iMechanica - a strange question about the Euler beam buckling? - Comments

Hi

Pu Zhang — Sun, 07 Dec 2008 19:44:10 -0500

The post-buckling analysis should be used to determine the deformation of the beam after the buckling.

Buckling Effect

siarheiarlou — Fri, 05 Dec 2008 18:42:04 -0500

Yes, and Timoshenko, Gere book (Mechanics of Materials): Buckling Theory Chapter.

This case is example of non linear relations between external force and deflections.

To understand it you may load a rod by force with eccentricity to longitudinal axis. And

to take into account bending like Force * eccentricity.

Re: a strange question about the Euler beam buckling

Zhigang Suo — Thu, 04 Dec 2008 21:54:33 -0500

The buckling analysis is carried out using a linearized equation, leading to an eigenvalue problem. The eigenvalue problem does not determine the amplitude. To determine the amplitude, you need to go back to the nonlinear equation. See p. 38-11, vol. II, the Feynman Lectures on Physics.

a strange question about the Euler beam buckling?

chunhui yang — Thu, 04 Dec 2008 18:28:02 -0500

I have a beam whose two ends are roller condition which means one end is zero moment and zero displacement and the other end is zero moment but can have the displacement only along the beam axis. Then the compression force is applied on this end along the beam axis.

>From the Euler formula it is y=Asin(n*3.14x/L). （n=1,2,3,4,5,......, L is the beam length) (y is the transverse displacemenr and x is the coordinate along the beam axis direction) But this A will never be determined from the boundary condition. Do you think this is strange? If I have a beam I want to calculate this beam's shape and I only know the elastic modulus and poisson ratio and the above boundary conditions ( two ends are roller condition which means one end is zero moment and zero displacement and the other end is zero moment) may I get the deformed solution of this beam? I think I cannot due to this strange constant A.

The below is the chinese translation of this problem:

大家都知道在1774年欧拉推导了著名的两端铰支细长压杆临界力的计算公式，通常成为欧拉公式。两端铰支就是说一端的水平和竖直位移都为零，而且在这端的弯矩也为零，另一端的水平位移和弯矩都为零。这时我在没有完全固定的那端施加一个挤压力，使这个梁发生侧向屈曲，好了通过边界条件欧拉得到了一个公式y=Asin(n*3.14x/L). 问题是如何确定这个A? 我就想知道在屈曲发生之前梁的具体形状。而且我认为从物理上来说边界条件已经给的够充分了应该可以做实验量出这个梁的此时的形状，为什么理论上就不行呢？