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Load and Stress Analysis 93
(a) For the cross section shown determine the maximum tensile and compressive
bending stresses and where they act.
(b) If the cross section was a solid circular rod of diameter, d = 1.25 in, determine
the magnitude of the maximum bending stress.

Solution (a) The reactions at O and the bending-moment diagrams in the xy and xz planes are
shown in Figs. 3–16b and c, respectively. The maximum moments in both planes occur
at O where
1 2
(M z ) O =− (50)8 =−1600 lbf-in (M y ) O = 100(8) = 800 lbf-in
2
The second moments of area in both planes are
1 3 4 1 3 4
I z = (0.75)1.5 = 0.2109 in I y = (1.5)0.75 = 0.05273 in
12 12
The maximum tensile stress occurs at point A, shown in Fig. 3–16a, where the maxi-
mum tensile stress is due to both moments. At A, y A = 0.75 in and z A = 0.375 in. Thus,
from Eq. (3–27)

−1600(0.75) 800(0.375)
Answer (σ x ) A =− + = 11 380 psi = 11.38 kpsi
0.2109 0.05273
The maximum compressive bending stress occurs at point B where, y B =−0.75 in and
z B =−0.375 in. Thus

−1600(−0.75) 800(−0.375)
Answer (σ x ) B =− + =−11 380 psi =−11.38 kpsi
0.2109 0.05273
(b) For a solid circular cross section of diameter, d = 1.25 in, the maximum bending
stress at end O is given by Eq. (3–28) as

32
Answer σ m = 800 + (−1600) 2 1/2 = 9329 psi = 9.329 kpsi
2
π(1.25) 3

Beams with Asymmetrical Sections 4
The bending stress equations, given by Eqs. (3–24) and (3–27), can also be applied to
beams having asymmetrical cross sections, provided the planes of bending coincide
with the area principal axes of the section. The method for determining the orientation
of the area principal axes and the values of the corresponding principal second-area
moments can be found in any statics book. If a section has an axis of symmetry, that
axis and its perpendicular axis are the area principal axes.
For example, consider a beam in bending, using an equal leg angle as shown in
Table A–6. Equation (3–27) cannot be used if the bending moments are resolved about
axis 1–1 and/or axis 2–2. However, Eq. (3–27) can be used if the moments are resolved

4 For further discussion, see Sec. 5.3, Richard G. Budynas, Advanced Strength and Applied Stress Analysis,
2nd ed.,McGraw-Hill, New York, 1999.