# Mechanics of Materials - Deflection of an L shaped beam

1. Nov 19, 2013

### chris627

1. The problem statement, all variables and given/known data

A. determine the deflection at point A in the X and Y direction.
B. determine the maximum normal stress in the beam
C. determine the maximum shear stress in the beam

2. Relevant equations
δ=(PL/EA)
(possibly?) δt=αΔτL
d2v/dx2 = M/EI
dM/dx = V
δabx*cosθ + δy*sinθ
σ=-My/I y- the distance from the neutral axis
τxy= VQ/IT

3. The attempt at a solution
I tried to split this problem into separate deflections.

First, the deflection due to the normal force:
δy= -(Pb)/(Ewd)

Next, the deflection due to the moment on the bar with width W:
Vx= (12Pab2)/(Edw3)

Finally, I related the deflection on the section with width t to the deflection on the bar with a width W:
My boundary conditions are as follows
dv/dx = -1/(dv/dxw) since they are perpendicular
V(0) = δy

Using these boundary conditions I found that when X=a.... this is gonna get messy.
Vy= 12/Edt3 [(-pa3/3)+(E2d2w2t2/144pb)-(pbt3/12w)]

My intuition tells me this is incorrect since there is an E2 term in there. I obtained the E2 term when I related the slopes of the t and w bars.

For V in the x direction...
Vx= (12pab2/2Edw2)

For parts B and C, I simply plugged in P for Vmax, and P*a for Mmax into the equations:
τxy= VQ/IT
σ=-My/I

2. Relevant equations
Are these the correct equations for deflection in the x and y? Do I need to account for the rotation of the beam, or is that already accounted for in these equations?

Thanks. Any help would be greatly appreciated.

Last edited: Nov 19, 2013
2. Nov 19, 2013

### PhanthomJay

Looks good
should be 6 not 12
seems like you should calculate the deflection as a simple cantilever, then add the deflection caused by the rotation at the corner joint
What's this?
where I = ?
this would be bending normal stress where? What about axial stress? I don't know how you can determine the max without knowing values for the givens
in calculating the deflection at the free end, you account for the deflection caused by rotation by calculating the rotation of the vertical piece at the corner joint, then geometrically find the deflection at the free end of the top piece which goes along for the ride so to speak.