We saw that the calculation of stresses and relative

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Unformatted text preview: s torsion formula analytically, starting with our own Assumptions 1 and 3 (see page 215), that is, cross sections remain planes and radial lines remain straight during small twists to circular bars. He also established that these assumptions are not valid for noncircular shafts. Augustin-Louis Cauchy, whose contributions to the mechanics of materials we have encountered in several chapters, was also interested in the torsion of rectangular bars. Cauchy showed that the cross section of a rectangular bar does not remain a plane. Rather, it warps owing to torsional loads. Jean Claude Saint-Venant proposed in 1855 the displacement behavior we encountered in Problem 5.53. Building on the observations of Coulomb, Duleau, and Cauchy, he developed torsion formulas for a variety of shapes. Saint-Venant’s assumed a displacement function that incorporates some features based on experience and empirical information but containing sufficient unknown parameters to satisfy equations of elasticity, an approach now called Saint-Venant’s semiinverse method. Ludwig Prandtl (1875-1953) is best known for his work in aerodynamics, but the German physicist’s interests ranged widely in engineering design. He originated boundary-layer theory in fluid mechanics. He also invented the wind tunnel and its use in airplane design. In 1903 Prandtl was studying the differential equations that describe the equilibrium of a soap film, a thin-walled membrane. He found that these are similar to torsion equations derived using Saint-Venant’s semi-inverse method. Today, Prandtl’s membrane analogy is used to obtain torsional rigidities for complex cross sections simply from experiments on soap films. Handbooks list torsional rigidities for variety of shapes, many of which were obtained from membrane analogy. We once more see that theory is the outcome of a serendipitous combination of experimental and analytical thinking. Printed from: http://www.me.mtu.edu/~mavable/MoM2nd.htm 5.6 CHAPTER CONNECTOR In this chapter we established formulas for torsional deformation and stresses in circular shafts. We saw that the calculation of stresses and relative deformation requires the calculation of the internal torque at a section. For statically determinate shafts, the internal torque can be calculated in either of two ways. In the first, we make an imaginary cut and draw an appropriate free-body diagram. In the second, we draw a torque diagram. In statically indeterminate single-axis shafts, the internal torque expression contains an unknown reaction torque, which has to be determined using the compatibility equation. For single-axis shafts, the relative rotation of a section at the right wall with respect to the rotation at the left wall is zero. This result is the compatibility equation. We also saw that torsional shear stress should be drawn on a stress element. This approach will be important in studying stress and strain transformation in future chapters. In Chapter 8, on stress transformation, we will first find torsional shear stress using the stress formula from this chapter. We then find stresses on inclined planes, including planes with maximum normal stress. In Chapter 9, on strain transformation, we will find the torsional shear strain and then consider strains in different coordinate systems, including coordinate systems in which the normal str...
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This note was uploaded on 04/08/2010 for the course ENGR 232 taught by Professor Smith during the Spring '10 term at Aarhus Universitet.

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