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Com,m/m & Soucrures Vol. 56, No. I, pp. I57- 176. 1995

00457949 94)00545-l

Copyright 1: 1995 Elsevier Sctence Ltd

Printed in Great Britain. All rights reserved

0045.7949/95 9.50 + 0.M)

COMPUTER ANALYSIS OF THIN WALLED

STRUCTURAL MEMBERS

J. P. Papangelis

and G.

J. Hancock

Centre for Advanced Structural Engineering, University of Sydney, NSW 2006, Australia

Received 24 January 1994)

Abstract-The calculation of the stresses and failure modes in thin-walled structural members is a complex

procedure. Structural designers will often need help in analysing these types of structures. A vehicle for

providing this help is the computer program developed for the microcomputer. In this paper, a computer

procedure is described for the cross-section analysis and elastic buckling analysis of thin-walled structural

members. The cross-section analysis calculates the section properties, warping displacements, and the

longitudinal and shear stresses for thin-walled open and closed cross-sections of any shape. The

longitudinal stresses are used to perform an elastic finite strip buckling analysis of thin-walled structural

members. The analysis can be done for a number of different buckle half-wavelengths of the member and

the load factor and buckled shape are output for each length. The analysis is performed by the

user-friendly computer program THIN-WALL, which is also described in the paper.

1. INTRODUCTION

Thin-walled structures can be used in a wide variety

of different applications in structural engineering.

Thin-walled structural systems include box and plate

girder bridges, roof sheeting, floor decking, and

frame structures such as houses, portal frame build-

ings and storage racks. Thin-walled sections are

composed of very slender plates and have modes of

failure and deformation which are usually quite

different from those of standard hot-rolled sections.

In addition, the stresses and failure modes in thin-

walled sections can be quite complex to predict.

Structural designers will often need help in the analy-

sis of thin-walled structures. A vehicle for providing

this help is the computer program developed for the

microcomputer. In this paper, a computer procedure

and program are described for the cross-section

analysis and elastic buckling analysis of thin-walied

structural members.

In order to use thin-walled structures efficiently, an

understanding of all the stresses caused by flexure and

torsion is required. The calculation of longitudinal

stresses caused by flexure is straightforward for all

sections, although for general cross-sections the cal-

culation of the required section properties can be

tedious. However the calculation of the longitudinal

stresses caused by torsion and the shear stresses

caused by flexure and torsion is more difficult to carry

out and manual methods usually require simplifying

assumptions.

Because thin-walled structural members are com-

posed of very slender plates, they may fail by local or

distortional buckling of the cross-section. Local

buckling assumes that the line junctions between

intersecting plates remain straight, whereas distor-

sional buckling involves movement of the line

junction between intersecting plates without a rigid

body rotation or translation of the cross-section. The

finite strip method of buckling analysis is a very

efficient tool for investigating the buckling behaviour

of thin-walled members in bending and compression.

In this paper, a general matrix method for

analysing the section properties and stresses in thin-

walled cross-sections of any shape is presented. The

analysis calculates the section properties, warping

displacements, and the longitudinal and shear stresses

caused by flexure and torsion. Also presented is a

finite strip method of analysis which can be used to

study the local, distortional and flexural-torsional

modes of buckling of thin-walled structural members

under longitudinal stress.

The analysis is performed by the user-friendly

computer program THIN-WALL, which is also de-

scribed in the paper. This program gives the struc-

tural designer the ability to analyse thin-walled

sections of any shape quickly and efficiently. Included

in THIN-WALL is a user-friendly data processor for

the creation of data, and a post-processor which

shows the results graphically on the screen. An

example is used to illustrate the program.

2. CROSS SECTION GEOMETRY

Before a computer analysis of a thin-walled section

can be commenced, the geometry of the cross-section

must be described to the computer. The method of

computer data input is similar to the data for a plane

structural framework being analysed on a computer.

To achieve this similarity, a thin-walled cross-section

should be divided into an assemblage of rectangular

elements, with the ends of the elements intersecting at

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158

J. P. Papangelis and G. J. Hancock

Fig. 1. Pi-section

5

- Node numbers

nodes. For example, the subdivision of a section

(called a pi-section) is shown in Fig. I. It has been

subdivided into five elements and six nodes.

The coordinates of the nodes of the cross-section

based on an arbitrary axis system (X,

Y) are

input

with the identification number for each node. The

rectangular elements of the cross-section are input

together with the identification number of the con-

nected nodes. A number of different element types

can be defined, and the effective thicknesses and elastic

moduli in flexure and shear, and Poissons ratio for

each type are also input. Any elements which form a

closed loop must be included in the data for the

section. The element number in the loop is preceded by

a negative sign if the order of the node numbers which

define the ends of the element is opposite to that for

a clockwise traverse around a loop.

