Lecture #22

Reading: 8:2

 

 

Combined States of Stress  (2D)

 

We want to be able to find the state of stress for any point in a two dimensional problem.  Since all forces are in a plane, the state of stress will also be two dimensional.  The first step in this type of problem is to use the method of sections to cut the member to isolate the forces acting on a cross section containing the point of interest.  Then we draw a free body diagram of part of the body and use equilibrium to determine the internal forces acting at the centroid of the cross section.  Finally, we use our stress equations to find the stress that each force creates at the point of interest and superimpose them on an infinitesimal element representing the point.

 

We recognize that there are only 3 internal reaction forces in two dimensions—a normal force, a shear force and a bending moment.

 

Therefore, the stress equations of interest are

 

s = P / A   s = M y / I        and        t = (V Q)/(I t)

 

In general, the stress that comes from moments are larger than those that come from forces.  When looking at the maximum normal stress, we want the cross section with the largest moment.  Also, the maximum normal stress will often occur where the

two normal stresses ( P/A and My/I ) add together (have the same sign) unless the distances from the centroid to the extreme fibers in each direction are different.  Then the maximum stress could occur where the two normal stresses oppose one another if y is larger for that point.

 

The stresses act on the cross section so the state of stress for an arbitrary point A will consist of at most two stresses.

 

 

 

If looking for the maximum shear stress, we use the same approach as for finding the maximum shear stress from the shear force in a beam as that is the only shear stress that occurs.

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