Timothy A. Philpot - Mechanics of materials _ an integrated learning system-John Wiley (2017)

15.3 principal Stresses in a Flexural member
Procedures for locating the critical sections of a beam (i.e., the maximum internal shear
force V and the maximum bending moment M) were presented in Chapter 7. Methods
for calculating the bending stress at any point in a beam were presented in Sections 8.3
and 8.4. Methods for determining the horizontal and transverse shear stresses at any
point in a beam were presented in Sections 9.5 through 9.7. However, the discussion of
stresses in beams is incomplete without a consideration of the principal and maximum
shear stresses that occur at the locations of maximum shear force and maximum bending
moment.
The normal stress caused by flexure is largest on either the top or bottom surfaces of
a beam, but the horizontal and transverse shear stress is zero at these locations. Consequently,
the tensile and compressive normal stresses on the top and bottom surfaces of
the beam are also principal stresses, and the corresponding maximum shear stress is
equal to one-half of the bending stress [i.e., τ max = (s p − 0)/2]. On the neutral surface, the
normal stress due to bending is zero; however, the largest horizontal and transverse shear
stresses usually occur at the neutral surface. In this instance, the principal and maximum
shear stresses are both equal to the horizontal shear stress. At points between these
extremes, one might well wonder whether there are combinations of normal and shear
stresses that create principal stresses larger than those at the extremes. Unfortunately, the
magnitude of the principal stresses throughout a cross section cannot be expressed as a
simple function of position for all sections; however, contemporary analytical software
often provides insight into the distribution of principal stresses by means of color-coded
stress contour plots.
Rectangular cross Sections
For beams with a rectangular cross section, the largest principal stress is usually the maximum
bending stress, which occurs on the top and bottom surfaces of the beam. The maximum
shear stress usually occurs at the same location and has a magnitude equal to one-half
of the bending stress. Although it may be of lesser intensity, the horizontal shear stress
(calculated from τ = VQ/It) at the neutral surface may also be a significant factor, particularly
for materials having a horizontal plane of weakness, such as a typical timber beam.
Flanged cross Sections
If the beam cross section is a flanged shape, then principal stresses at the junction between
flange and web must also be investigated. When a beam with a flanged cross section is
subjected to a combination of large V and large M, the bending and transverse shear stresses
that occur at the junction of the flange and the web sometimes produce principal stresses
that are greater than the maximum bending stress at the outermost surface of the flange. In
general, at any point in a beam, a combination of large V, M, Q, and y, together with a small
t, should suggest a check of the principal stresses at such a point. Otherwise, the maximum
bending stress will very likely be the principal stress, and the maximum in-plane shear
stress, will probably occur at the same point.
Stress Trajectories
Knowledge of the directions of the principal stresses also may aid in the prediction of
the direction of cracks in a brittle material (e.g., concrete) and thus may also aid in the
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