BS02-11

Section 2-11 Shear Stresses in Bending

When the shear force in a beam is not zero, a shear stress is developed of magnitude*


     tau = (V/(I*b))*INTEGRAL(y1,c,y*dA)         [2-33]

This stress occurs at distance y1 from the neutral surface as shown in Figure [2-16]. In this equation b is the width of the surface at y1, V is the shear force over the entire section, and I is the moment of inertia of the section about the neutral axis.

* For the complete development, see Joseph E. Shigley, "Applied Mechanics of Materials," pp. 149-158, McGraw-Hill Book Company, New York, 1976.

The integral, in equation [2-33], is the first moment of the area of the vertical face between y1 and c about the neutral axis. This moment is usually designated as Q. Thus


     Q = INTEGRAL(y1,c,y*dA)                     [2-34]

With this simplification, equation [2-33] may be written as


     tau = V * Q / (I * b)                       [2-35]

Equation [2-33] shows that the shear stress is maximum when y1 = 0 and zero when y1 = c for a rectangular section. Thus the shear stress is zero at the top and bottom of the beam, where the bending stress is maximum, and the shear stress is maximum at the neutral axis, where the bending stress (normal stress) is zero. Since horizontal shear stress is always accompanied by vertical shear stress, the distribution can be diagrammed as shown in Figure [2-16a].

Figure [2-16b] shows the shear-stress distribution for a rectangular section. Substitution of I for such a section in equation [2-33], with y1 = 0, gives the maximum


    tau_max = 3 * V / (2 * A)            [2-36]

For a solid circular beam, equation [2-33] gives


    tau_max = 4 * V / (3 * A)            [2-37]

and for a hollow circular section,


    tau_max = 2 * V / A                  [2-38]

A good approximation for structural W and S shapes is given by*


    tau_max = V / Aw                     [2-39]

where Aw is the area of the web only.

* The symbols W, S, and C are used in the structural industry to designate hot-rolled shapes. The symbol W designates a wide-flange beam, S an eye-beam, and C a channel section.

In the case of certain nonsymmetrical cross sections, the plane of the bending moment must pass through the shear center if twisting of the beam is to be avoided (Figure [2-17]). This problem usually arises when thin open sections are used for beams, and under these conditions local failure or buckling should also be investigated. Methods of locating the shear center are beyond the scope of this book.*

* See Crandall, Dahl, and Lardner, op. cit., p. 470

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Mechanical Engineering Design Section 2-11 Shear Stresses in Bending