Constant stress lines in a two-dimensional problem

[pedrok]

1. Problem statement
In a two-dimensional problem, σx=300xy and σy=300xy and τxy=0 at x=y=0. Determine lines of constant shear stress τxy in the x-y plane and plot them for τxy=100, 300, 400.

Homework Equations

dσx/dx + dτxy/dy + X = 0 [1]
dτxy/dx + dσy/dy + Y = 0 [2]

At least I think these are the ones I need.

The Attempt at a Solution

First off, when τxy is constant both dτxy/dx and dτxy/dy should be 0.
Further more, to simplify the problem I assume that the body forces X and Y are 0.

Then : dσx/dx = 300y and dσy/dy = 300x which can then be filled into equation [1] and [2] and I can then calculate for which values of x and y both terms end up as 0.

However if I do this, I (offcourse) end up with the equation y=x which seems a little simple to me and from this result I have no idea to plot lineS for the values of τxy as given in the assignment.

So actually, I think I'm either approaching this problem with a completely wrong method or I'm missing a part of the problem.

Any help or hints will be very much appreciated!

 

[Chestermiller]

I think the intention here was to solve for τ_xy under the assumption that X = Y =0. You need to use both the stress equilibrium equations to do this.

 

[pedrok]

I think I've solved it now thanks to your tip Chestermiller!

I shouldn't have analyzed the derivatives but just find an equation by integrating both stress equilibrium equations and then combining them, then set this equation equal to the given shear stress values and determine all values of x and y for which this is true.

In which I have succeeded in doing.

Thanks so much!