FINITE DEFORMATION AND STABILITY OFNONRECTANGULAR ELASTIC RIGID FRAME STRUCTURES

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Authors Qashu, Riyad K.

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8IU

FINITE DEFORMATION AND STABILITY OF NONRECTANGULAR

ELASTIC RIGID FRAME STRUCTURES

by

Riyad K. Qashu

A Dissertation Submitted to the Faculty of the

DEPARTMENT OF CIVIL ENGINEERING AND ENGINEERING MECHANICS

In Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY WITH A MAJOR IN CIVIL ENGINEERING

In the Graduate College

THE UNIVERSITY OF ARIZONA

1 9 8 0

t ,

THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE

As members of the Final Examination Committee, we certify that we have read

the dissertation prepared by ____ R_i~y_a_d __ K_. __ Q~a_s_h_u ______________________________ __

entitled Finite Deformation and Stability of Nonrectangular Elastic

Rigid Frame Structures.

and recommend that it be accepted as fulfilling the dissertation requirement

for the Degree of Doctor of Philosophy, Civil Engineering

Date

5'Pp Z 3 I Cfif2

Date

Date

Final approval and acceptance of this dissertation is contingent upon the candidate's submission of the final copy of the dissertation to the Graduate College .

I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation

STATEMENT BY AUTHOR

This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at The University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library.

Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author.

cPqsQa' SIGNED: / I VU b>\yv VA

ACKNOWLEDGEMENTS

I wish to express ray appreciation, gratitude and indebtedness

to Professor Donald A. DaDeppo for his guidance, encouragement, time,

and extensive knowledge of the problem to which he has generously

contributed during the course of this research.

I would like to extend my gratitude to Professor Ralph M.

Richard for his support and help to make available the necessary

computer funds.

Sincere thanks are due to Professor Daniel D. Evans for reading

the dissertation and valuable suggestions, the Department of Civil

Engineering for financial support, and to Miss Kathryn Warner for

excellent typing.

Much of the credit must be given to the author's wife, Fadia,

and daughter Ruba, who have given their fullest support and understanding

to this effort.

iii

TABLE OF CONTENTS

Page

LIST OF ILLUSTRATIONS v

ABSTRACT viii

1. INTRODUCTION 1

2. FORMULATION OF BEAM EQUATIONS 4

Strain Curvature Relations 4 Stress Resultants 9 Equilibrium Equations 11 Differential Equations for Beams 14 End Forces 15 Nondimensional Equations 17 Adaptation to Beam Under Uniform Load 19 Rate Equations 22

3. NUMERICAL SOLUTION 24

Problem P.l 25 Problem P. 2 29 Details of Solutions of Problems P.l and P.2 33 Rigid Frame Structures 38 Stability of Equilibrium 50 Computer Program Description 51

4. PRESENTATION AND DISCUSSION OF RESULTS 57

Test Cases 58 Examples 64

5. CONCLUSIONS AND FUTURE RESEARCH 91

NOMENCLATURE 95

BIBLIOGRAPHY 98

iv

LIST OF ILLUSTRATIONS

Figure Page

1 Deformation of Beam Segment in Bending 5

2 Centroidal Element in the Undeformed and Deformed Configurations 7

3 Segment of a Centroidal Element in the Deformed State with all the External Forces and Stress Resultants . . 12

4 Forces

5 Beam Under Uniformly Distributed Load 20

6 Plane Frame Composed of Three Segments in the Undeformed State 41

7 Load-peak Deflection Curve for a Gabled Frame with Snap Buckling 52

8 Load-peak Deflection for Gabled Frame with Bifurcation Buckling 53

9 Schematic Diagram of Computer Program 54

10 Error in Computation and Computer Cost vs. Number of Beam Subdivisions 59

11 Test Case 1 - Determinant versus Axial Load for a Cantilever Beam 60

12 Test Case 2 - Beam Subjected to Pure Bending 61

13 Test Case 3 - Load versus Downward Deflection of Point of Loading 62

14 Test Case 4 - Load versus Horizontal Deflection of Node 2 63

15 Buckling Load versus Span for Uniform Gabled Frames of Constant Rise and Built-in Support Under Three Downward Point Loads 65

v

vi

LIST OF ILLUSTRATIONS--continued

Figure Page

16 Load-peak Deflection for Uniform Gabled Frames of Constant Rise and Built-in Supports Under Three Downward Point Loads 66

17 Buckling Load versus Rise for Uniform Gabled Frames of Constant Span with Built-in Supports Under Three Downward Point Loads 67

18 Load versus Peak Deflection for Uniform Gabled Frames of Constant Span with Built-in Supports Under Three Downward Point Loads 68

19 Buckling Load versus Span for Gabled Frames with Built-in Supports Under Two Downward Loads Over the Columns 70

20 Load versus Peak Deflection for Uniform Gabled Frames with L=h and Built-in Supports 71

21 Load versus Peak Deflection for Uniform Gabled Frames with L=2h and Built-in Supports 72

22 Load versus Peak Deflection for Uniform Gabled Frames with L=3h and Built-in Supports 73

23 Buckling Load versus Rise for Uniform Gabled Frames with Built-in Supports, Subjected to a Uniformly Distributed Load 74

24 Load versus Peak Deflection for Uniform Gabled Frames with L=2h and Built-in Supports Under Uniformly Distributed Load 75

25 Load Deflection Curves for a Uniform Gabled Frame with Built-in Supports Under Asymmetric Distributed Load . 77

26 Buckling Load versus Distributed Load Intensity for a Uniform Gabled Frame with Built-in Supports Under Uniformly Distributed Load and Point Load at the Peak 78

27 Load-peak Deflection Curves for a Uniform Gabled Frame with Built-in Supports Under Uniformly Distributed Load and Point Load at the Peak 79

vii

LIST OF ILLUSTRATIONS--continued

Figure Page

28 Buckling Load versus Column Slope for a Uniform Frame with Built-in Supports Subjected to Uniformly-Distributed Load over its Entire Profile 80

29 Load versus Peak Deflection for Uniform Frame with Built-in Supports Under a Uniformly Distributed Load over the Entire Profile 81

30 Load versus Peak Deflection for a Uniform Gabled Frame with Built-in Supports and Side Overhangs Under Uniformly Distributed Load 82

31 Load Peak Deflection for a Uniform Gabled Bent Wire Side Overhang and Built-in Support Under Asymmetric Distributed Load 83

32 Buckling Load versus Eccentricity e for Asymmetric Uniform Frame with Built-in Supports Under Point Load at the Peak 84

33 Load versus Peak Deflection for Asymmetric Uniform Frames with Built-in Supports Under a Downward Point Load at the Peak 85

34 Load Deflection Curves for a Uniform Multibay Structure with Built-in Supports Under Gravity Loads 87

35 Load Deflection Curves for a Uniform Multibay Structure with Built-in Supports Under Gravity Loads and Outward Lateral Load 88

36 Load Deflection Curves for a Uniform Multibay Structure with Built-in Supports Under Gravity Loads and Inward Lateral Load 89

37 Load versus Peak Deflection for a Uniform Gabled Bent with Hinged and Built-in Supports Under Three Downward Point Loads 90

ABSTRACT

A numerical algorithm is presented to examine the in-plane

finite bending, buckling, and post-buckling behavior of rigidly jointed

elastic plane frame structures, under static, discrete and/or uniformly

distributed loads. The governing differential equations of the exact

theory of finite plane bending and stretching of beams, along with pre

scribed boundary conditions are solved numerically using the 4th order

Runge-Kutta integration scheme. A generalized Newton-Raphson iteration

procedure is used to systematically improve trial solutions of the

differential equations. Rate equations derived from the governing

equations are solved numerically to obtain instantaneous stiffness for

use in establishing equilibrium configurations, and in general, load-

deformation relationships through incremental loading. The method of

analysis is then used to determine the critical loads, wherein, the

effects of large deflections are taken into account. Computed and

known results are compared and found in good agreement, demonstrating

the accuracy of the formulation. The development is further demonstrat

ed by example problems in which instability occurred after the frames

have undergone large deformations. Bifurcation and snap-buckling modes

of instability as well as post-buckling behavior are investigated.

viii

CHAPTER 1

INTRODUCTION

The research discussed herein concerns the in-plane finite

bending, buckling, and post-buckling behavior of plane rigidly jointed

nonrectangular elastic frame structures under independent prescribed

distributed as well as concentrated loads.

