A timber beam A B of length L and rectangular cross section...

A timber beam $A B$ of length $L$ and rectangular cross section carries a uniformly distributed load $w$ and is supported as shown. (a) Show that the ratio $\tau_{m} / \sigma_{m}$ of the maximum values of the shearing and normal stresses in the beam is equal to $2 h / L$, where $h$ and $L$ are, respectively, the depth and the length of the beam. (b) Determine the depth $h$ and the width $b$ of the beam, knowing that $L=5 \mathrm{m}, w=8 \mathrm{kN} / \mathrm{m}, \tau_{m}=1.08 \mathrm{MPa},$ and $\sigma_{m}=12 \mathrm{MPa}$

A timber beam $A B$ of length $L$ and rectangular cross section carries a uniformly distributed load $w$ and is supported as shown.
(a) Show that the ratio $\tau_{m} / \sigma_{m}$ of the maximum values of the shearing and normal stresses in the beam is equal to $2 h / L$, where $h$ and $L$ are, respectively, the depth and the length of the beam.
(b) Determine the depth $h$ and the width $b$ of the beam, knowing that $L=5 \mathrm{m}, w=8 \mathrm{kN} / \mathrm{m}, \tau_{m}=1.08 \mathrm{MPa},$ and $\sigma_{m}=12 \mathrm{MPa}$

Key Concepts

Uniformly Distributed Load

A uniformly distributed load (UDL) is a load applied evenly across the entire length of the beam, producing continuous and predictable bending moments and shear forces. Under a UDL, the maximum bending moment and shear force can be derived from equilibrium conditions and are used in calculating the bending and shear stresses. This load type is commonly encountered in structural design because it simplifies the analysis of beam behavior under load.

Bending Stress in Beams

Bending stress is a normal stress induced in a beam when it is subjected to a bending moment. According to beam theory, this stress varies linearly with the distance from the neutral axis of the cross-section and is given by the formula ? = 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. In the context of a rectangular beam, the maximum bending stress occurs at the extreme fibers (at a distance h/2 from the neutral axis).

Shear Stress in Beams

Shear stress in a beam results from the internal shear force produced by external loading. In a rectangular cross-section, the shear stress does not distribute uniformly but reaches a maximum at the neutral axis, often described by the relation ? = (3/2) V/(b?h), where V is the shear force, and b?h is the area of the cross-section. Understanding this distribution is crucial when comparing shear stresses with bending stresses in beam design.

Sectional Properties of Rectangular Cross Sections

The geometric properties of a beam’s cross section, such as its moment of inertia (I) and area (A), are key to determining both bending and shear stresses. For a rectangular section, the moment of inertia is given by I = b?h³/12 and the area is A = b?h. These properties influence how the beam resists bending and shear, directly impacting the maximum stresses experienced under given loads.

Stress Ratio Analysis in Beams

The comparison between maximum shear stress and maximum bending (normal) stress in beams can provide insights into the efficiency and safety of a beam’s design. By deriving expressions for both stresses and their ratio, engineers can understand how geometric parameters, such as the depth and length of the beam, affect the stress distribution. This type of analysis is particularly useful in optimizing cross-sectional dimensions to meet specific design criteria.

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