Question 6.6: A channel section (C 10 × 15.3) is subjected to a bending mo...
A channel section (C 10 × 15.3) is subjected to a bending moment M = 15 k-in. oriented at an angle θ = 10° to the z axis (Fig. 6-23).
Calculate the bending stresses \sigma_{A} and \sigma_{B} at points A and B, respectively, and determine the position of the neutral axis.
Properties of the cross section. The centroid C is located on the axis of symmetry (the z axis) at a distance
c = 0.634 in.
from the back of the channel (Fig. 6-24).* The y and z axes are principal centroidal axes with moments of inertia
I_{y}=2.28 in.^{4} I_{z}=67.4 in.^{4}Also, the coordinates of points A and B are as follows:
y_{A} = 5.00 in. z_{A} = -2.600 in. + 0.634 in. = -1.966 in.
y_{B} = -5.00 in. z_{B} = 0.634 in.
Bending moments. The bending moments about the y and z axes (Fig. 6-24) are
M_{y} = M sin θ = (15 k-in.)(sin 10°) = 2.605 k-in.
M_{z} = M cos θ = (15 k-in.)(cos 10°) = 14.77 k-in.
Bending stresses. We now calculate the stress at point A from Eq. (6-38):
\sigma_{x}=\frac{M_{y}z}{I_{y}}-\frac{M_{z}y}{I_{z}}=\frac{(M\sin \theta)z}{I_{y}}-\frac{(M\cos\theta)y}{I_{z}} (6-38)
\sigma_{A}=\frac{M_{y}z_{A}}{}-\frac{M_{z}y_{A}}{I_{z}}
=\frac{(2.605 k-in.)(-1.966 in.)}{2.28 in.^{4}}-\frac{(14.77 k-in.)(5.00 in.)}{}
= -2246 psi – 1096 psi = -3340 psi
By a similar calculation we obtain the stress at point B:
\sigma_{B}=\frac{M_{y}z_{B}}{}-\frac{M_{z}y_{B}}{I_{z}}=\frac{(2.605 k-in.)(0.634 in.)}{}-\frac{(14.77 k-in.)(-5.00 in.)}{67.4 in.^{4}}
= 724 psi + 1096 psi = 1820 psi
These stresses are the maximum compressive and tensile stresses in the beam.
Neutral axis. The angle β that locates the neutral axis (Eq. 6-40) is found as follows:
\tan\beta=\frac{y}{z}=\frac{I_{z}}{I_{y}}\tan\theta (6-40)
\tan\beta=\frac{I_{z}}{I_{y}}\tan\theta=\frac{67.4 in.^{4}}{2.28 in.^{4}}\tan10^{\circ}=5.212 β = 79.1°
The neutral axis nn is shown in Fig. 6-24, and we see that points A and B are located at the farthest distances from the neutral axis, thus confirming that \sigma_{A} and \sigma_{B} are the largest stresses in the beam.
In this example, the angle β between the z axis and the neutral axis is much larger than the angle θ (Fig. 6-24) because the ratio I_{z} /I_{y} is large. The angle β varies from 0 to 79.1° as the angle θ varies from 0 to 10° . As discussed previously in Example 6-5 of Section 6.4, beams with large I_{z} /I_{y} ratios are very sensitive to the direction of loading. Thus, beams of this kind should be provided with lateral support to prevent excessive lateral deflections.
*See Table E-3, Appendix E, for dimensions and properties of channel sections.