Due to the relative economy of construction and the flexibility

in matching structure geometry to spacial occupancy and functional re

quirements, nonrectangular rigid frames, as exemplified by the gabled

bent, have been and will continue to be in widespread use. As with

most structures, nonrectangular frames are susceptible to fail by

buckling. Unbraced frames may undergo prebuckling deformations so

large as to require that one take into account prebuckling changes in

geometry in a stability analysis. Moreover, the prebuckling deforma

tions could be so large that a "second order" analysis based on con

ventional beam-column theory would not be adequate. Gabled bents are

in some way similar to arches and one can reasonably expect some simi

larities in structural behavior. Recent experience in analysis of

arches by Austin (1971, 1972, 1973), and by DaDeppo and Schmidt (1972,

1974) show that extreme caution must be exercised in any attempt to

extend results from stability analysis based on small deformation-small

rotation theory to situations in which large rotations and deformations

may occur. This is especially true for combined loading, as in the

case of own weight in conjunction of point loads.

1

2

The need for analysis of nonlinear deformation, buckling, and

post-buckling behavior of structures to attain a greater knowledge of

structural behavior and to provide a firm foundation for sound engineer

ing design has long been recognized. This is reflected in the litera

ture which contains many papers that deal with methods of analysis of

the nonlinear behavior of frames under various simplifying assumptions

in regard to magnitudes of deformations, magnitudes of rotations, load

ing, etc., for typical examples see Masur, Chang, and Donnell (1961).

A good summary of the state-of-the-art up to 1968 is given in Lee,

Manuel, and Rossow (1968), in which the authors present a method of

analyzing large deflections of frames subjected to a conservative

system of arbitrary discrete loads whose intensities are defined by a

single load parameter. The formulation is exact within the frame-work

of the Bsrnoulli-Euler theory for plane bending with inextensional

deformations. Because each member of the frame is also assumed to be

piecewise prismatic and the loads are discrete, it is possible to

express the solutions of the governing differential equations in terms

of elliptic integrals. The purpose of their paper was to present method

ology and, therefore, it contains few numerical results and there are

no applications to the problems intended for study in this research.

Moreover, the methods presented cannot be applied to the great majority

of problems that are considered in this research because analytical

solutions of the pertinent differential equations are not known.

A continuing study of the literature from 1968 to the present

time has not uncovered any extensive studies of the structures of

3

the type and under the conditions investigated in this research. In

addition no paper has been found that contains improvements on the

method of solution suggested by Lee et al. (1968) under the restrictions

employed in the study.

The method of Lee et al. (1968) or the finite element method

could be adopted to the structures under consideration to obtain approx

imate solutions. However, the primary objective of this research was

to obtain highly accurate results of permanent value which could be

used as a guide in the development and as a standard for evaluation of

approximate methods of analysis. A method of numerical analysis that

will yield the desired degree of accuracy and which appears to be

economically practical is developed in Chapters 2 and 3 of this

dissertation. The method is essentially one of direct numerical inte

gration of the governing differential equations, generating the finite

deformations of beam. A secondary objective of the research was to

conduct a limited systematic study of the response of nonrectangular

frames to develop insight with regard to their behavior. The results

of these numerical studies are presented in Chapter 4.

CHAPTER 2

FORMULATION OF BEAM EQUATIONS

An exact formulation of the basic governing beam equations will

be made within the limitations of the following assumptions:

1. The material of the frame is linearly elastic

and the moduli of elasticity in tension and

compression are equal.

2. Plane cross sections before bending remain plane

and normal to the longitudinal fibers after

bending, and bending occurs in a principal

plane.

3. Static loading system with the loading plane

coincides with the plane of bending.

4. Effects of shear, stress concentration, and

residual stresses are negligible.

5. Buckling is restricted to the plane of bending.

The problem is formulated as a system of simultaneous nonlinear equa

tions obtained by combining the general relations of deformations, with

the equilibrium equations obtained for each of the beam segments between

load points, and the prescribed boundary conditions.

Strain Curvature Relations

A segment of an initially straight beam is shown in its

undeformed and deformed states in Figure 1. The undeformed segment

4

Figure 1 Deformation of Beam Segment in Bending.

6

AB has length ds. After deformation, the deflected axis of the beam,

*

i.e., its elastic curve, is shown bent to a radius p . Its deformed

• ie length is ds and the central angle is d. An overall view of the

centroidal axis of a beam in the undeformed and deformed states is

shown in Figure 2.

The deformed length of the segment is

ds = ds + As (2.1)

By definition

As = ec ds (2.2)

where e is the strain in a beam fiber at the centroidal axis. c

Therefore

ds* = (1 + e ) ds (2.3)

or

*

ds = £ ds

where £ is the stretch of the line element,

Using Figure 2 we can write

* * dx = ds cos

or

*

dx = ? ds cosij> (2.4)

X

Undeformed Centroidal Line

Deformed Centroidal Line

Centroidal Element in the Undeformed and Deformed Configurations.

8

* *

dy = ds sin(f>

or

"k dy = £ ds sin (2.5)

The radius of curvature of the deformed centroidal axis is

i = K = % (2.6) p ds

From equations (2.3) and (2.6)

,, _ 1 1 d| .ldt * " 1 + E ds ~ C ds

P c (2.7)

From Figure 1 the length of the element in the deformed state at any

distance z from the centroidal line can be expressed as

ds = (p + z) d (2.8)

By application of the definition of axial strain to the line element

we can write

ds*z = (1 + ez) dsz (2.9)

where

ds = ds z

From equations (2.8) and (2.9)

•k

1 * - (e* * TT (2'10>

9

From equations (2.7) and (2.10)

*

1 • 4 %• * z inr t2-n) or

•

1 + E = 1 + E + z (2.12) z c ds '

Simplifying equation (2.12)

*

ez = ec + Z dT (2-13)

Stress Resultants

The one dimensional form of Hooke's Law is

a = E E (2.14)

where

a = normal stress at a point in the cross section

of a beam

E = Young's modulus

e = extensional strain at the same point where

normal stress is measured.

The normal force N and bending moment M acting on a cross sectional

area A at z can be expressed as

N = | °z dA = | E ez dA (2.15)

10

and

M = dA = E z e dA z

(2-16)

where dA is an element of area at distant z from the centroidal axis.

The value of from equation (2.13) substituted into equations

(2.15) and (2.16) yields

N = E(e + z ̂ -) dA v c ds J

= E E A + E ̂ f-c ds

z dA

But

z dA = 0

and therefore

N = E E A c (2.17)

In terms of the stretch £ = 1 + ecJequation (2.17) becomes

N = EA(C - 1) ( 2 . 1 8 )

11

M = | | E(ec + z ) z dA

d ds (2.19)

Equilibrium Equations

Figure 3 shows a deformed element of a beam in an equilibrium

position under the action of the stress resultant and distributed loads

and P per unit of deformed arc length which are normal and tangen

tial, respectively, to the element. Summation of forces in the normal

direction yields

* *

P^ ds + (N + dN) sin (^|—) + N sin

* *

- (Q + dQ) cos (^|-) + Q cos (^|-) = 0 (2.20)

12

Q+dQ

_j? ds N+dN

M+dM

Figure 3 Segment of a Centroidal Element in the Deformed State with all the External Forces and Stress Resultants.

