A W 200 ×31.3 ralled-steel beam is subjected to a couple M of moment 45 kN ·m. Knowing that E=20

step 1 For A, we have M sub Y equal to 240 times 10 to the power of 6, 3 .645 times 10 to the power of

A W 200 ×31.3 ralled-steel beam is subjected to a couple M...

A $\mathrm{W} 200 \times 31.3$ ralled-steel beam is subjected to a couple $\mathrm{M}$ of moment $45 \mathrm{kN} \cdot \mathrm{m}$. Knowing that $E=200 \mathrm{GPa}$ and $v=0.29$, deter mine $(a)$ the radius of curvature $\rho$ (b) the radius of curvature $\rho^{\prime}$ of a transverse cross section.

A $\mathrm{W} 200 \times 31.3$ ralled-steel beam is subjected to a couple $\mathrm{M}$ of moment $45 \mathrm{kN} \cdot \mathrm{m}$. Knowing that $E=200 \mathrm{GPa}$ and $v=0.29$, deter mine
$(a)$ the radius of curvature $\rho$
(b) the radius of curvature $\rho^{\prime}$ of a transverse cross section.

Key Concepts

Euler?Bernoulli Beam Theory

This theory forms the foundation of classical beam bending analysis. It assumes that plane cross?sections before bending remain plane and perpendicular to the neutral axis after bending. This assumption leads to the linear strain distribution through the depth of the beam, simplifying the relation between bending moment and curvature and allowing for the derivation of key equations in beam bending problems.

Moment–Curvature Relationship

This concept connects the applied bending moment in a beam to its curvature. According to this relationship, the curvature (reciprocal of the radius of curvature) is equal to the bending moment divided by the flexural rigidity (the product of the modulus of elasticity and the second moment of area). It is fundamental for determining how much a beam will bend under a given moment.

Flexural Rigidity

Flexural rigidity, represented by the product E*I (where E is the modulus of elasticity and I is the second moment of area), measures a beam’s resistance to bending. It is critical in the moment–curvature relationship; a higher flexural rigidity implies a beam that is stiffer and less prone to bending under the same moment.

Radius of Curvature

The radius of curvature quantifies the degree of bending of a beam. It is the reciprocal of the beam’s curvature. A large radius corresponds to a slight bend, while a small radius indicates a sharp bending. This concept is used to describe both longitudinal and transverse bending characteristics in a beam.

Anticlastic Curvature (Transverse Curvature due to Poisson’s Effect)

When a beam is bent, the effects of Poisson’s ratio can cause the cross-sectional plane to twist or curve in the transverse direction, a phenomenon known as anticlastic curvature. This secondary curvature arises because material contracts or expands laterally when stretched or compressed, modifying the overall deformation of the beam beyond the primary longitudinal bending.

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