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Stress Equations for Vertical vs. Horizontal Differential "Slices" of Beams

Stress Equations for Vertical vs. Horizontal Differential "Slices" of Beams

Stress Equations for Vertical vs. Horizontal Differential "Slices" of Beams

(OP)
I am a Structural Engineering master's student and last semester I took a course in Advanced Strength of Materials/Intro to Elasticity. I'm still on winter break and have been reviewing some of my material. We started the class by reviewing the derivations of equations from Strength of Materials such as beam bending and shear equations. On our first assignment we were asked to explain the derivations of these equations. The beam bending and shear equations are derived by taking a vertical differential slice of a beam and examining the resulting deformation, stress, and applied forces and relating them. I did this part fairly easily.

My professor also asked us to consider what would happen if we were to take a horizontal slice of the beam and do the same process. I am particularly having trouble with a pure bending moment. The Strength of Materials solution results in no stress or deformation except in the longitudinal direction of the beam (epsilon_xx and sigma_xx are the only non-zero strains and stresses). The implication was that by slicing the beam horizontally there would appear other stresses and strains - particularly normal stress and strain in the y-direction. This all assumes that the x-direction is along the length of the beam, y = 0 defines the neutral surface, and z is the axis of bending. I know that if I look at the beam end (the cross-section) I can determine a strain and then a stress in the z-direction. Does there exist a normal stress and strain in the y-direction? Any help with this or the similar shear problem would be greatly appreciated.

I also want to emphasize that this class is over but unfortunately the professor never released solutions. I am asking this because I'm curious and want to learn.

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