (18 Gauge) 1.547 2.95E5 3.444 3.445 222.63%
Steel (16 Gauge) 1.547 5.40E5 3.770 3.772 243.71%
Aluminum (20 Gauge) 1.547 1.24E5 2.047 2.048 132.35%
Aluminum (18 Gauge) 1.547 2.95E5 2.214 2.215 143.13%
Aluminum (16 Gauge) 1.547 5.40E5 2.329 2.330 150.55%
Wood (1x4) 1.547 0.123 8.192 8.438 529.51%
Note: Ieff is computed without Ieffpute, Ieff* is computed with Ieffpute.
For this comparison, the length of the reinforcing plated) will be 15 inches and
the plate is centered on the column (i.e., li = I3). The resulting computed buckling
loads are shown in Table 2.6.
13
Table 2.6 Column Capacity with Plate Reinforcement
Geometry
1 (in) li (in) h (in) 13 (in)
72 28.5 15 28.5
96 40.5 15 40.5
120 45.0 15 45.0
2x4
Column Length (in) 72 96 120
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 3404 30 1789 21 1100 17
Steel (18 Gauge) 3517 34 1832 24 1121 19
Steel (16 Gauge) 3577 36 1854 26 1132 20
Aluminum (20 Gauge) 3026 15 1642 11 1029 9
Aluminum (18 Gauge) 3118 19 1679 14 1047 11
Aluminum (16 Gauge) 3174 21 1701 15 1058 12
Wood (1x4) 3921 50 1979 34 1190 26
2x6
Column Length (in) 72 96 120
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 5072 23 2707 17 1679 13
Steel (18 Gauge) 5249 27 2774 20 1711 15
Steel (16 Gauge) 5347 30 2812 21 1729 17
Aluminum (20 Gauge) 4567 11 2505 8 1579 6
Aluminum (18 Gauge) 4682 14 2552 10 1602 8
Aluminum (16 Gauge) 4754 15 2581 11 1616 9
Wood (1x4) 5983 45 3047 31 1841 24
where A% =^SE. *100 and Pcbr is the buckling capacity of the reinforced
column found from equation 2.11.
14
An alternate reinforcement configuration is to wrap the timber column with
either steel or aluminum as shown in Figure 2.3.
1
1 i 2*P i3
Figure 2.3 Column Box Plate Reinforcement
The transformed moment of inertia is defined by
I
eff
=1
wood
+2
+Ieff^ +Ixx^)
wood
(2.14)
The plate inertia about the strong axis significantly increases the local inertia of the
section. The local section moment of inertias are presented in Table 2.7. The
resulting computed buckling loads are shown in Table 2.8 as well as the percentage
increases in buckling capacity. The 6 foot column with 16 gauge steel reinforcement
produces a 42% increase in the buckling capacity when compared to the Euler
buckling load. This is a 6% increase in the buckling load when compared to the plate
reinforcement capacity for the 15 inch plate reinforcement.
15
Table 2.7 Effective Section Inertia for 1650f1.4E MSR Rated Wood
with Box Plate Reinforcement
2x4 Section
Reinforcing Material Iwood (in4) Plate Ieff (in4) BOX Ieff* (in4) BOX Ieff/I\vood (%)
Steel (20 Gauge) 0.984 2.407 2.960 300.81%
Steel (18 Gauge) 0.984 2.881 3.619 367.78%
Steel (16 Gauge) 0.984 3.207 4.279 434.86%
Aluminum (20 Gauge) 0.984 1.484 1.679 170.63%
Aluminum (18 Gauge) 0.984 1.651 1.911 194.20%
Aluminum (16 Gauge) 0.984 1.766 2.143 217.78%
2x6 Section
Reinforcing Material Iwood (in4) Plate Ieff (in4) BOX Ieff* (in4) BOX Ieff/Iwood (%)
Steel (20 Gauge) 1.547 2.970 4.413 285.26%
Steel (18 Gauge) 1.547 3.444 5.369 347.06%
Steel (16 Gauge) 1.547 3.770 6.325 408.86%
Aluminum (20 Gauge) 1.547 2.047 2.555 165.16%
Aluminum (18 Gauge) 1.547 2.214 2.891 186.88%
Aluminum (16 Gauge) 1.547 2.329 3.228 208.66%
Note: Ieff is computed without Ieffpute, Ieff* is computed with Ieffpiate.
16
Table 2.8 Column Capacity with Box Plate Reinforcement
2x4
Column Length (in) 72 96 12 D
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 3532 35 1838 25 1124 19
Steel (18 Gauge) 3640 39 1877 27 1142 21
Steel (16 Gauge) 3717 42 1906 29 1156 22
Aluminum (20 Gauge) 3132 19 1685 14 1049 11
Aluminum (18 Gauge) 3237 23 1725 17 1069 13
Aluminum (16 Gauge) 3323 27 1759 19 1086 15
2x6
Column Length (in) 72 96 12 0
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 5504 34 2871 24 1758 18
Steel (18 Gauge) 5676 38 2935 27 1788 20
Steel (16 Gauge) 5801 41 2981 29 1810 22
Aluminum (20 Gauge) 4881 18 2631 13 1642 11
Aluminum (18 Gauge) 5042 22 2694 16 1672 13
Aluminum (16 Gauge) 5175 26 2746 18 1697 14
where A% = 100 and Pcbr is the buckling capacity of the reinforced
column found from equation 2.11.
2.4 Column Buckling Capacity with Reinforcement and with Reinforcement
Eccentricity
The process of building the truss assemblies with the reinforcement exactly
centered along the column length (as in the previous section) may not be realistic. The
manufacturing process most certainly may cause significant eccentricity in the
17
application of the reinforcement (i.e. h I3). This section will examine the effects of
eccentricities of 5,10, and 15 inches. The definition of the eccentricity is shown in
Figure 2.4. This analysis will use 1650f'1.4E MSR rated wood and a 15 inch
reinforcement as used in section 2.3.
ll* *vrr l >3 :
^ Sterns^ 2
1 ;
S'
Figure 2.4 Column Plate Reinforcement with Eccentricity
Tables 2.9 and 2.10 that follow show, for truss plates and box reinforcement
respectively, that increases in buckling capacities result even if the reinforcement is not
perfectly centered.
