emech9_04 - 9 Stresses Beams in Bending The organization of...

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9 Stresses: Beams in Bending The organization of this chapter mimics that of the last chapter on torsion of cir- cular shafts but the story about stresses in beams is longer, covers more territory, and is a bit more complex. In torsion of a circular shaft, the action was all shear; contiguous cross sections sheared over one another in their rotation about the axis of the shaft. Here, the major stresses induced due to bending are normal stresses of tension and compression. But the state of stress within the beam includes shear stresses due to the shear force in addition to the major normal stresses due to bending although the former are generally of smaller order when compared to the latter. Still, in some contexts shear components of stress must be considered if failure is to be avoided. Our study of the deflections of a shaft in torsion produced a relationship between the applied torque and the angular rotation of one end of the shaft about its longitudinal axis relative to the other end of the shaft. This had the form of a stiffness equation for a linear spring, or truss member loaded in tension, i.e., M T = ( GJ L ) φ is like F = ( A E L δ Similarly, the rate of rotation of circular cross sections was a constant along the shaft just as the rate of displacement if you like, u , the extensional strain x was constant along the truss member loaded solely at its ends. We will construct a similar relationship between the moment and the radius of curvature of the beam in bending as a step along the path to fixing the normal stress distribution. We must go further if we wish to determine the transverse dis- placement and slope of the beam’s longitudinal axis. The deflected shape will gen- erally vary as we move along the axis of the beam, and how it varies will depend upon how the loading is distributed over the span Note that we could have consid- ered a torque per unit length distributed over the shaft in torsion and made our life more complex – the rate of rotation, the d φ / dz would then not be constant along the shaft. In subsequent chapters, we derive and solve a differential equation for the transverse displacement as a function of position along the beam. Our exploration of the behavior of beams will include a look at how they might buckle . Buckling is a mode of failure that can occur when member loads are well below the yield or fracture strength. Our prediction of critical buckling loads will again come from a study of the deflections of the beam, but now we must consider the possibility of relatively large deflections .
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236 Chapter 9 In this chapter we construct relations for the normal and shear stress compo- nents at any point within the the beam’s cross-section. To do so, to resolve the indeterminacy we confronted back in chapter 3, we must first consider the defor- mation of the beam.
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emech9_04 - 9 Stresses Beams in Bending The organization of...

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