BUCKLING STRENGTH OF A TWO-BAR FRAME

WITH A SEMI-RIGID CONNECTION, VARIABLE BASE FIXITY, AND VARIABLE SIDESWAY RESTRAINT by

Miguel Ralph Casias B.S., University of Colorado, 1980

A thesis submitted to the Faculty of the Graduate School of'the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Civil Engineering

1986

This thesis for the Master of Science degree by Miguel Ralph Casias has been approved for the Department of Civil Engineering

by

Date

iii

Casias, Miguel Ralph (M.S., Civil Engineering)

Buckling Strength of a Two-Bar Frame with a Semi-Rigid Connection, Variable Base Fixity, and Variable Sideway Restraint

Report directed by Assistant Professor Andreas. Vlahinos

The buckling strength of a two-bar frame consisting of a beam and a column was investigated.

The column is attached to the beam and to its base with rotational springs whose stiffness could vary from zero to infinity. A spring attached to the end of the beam inhibited frame sidesway, and its stiffness varied from zero to infinity. Analyses were performed for varying stiffness of the springs. The results of the analyses showed that the buckling strength of the frame was dramatically increased when the stiffness of the springs was increased. In addition, it was noted that complete fixity was not necessary to significantly increase the buckling strength of the. frame.

ACKNOWLEDGMENTS

The author gratefully acknowledges the guidance and support of his professors, especially Assistant Professor Andreas Vlahinos. Also invaluable was the constant understanding and support of the author's wife, Theresa, and son, Timothy.

CONTENTS

CHAPTER

1. INTRODUCTION.............................. 1

Purpose of the Study..................... 1

Discussion on Semi-Rigid

Connections................................ 1

Discussion on Frame Sidesway................. 2

Discussion on Support Fixation............... 3

Scope of the Study........................... 3

2. MATHEMATICAL FORMULATION...................... 6

Development of the Basic

Beam-Column Equation...................... 6

Nondimensionalization of

Parameters................................. 8

Dimensional Boundary

Conditions................................. 9

Nondimensional Boundary

Conditions................................ II

Formulation of the Frame

Characteristic Matrix..................... 12

3. SOLUTION METHODOLOGY......................... 16

Solution Procedure.......................... 16

Computer Implementation..................... 17

Vi

CONTENTS Continued

CHAPTER

4. NUMERICAL RESULTS AND

DISCUSSION................................ 20

Effect of Support Fixation................. 20

Effect of Connection Flexibility........... 22

Effect of Bracing Stiffness....,........... 22

5. CONCLUSIONS............................ ... 31

REFERENCES......................................... 32

APPENDIX

A. PROGRAM "FRAME"................................ 34

B. SAMPLE "FRAME" COMPUTER OUTPUT................. 36

C. NOTATION.................................... 39

\

vii

TABLES

Table

1. Summary of Nondimensionalized

Parameters................................. 15

2. Critical Loads for Extreme Values

of Spring Stiffness........................ 21

3. Qcr versus ^ (K3 = 0, u = 1)................ 27

4. Qcr versus ^ (K3 = co , u = 1)................ 28

5. Qcr versus K3 ("jj = 0, y = 1)................ 29

viii

FIGURES

Figure

1. Two-Bar Frame.................................... 4

2. Differential Bending Element..................... 7

3. Joint Sign Convention........................... 11

4. Frame Characteristic Matrix..................... 14

5. FRAME Flowchart................................. 18

6 Qcr versus I2 (K3 = 0, y = 1)................... 23

7 QCr versus (K3 y = U...................... 24

8. Qcr versus K3 (8{ = 0, y = 1),

Linear Scale.................................. 25

9. Qcr versus K3 (8^ 0, y = 1),

Semi logarithmic Scale..................... 26

r

CHAPTER 1

INTRODUCTION Purpose of the Study

In practice, framed connections are typically assumed pinned or rigid (and they must be analyzed and designed accordingly). However, actual rigidly designed connections have some degree of flexibility. Likewise, actual pin designed connections have a certain amount of rigidity (References 2, 4, 5, 8, 11, 17).

A similar dilemma arises on the subject of frame sidesway. Despite reality, a frame typically must be assumed either restrained or unrestrained against sidesway. As a result, partial sidesway restraint is ignored.

