Mechanics Of Solids

The circumferential (hoop) strain for a thin cylindrical shell subjected to internal pressure ( ) is derived from the relationship between stress, strain, and material properties. For thin-walled cylinders, the relationship includes Young's modulus ( ) and Poisson's ratio ( ), and the expression is:

Here:
: Young's Modulus,
: Poisson's ratio,
: Internal pressure,
: Thickness of the shell.
This formula accounts for both the direct and lateral strain effects due to the applied pressure.

The strain energy ( ) stored in a shaft subjected to torque is proportional to:

where:
: Applied torque,
: Shear modulus,
: Polar moment of inertia.
For identical torque and material:

The section modulus ( ) for a circular section is calculated using the formula:

where:
: Moment of inertia for a circular section,
: Distance from the neutral axis to the outermost fiber.
Substitute the values:

For a simply supported beam with a point load W at its mid-span, the reaction forces at the supports create no bending moment at the supports. The bending moment is maximum at the center of the beam, where the load is applied, and zero at the supports.
Key moments for the beam:
• Maximum bending moment at mid-span: W L/4,
• Bending moment at supports: 0.

The relationship between E, K, and ν is derived from elasticity theory:
E = 3K(1 − 2ν)
where:
• E: Young’s Modulus of Elasticity,
• K: Bulk’s Modulus,
• ν: Poisson’s ratio.
This equation shows how the material’s resistance to deformation under pressure (K) and its lateral strain (ν) relate to its elastic modulus.

The force required to punch a hole is calculated using the formula:

where:
(diameter of the hole),
(thickness of the plate),
(shear strength of the material).
Substitute the values:

Coulomb’s Theory is another name for the Maximum Shear Stress Theory, which states that failure occurs in a material when the maximum shear stress reaches a critical value. This theory is widely used in soil mechanics and failure analysis of ductile materials.