The uniform brass bar A B has a rectangular cross section...

The uniform brass bar $A B$ has a rectangular cross section and is supported by pins and brackets as shown. Each end of the bar can rotate freely about a horizontal axis through the pin, but rotation about a vertical axis is prevented by the brackets. (a) Determine the ratio $b / d$ for which the factor of safety is the same about the horizontal and vertical axes. (b) Determine the factor of safety if $P=1.8$ kips, $L=7 \mathrm{ft}, d=1.5$ in., and $E=29 \times 10^{6} \mathrm{psi}$.

The uniform brass bar $A B$ has a rectangular cross section and is supported by pins and brackets as shown. Each end of the bar can rotate freely about a horizontal axis through the pin, but rotation about a vertical axis is prevented by the brackets. (a) Determine the ratio $b / d$ for which the factor of safety is the same about the horizontal and vertical axes.
(b) Determine the factor of safety if $P=1.8$ kips, $L=7 \mathrm{ft}, d=1.5$ in., and $E=29 \times 10^{6} \mathrm{psi}$.

Key Concepts

Orthogonal Bending (Bending about Different Axes)

When a beam can bend about two perpendicular axes (as in this case with a rectangular cross section), the bending behavior is different in each direction because of the difference in moments of inertia. Analyzing bending about both the horizontal and vertical axes and equating the associated factors of safety is necessary to achieve a design where neither mode of bending is disproportionately critical. This analysis can yield design ratios (such as b/d) that optimize the cross-sectional dimensions for balanced performance.

Factor of Safety

The factor of safety is a design parameter that provides a margin of safety by comparing the material’s strength (or yield moment) to the expected or applied stress (or moment). In bending problems, it is typically calculated as the ratio of the yield moment to the induced bending moment. Ensuring that the factor of safety meets design requirements is critical in preventing structural failure.

Moment of Inertia for Rectangular Cross Sections

For a rectangular cross section, the moment of inertia is a measure of the section’s resistance to bending about a given axis. The moment of inertia differs depending on whether the beam is bending about its horizontal axis (typically I = bd^3/12) or its vertical axis (typically I = b^3d/12). This difference is crucial when analyzing beams that may bend in more than one direction, as it directly affects the bending stress and, consequently, the factor of safety.

Bending Stress and Beam Theory

This concept involves understanding how bending moments induce stresses in a beam. When a beam is subjected to a bending moment, the fibers at the extreme distances from the neutral axis experience the greatest tensile or compressive stresses. The bending stress can be calculated using the relationship ? = M*y/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the cross?section about the axis of bending.

Section Modulus

The section modulus, defined as S = I/c (where c is the distance from the neutral axis to the outermost fiber), is a geometric property of a cross-section that directly relates the bending moment to the bending stress. A larger section modulus indicates that the beam can withstand a greater bending moment for a given level of stress. In design calculations, comparing the required section modulus to reach a certain safety factor is essential.

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