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5.7 Normal and Shear Stresses

When a beam is bent by transverse loads, usually both a bending moment M and a shear force V act on each cross section. The distribution of the normal stress associated with the bending moment is given by the flexure formula, Eq. (5.4):

where M and I are taken with respect to the z axis (Fig. 5.7).

In accordance with the assumptions of elementary bending, Eqs. (5.26) and (5.27), we omit the contribution of the shear strains to beam deformation in these calculations. However, shear stresses do exist, and the shearing forces are the resultant of the stresses. The shearing stress τxy acting at section mn, which is assumed to be uniformly distributed over the area b·dx, can be determined on the basis of equilibrium of forces acting on the shaded part of the beam element (Fig. 5.9). Here b is the width of the beam a distance y1 from the neutral axis, and dx is the length of the element. The distribution of normal stresses produced by M and M + dM is indicated in the figure. The normal force distributed over the left face mr on the shaded area A* is equal to

unequ261_232.jpg
FIGURE 5.9

Figure 5.9. (a) Beam segment for analyzing shear stress; (b) cross section of beam.

Similarly, an expression for the normal force on the right face ns may be written in terms of M + dM. The equilibrium of x-directed forces acting on the beam element is governed by

unequ261_233.jpg

from which we have

unequ261_234.jpg

After substituting in Eq. (5.29), we obtain the shear formula (also called the shear stress formula) for beams:

The integral represented by Q is the first moment of the shaded area A* with respect to the neutral axis z:

By definition, inline263_46.jpg is the distance from the neutral axis to the centroid of A*. In the case of sections of regular geometry, inline261_45.jpg provides a convenient means of calculating Q. The shear force acting across the width of the beam per unit length

is called the shear flow.

5.7.1 Rectangular Cross Section

In the case of a rectangular cross section of width b and depth 2h, the shear stress at y1 is

This equation shows that the shear stress varies parabolically with y1. It is zero when y1 = ±h, and has its maximum value at the neutral axis:

where A = 2bh is the area of the rectangular cross section. Note that the maximum shear stress (either horizontal or vertical: τxy = τyx) is 1.5 times larger than the average shear stress V/A. As observed in Section 5.4, for a thin rectangular beam, Eq. (5.42) is the exact distribution of shear stress. More generally, for wide rectangular sections and for other sections, Eq. (5.39) yields only approximate values of the shearing stress.

5.7.2 Various Cross Sections

Because the shear formula for beams is based on the flexure formula, the limitations of the bending formula apply when it is used. Problems involving various types of cross sections can be solved by following procedures identical to that for rectangular sections. Table 5.1 shows some typical cases. Observe that shear stress can always be expressed as a constant times the average shear stress (P/A), where the constant is a function of the cross-sectional form. Nevertheless, the maximum shear stress does not always occur at the neutral axis. For instance, in the case of a cross section having nonparallel sides, such as a triangular section, the maximum value of Q/b (and thus τxy) occurs at midheight, h/2, while the neutral axis is located at a distance h/3 from the base.

Table 5.1. Maximum Shearing Stress for Some Typical Beam Cross-Sectional Forms

Cross Section

Maximum Shearing Stress

Location

Rectangle

t0262_01.jpg

t0262_02.jpg

NA

Circle

t0262_03.jpg

t0262_04.jpg

NA

Hollow Circle

t0262_05.jpg

t0262_06.jpg

NA

Triangle

t0262_07.jpg

t0262_08.jpg

Halfway between top and bottom

Diamond

t0262_09.jpg

t0262_10.jpg

At h/8 above and below the NA

Notes: A, cross-sectional area; V, transverse shear force; NA, the neutral axis.

The following examples illustrate the application of the normal and shear stress formulas.

5.7.3 Beam of Constant Strength

When a beam is stressed to a constant permissible stress, σall throughout, then clearly the beam material is used to its greatest capacity. For a given material, such a design is of minimum weight. At any cross section, the required section modulus S is defined as

where M presents the bending moment on an arbitrary section. Tapered beams designed in this way are called beams of constant strength. Ultimately, the shear stress at those beam locations where the moment is small controls the design.

Examples of beams with uniform strength include leaf springs and certain cast machine elements. For a structural member, fabrication and design constraints make it impractical to produce a beam of constant stress. Hence, welded cover plates are often used for parts of prismatic beams where the moment is large—for instance, in a bridge girder. When the angle between the sides of a tapered beam is small, the flexure formula allows for little error. On the contrary, the results found by applying the shear stress formula may not be accurate enough for nonprismatic beams. Often, a modified form of this formula is used for design purposes. The exact distribution in a rectangular wedge is found by applying the theory of elasticity (Section 3.10).

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