An A-36 steel strip has an allowable bending stress of 165 MPa. If it is rolled up, determine th

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An A-36 steel strip has an allowable bending stress of 165...

An $A-36$ steel strip has an allowable bending stress of $165 \mathrm{MPa}$. If it is rolled up, determine the smallest radius $r$ of the spool if the strip has a width of $10 \mathrm{mm}$ and a thickness of $1.5 \mathrm{mm} .$ Also, find the corresponding maximum internal moment developed in the strip.

Key Concepts

Bending Stress

Bending stress is the internal stress induced in a material when an external moment or bending force is applied, causing it to bend. It is typically calculated using the formula ? = My/I, where M is the bending moment, y is the distance from the neutral axis to the point of interest, and I is the moment of inertia. This concept is crucial in ensuring that the induced stresses do not exceed the material’s allowable limits, thereby preventing structural failure.

Moment of Inertia

The moment of inertia is a geometric property that quantifies how a cross-sectional area of a beam or strip resists bending. For a rectangular section, it is calculated as I = (b*h^3)/12, where b is the width and h is the thickness. This property plays a central role in determining the bending stress distribution across the section, as it directly influences the stiffness of the cross-section.

Section Modulus

The section modulus is a derived property that represents the strength of a cross-section in bending. It is defined as S = I/y, where I is the moment of inertia and y is the distance from the neutral axis to the outermost fiber of the section. The section modulus provides a measure of the maximum bending moment a member can resist before yielding, linking material strength to geometric properties.

Allowable Bending Stress

The allowable bending stress is the maximum stress that a material is designed to withstand under bending loads, as specified in design codes or material standards. It incorporates safety factors to ensure that the structure remains within safe limits during service. Recognizing and adhering to this value is essential for preventing material failure and ensuring the reliability of the structure.

Bending Moment

The bending moment is the internal moment that develops within a structural element as a result of external forces or deformations. It creates a linear stress distribution through the depth of the element, with maximum compressive and tensile stresses occurring at the outermost fibers. Understanding how the bending moment interacts with the section modulus and bending stress is key to designing components that are both efficient and safe.

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