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Feb 18, 2019

2003/2 PAGES 1 7 RECEIVED 21. 9. 2002 ACCEPTED 15.11. 2002

12003 SLOVAK UNIVERSITY OF TECHNOLOGY

PSOTN M. - RAVINGER, J.

VON MISSES TRUSS WITHIMPERFECTION

ABSTRACT KEY WORDS

The von Misses truss is nearly 100 years old. Even so, it is still a subject of interest toresearchers. This truss is one of the best examples to explain different theoreticalapproaches, define the snap-through effect, illustrate interactive buckling, etc. Thepresented paper compares two alternative analytical solutions and a couple of numericalapproaches. The peculiarities of the effects of the initial imperfections are investigated.

Stability buckling post-buckling geometric non-linear theory initial imperfection finite element method

Ing. MARTIN PSOTN

Lecturer at the Department of Structural MechanicsResearch field: Post-Buckling Behaviour of Slender Webs.

Prof. Ing. JN RAVINGER, DrScProfessor at the Department of Structural MechanicsResearch field: Dynamic Post-Buckling Behaviour of Thin-Walled Structures. Stability of Structures.

Address: Department of Structural Mechanics, Faculty of Civil Engineering, Slovak University of Technology,813 68 Bratislava, Radlinskho 11, SlovakiaE-mails: [email protected], [email protected]

INTRODUCTION

Von Misses was a very active researcher at the beginning of thetwentieth century. He published a wide variety of articles orientedtowards the theory of structures. In addition to the world-famousHMH (Huber Misses Hencky) criteria for the plasticity ofmaterial, he investigated buckling problems as well. Timoschenko(1936) in his book The Theory of Elastic Stability mentioned thework of von Misses many times. It is not easy to say who first calleda two-hinged connected bar the von Misses truss. (Fig. 1). At thepresent time, the name von Misses truss is commonly used.Baant and Cedolin (1991) and Bittnar and ejnoha (1992) havesummarized a wide variety of solutions to this type of truss. Theauthors of the presented paper used these books as an introduction.

Analytical solution of the von Misses truss

Fig. 1 shows the von Misses truss. Using the effect of symmetry, thevon Misses truss can be simplified as a slope beam (Fig. 2.)

L L

H

2F

Fig. 1. Von Misses truss.

psotny.qxd 11.8.2004 21:01 Strnka 1

We assume the linear elastic material and prismatic beam (E Young modulus, A area of cross section, I moment of inertia).The strain of the beam can be written as

, (1)

where L is the span.The total potential energy isThe total potential energy is

, (2)

whereV, s, - are the volume of the beam, the length coordinate, and the

initial angle of the slope, respectively,F, , - the applied force, the vertical displacement, the angle after

deformation, respectively.

The integration over the volume has been divided into integrationover the cross section area and integration over the length of thebeam. We are able to explain the total potential energy as oneparameter function angle and the conditional equation we getas the extreme (minimum) of the total potential energy; therefore,

,

This equation can be arranged in the form

. (3)

For the given value of angle , we can evaluate the value of theapplied load. One hundred years ago there were neither computers nor calculators,and so it was appropriate to simplify Eq. (3) in the following way

. (4)

For the small values of angles and , we can assume

,

and finally we have a very simple equation

. (5)

Fig. 3 shows the results arranged in the form of the load versus thedisplacement for the initial angle of . To evaluate thevertical displacement, we can use the equation

.

This result shows the famous snap-through effect. According to an evaluation of a couple of examples, we can say thatthe differences between the results of Eqs. (3) and (5) are negligiblewhen the initial angle is .

Until now only the in-plane stiffness of girder has entered intothe solution. What happens if the normal force in the girder is higherthan the Euler elastic critical load ?

, (6)

whereEI is the bending stiffness of the girder,lcr is the buckling length.The equilibrium condition in the upper support of the girder gives

. (7)

From a comparison of (Eqs. (5), (6) and (7)), we have

, (8)

wherecr is the so-called critical angle, which means the angle when the

buckling of the girder occurs the local stability.

