Question

11.3-13 A horizontal beam AB has a port at end A and carries a load Qat end B, as shown in the figure part a. The beam is supported at C and D by two identical pinned-end columns of length L. Each column has flexural rigidity El. sliding sup- (a) Find an expression for the critical load Q (In other words, at what load Qr does the sys- tem collapse because of Euler buckling of the columns?) (b) Repeat part (a), but assume a pin support at A. Find an expression for the critical moment Mer (i.e., find the moment M at B at which the sys- tem collapses because of Euler buckling of the columns) A C D C D A 2d d 2d d d LL (b) (a) PROBLEM 11,3-13

Answer 1

Solution Part (a):

First find the reactions at C and D. So draw the free body diagram as shown in figure below,

A C D |RD d RD Kc L 777

Take equilibrium in vertical direction to the get following,

$R_C+R_D=Q\text{ \blank{ } ...(1) }$

Now take equilibrium of moments about point B,

$3dR_C+2dR_D=0\text{ \blank{ } ...(2) }$

Solve equations (1-2) to get, $R_C = -2Q,\blank{} R_D = 3Q$ .

So only column at D is under compression.

Since column is pin connected on both side, the Euler buckling load is given by,

$P_e=\frac{\pi^2EI}{L^2}$

To find the critical value of Q,

$R_D=P_e$

$\Rightarrow3Q=\frac{\pi^2EI}{L^2}$

$\Rightarrow Q=\frac{\pi^2EI}{3L^2}$

So the critical value of Q is

$Q_c=\frac{\pi^2EI}{3L^2}$  

Answer 2

give proper and fi=ull solution