To provide insight on the previously mentioned items and base fixity, a two-bar plane frame with a variable semi-rigid connection, variable base fixity, and variable sidesway restraint will be investigated.

Discussion on Semi-Rigid Connections

There has been considerable interest in semi-rigid connections for over 50 years. Because of

the numerous studies on the moment-relative rotation

characteristics of steel framing connections, a large amount of connection data is available.

Early contributors to flexible connection analysis include Rathburn (11), who used slope-deflection and moment distribution methods, and Batho

and Rowan (2) and Sourochnikoff (17), who proposed a beam line method. Modern contributors include DeFalco and Marino (4), who presented a modified effective column length method, and Fry and Morris (5), who presented an iterative analysis procedure.

Recent analysis procedures include those presented by Moncarz and Gerstle (9), who presented a matrix displacement method, and Simitses and Vlahinos (13, 14, 15) who presented a characteristic equationboundary condition method.

Discussion on Frame Sidesway

Braced frames are significantly more efficient than frames unbraced against sidesway. For example, as will be shown later, a two-bar braced frame has nearly 10 times the load-carrying capacity of an equivalent unbraced frame. Several approximate method of designing braced frames have been proposed.

Galambos (7) proposed an approximate method using diagonal bracing and the stiffness of brick walls (if available). Briswas (3) also proposed a scheme using

3

diagonal bracing. Salmon and Johnson (12) presented an approximate sidesway buckling scheme using diagonal bracing.

Discussion on Support Fixation

The buckling strength of a fixed-base frame is much greater than that of a pinned-base frame (References 1, 6, 8, 18). However, because they are often viewed as being too expensive, pinned-base frames are normally selected. An economical alternative between the two extremes, partial base fixity, will be explored in this study.

Scope of the Study

The elastic stability of the two-bar frame shown in Figure 1 will be investigated. Beam-column theory will serve as the basis. The equilibrium equations and associated boundary conditions will be derived. The critical load (approximately equal to the buckling load) will be derived by employing a pertubation method (17) based on the concept of the existence of an adjacent equilibrium position for either a bifurcation or a limit point). This method leads to a characteristic equation (the determinate of the frame matrix equals zero). Geometric parameters will be nondimensionalized, and a nondimensionalized characteristic equation in matrix form will be

Figure 1.

Two-Bar Frame

5,

developed in order to establish the critical loads.

For a given geometric parameter set, a unique critical frame buckling load will be calculated by setting the determinate of the characteristic frame matrix equal to zero. Nondimensionalized critical frame buckling load curves will then be developed by selecting a series of geometric parameter sets.

The frame consists of a column and a beam, connected at right angles to each other, with geometric properties EIi, and EI2 and i2, respectively. The

two members are assumed to be straight, slender, and piecewise prismatic. Rotational springs with stiffnesses and 32 connect the column to its base and to the beam. 6^ and maY vary from zero (pinned joint) to infinity (rigid joint). Spring k3, which is attached to the end of the beam, inhibits frame sidesway; and its stiffness can vary from zero (unbraced against sidesway) to infinity (braced against sidesway). The frame is subjected to a concentrated load Q which is applied concentrically at the top of the column.

CHAPTER 2

MATHEMATICAL FORMULATION Development of the Basic Beam-Column Equation

By making the following assumptions about the bending of an elastic member:

1. Plane sections remain plane after bending

2. Shear deformation can be ignored

it is possible to describe the curvature of a beam by the following equation:

C = _1 = ____wi,xx_____

R Tl + (wijX)2]3/2

Assuming that the deflection of the member is small, the (wijX)2 term is negligible compared to unity. As a result, the above equation reduces to:

C = w -

I,xx .

The bending moment (M) is related to curvature by El in the following manner:

51 = c (wi.,xx) (1-1)

The freebody of a differential bending element of length (dx) subjected to bending moment (m) and horizontal and vertical forces P and V, respectively, is shown in Figure 2.

