It was assumed in Sec. 4.1 B that the normal stresses σ, in a member in pure bending are negligi

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It was assumed in Sec. 4.1 B that the normal stresses σ, in...

It was assumed in Sec. $4.1 \mathrm{B}$ that the normal stresses $\sigma$, in a member in pure bending are negligible. For an initially straight elastic member of rectangular cross section, (a) derive an approximate expression thus, that $a_{y}$ can be neglected in all practical situations, (Hint: Consider the free-body diagram of the portion of beam located below the surface of ordinate $y$ and assume that the distribution of the stress $\sigma_{i}$ is still linear:)

Key Concepts

Negligibility of Secondary Stress Components

In the context of pure bending, certain stress components, such as additional normal stresses (often represented by parameters like a_y), are assumed to be negligible because their magnitudes are very small compared to the primary bending stresses. This simplification is justified by comparing these stresses through approximate analytical expressions derived from equilibrium considerations and is widely accepted in practice when the member’s deformations remain within the elastic range.

Neutral Axis and Plane Sections Remain Plane

The concept of the neutral axis is central to bending theory; it is the line within the cross-section where the normal stress is zero and around which the linear distribution of stress rotates. The assumption that plane sections remain plane after deformation implies that the strain, and hence the stress, varies linearly with the distance from this neutral axis. This fundamental idea is used to justify the neglect of transverse normal stress components in practical applications.

Free-Body Diagram Analysis

Analyzing a portion of the beam via a free?body diagram is an essential method for understanding the equilibrium of forces and moments. By isolating a section of the beam (for instance, the part below a certain ordinate), one can apply the principles of static equilibrium to derive expressions for internal stresses and moments. This approach is pivotal to validating the approximations used in bending theory and to quantifying terms that may be shown to be negligible.

Pure Bending Theory

Pure bending refers to the condition when a beam or structural member is subjected solely to bending moments without any accompanying shear forces. Under this assumption, the bending moment remains constant along the length of a segment, leading to a simplification in stress analysis where only the normal (axial) stresses due to bending need to be considered. Such an analysis forms the basis for classical beam theory and is foundational in understanding bending behavior in elastic members.

Linear Stress Distribution

In classical beam theory, it is assumed that plane sections remain plane and perpendicular to the neutral axis during bending. This assumption yields a linear variation in normal stress across the beam’s depth, with the maximum tensile and compressive stresses located at the outer fibers. The linear stress distribution simplifies the integration of stresses to determine overall moments and is critical in deriving approximate expressions for the stress parameters.

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