Mechanics of Materials - Deflection of an L shaped beam

[chris627]

Homework Statement

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

Homework Equations

δ=(PL/EA)
(possibly?) δt=αΔτL
d²v/dx² = M/EI
dM/dx = V
δ_ab=δ_x*cosθ + δ_y*sinθ
σ=-My/I y- the distance from the neutral axis
τxy= VQ/IT

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/dx_w) since they are perpendicular
V(0) = δy

Using these boundary conditions I found that when X=a... this is going to get messy.
V_y= 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...
V_x= (12pab²/2Edw²)


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

Homework 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.

 

[PhanthomJay]

Homework Statement

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

Homework Equations

δ=(PL/EA)
(possibly?) δt=αΔτL
d²v/dx² = M/EI
dM/dx = V
δ_ab=δ_x*cosθ + δ_y*sinθ
σ=-My/I y- the distance from the neutral axis
τxy= VQ/IT

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)

Looks good

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

should be 6 not 12

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/dx_w) since they are perpendicular
V(0) = δy

Using these boundary conditions I found that when X=a... this is going to get messy.
V_y= 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.

seems like you should calculate the deflection as a simple cantilever, then add the deflection caused by the rotation at the corner joint

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

What's this?

For parts B and C, I simply plugged in P for V_max, and P*a for M_max into the equations:
τ_xy= VQ/IT

where I = ?

σ=-My/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

Homework 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.

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.