7

dx dx V

Figure 2. Differential Bending Element

The three equilibrium equations for the

differential bending element are:

Horizontal: P + dP P = 0

Vertical: V + dV V = 0

Moment: M + dM M + (V + dV + V)dx

2

- (P + dP + P)dx tan0

Taking the derivatives of both sides of the equilibrium equations with respect to x and considering dx is small, the equations reduce to:

= 0

dx

s

dM + V PtanG = 0 dx

(1.2)

For small angles:

tan 0 = 0 = w-j_ ? x

After substituting and taking derivatives with respect to x, equation (1.2) becomes:

EIwi,xxxx Pwi,xx = ^ (1*3)

8

Equation 1.3 is an ordinary second-order linear differential equation with constant coefficients. The general solution (10* 17) is of the form:

Wi = A-qcos kÂ£X + A^2sin kx + A^gx + A^ (1.4)

Equation 1.4 describes the deflection of a beam

member and is called the basic beam-column equation.

The variable is a geometric parameter. The constants A^i* &2f an<^ Ai4 are determined by

considering the boundary conditions of the member.

Nondimensionalization of Parameters

The dimensional local coordinate Xi and the normal displacement wi defined in terms of nondimen-sional local coordinate and the nondimensional normal displacement Wj_ are:

y xi

1 'll

TT Wi

1 71

Using partial derivatives and the above relations* the following can be derived:

Wi v = 5W1 Â£* = /4-w- \ (Â£) = w-1,X 6x SX \f 1x/ 1 xx

Wi,XX = &E = (wi,xx> C4> = 4 wi,

<$x

XX

6X

Wi,XXX

= Wi-xx ii = (1.. Hi.) i.2.

Sx SXi i i,

XXX

From the preceding relations and definitions, the following beam-column relations can be expressed:

Deflection = w-Â£ =

Slope = 0 = wi fx

wi,X

9

Moment

El,

M

El-

= w

i >xx ]?i wi,XX

-i * X

Shear force EIj

- wi,xxx = J72 wi,xxx

= piWijX

- EIi Wh

ZI2

Vertical force = v = p-w. EI.w.

i i,x Â£-Liwl,xxx

riXXX

The vertical force (V) can be nondimension-

alized as follows:

V p.0,2

V EI^TJT2' = ElT" WX wxxx

e - wi,

If we define k,-

2 = -Piij.2

XXX

El-

(equals the

nondimensional critical buckling load, Qcr), then we get:

?! -

-ki Eli

h

Substituting Pj_ into the vertical force

equation previously defined, we obtain:

- k 2 El -,* Eli

V = 1 W 1 -

/72 i.X ^72 wi,XXX

^_ki2wi,x Wi,XXX^

Dimensional Boundary Conditions

Using local coordinate systems (xi and ^2 and normal displacements w^ and W2) as shown in Figure 1 and the joint sign convention as shown in Figure 3, the following boundary conditions define this particular frame:

1. Wj(0) = 0 (horizontal displacement of joint A equals zero)

10

2. M2 (0) = 0 (moment at joint C equals zero)

3. w2(0) = 0 (vertical displacement at joint C equals zero)

4. w2 dz) = 0 (neglecting the axial shortening of the column, the vertical displacement at joint B equals zero)

5. vi (/^i) + k3wi = 0 (the shear in the column

at joint B plus the force exerted by the

spring equal zero)

6. Mi () + M2(j?2) = P (the moment in the column

plus the moment in the beam are equal to zero

at joint B)

7. Mj(0) = 0]^ x (0)(the moment in the column is equal to the moment in the spring at joint A)

8 [wl,x^l> w2,x(/2)1P2 = -MiCii) (the

relative rotation times the spring stiffness is equal to the opposite of the column moment at joint B)

9. Q + PjCip = 0 (the applied load Q plus the column vertical load at joint B are equal to

zero)

11

Figure 3. Joint Sign Convention

Nondimensional Boundary Conditions

After substituting nondimensional terms for the dimensional ones and manipulating the previous boundary condition equations, the following nondimensional equations are obtained:

1. Wi(0) = 0

2.

3.

4.

5.

W2,XX<0) = 0 W2(0) = 0

W2(l) = 0

-ki2WijXxxx(i) + Vid) = ^

k # 3

where Ko = dimensionless parameter = 1___L_

3 EIi

wi,xx(1) + vW2,xx(I) = 9

ei, l\

where y = dimensionless parameter =

77 Eli

6.