( )crcrcrcr

tgEAl

EI cossinsin22

=

FN =sin

2

2

crcr l

EIN

=

)(EA

10

tgtgL

=

= 10

( ) tgEAF cossin =

11coscos

321 =

&

01coscos

321

1coscos

sin =+

FEA

01coscos

4coscos

32sin

2

2

=+

+ FEA

( ) 0cos

sin1coscos

cossincoscos

1coscos

2cos2

2

2

22

=+

+

+

FL

EAL

0=U

( )

tgtgFLEAL

FdsEAFdVU

L

V

=

=== 2

cos

0

2

1coscos

cos21

21

21

1coscos

cos

coscos =

=

L

LL

2003/2 PAGES 1 7

2 VON MISSES TRUSS WITH IMPERFECTION

H

F

E,A,I

Fig. 2. Simplification of the von Misses truss notation of thequantities.

L

psotny.qxd 11.8.2004 21:01 Strnka 2

From Eq. (8) we get

,(9)

where

is the slenderness,

is the radius of the inertia of the girder.

The validity of Eq. (9) gives

. (10)

We have defined * as the critical slenderness of the girder.For < * the effect of the local stability does not occur. The behaviorof the girder is 0 A, snap into B. Unloading path is B C andsnap into D (as shown in Fig. 3).For the presented example , we have

. The example for is presented

there. For this case we have

.

The behavior of the girder is 0 E, snap into F. Unloading path isF G and snap into H. Point E represents the bifurcation point (asillustrated in the example in Fig. 3).

Elastic critical load for the von Misses trussThe elastic critical load results from the linearised stability problem.In the case of the von Misses truss, the elastic critical load movesthe slope girder into a horizontal position.We have many possibilities for the solution of this problem.Case A the rectangular movement

, ,

In the table 1 we can compare the results according to the threevariants presented

3sinEAFcrA =tgL

EAFL

H ==cossin

12

cossin1

sin 20

EAFL==

EALF

EALS 1

cossin0

0 ==

sinF

S =

3

2

210*5916.0,0248.0,617.8

501

10coscos ===

=EAF

Larccr

50=49.2510cos1

* =

=

)10( =

*

cos1

=

AI

i =

il=

=

2

2

1

coscos

arccr

2003/2 PAGES 1 7

3VON MISSES TRUSS WITH IMPERFECTION

Fig. 3. The snap-through of the von Misses truss - analytical solution.

( )

cos1sin

cos0

=

=

EAF

LL

crB

sin

sin

2

2

tgEAF

tgLEALF

crC =

=

Case B a circular movement

Case C shortening in the horizontal position

L0 = L/cos H = L tg

L

L

LL . tg2

F

F

F

S

0

0

0

Angle The elastic critical load of the von Misses truss [multiplier EA*10-3]

Case A Case B Case C5 0.662 0.312 0.671

10 5.236 2.638 5.39915 40.01 20.63 45.31

Table 1

psotny.qxd 11.8.2004 21:02 Strnka 3

Significant differences between case B and cases A and C have beenobtained. The elastic critical load is unrealistically high compared tothe top of the curve represented in the nonlinear solution and thesnap-through effect (Fig. 3).

Numerical solution of the geometrically nonlinear problems

For the solution of the von Misses truss, the geometrically nonlineartheory must be used. In the analytical solution presented in theprevious part of this article, the geometrically nonlinear solutionwas satisfied by evaluating the strain (Eq. (1)) and the total potentialenergy (Eq. (2)) on the deformed shape of the truss. If we want toarrange the general solution (for example, the finite elementmethod), we must take into consideration the nonlinear terms in theevaluation of the strain (Fig. 4).

Fig. 4. Part of the girder., (11)

whereu is a function of the displacement in the direction of the axis of

the girder in-plane displacements,w is the function of the displacement perpendicular to the axis of

the girder the bending displacements, z is the thickness coordinate, and

the indexes denote the derivations.

The bending displacements are much bigger than the in-planedisplacements ; we can therefore ignore the in-planenonlinear term (underlined in Eq. (11)), and the strain is reduced to

(12)

In the case of the von Misses girder, in-plane displacements can playa crucial role. The effect of the in-plane nonlinear term is a partialproblem investigated in the presented article.

We assume the linear elastic material and the stresses are:

, (13)

the index 0 repres

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