Table 2.9 Column Capacity with Plate Reinforcement Eccentricity
2 x 4x72
Eccentricity in 5 10 15
Reinforcing Material PcBR lb A% P CBR lb A% PcBR lb A%
Steel (20 Gauge) 3350 28 3215 23 3051 16
Steel (18 Gauge) 3452 32 3292 26 3102 18
Steel (16 Gauge) 3506 34 3332 27 3129 19
Aluminum (20 Gauge) 3003 15 2942 12 2861 9
Aluminum (18 Gauge) 3089 18 3011 15 2910 11
Aluminum (16 Gauge) 3140 20 3052 16 2939 12
Wood (1x4) 3808 45 3550 35 3269 25
18
Table 2.9 Column Capacity with Plate Reinforcement Eccentricity (Continued)
2 x 4x96
Eccentricity in 5 10 15
Reinforcing Material PCBR lb A% PcBR lb A% PcBR lb A%
Steel (20 Gauge) 1778 21 1747 18 1703 15
Steel (18 Gauge) 1818 23 1782 21 1731 17
Steel (16 Gauge) 1840 25 1800 22 1745 18
Aluminum (20 Gauge) 1637 11 1622 10 1601 8
Aluminum (18 Gauge) 1672 13 1654 12 1627 10
Aluminum (16 Gauge) 1693 15 1673 13 1642 11
Wood (1x4) 1958 33 1900 29 1823 24
2 x 4x120
Eccentricity in 5 10 15
Reinforcing Material P CBR lb A% PcBR lb A% PcBR lb A%
Steel (20 Gauge) 1097 16 1087 15 1072 14
Steel (18 Gauge) 1117 18 1105 17 1088 15
Steel (16 Gauge) 1127 19 1115 18 1097 16
Aluminum (20 Gauge) 1027 9 1022 8 1015 7
Aluminum (18 Gauge) 1045 11 1039 10 1030 9
Aluminum (16 Gauge) 1055 12 1049 11 1038 10
Wood (1x4) 1184 25 1167 24 1141 21
2 x 6x72
Eccentricity in 5 10 15
Reinforcing Material P CBR lb A% PcBR lb A% PcBR lb A%
Steel (20 Gauge) 5015 22 4860 18 4663 13
Steel (18 Gauge) 5174 25 4984 21 4748 15
Steel (16 Gauge) 5263 28 5052 23. 4795 16
Aluminum (20 Gauge) 4543 10 4479 9 4392 7
Aluminum (18 Gauge) 4651 13 4568 11 4457 8
Aluminum (16 Gauge) 4718 14 4623 12 4496 9
Wood (1x4) 5829 41 5470 33 5068 23
19
Table 2.9 Column Capacity with Plate Reinforcement Eccentricity (Continued)
2 x 6x96
Eccentricity in 5 10 15
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 2694 16 2658 15 2605 12
Steel (18 Gauge) 2758 19 2714 17 2651 14
Steel (16 Gauge) 2794 20 2745 18 2676 15
Aluminum (20 Gauge) 2499 8 2484 7 2460 6
Aluminum (18 Gauge) 2544 10 2524 9 2495 8
Aluminum (16 Gauge) 2572 11 2549 10 2516 8
Wood (1x4) 3017 30 2936 27 2827 22
2 x 6x120
Eccentricity in 5 10 15
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 1675 13 1663 12 1645 11
Steel (18 Gauge) 1706 15 1692 14 1671 13
Steel (16 Gauge) 1724 16 1709 15 1685 14
Aluminum (20 Gauge) 1577 6 1572 6 1564 5
Aluminum (18 Gauge) 1600 8 1593 7 1583 7
Aluminum (16 Gauge) 1614 9 1606 8 1595 7
Wood (1x4) 1832 23 1808 22 1771 19
where A% = 100 and Pcbr is the buckling capacity of the reinforced
column found from equation 2.11.
20
Table 2.10 Column Capacity with Box Plate Reinforcement Eccentricity
2 x 4x72
Eccentricity in 5 10 15
Reinforcing Material PcBR lb A% P CBR lb A% P CBR lb A%
Steel (20 Gauge) 3466 32 3303 26 3109 19
Steel (18 Gauge) 3561 36 3373 29 3156 20
Steel (16 Gauge) 3630 38 3423 31 3188 22
Aluminum (20 Gauge) 3102 18 3022 15 2918 11
Aluminum (18 Gauge) 3198 22 3098 18 2971 13
Aluminum (16 Gauge) 3277 25 3159 20 3013 15
2 x 4x96
Eccentricity in 5 10 15
Reinforcing Material PcBR lb A% PcBR lb A% PcBR lb A%
Steel (20 Gauge) 1824 24 1787 21 1735 18
Steel (18 Gauge) 1862 26 1819 23 1760 19
Steel (16 Gauge) 1888 28 1842 25 1778 21
Aluminum (20 Gauge) 1678 14 1659 12 1631 11
Aluminum (18 Gauge) 1717 16 1693 15 1659 12
Aluminum (16 Gauge) 1749 19 1721 17 1682 14
2 x 4x120
Eccentricity in 5 10 15
Reinforcing Material PcBR lb A% PcBR lb A% PcBR lb A%
Steel (20 Gauge) 1120 19 1108 17 1090 15
Steel (18 Gauge) 1138 21 1125 19 1105 17
Steel (16 Gauge) 1151 22 1136 20 1115 18
Aluminum (20 Gauge) 1047 11 1041 10 1032 9
Aluminum (18 Gauge) 1067 13 1059 12 1048 11
Aluminum (16 Gauge) 1083 15 1074 14 1061 12
21
Table 2.10 Column Capacity with Box Plate Reinforcement Eccentricity (Continued)
2 x 6x72
Eccentricity in 5 10 15
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 5405 31 5159 25 4866 18
Steel (18 Gauge) 5558 35 5273 28 4941 20
Steel (16 Gauge) 5669 37 5354 30 4994 21
Aluminum (20 Gauge) 4836 17 4718 14 4564 11
Aluminum (18 Gauge) 4984 21 4836 17 4647 13
Aluminum (16 Gauge) 5106 24 4931 20 4713 14
2 x 6x96
Eccentricity in 5 10 15
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 2850 23 2794 21 2715 17
Steel (18 Gauge) 2911 26 2846 23 2756 19
Steel (16 Gauge) 2954 27 2883 24 2786 20
Aluminum (20 Gauge) 2621 13 2593 12 2552 10
Aluminum (18 Gauge) 2682 16 2647 14 2596 12
Aluminum (16 Gauge) 2731 18 2690 16 2631 13
2 x 6x120
Eccentricity in 5 10 15
Reinforcing Material Pcbr lb A% Pcbr lb A% Pcbr lb A%
Steel (20 Gauge) 1752 18 1734 17 1707 15
Steel (18 Gauge) 1781 20 1761 19 1731 17
Steel (16 Gauge) 1802 21 1780 20 1748 18
Aluminum (20 Gauge) 1638 10 1629 10 1615 9
Aluminum (18 Gauge) 1669 12 1657 12 1640 11
Aluminum (16 Gauge) 1693 14 1680 13 1660 12
where A% =^*100 and Pcbr is the buckling capacity of the reinforced
column found from equation 2.11.
