Lecture #18

Reading: 6:5

 

Unsymmetric Bending

 

When we derived the equation for bending stress, we assumed that the cross section was symmetric, i.e. I_xy = 0.  Actually, s = - M y / I  can be applied to any cross section with a bending moment as long as the moment is around a principal axis and I is a principal moment of inertia, i.e. I_xy = 0.

 

We could consider two different situations.

 

1. What if the cross section is symmetric, so we know where the principal axes are, but the moment vector is at an angle to the principal coordinate system?

 

Consider a rectangular cross section with a moment vector at an angle q from the principal x axis in the plane of the cross section.

 

If we break the moment vector into components about the x and y axes, we can apply the bending stress equation to each moment and use superposition to add the stresses together.

 

In the equation s = M y / I , we must recognize that each letter is a placeholder or a dummy variable.  The moment must be about a principal axis, the moment of inertia must be with respect to the same axis as the moment, and y represents a distance measured perpendicular to the axis the moment is around.

 

Consider the stress due to M_x.  The equation used will be s = M_x y / I_x .  There must be a sign placed on the stress based on whether the stress is tensile or compressive.  If we want, we can write a general equation where we allow y to be positive or negative based on the coordinate of the point where we want to calculate the stress. 

 

If you consider a point where y is positive, note that the moment M_x causes tension based on the given direction of the moment.  So in general,

 

             s = + M_x y / I_x .

 

 

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