13

*

from which, after dividing by ds and passing to the limit, one obtains

dQ_ = N + p (2.21) ds ds n

In a similar manner summation of forces in the tangential direction

yields

P ds* + (N + dN) cos (*|-) - N oos (^|-)

* *

+ (Q + dQ) sin (^|~) + Q sin = 0 (2.22)

from which

*

dN dd> ___ = _ Q _ P (2.23) ds ds

The condition of equilibrium of moment for the element is

•

dM + Pn ds* (̂ |-) - (Q + dQ) ds* = 0 (2.24)

• after dividing by ds and passing to the limit we obtain

^* = Q (2.25) ds

14

Differential Equations for Beams

Equations (2.4), (2.5), (2.18), (2.19), (2.21), (2.23) and

(2.25) can be written in terms of s and z, as

dx * 5? = ? cos •

iy > , * d? = 5 Sln *

M = - q S i l . 5 P ds ^ ds 4 t

4^ = N - + c P (2.26) ds ds ^ n v •*

N = EA (5 - 1)

M = EI ds

Equations (2.26) are the governing differential equations of a beam.

The quantities appearing in the differential equations are defined

in Figures 2 and 3 in which the beam is represented by its centroidal

line.

15

End Forces

In Figures 4(b) and 4(c), free body diagrams are shown for

segments cut from the beam shown in Figure 4(a). In these figures the

*

forces are positive as shown. Equilibrium at end A (x^,y^), Figure 4(b)

requires that

\ = t-Nl cos *1* " Q1 Sin Os-sj

Py = (_N1 Sin *1 + Q1 COS *1 )s=s (2.27)

M = (-M.) ej v l7s=s1

Similarly at end B (x2,y2), Figure 4(c)

P = (N, COS 2 + Q2 sin

16

M§s\

?>®s2

Undeformed Centroidal Line

Deformed Centroidal Line

(a) General View

N,

_j_fl\ Ni

\ y-

(b) End A (c) End B

Figure 4 Forces.

17

Nondimensional Equations

For convenience in the numerical work, equations (2.26), (2.27)

and (2.28) are nondimensionalized by introducing the following

nondimensional variables

2 2 NL QL - ML o o o

n " EI ' q EI ' m ~ EI o o o

3 P L 1 5 P + L * no to

Pn EI ' Pt EI o o

P L 2 P L 2 M L x o y o e o

Px = EI ' Py = EI ' me = EI (2.29) 1 o J1 o 1 o

P L 2 P L 2 M L x2 o V2 ° e2 °

Px_ = EI~ ' Py = ~~Ei ' me = ~EI 2 o J 2 o 2 o

** = T~ ' n = L~ ' ^ = * ' S = LoS o o

In the foregoing relations, Lq and Iq are conveniently chosen

reference length and moment of inertia, respectively. Substituting

the above relations into equations (2.26), (2.27) and (2.28) we obtain

£' = £ cos

n' = C sin

n1 = -q » - ? Pt

q' = n f + ? Pn (2.30)

18

m' = Cq

(j)' = m

5 = 1 + cx n

where

C1 ~ VLo A ' c2 Io/I » ( )' dS^

and

Px = (""j cos +! " 2^ 2 S=S2

py = (n2 sin 2 - q2 cos 2) ^ (2.32) 2 S=S2

m = M 2 S=S2

19

Adaptation to Beam Under Uniform Load

The loading of intensity A^ per unit undeformed length has

fixed orientation B in the plane as shown in Figure 5(a) in which the

beam is represented by its centroidal line. Quasi-static deformation

is assumed, therefore, the loads are considered as ;slowly varying

functions of time and derivatives with respect to position S along the

undeformed reference axis may be considered as partial derivatives with

respect to S of functions of position and time. In the equations that

follow the prime symbol is used to denote (partial) derivatives with

respect to S.

The governing equations are equations (2.30), (2.31) and (2.32).

To determine P and P„_ we resolve the elemental force A, ds into normal n t d

and tangential components along the axis of the beam, see Figure 5(b).

Thus,

*

P ds = C P ds = A, ds cos (d> - g) n n d J

or

C pn = Ad cos (2.33)

and

pt ds = £P^. ds = A^ ds sin ( - 3)

or

? pt = Ad sin ( - B) (2.34)

Uniform Load Intensity

(a) Deformed Centroidal Line w X -

(b) Components of ds

Figure 5 Beam Under Uniformly Distributed Load.

21

With the aid of the nondimensional uniform load intensity X

defined as

X = X, L 3/EI do o

the above equations can be written as

£ = X cos ( - 3) (2.35)

£ p = X sin (((> - 3) (2.36)

Substituting equations (2.35) and (2.36) into equation (2.30) one obtains

the final form of the governing equations as

= ? cos

n' = ? sin (|>

n' = -q ' - X sin ((|> - 3)

q1 = n ' + X cos ( - 3) (2.37)

m' = ? q

22

Rate Equations

In the analysis of quasi-static nonlinear response,one is

interested not only in the variables but in their rate of change as

well. The rates are of interest for two reasons. First, they enter

in the determination of the stability of an equilibrium configuration.

Second, certain differential coefficients related to the rates are

useful in the numerical solution of the governing equations. Differen

tial equations governing the rates are obtained by differentiating the

governing equations partially with respect to time.

In the equations that follow the dot symbol is used to denote

(partial) derivatives with respect to time. Thus, from equations

(2.37) we derive

= £ cos - t, 4> sin

• • •

n' = c sin (J) + ? cos 4>

n1 = -q cos ( - g) - A sin ( - g)

q1 = n 1 + n 4>' - A $ sin ( - g) + A cos ( - g) (2.38)

m' = h q + t. q

• • (J)' = c2 m

X, = n

23

where

«' - t«" - Js # '4E #5 = f"'

= rt'' = Js t|?' '-k # = f"'>

*. /-••»! 3 ,3n. 9 ,8n, , .»• n " ̂ ~ 3S (-¥t-) ~ dt ^ ̂ ' etc.,

In a similar way we can differentiate equations (2.31) and

(2.32) to obtain

px = (n2 sin^j - qj ̂ cos^ - n2 cos1 - qj sinifrp 1 S=S^

(2.39)

py = C-Hj costf^ - q2 ̂ sin^ - t\1 sin^ + q2 cos^) 1 S=Sj

m = (-m ) 1 1 S=S

and

Px = (~n2 ̂ 2 sin̂ 2 + q2 2 cos2 + n2 coŝ 2 + q2 sin(')2^ 2 S=S2

Py = n̂2 ̂ 2 cos

CHAPTER 3

NUMERICAL SOLUTION

In Chapter 2 the basic governing differential equations,

boundary conditions, and related rate equations for quasi static

deformation of a beam segment were derived. This chapter is concerned

with the development of the basic numerical procedure used to generate

solutions of the equations.

We begin with consideration of two fundamental problems. The

first problem (denoted P.l) is that of solving equations (2.37) and

evaluating the forces defined by equations (2.31) and (2.32) when E,,

r), and

the instantaneous stiffness matrix for the member and data that are

useful in the numerical computation of load-deflection behavior of

structure through incremental loading. The instantaneous stiffness

matrix plays a central role in the determination of the stability of

an equilibrium configuration.

Problem P.l

Let Sj and be the values of S at the "one" and "two" ends

of the beam segment, respectively, with < S < S2 and let

U, T[, , n, q, m)s=s = T̂ , 4^, n^ q2, 11^)

s=s — T12' ̂ 2' n2' ̂ 2' ̂ 2^

We understand the solutions of the governing equations to be

expressed as functions of S with rij, 1} n1, qj, and A appearing

as parameters. Thus,

5 ~ C(Sj ^1' *^1 * *"l' ^

~ n (^J ^1' ^1j ^1' ̂ 1' ̂ 1' ml' ̂

4>(S> *^1' ^1' ̂ 1' ml' ̂

^ — ^1' *^1' ^1' ̂ 1* ^

(3.1)

1 ^2' ̂ 1' ̂ 1' ̂ 1* ^1' "*1' ^

26

and

also

m = m(S; r^, 4>x, q^ n^, A)

P^ C^2> ^1' ̂ 1* ^1' ^13

P = P (5-^ V r n2, qr m , X) (3.2) yi yl —_

^ej^l' ̂ 1' *^1' ^1' ^1' "*2'

fx2 ' Px2(5r V *1' V V V x)

Py^ Py2^1' ̂ 1' ^1' ̂ 1' ^1 ' mi' ^ C^-3)

mQ̂ ^l> ^1' ^1' ̂ 1' ̂ 1' ̂

Given values for E,^, r^j

27

$2 $ C®2* ^1' ̂ 1' ̂ 1' ̂ 1' ̂ 1' ®]_» ^ (2*4)

where *12> anc* (S ; 5 , n , , n^, q^, m^, X) 2 2 1 1 1 1 1 1

and let 6n^,

Linear equations whose solutions yield first order approximations to

the corrections are derived by subtracting equations (3.5) from

equations (3.4), expanding the difference in Taylor's series and

retaining only the linear terms in the expansion, thus, we obtain

5 - '= (-M- ^ 6q + -2L 6m ) 2 2 8nl 1 **1 1 tal 1 q _ b-t>2

(a) , 3n x j- , 8ti r. f-z n n , = t i n i * s t t * sr 5mi'

The present improved approximations to n^, q^, m^, and further

improvements are obtained by additional application of the procedure.

