Strength Of Materials

Step 1: Understanding fatigue.
Fatigue is the failure of a material due to repeated or fluctuating stresses, often at stress levels much lower than the ultimate tensile strength.
Step 2: Definition of endurance limit.
The endurance limit is the maximum stress level below which a material can endure an infinite number of load cycles without fatigue failure.
Step 3: Eliminating incorrect options.
Elastic limit indicates the limit of elastic behavior, not fatigue resistance.
Young’s modulus measures stiffness, not fatigue strength.
Ultimate tensile strength represents maximum static strength, not cyclic performance.
Step 4: Conclusion.
Hence, the resistance of a material to fatigue is measured by its endurance limit.

Step 1: Understanding torsion in a circular shaft.
When a circular shaft is subjected to torsion, shear stress develops due to twisting.
Step 2: Variation of shear stress.
The shear stress at a distance from the center is given by: where is torque and is polar moment of inertia.
Step 3: Evaluating stress at centre and surface.
At the centre ( ): At the circumference ( ):
Step 4: Conclusion.
The shear stress varies from zero at the centre to maximum at the circumference.

Step 1: Writing Euler’s crippling load formula.
According to Euler’s theory, the critical (crippling) load for a column is given by: where is the effective length of the column.
Step 2: Identifying effective length.
For a column with one end fixed and the other end hinged, the effective length is:
Step 3: Substituting effective length.

Step 4: Conclusion.
The crippling load for the column is .

Step 1: Understanding strain energy.
Strain energy is the energy stored in a body due to deformation caused by an applied load.
Step 2: Defining proof resilience.
Proof resilience is defined as the maximum strain energy that a body can store without undergoing permanent deformation, i.e., within the elastic limit.
Step 3: Eliminating incorrect options.
Impact energy refers to energy due to sudden loading.
Proof stress is the stress corresponding to a specified permanent strain.
Modulus of resilience is the proof resilience per unit volume.
Step 4: Conclusion.
Therefore, the strain energy stored without permanent distortion is called proof resilience.

Step 1: Understanding the loading condition.
A cantilever beam is fixed at one end and free at the other. The uniformly distributed load acts over the entire length of the beam.
Step 2: Calculating the total load.
The total load acting on the beam is:
Step 3: Shear force at the fixed end.
The fixed end must balance the entire applied load. Hence, the shear force at the fixed end is equal to the total load:
Step 4: Conclusion.
The shear force at the fixed end of the cantilever beam is .

Step 1: Understanding the loading condition.
A simply supported beam carries a concentrated load at its mid-span. Due to symmetry, the reactions at both supports are equal.
Step 2: Calculating support reactions.
Each support reaction is:
Step 3: Finding maximum bending moment.
Maximum bending moment occurs at the centre of the beam:
Step 4: Conclusion.
The maximum bending moment is .

Step 1: Understanding Mohr’s circle.
Mohr’s circle is a graphical representation of the state of stress at a point. The principal stresses and lie at the extreme ends of the diameter of Mohr’s circle.
Step 2: Determining the center of Mohr’s circle.
The center of Mohr’s circle lies at the average of the principal stresses:
Step 3: Calculating the radius.
The radius of Mohr’s circle is half the difference of the principal stresses:
Step 4: Conclusion.
Hence, the radius of Mohr’s circle is .

Step 1: Understanding the beam condition.
The beam is fixed at both ends and carries a uniformly distributed load over its entire length. Fixed supports restrict both rotation and translation.
Step 2: Using standard deflection formula.
For a fixed beam under uniformly distributed load, the maximum deflection occurs at the centre of the beam.
Step 3: Writing the formula.
The maximum deflection is given by:
Step 4: Conclusion.
The maximum deflection of the beam is .