3. WARPING DISPLACEMENTS

3.1. Sectori al coordina te for open secti ons

When a thin-walled member is twisted, points on

the cross-section move parallel to the longitudinal

Z-axis. This phenomenon is called warping. The

(Shear Centre)

7

C (Centroid) x

Fig. 3. Shear flow along an element.

warping displacement w at a point in the cross-section

was derived by Vlasov [I] to be

d+

w = -Cq-

dz

(1)

where the term d+/dz is the angle of rotation 4 of the

cross-section per unit length commonly called the

twist. The sectorial coordinate a,, is the longitudinal

warping displacement resulting from unit negative

twist, and is defined by

an = a,,,, +

S

~0 ds

(2)

0

where the arbitrary constant a,,, is usually derived so

that the integral of the warping pattern over the

whole cross-section is zero:

Fig. 2. Normalized unit warping pattern for pi-section

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Computer analysis of

thin-walled structural members

159

s

,, dA = 0.

(3)

In addition, each element has an equation relating

A

the warping at each end to the twist of the section.

When this is achieved, the unit warping pattern is

This equation is derived from cqns (1) and (2) to be

d# L

ws -

WAC --

Pu ds.

(5)

sz o

termed normalized. The normalized warping pattern

for the pi-section shown in Fg. 1 is shown in Fig. 2.

The term p0 in eqn (2) is the perpendicular distance

from the shear centre to the tangent to the centreline

of the cross-section of an element dy located a

Since rhis equation can be written for each element,

distance s along the centreline of the cross-section, as

then there are five element compatibility equations

shown in Fig. 3. The shear centre is defined as the

for the pi-section

point in the plane of the cross-section through which

the shear force must act if no twisting of the section

is to take place.

w-n. --YF

2

I -

dt

3.2.

Computation of warping displacements of open

sections

The warping displacements at the ends of each

element are chosen as unknowns and hence the

number of simultaneous equations required to be

solved is twice the number of elements. For the

pi-section shown in Fig.

1.

there arc five clcments and

hence 10 warping displacements to be determined.

The warping displacements at the adjoining ends of

elements connected to the same node are equal,

therefore node compatibility equations can be written

J+,J

4

_MSjZ _?? F

dz

2

w

6

-Us= - F

da 3

(6)

)V*

w, =

d?

dz 4

to define this condition. Since there is atways one less

equation of warping compatibility at a node than the

number of elements, then the pi-section shown in

Fig. 4 has the following four node compatibility

u =_ ?F

1,O 19

dr7 5

equations

where

IL2 W)

w> =

I+-,

F/=

PO

ds

(7)

(4)

and L, is the length of element i.

Nine equations of warping compatibility have

been determined to solve for the unknown warping

Fig. 4. Warping of pi-section dissected into individual elements.

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160 J. P. Papangelis and G. J. Hancock

(i) Geometry

6.3 f -2.7

(ii) Warping Pattern (2 = 1)

(a) Open Cross-Section

-0.67

:

P

.67

--I

0.67

I

-0.67

(i) Geometry

(ii) Warping Pattern ($ = 1)

(b) Closed Cross-Section

Fig. 5. Warping of an open and a closed section.

displacements. The tenth equation results from a

rigid body translation of the whole cross-section in

the direction of the warping displacements. Any value

of translation can be used, however the value gener-

ally used produces a normalized warping pattern. A

normalized warping pattern is one for which the

warping stresses derived using this pattern produce

no net axial force when integrated over the section.

The normalized equation derived from eqn (3) is

given by

+/I,(

y)+/4~~7,%)

+A, (8)

where

A

s the area of element

i.

The 10 equations can be represented

notation as

[C,,.]{w} = -F {F)

z

in matrix

(9)

where {w} is the vector of warping displacements at

the ends of the elements, [C,,] is the matrix of

coefficients for warping displacements and {F} is a

vector of geometric constants. If the vector {w>

divided by the scalar

-d$ /dz is replaced by the

sectorial coordinate {c(,, , then

{%I =

[C,,

{Fi

(10)

where

iw> -g {F,).

(11)

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Computer analysis of thin-walled structural members

161

Equation (10) can be solved to determine the sectorial

coordinate at the ends of the elements.

3.3.