In the process of integrating the differential equations one

generates the values £2, ri2, 2, n2, q2, m2- Therefore, after having

calculated n^, q^, and m^, one has all of the numerical information

needed to evaluate the forces of restraint defined by equations (2.31)

and (2.32). A key step in the practical implementation of the

Newton-Raphson procedure is the evaluation of the differential

coefficients in equations (3.7), i. e. the evaluation of

(JL)

S=S,

etc.

S=S,

These coefficients enter quite naturally in the solution of the rate

boundary value problem P.2 which is considered next.

Problem P.2

Just as the rate differential equations are derived by partial

differentiation of the governing equations with respect to time, so

are solution functions for the rate differential equations derived by

differentiating solutions (3.1) partially with respect to time. Thus,

considering J, q^, m^, and A as functions of time one

obtains

30

i 3E • 3£ • 3£ r 3E • 3E • 3E • 3E r C ~ 3?1 C1 + 3n1 nl + 3^ *1 + 3n^ "l + 3qx ql + 3m1 mi + 3A

* _ 3TI T . 3TI • 3I) ; 3n * , 3n • 3n * 3TI • n 3ex ?1 dr)1 nl + 3 2> and x-

Substitution of the results into the force rate equations (2.39) and

(2.40), then completes the solution of problem P.2. Details are

presented later in this chapter. At this point it may be observed

• •

that the coefficients of n^, q^, and m^ in the expressions for

• •

£, n, and (j> in equations (3.8) evaluated at S=S2 are the same coeffi

cients in equations (3. 7) needed in applying the Newton-Raphson method

to the solution of P.l. The key to completing the solution lies in

31

the evaluation of the differential coefficients that appear in

equations (3.8). Differential equations and initial conditions for

the determination of these differential coefficients are obtained by

substituting equations (3.8) into the the rate equations (2.38),

demanding equality for all values of ri^, n^, q1? m^, and X, and

making use of the fact that equations (3.8) reduce to identities when

S = i.e.,

• • • • • •

K = Sj, n = Tij, = » etc.

The equations for the determination of the partial derivatives

with respect to are obtained by differentiating equations (2.37)

partially with respect to Thus,

, 8? ' 3C . 34> • . = ~*F~ cos + ~ ? "ar sin * a?i n1 3^

, 3n ̂ 1 , 3d> t-w? " jqsin •+ c 117cos •

9 • _ 3m " d2 3^

9? _ 9n K1 C1

The corresponding initial conditions are

33

= i r_M_i = r-^ZL") _ r • ) = r 3 c ? ) = = 0 @ S = S l3* J L* 8$^ 3(J>1 34>1 l8*1J 3«f>1 & &1

t-lr' - (4f) - ̂' Hir> = C-lt-5 = 0 s s=sj

Details of Solutions of Problems P.l and P.2

Given the initial values rij, j, , and

A we can integrate the governing differential equations (2.37) along

with equations (3.9) and the corresponding equations for the derivation

with respect to j, n^, q 3̂ and A simultaneously to obtain

numerical values of solutions (3.1) and the partial derivatives that

appear in equations (3.8). Evaluating these solutions at S=S2> yields

fell Tel") f G ̂ estimates of the variables > $2 anc* t̂ le differential

coefficients appear in equations (3.7), which may then be solved for

the corrections 61^, Sq^, and Sm^.

To satisfy static equilibrium, we must solve equations (2.39)

and (2.40). Matrix algebra is employed in treating remaining elements

of Problem P.2. Toward their end, define matrices as follows

. {zL}=

Z1

Z 2

35

[u2i] -

[U2 2 ] -

U41 U42 U43_

U51 U52 U53

U61 U62 U63

U44 U45 U46

U54 U55 U56

U64 U65 U66

U 47

' 'V =

36

[A2 2 ] -

+cos(|> + sin 0

+sin(fi -cos^i 0

0 0 1 S=S,

[Bn ]

0

0

0

+n sin4> - q cos

-n cosij) - q sin

0 -> S=S1

[B 223

0

0

0

-n sin + q cos

+n c.os (3.11)

and from equation (3.8) evaluated at S=S,

{Z2> = [U113(Z1> + [U12]{r1J + X{U13J (3.12)

{r2) [U21]{Z1} + [U22]{r1J + MU23> (3.13)

37

Equations (3.12) and (3.13) can be solved to obtain {r^} and {r2} in

terms of {Z2> and X. Then

{V = ^ll^V + [ V12 ] {V + ^ { V13 } ( 3-1 4 )

'{T2} = tV21 {̂̂ l} + £V22]{̂ 2} + *{V23} C3'15;)

where

[Vn] = [v12] = [uI2]"x , [V1 S ] = -CU 123_1 (U13)

CV213 = tU21"' " tU22-"-U12^ Ûll^ ' ̂V22^ = Û22 Û12"'

and

*V23* = *U23* " '-lJ22^'-l,12^ *U13*'

Substitution of (3.14) into (3.10) and (3.15) into (3.11) yields

{Pl} = ([Bn] + [An][Vn]){Z1} + [An][V12]{Z2} + X[An]{V13> (3.16)

{P2} = CA22 V̂2lJ{V + (tB22] + tA22 V̂22]) {V + Â22]{V23} C3"17)

38

which can be written in the form

= + J{cL» (3.18)

where

+ A JL1

JL2

and

CKLII] Din] - [Ku2] = EaJCV^L «L1} =CAlP{Vl3»

^"KL21"^ '-A22-"-V21-' ' '"KL22"' B̂22"^ + Â22 V̂22-'' *CL2^ l-A22 V̂23^

[Kl] is the instantaneous or tangential stiffness matrix of beam

segment calculated in the local £, n coordinates. [K ] is Li

necessarily symmetric if the applied loads on the beam segment are

conservative.

Rigid Frame Structures

In order to determine the instantaneous stiffness matrix for

a complete rigid frame structure, a common datum must be established

for the unassembled structural elements so that all the displacements

and their corresponding forces may be referred to a common coordinate

system. The datum system is selected in such a way the coordinates

of points on the structure can easily be found.

39

Since the element force vector {p^}* the stiffness matrix

[K^] arid vector {C^} are initially calculated in local coordinates,

suitably oriented to minimize the computing effort, it is necessary

to introduce transformation matrices changing the frame of reference

from a local to global coordinate system. The first step in deriving

such transformation is to obtain matrix relationships between the

• •

element forces {p^Jj the positions and the rates {p^} and {Z

in local coordinates and element forces {p}, positions {Z} and their

• • rates {p} and {Z}, respectively, in the global coordinate system.

Such relationships are derived in (Prezmieniecki 1968, pp. 67-69) as

-Cc} = 0]T{Cl} (3.19)

{p} = 0]T{pL) • (3.20)

{p} = WT{pL> (3.21)

{ZL> = [*]{Z} (3.22)

{ZL> = [*]{Z} (3.23)

[K] = [*]T[Kl][*] (3.24)

where [ij;] is a matrix of coefficients that has the property (of

orthonormality)

WW' - [ I ]

40

T where [ifi] is the transpose of

The matrix [ty] is obtained by resolving global displacements

in the direction of local coordinates. The elements of [1//] are found

to be nothing more than the direction cosines of the angles between

the local and the global coordinate systems.

