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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.
AugustinLouis 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 SaintVenant 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. SaintVenantâ€™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 SaintVenantâ€™s semiinverse method.
Ludwig Prandtl (18751953) is best known for his work in aerodynamics, but the German physicistâ€™s interests ranged widely
in engineering design. He originated boundarylayer 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 thinwalled
membrane. He found that these are similar to torsion equations derived using SaintVenantâ€™s semiinverse method. Today,
Prandtlâ€™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 freebody
diagram. In the second, we draw a torque diagram. In statically indeterminate singleaxis shafts, the internal torque expression
contains an unknown reaction torque, which has to be determined using the compatibility equation. For singleaxis 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.
 Spring '10
 SMITH
 Torsion

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