Computat ion of w arpi ng displacements of cl osed

sections

Closed thin-walled sections, although similar to

open thin-walled sections in their flexural behaviour,

are distinctly different in their torsional behaviour.

The difference results mainly from the restraint to

warping provided by the closure of loops in the

cross-section to form boxes. As an example, a rec-

tangular cross-section has been drawn in Fig. 5. In

Fig. S(a)(i) it can be seen that there is a slit along the

centre of one side of the section and in Fig. S(b)(i) a

closed box section is formed. When subjected to a

twist, the open section warps significantly, as shown

in Fig. S(a)(ii). The warping pattern is derived by

integration of eqns (1) and (2) around the section to

give

(0 p

(iii) My

(iii) T,

d4 .

w, = wo - -

s

z o

PO ds

(12)

where w, is the warping displacement at a distance s

along the centreline of the section and w, is the

warping displacement at s = 0. It can be seen that the

difference in the warping displacements at the adja-

cent edges of the slit is very large.

The closed section however, has such a differential

warping displacement prevented since the slit does

not exist. Shear straining therefore takes place in the

elements of the box to compensate for the geometric

warping. This shear straining produces a shear flow

(SF) around the box. Shear flow is a term used to

describe the shear force per unit length acting along

an element of a cross-section as shown in Fig. 3. If

the warping resulting from the shear straining is

included in eqn (12), the equation for warping

becomes

(a) Longitudinal Stresses

(ii) M,

(iv) B

(ii) Vy

(iv) Tw

(b) Shear Stresses

Fig. 6. Stresses in a channel section.

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162


in which G is the shear modulus. Applying eqn (13)

around the whole box so that M,< uaO ields

9 ds.

(14)

The area swept out by the radius vector

p,,

is equal

to twice the area of the loop, hence

(15)

where A, is the area of the loop of the box. Equation

(15) can be used to solve for the shear flow around

closed sections resulting from twist. This calculation

is described in Section 7. Once the shear flow has been

(9 p

(iii) M

(0 VI

determined, it is included in eqn (6) by modifying eqn

(7) as follows:

L,

SF)

s

L,

F, =

p,,

ds -

s

zds

d4

(16)

0

-

dz

Figure 5(b) (ii) shows the warping of the closed box

derived using eqns (13) and (15). It is clearly much

less than the warping of the open section and is

generally regarded as insignificant except for heavy

box beams. Consequently, restraint of warping in

closed sections normally produces very little increase

in torsional stiffness and so the bimoment and warp-

ing torque are generally taken as negligible for closed

sections. Sections with a combination of open and

closed parts may display significant warping in some

parts of the section. A recent example of this is the

hollow flange beam which is essentially an I-section

with tubular flanges.

la) Longitudinal Stresses

(ii) M

(iv) B

(ii)

Vv

(iii) Tu

Ib> Shear Stresses

Fig. 7. Stresses in a box section.

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164


the principal second moments of area I, and I

and the section moduli Z, and Z,. Figure 8 shows the

axis systems used for the calculations described

above.

By using these cross-section properties it is possible

to calculate the longitudinal stresses at any point

(x,y) in the cross-section resulting from an axial

force P applied through the centroid and bending

moments M, and MY applied about the principal axes

from

(19)

Since the coordinates of the nodes are known, the

stresses are generally calculated at these points.

5.2,

Bimoment

Vlasov [l] defined the bimoment

B

as the product

of the longitudinal warping stressf;,, with the sectorial

co-ordinate c(, integrated over the area of the cross-

section as given by

B = _(,,a, dA.

(20)

The name bimoment results from the pair of equal

and opposite flange moments caused by restrained

warping of an I-section. The longitudinal warping

stress at a point in the cross-section can be calculated

from

_Nodc I

f = _E,?.

I

dz2

(21)

Equation (21) is derived by multiplying the elastic

modulus

E

with the warping normal strain at a point

in the cross-section. The warping normal strain is

calculated by differentiation of the warping displace-

ment given by eqn (1) with respect to z. Substituting

eqn (21) into eqn (20) yields

B = -El,*

dz2

(22)

in which the warping section constant is given by

I,, =

s

: dA.

(23)

A

Combining eqns (21) and (22) yields

Equation (24) allows the calculation of the longitudi-

nal stress distribution in a cross-section in which a

bimoment acts. Equation (24) can be written in

matrix notation as

@

Node 5

Node 3

SF,,

(25)

Node 6

Fig. 9. Unit length of pi-section (dissected to show shear forces at nodes)

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165

where {f;,} is the vector of longitudinal warping

stresses, and {c(} is the vector of sectorial coordinates

obtained from eqn (IO). The longitudinal stresses

contained in {JV} can be added to the longitudinal

stresses calculated from eqn (19) to obtain the total

longitudinal stress at each node.