22
The net resulting decrease in buckling capacity (compared to the perfectly
centered reinforcement h = I3) due to the eccentricity of the reinforcement (h ^ I3) is
shown in Table 2.11.
Table 2.11 Percentage Decrease in Capacity Due to Eccentricity
Eccentricity = 5 in
2x4
Column Length (in) 72 (Plate) 96 (Plate) 120 (Plate) 72 (Box) 96 (Box) 120 (Box)
Steel (20 Gauge) 2 1 0 3 1 0
Steel (18 Gauge) 2 1 0 3 1 0
Steel (16 Gauge) 3 1 0 3 1 1
Aluminum (20 Gauge) 1 0 0 1 0 0
Aluminum (18 Gauge) 1 0 0 1 1 0
Aluminum (16 Gauge) 1 1 0 2 1 0
Wood (1x4) 4 1 1 N/A N/A N/A
Eccentricity = 10 in
2x4
Column Length (in) 72 (Plate) 96 (Plate) 120 (Plate) 72 (Box) 96 (Box) 120 (Box)
Steel (20 Gauge) 7 3 1 9 3 2
Steel (18 Gauge) 9 3 2 10 4 2
Steel (16 Gauge) 9 4 2 11 4 2
Aluminum (20 Gauge) 3 1 1 4 2 1
Aluminum (18 Gauge) 4 2 1 5 2 1
Aluminum (16 Gauge) 5 2 1 6 3 1
Wood (1x4) 14 5 2 N/A N/A N/A
Note: Percentage decrease is defined as A%e(.c=0 A%ecci0
23
Table 2.11 Percentage Decrease in Capacity Due to Eccentricity (Continued)
Eccentricity = 15 in
2x4
Column Length (in) 72 96 120 72 96 120
(Plate) (Plate) (Plate) (Box) (Box) (Box)
Steel (20 Gauge) 13 6 3 16 7 4
Steel (18 Gauge) 16 7 3 18 8 4
Steel (16 Gauge) 17 7 4 20 9 4
Aluminum (20 Gauge) 6 3 1 8 4 2
Aluminum (18 Gauge) 8 4 2 10 4 2
Aluminum (16 Gauge) 9 4 2 12 5 3
Wood (1x4) 25 11 5 N/A N/A N/A
Eccentricity = 5 in
2x6
Column Length (in) 72 (Plate) 96 (Plate) 120 (Plate) 72 (Box) 96 (Box) 120 (Box)
Steel (20 Gauge) 1 1 0 2 1 0
Steel (18 Gauge) 2 1 0 3 1 0
Steel (16 Gauge) 2 1 0 3 1 1
Aluminum (20 Gauge) 1 0 0 1 0 0
Aluminum (18 Gauge) 1 0 0 1 1 0
Aluminum (16 Gauge) 1 0 0 2 1 0
Wood (1x4) 4 1 1 N/A N/A N/A
Note: Percentage decrease is defined as A%ecc=0 A%ecc>0
24
Table 2.11 Percentage Decrease in Capacity Due to Eccentricity (Continued)
Eccentricity = 10 in
2x6
Column Length (in) 72 (Plate) 96 (Plate) 120 (Plate) 72 (Box) 96 (Box) 120 (Box)
Steel (20 Gauge) 5 2 1 8 3 2
Steel (18 Gauge) 6 3 1 10 4 2
Steel (16 Gauge) 7 3 1 11 4 2
Aluminum (20 Gauge) 2 1 0 4 2 1
Aluminum (18 Gauge) 3 1 1 5 2 1
Aluminum (16 Gauge) 3 1 1 6 2 1
Wood (1x4) 12 5 2 N/A N/A N/A
Eccentricity = 15 in
2x6
Column Length (in) 72 96 120 72 96 120
(Plate) (Plate) (Plate) (Box) (Box) (Box)
Steel (20 Gauge) 10 4 2 15 7 3
Steel (18 Gauge) 12 5 3 18 8 4
Steel (16 Gauge) 14 6 3 20 8 4
Aluminum (20 Gauge) 4 2 1 8 3 2
Aluminum (18 Gauge) 5 2 1 10 4 2
Aluminum (16 Gauge) 6 3 1 11 5 3
Wood (1x4) 22 9 5 N/A N/A N/A
Note: Percentage decrease is defined as A%ecc^ A%ecc>)
25
The predicted decrease in column capacity which has been summarized in
Table 2.11 shows less than a 5% decrease for eccentricities not exceeding 5 inches for
all configurations presented. A 5 inch limit on the eccentricity of the reinforcement is
not unreasonable either in the manufacturing process, nor in the field, and therefore
should be considered a maximum allowable limit.