12

1 W1,XX(0)

where

exwi,x(0) = 0

dimensionless parameter

8.

W1,XX(X) + 02^*1,x^1) W2,X^1)^ -where 82 = dimensionless parameter

9.

gill

El!

Ball

El!

Formation of the Frame Characteristic Matrix

For the column portion of the two-bar frame, the basic beam-column equation derived earlier will be used to create the column nondimensional characteristic equations. Expressing the equation in nondimensional terms and taking derivatives, the following equations are derived:

W! (X) = A!!cos kiX + A^sin kiX + A^X + A^ WijX(X) = -k! (A! !Sin k!X A^cos kiX) + A! 3 Wi^xCX) = -k^ (A! jcos k!X + A!2sin k!X)

W1,XXX^X) kl3
For the beam portion of the two-bar frame, it is not known whether the beam will be subjected to axial tension or compression. Therefore, the beam nondimensional characteristic equations must be developed. An approximate expression (neglecting the axial force) of the equilibrium equation 1.3 is the following:

EIW2,XXXX = 0

13

The solution of this equation will define the deflection of the beam.

Integration of the equilibrium equation gives:

/w2 XXXX = j 0 ~>W2,XXX = A" 2 i Taking successive integrations, the following equations are obtained:

W2,XX
W2>XCX) = A'21X + a'22x + a23

W 2(X) = A21x3 + A22X2 + A23X + A24

Introducing nondimensional boundary equations 2 and 3:

w2,xx(0) = ^ * a22 = 0

W2(0) = 0 > A24 = 0

Therefore, the following nondimensional characteristic equations define the beam: w2(x) = a21x3 + a23x W2,x(x) = 3A21x2 + A2 3 W2,XX^X^ = 6a21X

Applying the nondimensional boundary conditions to the column and beam nondimensional characteristic equations creates the frame characteristic equation. Figure 4 shows the frame characteristic equation written in the matrix form [N]{A} = 0. Matrix [n] is in terms of five variables, "32 K3 y, and ,

the vector {A} represents the coefficients of the

/

general solution of the equilibrium differential equations.

and

1 0 0 1 0 0

0 0 0 0 1 1

l^cos k^ 2 -kj sin ki 0 0 6y 0

geos kj K^sin kj k3 kx2 ^3 0 0

-k 2 fcl -kill -"01 0 0 0

N61 CM VO *5 02 0 -3 32 I;

A11 a12 A13 a14 A21 A23

n61 = (~k132sin M kj_2cos k^)

n62 = (^1 32cos ^1 kj.2sin k^)

Figure 4

Frame Characteristic Matrix

15

A summary of nondimensionalized parameters is given in table 1.

Table 1. Summary of Nondimensionalized Parameters

CHAPTER 3

SOLUTION METHODOLOGY Solution Procedure

When the determinate of the frame characteristic matrix is equal to zero, the frame has buckled. Therefore, if values of "gj "g2, K3 / and y are selected, the associated critical buckling value kj (that causes the determinate to equal zero) can be found.

Thus, the following solution procedure is proposed:

1. Select values for parameters "gj. f2 > K3 and y .

2. Estimate the critical value of ki .

3. Assemble the frame characteristic matrix (see Figure 4).

4. Using ki as a variable, solve the equation f(B1 02 t K3 y k^) = 0 where f is the determinate of the frame characteristic matrix. (Successive approximations will converge on the critical value ofk^ .)

17

Computer Implementation

The iterative method of solving for k^ used University of Colorado at Denver IMSLIB programs ZFALSE and LINV3F. These programs were used as subroutines in a Fortran program (FRAME), written to solve for kj Values of k .* , 3.,, and y were read into FRAME.

Using these values, an approximate value of k^ and upper and lower limit values of kj, ZFALSE converged on the value of that made the determinate of the frame characteristic matrix equal to zero. (LINV3F computed the determinate of the frame characteristic matrix.)

A flowchart of FRAME is in Figure 5. The complete FRAME program is given in Appendix A.

18

Figure 5. FRAME Flowchart

19

(previous page)

Figure 5. FRAME Flowchart Continued

yes

CHAPTER 4

NUMERICAL RESULTS AND DISCUSSION

Studies investigating the buckling strength for varying values of ad Kj were performed.