6. SHEAR STRESSES IN OPEN SECTIONS

6.1.

Shear low in an element of a cross-section

If a cross-section is subjected to shear forces V, and

yV and a warping torque T,,., a distribution of shear

flows is set up along the elements of the cross-section.

In order to determine the shear flow distribution

throughout the whole cross-section, it is first necess-

ary to determine the shear flow distribution within

each element. The shear flow is derived by integrating

the differential equation of longitudinal equilibrium

(26)

with respect to s along the element shown in Fig. 3.

The value off is the longitudinal stress at the point

s along the element and is given by

(27)

wheref;,., is the warping stress at end A of the element

and can be derived from eqn (25). The result of the

integration of eqn (26) along the element is

SF,- =,.,= KC,+ VyD,+T,,.E,

where

xt ds

D,= -f

s

yt

ds

II 0

5

E,=-; t

[S

s

L

p ds ds + tlnA

II 0

0

s 1

ds

0

Fig. 10. Closed pi-section.

(28)

(29)

Fig. 11.Strip subdivision of an I-section

6.2. Shear fiow in complete section

The shear flows can be determined for a given set

of applied shears and a torque at each end of each

element of the cross-section, as shown in Fig. 9.

This is analogous to determining the bending

moment at each end of a beam in a frame analysis.

It is sufficient to determine the shear flow at each

end of an element, however additional nodes can

be added if the shear flow is required at points

along elements. This increases the number of

elements. For the final number of elements chosen,

there are two unknown shear flows for each element

and hence the number of simultaneous equations

to be solved is twice the number of elements.

For example, the pi-section shown in Fig. 1 has

five elements and consequently 10 unknown shear

flows.

In order to set up the simuitaneous equa-

tions which are required to be solved for the

unknown shear flows, longitudinal equilibrium of

the

the

of

by

nodes and elements is considered. Since

pisection has six nodes, there are six equations

longitudinal equilibrium at the nodes given

-SF, =0

SF,-SF,-SF,=0

SF,-SF,-SF,=0

(30)

-SF,=0

-SF,=0

-SF,, = 0.

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X

Z

(a) In-Plane (Membrane) Displacements

Cubic

(b) Out-of-Plane (Flexural) Displacements

Fig. 12. Displacement fields of strip.

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167

Similarly there are five elements and hence five longitudinal equilibrium for the additional element.

element equilibrium equations given by

The final equation is not based on equilibrium but

uses eqn (15) for warping compatibility around the

-SF, + SF, = v,C, + V,.D, +

T,,.E,

loop as follows.

Firstly,

-SF,+SF,= v,C,+

l(,Dz+ T,,E,

resistance

loops can

the torsion constant JL for the torsional

caused by shear flow around the closed

be defined by

-SF5+SF,= v,C,+ V,vDx+T,,.E,

(31)

-SF, +

SF, =

V, C, + vv D, + T,,,E4

-SF, + SF,, = v,C, + v,.D, + T,

(32) Equation (34) relates the shear flow around a closed

loop to the Bredt-Batho component of total torque.

in which {SF} is the vector of shear flows at the ends

The shear flow at any point s along an element can

of the elements, [GP] is a matrix of constants C,,

D,,

be found from eqns (26) and (27) as

E, defined by eqn (29), [Cs,] is the matrix of co-

efficients of the shear flows in the linear equations,

and the vector {VT

}

contains

V,, I , T,..

SF=SF,-~~~~tds-~~~yrds

7. SHEAR STRESSES IN CLOSED SECTIONS

If the pi-section shown in Fig. 1 has an element

added between nodes 5 and 6 as shown in Fig. 10, the

section becomes a box-section in the form of a closed

T.

J

2

I,, 0 t 0

.I s

p0 ds ds + c(,~

s

1

ds

. (35)

0

pi-section. The addition of the extra element creates

two additional unknown shear flows and so two extra

Substituting eqn (35) into eqn (34) gives

equations are required to solve for these extra un-

knowns. Only one equation of longitudinal equi-

librium is available and this is the equation of

SF, L,

,$, +-=

V,G, + k,H, + T,K,

(36)

I

Fig. 13. Longitudinal stress distribution in a strip.

CAS 5611 L

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168

J. P.

Papangelis and G. J. Hancock

where n is the number of loop elements, and

xt

ds ds

yt d.9 ds

(37)

K,= -+.