2.5 Effects of Reinforcement Length on Column Buckling Capacity
For a given column length, the column capacity may be incrementally increased
by changing the length of the reinforcement. This section presents the column
capacity for a range of reinforcement lengths without reinforcement eccentricity
present. The 1650f1.4E MSR rated wood has been used for this study.
The buckling capacity of 6, 8, and 10 foot columns with reinforcement lengths
of 5 to 18 inches is shown in Figures 2.5 through 2.10. These figures show the
predicted column capacity Pcbr of the reinforced column found by solving
equation 2.11. The percentage increase in buckling capacity as defined in Tables 2.9
and 2.10 is shown in Figures 2.11 through 2.16. The Euler buckling load of an
unreinforced column has been included for reference purposes.
26
Buckling Capacity lb Buckling Capacity lb
Reinforcement Length in.
Reinforcement Length in.
Figure 2.5 Buckling Capacity of a 6 Foot 2x4 1650f1.4E Reinforced Column
27
Buckling Capacity lb Buckling Capacity lb
2200
2100
2000
1900
1800
1600
1500
11.0 13.0
Reinforcement Length in.
19.0
Reinforcement Length in.
Figure 2.6 Buckling Capacity of an 8 Foot 2x4 1650f1.4E Reinforced Column
28
Buckling Capacity lb Buckling Capacity lb
1300
Reinforcement Length in.
Reinforcement Length in.
Figure 2.7 Buckling Capacity of a 10 Foot 2x4 1650f1.4E Reinforced Column
29
Buckling Capacity lb Buckling Capacity lb
Reinforcement Length in.
Reinforcement Length in.
Figure 2.8 Buckling Capacity of a 6 Foot 2x6 1650fl ,4E Reinforced Column
30
Buckling Capacity lb Buckling Capacity lb
Reinforcement Length in.
Reinforcement Length in.
Figure 2.9 Buckling Capacity of an 8 Foot 2x6 1650fl 4E Reinforced Column
31
Buckling Capacity lb Budding Capacity lb
Reinforcement Length in.
Reinforcement Length in.
Figure 2.10 Buckling Capacity of a 10 Foot 2x6 1650f1.4E Reinforced Column
32
in Buckling Capacity Percent Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.11 A% for a 6 Foot 2x4 1650fl 4E Reinforced Column
33
in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.12 A% for an 8 Foot 2x4 1650f1.4E Reinforced Column
34
in Buckling Capacity Percentage Increase in Buckling Capacity
35.00
30.00
25.00
20.00
15.00
10.00
0.00
" Steel Plate 20 Gauge
 Steel Plate 18 Gauge
Steel Plate 16 Gauge
 Steel Box 20 Gauge
 Steel Box 18 Gauge
* Steel Box 16 Gauge
Wood Plate 1x4
110 13 0
Reinforcement Length in.
Reinforcement Length in.
Figure 2.13 A% for a 10 Foot 2x4 1650f1.4E Reinforced Column
35
Percentage Increase in Buckling Capacity Percentage Increase in Budding Capacity
60.00
Reinforcement Length in.
Reinforcement Length in.
Figure 2.14 A% for a 6 Foot 2x6 1650fl 4E Reinforced Column
36
in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Figure 2.15 A% for an 8 Foot 2x6 1650fl 4E Reinforced Column
37
in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.16 A% for a 10 Foot 2x6 1650f1.4E Reinforced Column
38
2.6 Effects of Timber Modulus of Elasticity on Column Buckling Capacity
The range of values of the modulus of elasticity in wood used in construction
includes 0.8E6 psi to 2.6E6 psi, with the normal value being approximately 1.4E6 psi.
The effect of the modulus therefore bears investigation.
The basic buckling capacity of the column is linearly related to the modulus of
elasticity in the Euler buckling equation. However, the increase in the buckling
capacity of the reinforced column is dependent upon not only the column modulus, but
also the modulus of the reinforcing member as well as the cross sectional properties of
that member.
The highest modulus of elasticity available for construction purposes according
to the NDS (Ref. 2) is 2.6E6 psi which will be used to evaluate this effect.
The buckling capacity of a reinforced 3300f2.6E MSR rated column was
computed for column lengths of 6, 8, and 10 feet. The percentage increase in buckling
capacity is shown in Figures 2.17 through 2.22. These figures may be compared to
Figures 2.11 through 2.16 in section 2.5. It should be noted the percentage increase in
the buckling capacity of the steel and aluminum reinforced columns is less for the
higher modulus. The net decrease in performance of the column is due to the smaller
ratio of the column modulus to the modulus of the reinforcing material. Also, the
wood reinforcement is invariant since the increase in buckling capacity is due only to
39
the cross section properties of the material, since both the plate and the column are
assumed to be of the same material.
40
Percentage Increase in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.17 A% for a 6 Foot 2x4 3300f2.6E Reinforced Column
41
Percentage Increase in Buckling Capacity Percentage Increase in Buckling Capacity
45.00
Reinforcement Length in.
Reinforcement Length in.
Figure 2.18 A% for an 8 Foot 2x4 3300f2.6E Reinforced Column
42
Percentage Increase in Buckling Capacity Percentage Increase in Buckling Capacity
35.00
30.00
20.00
10.00
1 1 Steel Plate 20 Gauge Steel Plate 18 Gauge Steel Plate 16 Gauge Steel Box 20 Gauge Steel Box 18 Gauge Steel Box 16 Gauge Wood Plate 1x4
Euler Buckling Load 1753 lb
5.0
7.0
9.0
11.0 13.0
Reinforcement Length in.
15.0
17.0
19.0
Reinforcement Length in.