(The value of p was equal to 1 in all cases.) Graphs summarizing the results of these studies are given in Figures 6 through 9. Table 2 shows the critical buckling load values (Qcr) for the extreme values, of 01, 02 / an<3 K3 .

Effect of Support Fixation

The results of this study clearly show that the buckling strength of a fixed-base frame is considerably higher than an equivalent pinned-base frame. For example, making buckling strength comparisons from Table 2:

Frame D = 4.25 x Frame B

Frame G = 2.04 x Frame E

Frame H = 1.94 x Frame F

Despite the superiority of fixed-base frames, pinned column bases are specified (in current design practice) for most structures. (Fixed-base construction costs are usually very high.) However,

Table 2. Critical Loads for Extreme Values of Spring Stiffness

ft @2 Qcr

n 0 f ^ 0 1 a

n u o 1.42

U 0 2.47 /77T C

o 1 ^ 6.03 D

0 1 E 907 777/7

oo u o [ ^ 13.88 4 F

o 0 n -

oo Pj 26.95

22

it is not necessary to completely prevent the base of a frame from rotating in order to achieve the buckling strength of a fixed-base frame. There is a critical rotational restraint that increases the buckling strength of pinned-base to almost that of a fixed-base. For instance, Qcr equals 5.93 ("g2 = 00 , 3 = 0) and

26.46 (= oo, = co) for = 100. These values are

less than 2 percent lower than the fixed-base values (6.03 and 26.95).

Effect of Connection Flexibility

This study also shows that rigid connections substantially increase the buckling strength of a frame. The increase in strength (from Table 2) can be shown as follows;

Frame D = 2.44 x Frame C

Frame F = 1.41 x Frame E

Frame H = 1.34 x Frame G

A critical rotational restraint, also, exists for framed connections. Figures 6 and 7 show that there is negligible increase in buckling strength past

?2 = I-

Effect of Bracing Stiffness

Figures 8 and 9 show the importance of frame bracing on buckling strength. The value of qcr ($2 = 100) increased from 1.41 to 13.80 or an

oo

23

.01 0.1 1 10 100

&2 ~ &zÂ£\ Ell

Figure 6. Qcr versus 32 (K3 = 0, y = 1)

1000

24

0.01 0.1 1 10 100 1000

0z = @z i. |

Eli

Figure 7. Qcr versus $2 (K3

= 00 u = l)

25

lo

Eli

Figure 8. Q versus K (8, = 0, y = 1), cr 3i

Linear Scale

/

Ocr- Ocrif

26

K,= K ,1* El,

Figure 9. Qcr versus K3 (61 = 0, y = 1), Semilogarithmic Scale

Table 3

Qcr versus g2 (Kg

0. V 1)

Bi b2 Qcr

0 0.00

0.1 0.09

1 0.60

0 5 1.12

10 1.25

100 1.40

500 1.42

1000 1.42

0 1.73

0.1 1.89

1 2.79

5 5 3.84

10 4.13

100 4.47

500 4.50

1000 4.51

0 2.04

0.1 2.21

1 3.21

10 5 4.39

10 4.72

100 5.11

500 5.15

1000 5.15

0 2.42

0.1 2.61

1 3.70

100 5 5.04

10 5.42

100 5.87

500 5.92

1000 5.93

28

Table 4. Qcr versus 02 CK3 = oo y = 1)

Pi e2 Qcr

0 9.87

r1 ts 10.06

1 11.21

0 5 12.75

10 13.23

100 13.81

500 13.87

1000 13.88

0 15.28

0.1 15.52

1 17.05

5 5 19.13

10 19.79

100 20.60

500 20.68

1000 20.70

10 0 0.1 1 5 10 100 500 1000 17.07 17.34 19.00 21.28 22.01 22.91 23.01 23.02

0 19.79

0.1 20.09

1 21.91

100 5 24.48

10 25.31

100 26.34

500 26.45

1000 26.46

29

Table 5. QCr versus K3 (0^ = 0, y = 1)

32 k3 Qcr

0 0.00

0.1 0.10

0 1 1.00

5 5.00

9.87 9.87

9.87 9.87

0.1 100 10.06

500 10.06

1000 10.06

0 1.13

0.1 1.21

1 2.08

5 5 5.85

9.87 9.87

100 12.71

500 12.74

1000 12.75

10 0 0.1 1 5 9.87 100 500 1000 1.26 1.35 2.21 5.92 9.87 13.16 13.22 13.22

0 1.41

r1 1.50

1 2.34

100 5 6.00

9.87 9.87

100 13.72

500 13.79

1000 13.80

30

increase of 979 percent. A critical spring stiffness also applies to frame bracing. From Figures 8 and 9, it can be seen that additional bracing beyond K3 = 40 does not significantly improve buckling strength.