L

The + or - sign depends on the direction of

integration with respect to s relative to a clockwise

rotation around loop i. Further, the term containing

the warping torque T,, in eqn (35) has been deleted

when deriving eqn (361, since the warping torque has

been taken as zero for a closed section, and only the

torque

TL

due to shear flow around the loops is

considered.

The complete set of equations to be solved can be

expressed in matrix format as

[G,I{W = [GJl{

T

1

(38)

where the vector {VT } now contains V,, V, and the

uniform torque which is the loop torque

TL

It can be

seen that three components of shear flow can be

determined from eqn (38). The shear flow component

due to the uniform torque can be substituted into eqn

(16) to solve for the warping displacements.

8. FINITE STRIP BUCKLING ANALYSIS

8.1. General

The finite strip method is a variant of the finite

element method. The finite strip method differs from

the finite element rnethod in the way a thin-walled

structure is subdivided for analysis, and the displace-

ment functions used to describe the deformations of

the elements or strips. In the finite strip method, the

thin-walled section, such as the I-section in Fig. 11,

is subdivided into longitudinal strips. The displace-

ment functions used to describe the variation in the

longitudinal direction are assumed to be harmonic

functions, whereas polynomial functions are used to

describe the variation in the transverse direction. The

finite strip method generally has an order of magni-

tude fewer equations to be solved for a given problem

than the finite element method.

The finite strip buckling analysis can be rep-

resented in matrix format by

[Kl{~) - Wl{D) =

0

(39)

where [K] and [G] are the stiffness and stability

matrices of the thin-walled structure and A is the load

factor against buckling under the initially assumed

applied stress used to assemble the matrix [Cl. The

values of /I for which the determinant of the co-

efficients of {D} in eqn (39) vanishes are called the

eigenvalues. The corresponding vaues of {D} are

called the eigenvectors, which are the buckling modes.

8.2.

StifSness matrix

Two assumptions are required in the derivation of

the stiffness matrix for the analysis of thin-walled

sections by the finite strip method. The first is that the

plates behave according to some plate theory. The

second assumption concerns the displacement fields

used in the analysis of a strip. The plate theory and

displacement fields used in this paper are the same as

those presented by Cheung [3].

The displacement field in the plane of a strip

(membrane displacements) is shown in Fig. 12(a).

Fig. 14. Z-Beam geometry and restraints.

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169

The deflected shape in the plane of the strip consists

of the component 6, in the x direction and the

component 6; in the z direction. The 6, component is

represented by a sine curve in the z direction and a

linear variation in the x direction. The 6: component

consists of a cosine curve in the z direction and a

linear variation in the x direction. The latter assump-

tion is equivalent to the engineering bending theory

assumption of plane strain, although shear straining

under a constant shear flow is included. The displace-

ment field normal to the plane of the strip is shown

in Fig. 12(b). The deflections S, normal to the plane

of the strip are assumed to be represented by a cubic

polynomial in the x direction and a sine curve in the

z direction.

The lines A, B, and

A, B2

of the strip shown in Fig.

12 are called the nodal lines. The vector {D} in eqn

(39) contains the displacements in the x, y and z

directions and the rotations 8, about the z-axis of the

nodal lines. The displacements in the Cartesian direc-

tions and the rotations about the z-axis are assumed

to be compatible between adjoining strips along these

lines. The lines

A, A,

and

B, B2

define the ends of a

strip. The displacements of the lines

A, A,

and

B, B2

are limited by assuming a Fourier series displacement

function in the longitudinal direction.

The mathematical formulation of the displacement

functions used in this paper is presented in Ref. [4].

The stiffness matrix for the finite strip analysis of a

thin-walled section can be found in Ref. [3]. Alterna-

tively, a derivation of the stiffness matrix for a strip,

using orthotropic plate theory and the displacement

fields assumed herein, can be found in Ref. [4].

8.3. Stability matrix

The strip is subjected to an initial stress g that acts

in the z direction and varies linearly from 6, on nodal

line A, B, to rr2 on nodal line A, B2, as shown

in Fig. 13. The potential energy U resulting from

the longitudinal in-plane forces can be calculated

using

(40)

in which V is the volume, and

u

=o,+(u,-u,) ;

0

1.6

201.4

X

(Dimensions in mm)

{a) Z-Section Purlin

(b) Computer Model

Fig. 15. Z-Section geometry.

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170 J. P. Papangelis and G. J. Hancock

where b is the width of the strip, and t is the nonlinear

component of the strain in the longitudinal direction

as derived by Novozhilov [5].

