Now repeat this process for M_y to find

 

             s = - M_y x / I_y .

 

Using superposition, we write

 

             s = + M_x y / I_x - M_y x / I_y .

 

Now, we have an equation which is good for any point in the cross section.  We only need to substitute the appropriate coordinate values of the point where we want to find the stress.

 

This general equation allows for the determination of the neutral axis, the line of zero stress in the cross section.  By setting our general stress equation equal to zero, we get the equation of a line.

 

             0 = + M_x y / I_x - M_y x / I_y         or          y = (( M_y I_x )/( M_x I_y )) x

 

We can plot the line or determine the angle it makes with the coordinate axes.

 

We can also see where the maximum stress occurs from the general equation by finding the point farthest away from the neutral axis.

 

In this case, the maximum occurs at points A and B where the stress at A is tensile and the stress at B is compressive.

 

If we are only interested in finding the stress at one point, we do not need to develop a general expression.  We can instead take all placeholders in the equation 

s = M y / I  to be positive and place a sign on the stress depending on whether it is tensile or compressive at the point.  Again, the sign of the stress can be determined from the direction of the moment and the location of the point relative to the axis the moment goes around.

 

 

For example, the stress at B could be written as

 

             s = - M_x (h/2) / I_x - M_y (b/2) / I_y

 

The difference is that we did not write a general equation and take the x and y values to the be coordinates of the point of interest.

 

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