A short length of a W 8 ×31 rolled-steel shape supports a...

A short length of a $\mathrm{W} 8 \times 31$ rolled-steel shape supports a rigid plate on which two loads $\mathbf{P}$ and $\mathbf{Q}$ are applied as shown. The strains at two points $A$ and $B$ on the centerline of the outer faces of the flanges have been measured and found to be \[\varepsilon_{A}=-550 \times 10^{-6} \text {in./in. } \quad \varepsilon_{B}=-680 \times 10^{-6} \text {in./in. }\] Knowing that $E=29 \times 10^{6}$ psi, determine the magnitude of each load.

A short length of a $\mathrm{W} 8 \times 31$ rolled-steel shape supports a rigid plate on which two loads $\mathbf{P}$ and $\mathbf{Q}$ are applied as shown. The strains at two points $A$ and $B$ on the centerline of the outer faces of the flanges have been measured and found to be
\[\varepsilon_{A}=-550 \times 10^{-6} \text {in./in. } \quad \varepsilon_{B}=-680 \times 10^{-6} \text {in./in. }\]
Knowing that $E=29 \times 10^{6}$ psi, determine the magnitude of each load.

Key Concepts

Section Properties

Section properties, such as the moment of inertia and the distance from the neutral axis (c), characterize the geometric resistance of a beam to bending. These properties are essential in the flexure formula and play a key role in linking the applied bending moments with the resulting strains, allowing for the design and analysis of structural elements.

Strain Distribution in Beams

Under bending, the strain distribution across the cross-section of a beam is linear, with zero strain at the neutral axis and maximum compressive and tensile strains at the extreme fibers. This linear variation is used to calculate bending moments from strain measurements, enabling the determination of applied loads on the beam.

Hooke’s Law

Hooke’s law states that, within the elastic limit, the stress in a material is directly proportional to the strain, expressed as ? = E·?. This relationship provides a means to relate measured strains to corresponding stresses, and thereby connect experimental data with theoretical predictions in beam theory and structural analysis.

Flexure Formula

The flexure formula, ? = M·c/I (and its strain version ? = M·c/(I·E)), relates the bending stress (or strain) at a distance c from the neutral axis to the bending moment M, the moment of inertia I, and the elastic modulus E of the material. This formula is crucial for determining how internal stresses and strains vary across the sectional height of a beam when subjected to bending loads.

Beam Bending Theory

Beam bending theory describes how beams deform under loads, resulting in curvature that creates tensile and compressive strains on opposite sides of the neutral axis. This theory is central in structural analysis for predicting the response of beam?like elements to applied forces and moments, and it establishes the relationship between bending moments, beam curvature, and the resulting strain distribution.

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