The simply supported beam is made of timber that has an...

The simply supported beam is made of timber that has an allowable bending stress of $\sigma_{\text {allow }}=1.20 \mathrm{ksi}$ and an allowable shear stress of $\tau_{\text {allow }}=100$ psi. Determine its smallest dimensions to the nearest $\frac{1}{8}$ in. if it is rectangular and has a height-to-width ratio of 1.5.

Key Concepts

Shear Stress

Shear stress in a beam arises due to the transverse load applied to it and is calculated by dividing the shear force by the cross-sectional area, usually adjusted for the distribution of stress in the beam's profile. In rectangular cross-sections, this stress is highest near the supports and can govern the design when the material’s permissible shear stress is reached before the allowable bending stress.

Simply Supported Beam

A simply supported beam is a basic structural element that is supported at its two ends without any moment restraint at the supports. This configuration leads to a bending moment that is highest at the mid-span and zero at the supports, and it is a common element in structural analysis and design, particularly in applications such as bridges and floor systems.

Bending Stress

Bending stress is generated in a beam when it is subjected to lateral loads that cause it to bend. This stress is typically calculated using the bending equation (? = 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. The allowable bending stress limits help ensure that the material does not fail in tension or compression within the beam.

Rectangular Cross-Section Properties

A rectangular cross-section is commonly used in beam design because of its simplicity and predictable structural behavior. Key properties include the moment of inertia, which affects the bending resistance, and the section modulus, which relates directly to the bending stress. These geometrical properties are crucial in determining the beam’s capacity to resist bending and shear forces.

Allowable Stress

Allowable stress is the maximum stress that a material is permitted to experience under service conditions without risking failure. In structural engineering, these limits are established for different types of loading, such as bending and shear, ensuring that the design remains within safe performance parameters under expected loads.

Aspect Ratio in Cross-Section Design

The aspect ratio in cross-section design refers to the ratio between different dimensions of the section, such as height to width. When a specific ratio is prescribed, it imposes a geometric constraint that affects the overall design and performance, influencing both the structural efficiency and the material distribution within the beam.

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