[K] is called the instantaneous stiffness matrix of segment of

the beam in global coordinates. [K] is necessarily symmetric if the

applied loads on the beam segment are conservative.

Consider a frame composed of several beam segments rigidly

joined at their ends. The field variables are assumed to be continuous

over each beam segment. Therefore, concentrated forces may act only at

joints which are also referred herein as nodes or nodal points.

Figure 6 shows a section of a frame composed of 3 segments. Given the

position of the nodes {Z}, the distributed load parameter A, and their

respective rates, we want to determine the forces that must be applied

at the nodes and their respective rates for quasi static deformation.

Obviously the single segment analysis presented above may be applied

to the isolated segments of the beam to obtain the required end forces

[p , p , m], and corresponding rates in terms of displacement rates of x y

the nodes and load rates over the segments. For simplicity the analysis

will be described with reference to the three members plane frame shown

in Figure 6.

41

Figure 6. Plane frame composed of three segments in the undeformed state.

42

As indicated in Figure 6 the "2" end of segment (j-1) and the

"1" end of segments (j) and (j+1) have a common node (j). Equilibrium

of node (j) requires that

(j)_

Px = Px2(j-1) + Pxl(j) + Pxl(j+1)

(j)

py = py2(j-i) + pyl (j) + pyl (j+l) (3'25

m = m _ ,. +m1/..-v+m.,. ... e e2(j-l) el (j) el(j + l)

where the superscript in parenthesis indicates the node and the

subscript in parenthesis indicates the connecting segment.

Let

{pl(j)} ^xl' Pyl' mel"^ r. yl- (j)

{p2(j)} = Cpx2' Py2' me2](j) (3.26)

(1 ) (1 ) (1 ) (2 ) (2 ) (2 ) {P) = [ P X . P y ' ̂ » P X » P y > ^ , • • • ]

where the force vector {p} is a function of the position vector {Z}

of the nodes, and the distributed load intensity X.

43

Then according to equation (3.25)

p(j:) P2(j-1) + pl(j) + Pl(j+1) (3-27)

and for simplicity letting j = 2

^1(1)' P2 (1) + Pl(2) + P1 (3) ' P2 (2) ' P2 (3) ' "(3'28)

Differentiating equation (3.28) with respect to time yields

{P> P̂l(l)' P2 (1) + P1 (2) + P1 (3) ' P2 (2) ' P2 (3 ) ' ' ' ̂ (3-29)

Let (Z) be the column position vector for the node points arranged so

that

mT - n"' . 5 f 2 ' , • ( 2 ) , S ( 3 \ n ( 3 ) , • « . .. . ] (J.S0)

Then because of the rigid connection of joint j,

rj-(j) ;(j) ;(j)-i _ rf ; * i U , T) , J Le2(;j_1-), ^2 (j-1) ' 2 (j -1)

= ^l(j)' ^(j)' *l(j) ]

= ^l(j + l)' nl Cj + 1) ' *l(j + l)^ (3.31)

44

or

r^Cj) *Cj) ;(j)-j _

{C} Ĉl(l)' C2(l) + Cl(2) + C1 (3) ' C2(2) ' C2(3)^

45

[K]is necessarily symmetric, if the applied loads are conservative.

The matrix [K] is singular in the absence of prescribed boundary

displacement conditions. At least three appropriately nodal displace

ments must be prescribed to restrict rigid body motion of a rigidly

jointed plane structure.

In the preceding analysis the load parameter A, the nodal

position coordinates and their rates were considered an indepen

dent prescribed quantities, and the nodal forces and their rates

treated as unknowns to be determined. The more general situation (which

represents the rule rather than the exception) encountered in structural

analysis is that in which some of the nodal forces and their rates are

prescribed. It is understood that where a nodal force component is

prescribed the corresponding final position coordinate or its rate is

to be determined. The analytical problem to be solved may be formulated

as follows. In the equilibrium equations

p. = f.(Zk, A) i,k = 1, 2, ..., n (3.34)

where p^ is the nodal force component in the direction of Z^, let

P°i = Pi 1 / i - 1, 2, Hj

46

Z, = Z. ok 1

Psk

k - 1 - j i + 1 ̂ ^ 2 j • • • ̂ n

With the foregoing notations the equations of equilibrium can be

rewritten as

p . = f. (Z ., Z . , A) *oi si' ok'

(3.35)

Psk = fk(Zsi' Zok'

The problem is: given the PQ̂ > Z ̂ and A, and their respective rates,

determine the Zĝ and pĝ and their respective rates.

The rate problem is most easily dealt with and is considered

first. Differentiating the equations of equilibrium (3.35) with respect

to time yields equations that are linear in the rates and which may be

written in matrix form as

[K11]{V + [K12]{V = {V = {Po} " *{Co} (5-56)

[KnJ{Zs) • [K22]{Zo> = (Fs) = {ps} - A(Cs} (3.37)

47

or, in partitioned form as

K11 | K12

K21 I K22

(3.38)

or

[K]{Z> + A{C} = {p} (3.39)

From an examination of the equations of equilibrium it is clear

that the coefficients in the stiffness matrix [K], and therefore, the

coefficients in E^j], are the partial derivatives of the functions f^

with respect to the in equation (3.34), and the coefficients in (C)

are the partial derivatives of the f^ with respect to A. All of the

coefficients are understood to be evaluated for the value of A and the

values of the position coordinates that correspond to the current

equilibrium configuration. Solution for the desired rates is now

straightforward if [KJJ] is nonsingular equation (3.36) yields

{V = EKJJ]"1 ({Fq} " [K12]{Zq}) (3.40)

which when substituted into equation (3.37) yields the force rates

48

position variables Zg, the related deformations, and stress resultants

used in computing [k] and {C} are evaluated in the true equilibrium

position which satisfies the given conditions at the nodes and the

governing differential equations for all the frame elements.

If {Zg } is an approximate solution for the unknown position

elements, the Newton-Raphson method can be used to obtain an improved (a)

approximation. Corresponding to {Z } and the prescribed {Z } and A, (a) S °

we can calculate {p }, the applied force vector that is required to

satisfy all the equilbrium conditions in the problem with position (a)

{Z } and {Z }. S ° r \ (a)

Let {6Z } be the correction such that s

(a) (a) {6Zs } + {Zs } = {Zg} (3.41)

(a) (a) is the correct position. If {p - p } is small, we expect {6Z }

(a) ° ° S

to be small. Consider {p } as a function of the elements of {Z }, , ̂ o , s (a) (a)

expand {p - p } in a Taylor's series about {Z } and retain only ° ° (a) S

terms linear in {6Z } to obtain s

(a) (a) (a) (a) {po " PG > = {Spo > = [Kn ]{

49

is found. The process can be repeated to obtain the {Zg} to any desired

degree of accuracy.

The rate equation (3.39) provide the basis for a step-by-step

analysis of finite deformation of a rigidly connected frame structure

under prescribed loading. Assume that the equilibrium position and

stress resultants at the nodes consistent with the applied loads {pQ}j

A, and the prescribed displacements have been calculated. Then

for each element of the frame we can calculate the instantaneous stiff

ness matrix [K^]> and the structural stiffness matrix [K] appearing in

equation (3.39). In the passage of "small interval of time",

{Zq} and X change in small known amounts

{6po> = 6t{po} (3.44)

{

50

Having {SZo} and {6Zs>, one can calculate {6r^} for each element

of the frame. Then under the prescribed loading and positions,

{p +

indefinite. Thompson (1963) calls the equilibrium state for which the

stiffness matrix first becomes positive semi-definite the primary crit

ical equilibrium state. The corresponding load is understood to be the

critical or buckling load.

In the general case the primary critical equilibrium state is

associated with snap-buckling condition, which, for one parameter load

ing, is characterized by simple maximum on the load-deflection curve,

point C, Figure 7. The load at which the snap buckling is attained

identifies a transition state from stable to unstable equilbrium.

A bifurcation point, point b in Figure 8, is established where

a secondary equilibrium path intersects the primary equilibrium path.

At such a point equation (3.36) has two solutions, one corresponding to

path A and one corresponding to path B.