The method for the calculation of the stability

matrix from the potential energy given by eqn (40) is

set out in Ref. [3]. The stability matrix for flexural

displacements was first presented by Przemieniecki [6]

and the stability matrix for membrane displacements

was first developed by Plank and Wittrick [7]. The

matrices used in the analyses herein are based on the

same assumptions as those in Refs [6] and [7] and are

presented in Ref. [4].

8.4.

Solution

In program THIN-WALL, the eigenvalue prob-

lem is solved by using the direct eigenvalue rou-

tine set out in Ref. [8]. The method involves

computation of the Sturm sequence count to isolate

each particular eigenvalue by bisection. The eigenvec-

tor corresponding to each particular value of 1.

is computed using four iterations of inverse

iteration.

8.5.

Buckle half-wavelengths

The finite stiip buckling analysis does not dis-

tinguish between local, distortional or flexural-tor-

sional modes. However, for short wavelength

buckling modes such as local and distortional, where

multiple half-wavelengths occur within the length of

the thin-walled member, then the analysis applies to

one half-wavelength and the assumption of simply

supported ends is usually valid at the ends of the

half-wavelength being considered. Hence, the analy-

sis must normally be repeated over a range of lengths

corresponding to buckle half-wavelengths, especially

for short wavelength buckling modes such as local

and distortional.

9. PROGRAM THIN WALL

9.1.

Introduction

THIN-WALL [9] is a user friendly computer pro-

gram for calculating the section properties and

stresses in thin-walled cross-sections of general

geometry. The input data for the program includes

the geometry of the cross-section and the stress-resul-

tants acting at the cross-section. The program calcu-

lates the section properties, sectorial coordinate, and

the longitudinal and shear stresses for the cross-

section.

THIN-WALL can also perform an elastic finite

strip buckling analysis of thin-walled structures sub-

jected to longitudinal stress. The structure being

analysed may be a folded plate system, a stiffened

plate or a thin-walled structural member, but it must

be uniform in thickness in the longitudinal direction

TH I N-WALL - Crosssection analysis and finiie ship budding ;

STANDARD

SECTIONS

,

latysis

Fig. 16.

Standard sections menu

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171

. TtiIN WALL Cross-sectibn analysis and finite strip buckling analysis : 1

.L;

.ZED: .

L ,

_ 9ci

~,(..&

,_

.,:;:

B ,,ODESmHu,u

*. I

Z-SECTION IN BENDING

node

no

Y

restraints

(l-free, Q:fixed)

long

QX 6Y 62 02

stress

1

2

3

4

5

6

7

8

9

10

-77.4

-76.9585

-75.7E122

-73.81956

-71.6

-55.15

-38.7

-22.25881

-5.8

86.5 1 1 1 1

94.9

1 1 1 1

97.11957 1 1 1 1

99.00122

1 1 1 1

180.2585 1 1 1 1

100.7 1 1 1 1

10B .7 1 1 1 1

MB.7 1 1 1 1

188.7 1 1 1 1

180.7 1 1 1 1

and

simply supported at its ends. The longitudinal can be done for a number of different buckle half-

edges may be simply supported, clamped or free wavelengths of the structure and the load factor and

Fig. 17. Node data input.

along the full length of the plate system. The analysis

buckled shape are output for each length.

PRIMMY DATA

SECTION PROPERTIES

NODEDATA

ELEMENTDATA

ELEMENT ROPERTIES

A = 591.05

xc = -1.1715 CLOSEDLOOP DhTA

Yc 1.363 STRESS RESULTFlNTS

Ix = 3748780 2x = 36729

HALF-WAUELENGTHS

Iy = 657050

Zy = 8619.4 px = -17.586

py = -17.189 WC1RPNG

Ixy =

-11604ElO LONGSTRESS

xo = 2.2624 SHEAR FLOW

Ix = 41357m 2x = 34887 yo = 8.2955 BUCKLINGMODE

Iy = 269990

Zy = 5950.5 GRAPHICS

a 18.447

J = 504.37

Iw = 4.5157E+09

UNIUERSITY OF SYDNEY CENTREFOR ADVANCEDTRUCTURAL NGINEERING

Fig. 18. Section properties

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172

J. . apangelis and G. J. Hancock

ZED

TH IN-WALL - Crosssection analysis and finite strip buckling analysis

:

SECTION IN BENDING

I

SECTORIfiLOORDINATE

SECTION

NODES

ELENENTS

TYPES

CLOSED LOOPS

WARPING

LONG STRESS

SHEAR

FLOW

BUCKLED SHAPE

GRAPH

AXES

UNIUERSITY F SYDNEY CENTRE FOR FlDUANCED TRUCTURAL ENGINEERING

Fig. 19. Warping pattern.

