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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.
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.)
Fig. 1. Von Misses truss.
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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
where L is the span.The total potential energy isThe total potential energy is
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
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
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
For the small values of angles and , we can assume
and finally we have a very simple equation
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 ?
whereEI is the bending stiffness of the girder,lcr is the buckling length.The equilibrium condition in the upper support of the girder gives
From a comparison of (Eqs. (5), (6) and (7)), we have
wherecr is the so-called critical angle, which means the angle when the
buckling of the girder occurs the local stability.
( ) tgEAF cossin =
( ) 0cos
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2 VON MISSES TRUSS WITH IMPERFECTION
Fig. 2. Simplification of the von Misses truss notation of thequantities.
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From Eq. (8) we get
is the slenderness,
is the radius of the inertia of the girder.
The validity of Eq. (9) gives
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
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3VON MISSES TRUSS WITH IMPERFECTION
Fig. 3. The snap-through of the von Misses truss - analytical solution.
Case B a circular movement
Case C shortening in the horizontal position
L0 = L/cos H = L tg
LL . tg2
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
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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
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:
the index 0 repres