Figure 2.19 A% for a 10 Foot 2x4 3300f2.6E Reinforced Column
43
Percentage Increase in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.20 A% for a 6 Foot 2x6 3300f2.6E Reinforced Column
44
Percentage Increase in Buckling Capacity Percentage Increase in Buckling Capacity
Reinforcement Length in.
Reinforcement Length in.
Figure 2.21 A% for an 8 Foot 2x6 3300f2.6E Reinforced Column
45
Percentage Increase in Buckling Capacity Percentage Increase tn Buckling Capacity
Reinforcement Length in.
Figure 2.22 A% for a 10 Foot 2x6 3300f2.6E Reinforced Column
46
2.7 Chapter Summary
This chapter has presented the improved timber column theoretical
development. It has shown conclusively that reinforcement of the midheight of the
column produces an increase in column capacity. The increase in column capacity is
primarily due to the increase in the effective moment of inertia at the midheight and to
the length of the reinforcement. Installing the reinforcement eccentric to the midheight
of the column causes little change (compared to perfectly centered reinforcement) so
long as the eccentricity is less than 5 inches. Finally, an examination of the data
presented reveals the softer and shorter columns receive the highest percentage
increase in the buckling capacity of the column.
47
3. Finite Element Analysis
Finite element analysis was used to validate the strength increases predicted by
the analytical method presented in Chapter 2. The numerical computations were made
with an industry standard finite element code named ANSYS (Ref. 4).
Using finite element analysis, the stiffness properties of the reinforcing plate
were examined to validate the effective plate area chosen for these studies. Secondly,
the column buckling capacities for the reinforced members found by the previous
method for cases with and without eccentricity were validated. Finally, the internal
compressive stresses of the reinforced column were investigated.
3.1 Validation of Plate Stiffness
The metal plates which have been selected to reinforce the timber columns are
perforated through a stamping process, resulting in teeth that project at right angles
to the plate. These perforations allow the plate to be pressed on to the wood and form
part of the load path. These perforations also reduce the effective cross sectional area
of the plate.
The reinforcing plates used in the timber construction industry come in several
thickness and widths which are presented in Chapter 2 (Table 2.4). These plates are
available in a variety of lengths. A typical plate is shown in Figure 3.1.
48
Figure 3.1 Typical Reinforcing Plate Configuration
The numerical model of the plate was made using the planes of symmetry. The
reduced model with the assumed boundary conditions is shown in Figure 3.2.
49
Figure 3.2 Reduced Analytical Model
The element chosen for this analysis was an 8 node twodimensional finite
element CANSYS element PLANE82). A tensile load was applied as shown in Figure
3.2 and the mean deflection was computed. This displacement was compared to the
mean displacement for a nonperforated plate and found to be 65.9% less.
The major uncertainty in this analysis is the true load path in the plate. The
assumed boundary conditions do not represent the actual loading conditions since the
prongs formed by the stamping process are a major part of the load transfer. The load
distribution may not be uniform since the materials are bonded only by friction.
Therefore, only 80% of the effective area (53%) was chosen to be used in the
computation of the reinforcement inertia of the plates presented in Chapter 2.
50
The 53% effective area represents the effective area derived through the finite
element analysis. This effective area represents an unrealistic level of accuracy to used
in design. Therefore, in section 3.3 of this chapter, the following chapters, and
appendices, an effective area of 50% was chosen to be used.
3.2 Validation of Column Buckling Capacity
The column buckling capacities presented in Chapter 2 required independent
validation to insure the methodology was correct. A finite element model was
assembled using an elastic 2D beam finite element (ANSYS element BEAM3). The
analysis model and the boundary conditions are shown in Figure 3.3 below.
The timber column elements and the steel reinforcing elements were modeled
separately. The buckling analysis was performed using a 6 foot timber column
reinforced with a 15 inch long 16 gauge steel plate. The method presented in Chapter
2 predicts exactly the buckling capacities computed by the finite element code as
shown in Table 3.1.
Figure 3.3 Beam Finite Element Model
51
Table 3.1 Comparison of Theoretical and FEM Buckling Capacities
Eccentricity in. Equation 2.11 lb Finite Element Solution lb
0 3577 3577
15 3129 3128
3.3 Detailed Column Buckling Analysis
The purpose of this section is to develop an understanding of the internal
compressive stress distribution in the reinforced timber column.
First a numerical model of a steel reinforced timber column was made using
8 node 3D finite elements with 6 degrees of freedom at each node CAN SYS element
SOLID73). The model with the assumed boundary conditions is shown in Figure 3.4
below.
Effective Area = 50%,
Reinforcement Modulus = 26E6 psi
Column Modulus = 1.4E6 psi
Plate length = 7.5 in
Column Length = 36 in
Figure 3.4 3D Finite Element Model
52
The computed buckling load for the 3D finite element model matched the
predicted capacity of 3555 lb from equation 2.11.
The internal stresses at the reinforced section proved to be quite high.
However, the definition of the mesh size for the 3D model limited the accuracy of the
solution.
The internal stresses were determined using 2D plane stress finite elements
(ANSYS element PLANE42) in the configurations shown in Figure 3.5.
\*~ 75 1 h 75 *\
a) Reinforcement without prongs b) Reinforcement with prongs
Reinforcement Modulus = 29E6 psi
Column Modulus = 1 4E6 psi
Figure 3.5 Plane Stress Finite Element Configurations
The first model (a) was constructed to be consistent with the beam model.
Model (b) was based on the actual plate dimensions. The resulting internal
compressive stresses for the two configurations are presented in Figure 3.6. This
graph indicates that there may be high internal stresses at the timber/reinforcement
53
interface. These internal stresses may be the limiting factor in column capacity, rather
than buckling being the limiting factor. The actual material stresses will lie between
the two limiting cases depicted in Figure 3.6.
Figure 3.6 Internal Compressive Stress for Reinforced Plates with and without Prongs
When the modulus of elasticity of the reinforcement is very large in comparison
to the modulus of elasticity of the wood column, the resulting internal compressive
stresses become quite high. However, if a material with a smaller effective modulus
(aluminum or wood) is used as a reinforcement, then the internal stresses are smaller.