Computer Program Description

The solution procedure developed above was programmed in

FORTRAN IV for batch processing on the CYBER 175 University Computer

Systems at the University of Arizona. A general schematic of the

computer code written for this purpose is shown in Figure 9. In

applying the program, one inputs:

a. Structure parameters - one card containing the

number of elements M, the number of joints NJ,

and the number of supports NS.

b. Joint coordinates - a set of cards (NJ cards total)

containing the data required to specify the coordinates

of the joints of the plane frame. Each card in this

52

12

11

10

9

8

7

6

5

4

P,A

3 h/2

2

2h 1

0 2 4 6 1 . 0 8 A/h

Figure 7. Load-peak deflection curve for a gabled frame with snap buckling.

53

13

10 Path B Path A

h/2 CM rC Cu

-04 .08 .12 .16 .20

A/h

Figure 8. Load-peak deflection for gabled frame with bifurcation buckling.

54

BEGIN

CINPUT DATA )

V 1 = 0

I 1 = 1 + 1

INTEG. EQUS. 2.37 § 3.9-

CALCULATE [K].S(p^}. 1 * o 1

ASSEMBLE STRUCTURE STIFF. MATRIX [K] § NODAL FORCES {pOO} ro

I = NO. OF ELEM

K| < 0

PRINT RESULT

Sr = f05Zs)

1

II

rl + 6rl

ri rl + 6rl

1 6r! =

V12 6Z2

z. -s Z W s

= 6Z s

Z = s

Z + s

6z(a) s

1 6p *0

= K 6Z^ s

1 6p ro = Po -P

(a) ro

Z = s

Z + s 6Z s

I 6p = K 6Z s

1

ii o P

+ 1 o 6p ro

Figure 9. Schematic diagram of computer program.

set contains a joint number J, the £ coordinate

X(J) of the joint, and the r) coordinate H(J) of

the joint.

Member designation and properties - a set of cards

(M cards total) containing member data. On each

card, the member number I is listed first, followed

by the joint IP number IP (I) and the joint IQ number

IQ(I) for the two ends of the member, IP-IQ being

the local £ axis of the member. The choice of the

end of the member that is to be the IP end that

which is to be IQ end is arbitrary, except at the

supports, where the IP end is always taken to be

at the support. The next two items on each card, the

cross-sectional area ratio A/A , and the moment o

of intertia ratio I/I of the cross-section. o

Joint restraint list in a set of cards (a total of

NJ cards) containing joint restraint data. Each

card contains a joint number J and three code

numbers which indicate the condition of restraint

at that joint. The terms KK(3J-2), KK(3J-1) and

KK(3J) denote the restraint in the £, TI and

rotation in the z sense, respectively at joint J.

The convention adapted in this program is the

following: if the restraint exists, i.e. if

displacement is prevented the integer 1 is assigned

56

as the value of KK, and if there is no restraint a

value of zero is assigned.

e. Loads applied at joints - a set of NJ cards. Each

card contains a joint number J and the three actions

applied at the joint. These actions are the applied

force rates in the £, n directions and the couple in

the z sense.

f. Distributed loads a set of total of M cards. Each

card contains a member number I, the distributed

load rate, and angle 3(radians) which described the

fixed orientation of the load measured in local

coordinates.

A consistent nondimensionalized system of units is used for the

input data.

The procedure described in Cook (1974) is used to treat sup

pressed degrees of freedom . For example, if a zero displacement z^

is to be imposed as in the case of rigid support, the diagonal term

K ̂of the stiffness matrix [K] is augmented by a number, which is

several orders of magnitude larger than k^, e.g. k^^ * 10"^.

Physically this device amounts to adding to the structure a spring of

high stiffness that resists the displacement z^.

Gaussian elimination scheme is used to compute the determinant

of the instantaneous stiffness matrix, and to solve the governing

structure rate equations, using single precision arithmetics.

CHAPTER 4

PRESENTATION AND DISCUSSION OF RESULTS

In this chapter quantitative results will be presented for a

number of frame structures. The discussion of the results will be made

on a case by case basis; however, several examples with a common param

eter (i.e., span, rise, and/or loading choice) will be considered in

groups in an attempt to draw conclusions regarding effects of variations

of parameters on buckling loads and structural behavior.

The choice of load-shape configurations of frames studied herein

is based on the importance and applicability of such structures. It is

important to mention that neither the theoretical development nor the

computer program written for the purpose of this research are limited

to the cases presented. The numerical studies were limited to the cases

presented here in order to keep computer cost and total work effort

within reasonable bounds.

The available methodology can be readily extended to more

complicated structures without major modifications.

Since the extent of the numerical calculations, accuracy of

results and cost of computation depend on the increment of arc length

used in numerical integration of the governing differential equations,

discussion in this chapter begins with consideration of the problem of

selecting the increment. Then, for the purpose of comparison with

known results, the results of four test cases are presented. The

57

58

remainder of the chapter is devoted to discussion of the results of

examples solved.

The view taken in arriving at a choice for the increment was

that a balance should be attained between accuracy and cost. Toward

this end numerical experiments were run for the case of a cantilever

beam under uniformly distributed load. The beam was subjected to loads

that produced tip deflection of approximately 0.9Lq, where Lq is the

length of the beam. For the purpose of comparison, the results obtained

for 100 equal subdivision (AS = 0.01) were considered to be "exact".

The results of the experiments are summarized in Figure 10. Relative to

the results for 100 subdivisions the error (of tip force resultants) for

just five subdivisions is within 0.35%but the cost of computer is l/8th.

In view of this, all of the remaining examples were solved using

AS = 0.2.

Test Cases

Figure 11 shows test case 1, where the buckling loads for the

cantilever beam are found in agreement with the literature Bleich (1952).

In test case 2, the same beam is subjected to pure bending by

couples. The deformed beam as represented by its centroidal line

(Figure 12) formed a circular arc, in exact agreement with the theory

of beams under pure bending (Popov, 1976).

Test cases 3 and 4, Figures 13 and 14, respectively, focus on

the complete behavior of two frames as represented by load-deflection

curves as predicted in this study. The curves are compared to those

obtained by Lee, Manuel , and Rossow (1968). The results are in good

59

.5

.4

. 1

Cost

5

4

4-> tn o u f= o +J - CO

2 4-> 3 C. e o u

1

0 10 20 30 40 50

Number of subdivisions of Beam (1/AS)

Figure 10. Error in computation and computer cost vs. number of beam subdivisions.

19

15

11

EI

7

3

2.47 16.5 21.5 0

PL /EI 1

5

Figure 11. Test Case 1 - Determinant versus axial load for a cantilever beam.

61

y i

ML

0. 0 0 . 2 0.4 0 . 6 0 . 8

x/L o

Figure 12. Test Case 2 - Beam subjected to pure bending.

62

This study

Lee et al. (1968)

. 2L, • 8L,

0 0 . 1 0 . 2 0.3 0.4 0.5

A/L o

Figure 13. Test Case 3 - Load versus downward deflection of point of loading.

63

20 This study

Lee et al. (1968)

16

12

8

4

0 0 . 0 6 0 . 1 8 0 . 1 2 0.24 0.30

VL0

Figure 14. Test Case 4 - Load versus horizontal deflection of node 2.

agreement. For the L-frame both calculations predicted snap buckling.

2 In this study at P = 18.03 EI/Lq and in the work of Lee et al. at

2 P = 18.5 EI/Lq • For the portal frame both studies predicted the first

2 buckling load by bifurcation at P = 1.5720 EI/Lq . When the frame is

restrained against side sway, the study predicted snap buckling at

2 2 P = 15.38 EI/Lq , whereas snap theory of buckling at P = 14.9 EI/LQ .

Examples

Buckling load versus span (L/h) are shown in Figure 15, for a

group of gabled frames with members of identical uniform cross-section,

built-in supports, constant rise (r = h/2), and subjected to three

downward point loads as shown (from this point on these frames are

described as being uniform). As L/h increases the buckling load gener

ally decreases. On the plot two intersecting graphs can be identified.

