If our point of interest is on the boundary of the cross section, the state of stress will be two dimensional.  The most difficult part of drawing the two dimensional state of stress is making sure your element orientation is understood.  You will note that I always look at the element from the outside of the member and letter the element both on the member and the state of stress.

 

When considering a two dimensional state of stress for a point in a three dimensional structure, if the point of interest is on one of the principal axes of the cross section, you will note that one of the shear forces and one of the bending moments will give a zero stress at the point.  The force and moment associated with the axis that the point is on will give zero stress and can be ignored. 

 

The the cross section to the left, the shear stress from V_x at A is zero because the point is at the top/bottom of the cross section with respect to the direction of the force so Q will be zero in 

(VQ)/(It).

 

The normal stress from M_x is zero at A because the point A is on the centroidal axis that the moment goes around so the perpendicular distance y will be zero in (M y)/I.

 

The state of stress for point A can be seen below with the equations used to calculate each.

 

Again, if this structure were a pressure vessel, then the hoop and longitudinal stresses from the internal stresses must be calculated and superimposed over the state of stress from the externally applied forces.  In this case, there will be normal stresses in both directions on the element.  Otherwise, we only have one normal stress.

Lecture #23, page 2                 Back