Another interesting item on frame bracing revealed by Figures 8 and 9 is that the curves seem to be divided into two separate regions (K3 < ir2 = 9.87 and > 9.87). For values of near 9.87, the effect of 82 is negligible. In addition, the graphs show that the effect of on buckling strength is much less for values of less than 9.87 than for values greater.

CHAPTER 5

CONCLUSIONS

This study has shown the importance of flexibl semi-rigid connections, the support fixation, and bracing stiffness' on buckling strength. A severe penalty, in terms of reduced buckling strength capacity, is paid whenever pinned connections, pinned bases, and sway frames are designed. The buckling strength of frames can be enhanced tremendously by providing rotational restraint at connections and sidesway restraint. In all three cases, there is a "critical stiffness above which there is no

significant increase in buckling strength.

REFERENCES

1. Appletauer, J. W., and T. A. Bartha, Discussion of "Influence of Partial Base Fixity on Frame Stability," by T. V. Galambos, Journal of Structural Division, ASCE, Vol. 87, No. ST2, February 19 61.

2. Batho, C., and H. C. Rowan, "Investigation on Beam and Stanchion Connections, Second Report of the Steel Structure Research Committee, HMSO, London,

19 34.

3. Briswas, M., "Threshold Bracing Stiffness for Two Story Frames, Third International Colloquium on Stability of Metal Structures, Toronto, Canada,

May 1983.

4. DeFalco, F., and F. J. Marino, "Column Stability

in Type 2 Construction, AISC Engineering Journal, Vol. 3, No. 2, 1966.

5. Frye, J. M., and G. A. Morris, "Analysis of

Flexibly Connected Steel Frames, Canadian Journal of Civil Engineering, Vol. 2, 1975.

6. Galambos, T. V., "Influence of Partial Base Fixity on Frame Stability, Journal of Structural Division, ASCE, Vol. 86, No. ST5, May 1960.

7. Galambos, T. V., "Lateral Support for Tier Building Frames, AISC Engineering Journal, January 1964.

8. Salmon, C. G., L. Schlenker, and B. G. Johnston, "Moment Rotation Characteristics of Column Anchorages, Proceedings, ASCE, Vol. 81, No. ST3, April 1955.

9. Moncarz, P. D., and K.. H. Gerstle, "Steel Frames with Nonlinear Connections, Journal of Structural Division, ASCE, Vol. 107, No. ST8, 1981.

10. Rabenstein, Albert L., Elementary Differential Equations with Linear Algebra, second edition, Academic Press, New York, p. 201, 1975.

33

REFERENCES Continued

11. Rathburn, J. E., "Elastic Properties of Rivited Connections, Transactions of American Society of Civil Engineers, Vol. 101, 1936.

12. Salmon, C. G., and J. E. Johnson, Steel Structures

Design and Behavior, second edition, Harper and Rowe, p. 858-867, 1980.

13. Simitses, G. J., and A. S. Vlahinos, "Connection Flexibility and Steel Frames, Proceedings, ASCE Conference, Detroit, Michigan, October 1985.

14. Simitses, G. J., and A. S. Vlahinos, "Stability Analysis of a Semi-Rigidly Connected Simple Frame, Journal of Constructional Steel Research, Vol. 2, No. 3* September 1982.

15. Simitses, G. J., and A. S. Vlahinos, Steel Frame Structures Stability and Strength, Elsevier, Applied Science Publishers, New York, p. 115-152, 1985.

16. Sourochnikoff, B., "Wing Stresses in Semi-Rigid Connections of Steel Framework, Transactions of American Society of Civil Engineers, Vol. 114,

1949.