9.2. ata input

section or for the modification of data for

an

existing section. Help screens

provide

In THIN-WALL a user-friendly data processor

advice concerning the nature of the data re-

can be used for the creation of data for a new

quired, and a facility exists for the automatic

THIN-WALL - Cross-section analysii and finite strip bucking anatysis

ZED

Z SECTION IN BENDING

I

1 SECTION

NODES

ELENENTS

TYPES

CLOSED LOOPS

WARPING

LONG STRESS

SHEAR FLOW

BUCKLED SHAPE

tlax tress = 274.05 at node 22

flin tress = 266.8 at node 10

UNIVERSITY F SYDNEY

CENTRE FOR ADUANCED STRUCTURFlLNGINEERING

Fig. 20. Longitudinal stress distribution

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173

TH I N-WALL - Cross-section atklysid&dinite strip btidditig analysis

;_

ED

~-~E~~~INBENDING

2:

I

I

TYPES

CLOSED LOOPS

WARPING

BUCKLED SHAPE

GRAPH

Load factor = 1.248 Mode = 1

Half-wauelength l M

AXES

UNIUERSITY F SYDNEY

CENTRE FOR ADVANCED TRUCTURAL NGINEERING

Fig. 21. Buckling mode for L = 100 mm.

generation of the node and element numbers for 9.3. Analysis

many standard sections. The data can easily THIN-WALL is based on the cross-section analy-

be checked by drawing the section on the

sis described in Sections 3, 5-7, and on the finite strip

screen. buckling analysis described in Section 8.

LONG STRESS

1 SHEAR FLOW

Load factor = -8769

Node = 1

Half-uauelength 660 AXES

UNIUERSITY F SYDNEY

CENTRE FOR ADUANCED STRUCTURRL NGINEERING


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174


9.4.

esults

THIN-WALL can display all the data and results

on the screen in tabular or graphic format. The data

and results can also be written to output files. The

graphics results include plots of sectorial coordinate,

longitudinal and shear stress distribution, and the

buckled shape. A plot of maximum buckling stress vs

buckle half-wavelength can also be shown.

10 EXAMPLE

10.1. Data

The beam shown in Fig. 14 was analysed to

determine the stress resultants at the central cross-

section. The span of the beam is 6 m with two 5 kN

loads acting down at the third points. Lateral re-

straints exist at the load points and these are located

at the centroid of the section. At the ends, lateral and

torsional restraint is provided, and the beam is simply

supported for bending in the plane of the web which

is the plane of loading. It is assumed that the load is

applied in the plane of the shear centre and hence no

torque is applied to the beam. In practice, a lateral

and torsional restraint should be applied at each load

point to prevent twisting of the section if the load

becomes eccentric from the shear centre.

The section of the beam is a Z-section whose

geometry is shown in Fig. 15(a). The section has been

sub-divided into 30 elements (strips), as shown in

Fig. 15(b). Each flat plate consists of four elements,

and the corner radii have been approximated by four

flat elements. The node coordinates were generated

by choosing the Z-section from the list of standard

sections shown in Fig. 16. Figure 17 shows the screen

for node data input for the Z-section with the

automatically generated node coordinates included.

The beam was analysed using a matrix displace-

ment analysis of frames composed of thin-walled

members described in Ref. [IO]. The analysis pro-

duced the following stress resultants

P=O

MY=

-10.5 x 1ONmm

M, =0.238 x 1ONmm

B=O

V,=

-119N

v> = -5240N

T,, = 0

T,v = 0.

10.2. Cross-section analysis

The section properties of the Z-section from the

cross-section analysis are shown in Fig. 18, where /3,

and /$ are monosymmetry parameters, and x0 and .yO

are the coordinates of the shear centre with respect to

TH I N-WALL - Crosssection anaiysii and linite ship bucfding analysis

s

ZED

Z-SECTION PJ BENDING

TYPES

I

CLOSED LOOPS

SHEAR

FLOW

BUCKLED SHfiPE

GRAPH

Load factor = .124 Node = 1

Half-uauelenyth = BOOI

UNIVERSITY OF SYDNEY

CENTRE FOR ADUANCEDSTRUCTURAL ENGINEERING


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175

tMx1nun

STRESS

IN

SECTION

AT

BUCKLING

1

-r-

:.