The high curve (L/h < 2.25) applies to frames for which buckling is in

a bifurcation mode. And the lower curve (L/h > 2.25) applies to frames

that undergo snap buckling. At L/h = 2.25 either mode of buckling can

occur. On Figure 16, load versus downward deflection of the peak (later

on termed peak deflection) are plotted. One can notice that in general,

the deformations are smaller when the bifurcation mode of buckling is

dominant.

Figure 17 shows buckling load versus rise (r/h) for a uniform

gabled bent subjected to three downward point loads. In contrast to

the previous case the buckling load increases as rise increases until

r = 0.25h, and for r < 0.25h the snap buckling mode dominates. As the

rise continues to increase (r > 0.25h) the buckling load decreases and

65

Bifurcation buckling mode

Snap buckling mode

P/2

P/4 P/4 h/2

7/77

-i

5

-T"

3

T

4

L/h

Figure 15. Buckling load versus span for uniform gabled frames of constant rise and built-in support under three downward point loads.

66

12

2h

2. 2h 11 -

2.30h

10 .

2. 4h

2.6h

L=3h

P/2 ,A

P/4 P/4 h/2

Bifurcation

Snap buckling

1.25 0.25 0.50 0.75

A/h

Figure 16. Load-peak deflection for uniform gabled frames of constant rise and built-in supports under three downward point loads.

67

12

Bifurcation mode

Snap buckling mode 11

P/2

P/4 P /4

1 0

7777 7/7 2h

9

0.5 0 .4 0.3 0 . 2 0 0. 1

r/h

Figure 17. Buckling load versus rise for uniform gabled frames of constant span with built-in supports under three downward point loads.

68

13 n

1 2 -

r/h=0

CN

P/2 , A

P/4 P/4

77" 2h

Figure 18. Load versus peak deflection for uniform gabled frames of constant span with built-in supports under three downward point loads.

69

bifurcation buckling occurs. Figure 18 shows the load versus peak

deflection for the range of frames discussed above.

Buckling load for a range of gabled frames under concentrated

loads over the columns, as shown in Figure 19, is independent of the

span, rise and the properties of the connecting beams. Bifurcation

buckling occurs in all the cases studied. The above result confirms

the findings of Masur, Chang and Donnell (1961).

Figures 20, 21 and 22 show load versus peak deflection for gable

bents with several combinations of load and geometry. In all, four

cases of loading were studied in conjunction with three different frames.

The loading cases are as shown in Figure 20. Buckling is in a bifurca

tion mode in all of the loading cases for the frame in Figure 20, for

which L = h. Although there is a wide variation in the peak deflection

at buckling, the buckling loads are reasonably close, especially when

the variety of loading is considered. In Figure 21, where L = 2h, the

buckling loads for cases b,d and c are still close, and the buckling is

still by bifurcation. In load case a, very large prebuckling deforma

tions are realized, and buckling is by snap through. For the frame in

Figure 22 where L = 3h, only load cases b and d are considered. The

buckling load for case b is slightly higher than d, and in both cases

failure is by snap buckling.

Figures 23 and 24 summarize the results obtained in a study to

determine how the buckling load and buckling mode of a gabled bent under

uniformly distributed load vary with the rise of the rafters. As

indicated in Figure 24, either snap buckling or bifurcation buckling may

occur.

70

13

12 "

w CM

JS 11 "

u o

1 0 -

9 - ' / / /

_ a r

-T~

3

i •>

4 5

L/h

Figure 19. Buckling load versus span for gabled frames with built-in supports under two downward loads over the columns.

a. r=h/2, except for L/h=5, r=h and I,=31 . D C

71

13

1 2 •

11 -

10 •

9 "

8 "

5 -

1 -

^P/4 (b)

P/2 (c)

X Bifurcation point Snap-buckling point

P = distributed load resultant for case d

0.04 0 . 0 8 0 . 1 2 0 . 1 6 0 . 2 0

A/h

Figure 20. Load versus peak deflection for uniform gabled frames with L=h and built-in supports.

72

12

10

UJ

0 0 . 2 0.4 0 . 6 0 . 8 1 . 0

A/h

Figure 21. Load versus peak deflection for uniform gabled frames with L=2h and built-in supports.

73

6

5

4

3

2

1

0 0. 25 0.50 0.75 1 . 0 1.25

A/h

Figure 22. Load versus peak deflection for uniform gabled frames with L=3h and built-in support.

13

1 2 -

2h

11-

u o

10 .

W = distributed load resultant

0 0.1 0 . 2 0.3 0.4 0.5

r/h

Figure 23. Buckling load versus rise for uniform gabled frames with built-in supports, subjected to a uniformly distributed load.

75

r/h=0

0 . 2

Figure 24.

0.4

—i—

0 . 6

—r~

0 . 8 1 . 0

A/h

Load versus peak deflection for uniform gabled frames with L=2h and built-in supports under uniformly distributed load.

76

The load deflection behavior of the gabled bent with r = 0.5h

under asymmetric distributed load is shown in Figure 25.

The results of a study of the bending and buckling of a gabled

bent under a combination of an independently prescribed distributed

load and concentrated load at the peak are displayed in Figures 26 and

27. The total load at which buckling occurs attains its maximum value

3 at a relatively small distributed load =0.1 El/h ). This may be

attributed to slight prebending of the structure as noted in Bleich

(1952). The critical point load decreases as the distributed load

increases.

The effect of variation in geometry on load-deflection behavior,

buckling load, and buckling mode for a frame are shown in Figures 28

and 29. The results clearly show that a knowledge of structure geometry

is insufficient to predict buckling modes. This is an important point

to keep in mind in an approximate analysis based on assumed buckling

mode shapes.

Figures 30 and 31 show how the load-deflection behavior and

buckling loads of a symmetric gabled bent with overhangs varies with

the distribution of symmetrically and asymmetrically placed distributed

loads.

Figures 32 and 33 summarize, the results obtained for an asym

metric frame, under a downward point load at the peak. The asymmetry

of the frame geometry seems to have little effect on the buckling load

when the eccentricity e is small (e 0.4h). The buckling load starts

to increase with e _> 0.4h. This may be attributed to the load being

closer to the centroidal axis of the column beneath, minimizing the

77

10 i

h/2 i—i UJ IN

2h

deflected position @ critical load (true scale)

0 0 . 2 0.4 0 . 6 0 . 8 1 . 0

A/h or (radians)

Figure 25. Load deflection curves for a uniform gabled frame with built-in supports under asymmetric distributed load.

h/2

3h

W=total resultant force

—i 1 1 1 1

0-2 0.4 0.6 0.8 1.0

Adh3/EI

Figure 26. Buckling load versus distributed load intensity for a uniform gabled frame with built-in supports under uniformly distributed load and point load at the peak.

h/2

3h

0.4

0 . 6

0.4 0.8 1.2 1.6 2.0

A/h

Figure 27. Load-peak deflection curves for a uniform gabled frame with built-in supports under uniformly distributed load and point load at the peak.

80

10

t—i w

U a

r<

h/2

0 0.25 0.50 0.75 1 . 0

tan y

Figure 28. Buckling load versus column slope for a uniform frame with built-in supports subjected to uniformly distributed load over its entire profile.

81

24

0.245 W = total force resultant 22

20 .64

.46 18

16

12

h/2

.15

-.015 0 .015 .03 . 045 . 0 6

A/h § (Y=0, .245, .46, .64)

Figure 29. Load versus peak deflection for uniform frame with built-in supports under a uniformly distributed load over the entire profile.

82

10

W = distributed load resultant

CM

s

rrr TTT 75h 3h . 7 5h

0.4 0 . 8

Figure 30. Load versus peak deflection for a uniform gabled frame with built-in supports and side overhangs under uniformly distributed load.

83

18

15

12

3h

9

6

3

Deflected position @ critical load (true scale)

0 2 4 6 8 1 . 0

A/h

Figure 31. Load peak deflection for a uniform gabled bent wire side overhang and built-in support under asymmetric distributed load.

84

10

w CM

u u

h/2

2h

0 0.2 0.4 0.6 0.8 1.0

e/h

Figure 32. Buckling load versus eccentricity e for asymmetric uniform frame with built-in supports under point load at the peak.