54
Therefore, columns reinforced with either 1x4 wood plates or aluminum plates were
investigated.
A comparison of the maximum internal stresses in the wood for certain
reinforcement types is shown in Table 3.2.
Table 3.2 Maximum Internal Compressive Stress
Steel Plate without prongs psi Steel Plate with prongs psi Aluminum Plate without prongs psi Wood Plate psi
5500 1500 1800 780
The internal stresses found through the analysis of the configurations presented
in Table 3.2 show the compressive capacity of the wood and the stiffness of the
reinforcement may control the actual buckling capacity of the reinforced timber
column. While the finite element models used in the analyses presented in this section
have investigated two idealized representations to obtain bounding compressive stress
levels, the actual load paths are quite different. The loads are transferred between the
reinforcement and the wood by prongs which extend into the wood only 3/8 of an
inch. These prongs are spaced as shown in Figure 3.1. The complexity of the material
interface, variety of material properties, and resulting internal load paths make it
unreasonable to attempt to solve for the actual internal compressive stresses.
While the buckling capacity of the reinforced column may be controlled by the
compressive strength of the material, verification may be obtained only through test.
55
4.
Practical Considerations
This chapter will present some potential practical limitations for the use of this
method, review the National Design Specification requirements (Ref. 2), and present a
modified form of the NDS buckling equation to be used in the design of reinforced
timber columns.
4.1 Limitations
The boundary conditions, load eccentricity, slenderness ratios, and maximum
internal stresses define the major limitations of this method.
All of the data presented within this document assumes pinned end
connections. As a result, the effective length modifier k is always taken as 1.0.
However, in practice, the ends of the timber columns are never truly pinned and are
never fixed. Therefore, an appropriate effective length must be used when with the
method presented in Chapter 2. The typical value chosen by manufacturers of timber
truss systems is 0.8.
The connections produced when using the truss plates have some ability to
transmit a moment. The addition of an eccentric load would have the tendency to
reduce the buckling capacity of the column. The methodology presented in Chapter 2
was not developed to include this effect.
56
The slenderness ratio for the columns used in this document range from 48 for
a 6 foot member to 80 for a 10 foot member. The NDS requires the slenderness ratio
not to exceed 50, except during construction. The present code would need to be
revised to permit higher slenderness ratios for the reinforced timber columns.
Finally, the uncertainty in defining the true distribution of stress internally in
the column has been shown in Chapter 3. These internal stresses may indeed limit the
actual capacity of the reinforced column and may only be resolved through test.
4.2 National Design Specification Buckling
The axial capacity of a timber column is defined by the NDS (Ref 2) as
PNDS =AcFcCDCMCtCFCp (4.1)
where Ac = Area in2
Fc = Compressive capacity psi
CD = Load duration factor
CM = Wet service factor
Ct = Temperature factor
CF = Size factor
Cp = Column Stability factor
57
The only terms in equation 4.1 which are of interest in this study are Ac Fe ,
and Cp. These terms define the nominal buckling capacity of the timber columns.
The remaining terms account for environmental and material degradation in the
manufacturing process and will be taken as 1.0 in this document. In normal design
however, these factors are important and must be included.
The column stability factor is defined by the NDS as
where
F =FCC CC
psi
_^Â£(CmC,)
psi
(4.2)
le = Effective column length in
d = Column depth in
= 0.3 for visually graded lumber,
0.418 for products with COVe <0.11
c = 0.8 sawn lumber
The reduced NDS maximum column capacity reduces to
^nds dcFcCp
(4.3)
58
The timber truss manufacturers normally use a MSR rated wood of 1650fl ,4E
in fabrication. The resulting timber allowable loads are shown in Table 4.1. and
compared to the Euler buckling load.
Table 4.1 Comparison ofNDS Column Capacity with Euler Buckling Load
2x4
Length Pe Pnds Po/Pnds
ft lb lb
6 2623 1290 2.03
8 1475 737 2.00
10 944 475 1.99
2x6
Length Pe Pnds Pe/pNDS
ft lb lb
6 4123 2027 2.03
8 2319 1158 2.00
10 1484 746 1.99
It is important to notice the NDS column capacity equation includes a factor of
safety of 2.0 for the range of lengths currently under investigation. This factor of
safety should be maintained for the reinforced column.
4.3 NDS Modified Buckling Capacity
The previous section presented the definition of the NDS column compressive
capacity. It was shown that the stability factor Cp, in general produces a factor of
safety of 2 over the Euler buckling load. Any modification to the NDS requirements
59
to incorporate the reinforced column capacity should maintain a similar factor of
safety.
The column stability factor may be modified to account for the increase in
column capacity obtained from the reinforced section by modifying the Cp equation.
C; (4.4)
where CBR =0.009
P, P
Fcbr = 100 = Column Buckling Factor
e
PCBR = Reinforced column capacity found using Eqn. 2.11
Pe = Euler buckling load found using Eqn. 2.12
The factor CBR accounts for the uncertainty inherent in the analysis procedure
and the lack of experimental validation. The factor Fcm is the percentage increase in
column capacity due to reinforcement described in Chapter 2. The modified NDS
equation then becomes
Puds =AeFeC'P (4.5)
The reinforcing plate may be the selected by solving for the required column
buckling factor Fcbr
CBR,
Rc (flirtd
4>,
CBR
* Re quired
1
(4.6)
60
The required reinforcing material and material length may be found directly by
solving equation 2.11. However, equation 2.11 is difficult transcendental equation and
it is unrealistic to be used by the normal design engineer. Therefore, the factor FCBR
has been found for a wide range of material strengths and sizes. These factors are
presented in the appendices.
As an example, given a 6 foot MSR rated 1650f1.4E member, find the
required column bucking factor and select a steel plate to increase the capacity of the
column to 1500 lb.
Using equation 4.6 and the NDS available strength provided in Table 4.1 of
1290 lb, the required column buckling factor is 18. Using Figure 4.1 which has been
reproduced from the appendices, an 9 inch 16 gauge plate will produce the required
column buckling factor.