,..

.

I ,

: :

:

:

,...I...:__

i :

.i...i..

:

._..i._.i..

: :

: .:..

.i...:,,

:

.:

..I._

:

.i .:

: :

\

:

. .y...

.

: : ::

: : :.

:..:..:.I

: :

i ::

. . . . . .

: : ::

-ii

: : ::

..: .._

: : :.

: : :.

.;.: . . .

: :

_:

: ::..

5000

1

BUCKLE HALF-WAVELENGTH

Fig. 24. Maximum buckling stress vs buckle half-wavelength.

the

centroid,

as shown in Fig. 8. The sectorial coor-

dinate is plotted in Fig. 19, and the longitudinal

stresses computed are plotted in Fig. 20. The stress

distribution has a uniform stress in both the top and

bottom flanges and the beam simply deflects verti-

cally with no twisting or lateral deflection.

10.3.

Buckling analysis

procedure. Structural designers will often need help

in analysing these types of structures. A vehicle for

providing this help is computer programs developed

for the microcomputer. In this paper a computer

procedure and program for the cross-section analysis

and buckling analysis of thin-walled structural

members has been presented.

The buckling modes of the Z-section for various

buckle half-wavelengths are shown in Figs 21-23. The

local buckling mode in Fig. 21 consists of defor-

mation of the web, flange and lip elements without

movement of the line junctions between the flange

and web and the flange and lip stiffener. The buckling

mode in Fig. 22 is a distortional buckling mode since

movement of the line junction between the flange and

lip stiffener occurs without a rigid body rotation or

translation of the cross-section. At long wavelengths,

the beam buckles in a flexural-torsional mode, as

shown in Fig. 23.

A general matrix method for analysing the section

properties and stresses in thin-walled open and closed

cross-sections of any shape has been presented. The

analysis calculates the section properties, warping

displacements, and the longitudinal and shear stresses

caused by flexure and torsion. A finite strip method

of analysis which can be used to study the local,

distortional and flexural-torsional modes of buckling

of thin-walled structures under longitudinal stress

was also presented.

Figure 24 shows the variation of the maximum

buckling stress with the buckle half-wavelength for

the beam. A minimum occurs in the curve at half-

wavelengths of 100 and 600 mm. These minimum

points represent the local and distortional buckling

modes shown in Figs 21 and 22, respectively.

The cross-section analysis and finite strip buckling

analysis are performed by the user-friendly computer

program THIN-WALL. This program gives the

structural designer the ability to analyse thin-walled

sections of any shape quickly and efficiently. The user

can take advantage of a user-friendly data processor

for the creation of data, and a post-processor which

can show the results graphically on the screen. An

example of a Z-section in bending was used to

illustrate the program.

11. CONCLUSIONS

REFERENCES

The calculation of the stresses and failure modes

1. V. Z. Vlasov, Thin- Wall ed Elastic Beams. Israel

in thin-walled structural members is a complex

Program for Scientific Translations, Jerusalem (1961).

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176


5.

6.

C. F. Kollbrunner and K. Basler, Torsion in Str uctures,

2nd Edn. Springer, Berlin (1969).

Y. K. Cheung,

Finit e Stri p Method i n Structural Analy -

sis. Pergamon Press, New York (1976).

G. J. Hancock. Local, distortional and lateral

buckling

of I-Beams. Research Report R312,

School of Civil Engineering, University of Sydney

(1977).

?. Novozhilov, Foundati ons of the Non-l inear Theory

of Efasticitv.

Gravlock Press, Rochester, NY

(i953). . -

J. S. Przemieniecki, Finite element structural analysis of

local instability. Am. Inst. Aeronaut. Astr onauf. J. 11,

33-39 (1973).

7. R. J. Plank and W. H. Wittrick, Buckling under com-

bined loading of thin, flat-walled structures by a com-

plex finite strip method. In t. J. namer. M eth. Engng 8,

323-329 (1974).

8. G. J. Hancock, Structural buckling and vibration analy-

ses on microcomputers.

Ciu il Engng Trans. Inst. Engrs,

Aust ral ia CE26, 327-332 1974).

9.

THIN-WALL,

Cross-Secti on Ana ly sis and Fini t e Str ip

Buckli ng Anal ysis of Thi n- Wall ed Structures, Users

M anual. Centre for Advanced Structural Engineering,

University of Sydney (1994).

10. G. J. Hancock, Design of Cold- Formed Steel Str uctures,

2nd Edn. Australian Institute of Steel Construction,

Sydney ( 1994).

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