17. Vlahinos, A. S., C. V. Smith, Jr., and G. J. Simitses, "A Nonlinear Solution Scheme for Multi-Story, Multi-Bay Plane Frames, "

International Journal of Computers and Structures, Vol. 22, p. 1025-1045, 1986.

18. Vlahinos, A. S., "The Effect of Support Fixation

on Buckling Strength of Frames, Proceedings,

1986 Technical Session of Structural Stability Research Council, Washington, D.C., April 1986.

APPENDIX A

onoo no

35

PROGRAM FRAME

28

29

30

SO

70

40

ao

90

PROGRAM FRAME

X1=SQUARE ROOT OF (P1 *U *L1 ) / (EI1 ) X3o=(K3L1*L1*L1)/(E*I1)

NU=(EI2/L2)*(L1FE*I1)

B1B=(B1L1)/(E*I1)

32B=(a2*L1)/(E*I1)

INTEGER NSIG, N/ ITMAx, IER RcAL ft EPS, EPS2, ETA, X(1)

EXTERNAL F

COMMON X3d, NU, B1B, 82B, C(4,6)

DIMENSION WKAREA(6J, 8(6)

REAL XI, K3B, NU, 318, 828, QET OPEN (UNIT a 5, FILE => OATA1 STATUS OPEN (UNIT a 6, FILE = '0UT2', STATUS *

WRITE(6,28) '

F0RNAT(X38,9X,NU,10X,818,9X,'82B',9X,'X1',1QX,'QCR*) HR IT E (6,29)

C-'?2xi'----------2x^-----"----~) 'ZX' '/2X,'

REAOCS, 30) NCARO FORMAT(13)

X1 = 3.2

'OLD') NE* )

DO 40, II 3 1,NCARO READ (5,50) X3B, NU,

818, B2S

FORMATUF10.4)

EPS 1.0E-8 NSIG = 8 XAPP XT XL a 3.0 XR a 5.5 UMAX* 100

CALL IFALSE (F, EPS, NSIG, XL, XR, XAPP, UMAX, IER)

X1 a XAPP OCR* X1*X1

HRITE(4, 70) X3B, NU, B1B, 828, X1, OCR

FOSmAT(F10.3,2X,F10.5,2x,F10.5,2X,F10.5,2X,F10.5,2X,F10.5)

CONTINUE

CLOSE(5)

SE(6)

END

SUBROUTINE OETE COMMON X3B, NU/

(DET, X1) 818, 82B/

DIMENSION WXAREACo), 8(o) REAL Xl,X3B,NU,B1d,83B REAL 01, D2

INTEGER IJOd, N, IA, IER IA a i IJOd = 4 01 = 1.0 00 90, J a 1,6 DO 80, X 1,6 C(J, X) a 0.0 CONTINUE CONTINUE

* 1:8

C(6,6)

#1:1] =a 1:3

C(4,1) C C 4,2) C(4,3) C(4,4) C(3,1)

{\m

C(5,1) C(5,2)

X3 B*COS(X1)

X38*SIN(X1)

X38-(X1X1)

X38

-(X1*X1)*C0S
-(X1*X1)*SIN(X1)

6*NU -(X1*X1)

_____ -Xl*d1d

C0S(X1)-
C(6^5> -3*828 C(6,6) -828 N 6 IA* 6

CALL LINV3FCC, 8, IJ08,

QET* 01*2**02 RETURN ENO

REAL FUNCTION F(X1)

COMMON X3B, NU, 818, 828, C(6,6)

REAL XI, K38, NU, 818, d2B CALL OETE(OET, X1)

F OET RETURN ENO

IA, 01, 02, HKAREA, IER)

APPENDIX B

04

03

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B1B B2B K1 QCR

APPENDIX C

NOTATION

Coefficient of general solution to equilibrium differential equation of member 1

Bending stiffness of member i

Q/l2

Eli

k3^i3 El i

Spring.stiffness at end of beam Length of member i

Bending moment in member i

Axial force in member i

Concentrated load applied on column

Critical load obtained by linear theory for special geometries

Qcr h2 EIi

Shearing force of member i

wi/ii

In-plane normal displacement component along member i

Axial coordinate of member i

41

NOTATION Continued

ei Rotational spring stiffness for spring i