Assume that if the shear stress in steel exceeds about 4.00 ×10^8 N / m^2 , the steel ruptures.

step 1 So for Part A, we know that the force needed to shear the bolt through its cross -sectional area

Assume that if the shear stress in steel exceeds about 4.00...

Assume that if the shear stress in steel exceeds about $4.00 \times 10^{8} \mathrm{N} / \mathrm{m}^{2}$ , the steel ruptures. Determine the shearing force necessary to (a) shear a steel bolt 1.00 $\mathrm{cm}$ in diameter and $(\mathrm{b})$ punch a 1.00 -cm-diameter hole in a steel plate 0.500 $\mathrm{cm}$ thick.

Key Concepts

Shear Stress

Shear stress is defined as the force per unit area acting parallel to the surface of a material. It measures the internal forces that arise when material layers slide relative to each other. Understanding shear stress is critical in mechanical and structural analyses to determine when a material will fail under applied loads.

Shear Force

Shear force is the external force that produces a shear stress within a material. It is related to shear stress by the relationship F = (shear stress) × (area over which the force acts). This relationship is essential for calculating how much force is needed to overcome the material's resistance before failure occurs.

Effective Shear Area

The effective shear area is the area over which the shear force is distributed. In engineering applications, this area must be carefully determined based on the geometry of the component. For example, when shearing a bolt, the area is the cross-sectional area of the bolt, while for punching holes, it is the product of the hole's perimeter and the plate thickness.

Geometric Relationships in Circular Sections

Geometric formulas, such as the area of a circle (A = ?d²/4) and the circumference (C = ?d), are used to calculate the effective shear area in circular components. These relationships are crucial when dealing with problems involving cylindrical objects or circular holes, as they allow for the accurate determination of the surface area resisting the applied shear force.

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