85

10 Deflected position at critical load and e=.2h (true scale)

e/h=0

CM •C a,

h/2

h+e h-e

0 0 . 2 0.4 0 . 6 0 . 8 1 . 0

A/h

Figure 33. Load versus peak deflection for asymmetric uniform frames with built-in supports under a downward point load at the peak.

86

beam bending effect of the rafters. The deflected pattern shown in

Figure 33 is for a frame with e = 0.2h under critical load.

Figures 34, 35 and 36 show the results of analysis of a two bay

structure in the form of a gabled bent attached to two story rectangular

frame under three combinations of gravity and lateral loads. In all

three cases the structure experiences very large prebuckling swaying

displacement toward the rectangular frame, regardless of the sense of

the lateral load.

Figure 37 shows how the response of a gabled bent varies with

2 support conditions. The buckling load is as low as P = 3 El/h in the

case of pin supports, whereas in the case of built-in supports

2 P = 12 El/h . In both cases the buckling is in a bifurcation mode.

87

3.25

3.00

2.75 Deflected position @ maximum load (true scale)

2.50

2.25

2 . 0 0

1.75

1.50

1.25

1 . 0 0 h/2

h/2 I _ _ 0.75 -

3h 2h

0.25

0 0.25 0.50 0.75 1 . 0 1.25

Aj/h, A2/h

Figure 34. Load deflection curves for a uniform multibay structure with built-in supports under gravity loads.

88

*

to f—

r<

0 . 8 h/2

h/2

0.4

3h 2h

0 0.25 0.50 0.75 1 . 0 1.25

Aj/h, A2/h

Figure 35. Load deflection curves for a uniform multibay structure with built-in supports under gravity loads and outward lateral load.

89

h/2

h/2

3h 2h

125 0 O.SO 0.75 1 . 0 1.25

Aj/h, &2/h

Figure 36. Load deflection curves for a uniform multibay structure with built-in supports under gravity loads and in-ward lateral load.

90

12

Built-in supports

P/2 ,A

P/4 P/4 h/2 i—( w

CM

Pin supports

0 .02 .04 .06 .08 1.0

A/h

Figure 37. Load versus peak deflection for a uniform gabled bent with hinged and built-in supports under three downward point loads.

CHAPTER 5

CONCLUSIONS AND FUTURE RESEARCH

A highly accurate, in principle exact, numerical method of

analyzing the quasi-static bending and buckling of plane rigid frame

structures has been developed in here. The methodology was checked

against four known cases and all the values obtained were in satisfac

tory agreement. The development was further demonstrated by its

application to the analysis of approximately fifty example problems

from which the following conclusions can be drawn:

1. This method of analysis provides a feasible way

for studying the finite deformation buckling and

post-buckling behavior of frame structures. Using

a reduced number of beam subdivisions (integration

interval AS = 0.2) proved to give acceptable

accuracy of results at a reasonable computer cost.

2. The availability of high speed computers with

large core memory permits studying nonlinear

structural elastic behavior conveniently using

this method.

3. Although the numerical examples deal with frames

consisting of prismatic members, the method is

equally applied for obtaining solutions to problems

involving non-prismatic members.

91

92

4. It has been shown in Chapter 4 that under the action of

concentrated independently prescribed loads a given framed

structure may lose its stability either by snap-buckling or

by bifurcation. Both such cases are shown on the actual

solutions of examples. The buckling mode (and load) is

sensitive to the relative intensity of the prescribed loads.

5. Moving along the primary equilibrium path (load-deflection

diagram) with increasing load, the determinant of the

instantaneous stiffness matrix changes in sign from

positive to negative on passing through the critical

equilibrium state. In practice it is not possible to

determine the value of load required to produce an exact

singularity of the stiffness matrix, but only values

producing relatively small positive and negative deter

minants. Once the determinant has changed sign, the load

increment can be made smaller and answers refined.

6. The required computation time is sensitive to load

increments and convergence tolerance. Throughout this

study, the "best rates" of convergence were achieved by

choosing (by experiment) a load increment parameter and

convergence tolerance at the beginning of loading history

and holding them fixed throughout the loading process.

7. It has been demonstrated, that nonrectangular frames

structures may undergo large deformations prior to

buckling, demonstrating that analysis based on small

deflection-rotation theory can be erroneous.

93

8. It appears that where buckling is in a bifurcation mode

the associated prebuckling deformations are generally

smaller than the deformations associated with instability

is in snap-buckling mode.

9. Great caution has to be taken when replacing distributed

loads by equivalent point loads for the purpose of analysis.

As demonstrated in Figures 20-22, the results can be con

siderably different, especially when only few point loads

are used.

10. It was shown in Figure 32 that slight asymmetry in a gabled

bent has little effect on the magnitude of the buckling

load, however, large asymmetry could have considerable

effects.

11. The buckling load of gabled bents under equal downward

concentrated loads applied directly over the columns is

independent of the span, rise, and the properties of the

beams, as demonstrated in Figure 19. The critical load

is considerably reduced when the applied loads are moved

away from the column, see cases a and c in Figure 21.

12. Generally, buckling loads of gabled bents are reduced

in the presence of initial bending moments, as in the

case of frame own weight in conjunction of concentrated

loads. However, small moments can have beneficial effects

upon the critical load as shown in Figure 26.

94

The present investigation has not been exhaustive, as already

indicated. Hence, if future research and extensions of present develop

ment are to be carried out, some of the more obvious areas are as

follows:

1. Cost comparison of the method developed in this study

and the finite element method.

2. Stability of rigidly connected plane frames with

nonprismatic members under combinations of static

loading.

3. Stability of braced frames.

4. Large elastic-plastic bending.

5. Adaptation of this method of analysis for design.

NOMENCLATURE

A cross-sectional area

A reference cross-sectional area 0

[A] matrix

[B] matrix

2 Cj extensional stiffness parameter: Cj=Io/Lq A

C£ rotation stiffness parameter: C2=IQ/I

{C},{CT} vectors LI

E Young's modulus

e eccentricity

h height of gabled bent

1 moment of inertia

I reference moment of inertia o

stiffness matrix

L span of gabled bent

M,M-,M_,M ,M bending moment 1 el e2

m,m.,m_,m ,m nondimensional bending moment: m=M L /EI 1' 2 e^ e2 6 0 0

N,N^,N2 internal normal force

n,n^,n2 nondimensional internal normal force:

n=NL 2/EI o o

P,P ,P ,P ,P nodal forces xi x2 V y2

2 P > P Y > P Y > P V > Pv nondimensional nodal forces: P = P L Q

kl 2 J 1 J 2

95

96

Pn,P^ normal and tangential components of distributed

load intensity

Pn>Pt nondimensional normal and tangential components of

3 distributed load intensity: p = P L /EI rn n o o

P critical load cr

2 p nondimensional critical load: p = P L /EI rcr cr cr o o

Q,Ql,Q2 shear force

2 q , q^,q2 nondimensional shear force: q = Q L q / E I q

r rise of the gabled bent r

{r} internal nodal forces vector: {r} = Jq

s arc length lm

S nondimensional arc length: S = s/L o

t time

[U] matrix

[V] matrix

x,y rectangular cartesian coordinates

z distance from centroidal axis

J

{Z} position vector: {Z} =

A displacement

6 variations

3 distributed load orientation

E.E axial strain ' c

C stretch

£,n nondimensional cartesian coordinates

K = X/Lq , N = y/LQ

97

K curvature

X^ distributed load intensity

3 X nondimensional distributed load intensity: X = X ^ L ^ / E I q

p* radius of curvature: p* = 1/K

a normal stress

4>, * rotation (radians)

M coordinate transformation matrix

( )• partial differentiation with respect to the arc length

( )' = 30/3S

( ) partial differentiation with respect to time

( )" = 30/3t

[ ] square matrix

{ } vector matrix

Subscripts

1,2 @ "1" end and "2" end , respectively

c @ the centroid

cr critical

e applied externally

L local coordinates

n, t normal and tangential components

o prescribed

s unknown

x,y in the x and in the y direction

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