61
Figure 4.1 Column Buckling Factor FCbr for a 6 foot 2x4
Column with a 1.4E6 psi Modulus of Elasticity
The appendices contain column buckling factors for a range of material
properties, reinforcement lengths, column lengths, and reinforcement configurations.
These curves are provided to assist the design engineer in selecting quickly the
required reinforcement.
62
5.
Recommendations and Conclusions
This study has concerned improvement of timber column buckling strength
through reinforcement of the midheight of the column. An analytical method was
developed and was validated by numerical methods to assess strength increases. This
method has been Used to produce factors which are easy to use by the typical design
engineer to select the required column reinforcement.
Since the reinforcement materials are on hand in most truss manufacturing
operations, this method produces a practical and inexpensive alternative to lateral
braces for meeting the NDS column stability requirements. However, while the
method presented in this document offers a practical alternative design technique, the
uncertainty in the distribution of the internal stress and the uniqueness of the material
require validation through testing. Further, since this method, in general, operates
outside of the NDS slenderness requirements, changes to the existing code are
required for this method to be used in practice.
63
Appendix A Column Buckling Factor Notes
The purpose of this appendix is to document the assumptions which have been
used to form the column buckling reinforcement factors Fcbr found in Appendices B
through M. This appendix is intended to reinforce and supplement the main body of
this document.
The column buckling factor is defined in chapter 4 as
Fcb* =Fm ~P *100 (A.1)
e
where Pcbr is the critical buckling load found from equation 2.11
Pe is the Euler buckling load
The primary components which effect the critical buckling load in equation
2.11 are the column and reinforcement modulus of elasticity, column length, column
cross sectional properties, reinforcement configuration, and reinforcement length.
The column modulus of elasticity presented on the figures of the following
appendices should be treated as the modified modulus E as defined by Part II,
Table 2.3.1 of the National Design Specification. The MSR rated wood nominal
elastic moduli as presented by the NDS were used to develop the column buckling
factors and are presented in Table A. 1.
The column lengths presented in the appendices should be treated as the
effective column length.
64
Table A. 1 Selected Wood Material Properties *
Commercial Grade Allowable Bending Stress Fb (psi) Allowable Tension parallel to grain Ft (psi) Allowable Compression parallel to grain Fc (psi) Modulus of Elasticity E (psi)
900f1.0E 900 350 1050 1,000,000
1200f1.2E 1200 600 1400 1,200,000
1650f1.4E 1650 1020 1700 1,400,000
1800f1.6E 1800 1175 1750 1,600,000
2100f1.8E 2100 1575 1875 1,800,000
2400f2.0E 2400 1925 1975 2,000,000
2700f2.2E 2700 2150 2100 2,200,000
3000f2.4E 3000 2400 2200 2,400,000
3300f2.6E 3300 2650 2325 2,600,000
* Note: National Design Specification (Ref. 2)
The appendices use the nominal cross sectional properties as presented in the
NDS and shown in Table A.2.
Table A. 2 Section Properties of Selected Standard Dressed Sawn Lumber
Nominal Size Dressed Size (in) Cross Sectional Area (in2) XX axis Moment oflnertia Ixx (in4) YY axis Moment oflnertia Iw (in4)
1x4 3/4 x 31/2 2.625 2.680 0.123
2x4 11/2 x 31/2 5.250 5.359 0.984
2x6 11/2 x 51/2 8.250 20.80 1.547
The steel (29.0E6 psi) and aluminum (10.2E6 psi) reinforcement elastic moduli
presented in Chapter 2 of this document were used for all the computations of these
65
appendices. The wood plate reinforcement was always assumed to have the same
elastic modulus as the column.
An effective cross sectional area of 50% assumed for the metal reinforcement
used through out the appendices. The resulting reinforced section transformed inertia
for each of the columns investigated is presented in Tables A. 3 through A. 11.
Table A.3 Effective Section Inertia for a Nominal Column Modulus of 1.0E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate Box Plate Plate Box Plate
(in4) (in4) (in4) (in4)
Steel (20 Gauge) 2.863 3.627 3.426 5.365
Steel (18 Gauge) 3.490 4.509 4.053 6.639
Steel (16 Gauge) 3.920 5.391 4.483 7.914
Aluminum (20 Gauge) 1.645 1.914 2.208 2.890
Aluminum (18 Gauge) 1.865 2.224 2.428 3.338
Aluminum (16 Gauge) 2.017 2.534 2.580 3.786
Wood (1x4) 7.875 N/A 8.438 N/A
Table A.4 Effective Section Inertia for a Nominal Column Modulus of 1.2E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 2.550 3.187 3.113 4.729
Steel (18 Gauge) 3.072 3.922 3.635 5.790
Steel (16 Gauge) 3.431 4.657 3.994 6.852
Aluminum (20 Gauge) 1.535 1.759 2.098 2.666
Aluminum (18 Gauge) 1.718 2.017 2.281 3.094
Aluminum (16 Gauge) 1.845 2.276 2.408 3.413
Wood (1x4) 7.875 N/A 8.438 N/A
66
Table A.5 Effective Section Inertia for a Nominal Column Modulus of 1.4E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 2.326 2.872 2.889 4.274
Steel (18 Gauge) 2.774 3.502 3.337 5.184
Steel (16 Gauge) 3.081 4.132 3.644 6.095
Aluminum (20 Gauge) 1.456 1.648 2.019 2.506
Aluminum (18 Gauge) 1.614 1.870 2.177 2.826
Aluminum (16 Gauge) 1.722 2.091 2.285 3.147
