Lecture #23

Reading:  8:2

 

 

Combined States of Stress  (3D)

 

Again, we want to be able to determine the state of stress at a point, but now for a three dimensional problem.  The procedure is basically the same as for the two dimensional problem in that we must first use the method of sections to isolate a cross section that contains the point of interest.  Then we draw a free body diagram of part of the member.  However, now we have six possible reactions acting at the centroid of the cross section ( a normal force, two shear forces, a torque and two bending moments).  We use equilibrium to determine the reactions then apply the stress equations to find the stresses at the point of interest.

 

Our stress equations are

 

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

 

t = T r / J   and        t = T / ( 2 A_m t )

 

If we have a pressure vessel, we will also need

 

s_H = (p_i r_i)/ t          and        s_L = (p_i r_i)/(2 t)

 

Then we draw the stresses on an infinitesimal element representing the point.

 

 

For a general point on the cross section, the state of stress is three dimensional and will include three stresses as shown below for point A.

 

 

We are really most interested in the point that has the most critical state of stress.  Remember that the largest stresses come from the moments.  Therefore, it makes sense that the most critical point will be on the boundary of the member since the maximum stress from all moments are as far as possible from the centroid.

A

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