Wood (1x4) 7.875 N/A 8.438 N/A
Table A. 6 Effective Section Inertia for a Nominal Column Modulus of 1.6E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate Box Plate Plate Box Plate
(in4) (in4) (in4) (in4)
Steel (20 Gauge) 2.159 2.636 2.722 3.933
Steel (18 Gauge) 2.550 3.187 3.113 4.729
Steel (16 Grauge) 2.819 3.739 3.382 5.526
Aluminum (20 Gauge) 1.391 1.565 1.960 2.386
Aluminum (18 Gauge) 1.535 1.759 2.098 2.666
Aluminum (16 Gauge) 1.630 1.953 2.193 2.947
Wood (1x4) 7.875 N/A 8.438 N/A
67
Table A. 7 Effective Section Inertia for a Nominal Column Modulus of 1.8E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 2.028 2.453 2.591 3.668
Steel (18 Gauge) 2.376 2.942 2.939 4.376
Steel (16 Gauge) 2.615 3.433 3.178 5.084
Aluminum (20 Gauge) 1.351 1.501 1.914 2.293
Aluminum (18 Gauge) 1.474 1.673 2.037 2.542
Aluminum (16 Gauge) 1.558 1.845 2.121 2.791
Wood (1x4) 7.875 N/A 8.438 N/A
Table A. 8 Effective Section Inertia for a Nominal Column Modulus of 2.0E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 1.924 2.306 2.487 3.456
Steel (18 Gauge) 2.237 2.747 2.800 4.093
Steel (16 Gauge) 2.452 3.188 3.015 4.730
Aluminum (20 Gauge) L315 1.449 1.878 2.219
Aluminum (18 Gauge) 1.425 1.604 1.988 2.443
Aluminum (16 Gauge) 1.500 1.759 2.063 2.667
Wood (1x4) 7.875 N/A 8.438 N/A
68
Table A. 9 Effective Section Inertia for a Nominal Column Modulus of 2.2E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 1.838 2.186 2.401 3.283
Steel (18 Gauge) 2.123 2.586 2.686 3.862
Steel (16 Gauge) 2.319 2.987 2.882 4.441
Aluminum (20 Gauge) 1.284 1.407 1.847 2.157
Aluminum (18 Gauge) 1.385 1.548 1.948 2.361
Aluminum (16 Gauge) 1.453 1.689 2.016 2.565
Wood (1x4) 7.875 N/A 8.438 N/A
Table A. 10 Effective Section Inertia for a Nominal Column Modulus of 2.4E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 1.767 2.085 2.330 3.138
Steel (18 Gauge) 2.028 2.453 2.591 3.669
Steel (16 Gauge) 2.207 2.820 2.770 4.200
Aluminum (20 Gauge) 1.259 1.371 1.822 2.107
Aluminum (18 Gauge) 1.351 1.501 1.914 2.293
Aluminum (16 Gauge) 1.414 1.630 1.977 2.480
Wood (1x4) 7.875 N/A 8.438 N/A
69
Table A. 11 Effective Section Inertia for a Nominal Column Modulus of 2.6E6 psi
Nominal Column Size 2x4 2x6
Reinforcing Material Plate (in4) Box Plate (in4) Plate (in4) Box Plate (in4)
Steel (20 Gauge) 1.707 2.000 2.670 3.016
Steel (18 Gauge) 1.948 2.340 2.511 3.505
Steel (16 Gauge) 2.113 2.679 2.676 3.996
Aluminum (20 Gauge) 1.238 1.342 1.801 2.064
Aluminum (18 Gauge) 1.323 1.461 1.886 2.236
Aluminum (16 Gauge) 1.381 1.580 1.944 2.408
Wood (1x4) 7.875 N/A 8.438 N/A
Finally, the figures provided in the following appendices do not contain all
possible values of the modified timber modulus of elasticity. It is recommended that
either linear interpolation be used, or to be conservative, use the figure for the higher
modulus since the values of the column buckling factor are lower for higher modulus
values.
In the event alternative reinforcement configurations are used, the effective
section modulus tables will be useful in determining the net improvement in the
buckling capacity for the new section. It should be noted, however, while higher
modulus values will increase the theoretical buckling capacity, the internal stresses in
the material may exceed the allowable stress limit. Therefore, care in the selection of
alternatives and additional testing is suggested.
70
As a final note, the buckling solutions were obtained numerically using
Mathcad 5,0 Plus (Ref. 3), a commercially available symbolic equation solution
package.
71
Appendix B Column with Steel Reinforcement 2x4x72
This section contains 2x4 columns with a length of 6 feet with steel
reinforcement. The column modulus of elasticity ranges from 1.0E6 psi to 2.6E6 psi.
The Euler buckling load of the unreinforced column has been included for reference
purposes only.
72
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B. 1 Column Buckling Factor for a 6 Foot 2x4 Column with a
1.0E6 psi Modulus of Elasticity Steel and Wood Reinforcement
73
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.2 Column Buckling Factor for a 6 Foot 2x4 Column with a
1.2E6 psi Modulus of Elasticity Steel and Wood Reinforcement
74
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.3 Column Buckling Factor for a 6 Foot 2x4 Column with
a 1.4E6 psi Modulus of Elasticity Steel and Wood Reinforcement
75
Column Budding Factor (Fcbr)
70.00
0.00 
5.0 7.0 9.0 11.0 13.0 15.0 17.0 19.0
Reinforcement Length in.
Figure B.4 Column Buckling Factor for a 6 Foot 2x4 Column with a
1.6E6 psi Modulus of Elasticity Steel and Wood Reinforcement
76
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.5 Column Buckling Factor for a 6 Foot 2x4 Column with
a 1.8E6 psi Modulus of Elasticity Steel and Wood Reinforcement
77
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.6 Column Buckling Factor for a 6 Foot 2x4 Column with a
2.0E6 psi Modulus of Elasticity Steel and Wood Reinforcement
78
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.7 Column Buckling Factor for a 6 Foot 2x4 Column with
a 2.2E6 psi Modulus of Elasticity Steel and Wood Reinforcement
79
Column Buckling Factor (Fcbr)
Reinforcement Length in.
Figure B.8 Column Buckling Factor for a 6 Foot 2x4 Column with a
2.4E6 psi Modulus of Elasticity Steel and Wood Reinforcement
80
