Strength Of Materials

Step 1: Identify the support reactions.
The beam is simply supported at points A and B, and a concentrated load of 50 kN acts at point C.
Step 2: Calculate reactions using equilibrium equations.
Taking moments and vertical force equilibrium, the reaction at support A is found to be 37.5 kN, while the reaction at B is 12.5 kN.
Step 3: Determine shear force at point C.
Shear force at point C just to the left of the applied load is equal to the reaction at A.

Step 4: Conclusion.
Therefore, the shear developed at point C is 37.5 kN.

Step 1: Compare tension and compression behavior.
A member in tension fails mainly by yielding (or rupture), so its capacity depends mostly on material strength and area.
A member in compression may fail either by crushing/yielding (short column) or by buckling (long column).
Step 2: Key idea for “same load in compression and tension”.
If a column can take the same load in compression as in tension, it means the column is not losing capacity due to buckling.
That happens when the column is short (small slenderness ratio), so failure is by crushing/yielding similar to tension capacity (area-based).
Step 3: Analyze options.
(A) Intermediate column: Buckling effects start becoming important, so compression capacity reduces.
(B) Long column: Buckling governs, so compression capacity is much less than tension capacity.
(C) Short column: Correct, buckling is negligible, so compression capacity can be comparable to tension capacity.
(D) Cannot be determined: Not required because classification is clear from the concept.
Step 4: Conclusion.
Hence, the column is a short column.

Step 1: Understand the term “permanent deformation.”
Permanent deformation refers to deformation that remains even after the removal of load.
Step 2: Definition of creep.
Creep is the time-dependent permanent deformation of a material when subjected to a constant load or stress over a long period of time.
Step 3: Analyze the options.
(A) Strain: Strain is a general measure of deformation and can be elastic or plastic, not necessarily permanent.
(B) Extent: Not a standard term used in strength of materials.
(C) Creep: Correct, as it represents permanent dimensional change under sustained loading.
(D) Ambit: Not related to deformation mechanics.
Step 4: Conclusion.
Therefore, permanent dimension change due to loading is called creep.

Step 1: Understand thermal expansion.
When a material is heated, it tends to expand. Thermal stress develops only if this expansion is restrained.
Step 2: Analyze the given condition.
The steel bar is stated to be free to expand. This means there is no external restraint preventing its thermal expansion.
Step 3: Apply the concept of thermal stress.
Thermal stress is given by:

This formula applies only when expansion is fully or partially restrained.
Since the bar is free, no resistance is offered against expansion.
Step 4: Conclusion.
As there is no restraint, no stress is developed in the bar.

Step 1: Understanding the ideal truss assumption.
In the analysis of trusses, certain ideal assumptions are made to simplify calculations. One key assumption is related to the nature of joints between members.
Step 2: Nature of joints in a truss.
Truss members are assumed to be connected by smooth pin joints.
Smooth pins allow rotation but do not resist moments, ensuring that only axial forces (tension or compression) act in the members.
Step 3: Analysis of options.
(A) Rough pins: Incorrect, because rough pins would resist rotation and introduce moments.
(B) Smooth pins: Correct, as this assumption ensures members carry only axial forces.
(C) Either of them: Incorrect, since the assumption is specific to smooth pins.
(D) Neither of them: Incorrect, as pin joints are essential to truss behavior.
Step 4: Conclusion.
Hence, in a truss, members are assumed to be joined by smooth pins.

Step 1: Recall the boundary condition of a simply supported beam.
In a simply supported beam, the supports are usually a pin support and a roller support.
These supports provide reactions (vertical and/or horizontal), but they do not resist a moment (i.e., they cannot provide a fixing moment).
So, the bending moment at the supports is zero unless an external couple (moment) is applied exactly at that end.
Step 2: Apply the concept to the given statement.
At the end (support) of a simply supported beam:
- If no couple is applied at the end, the support cannot develop a resisting moment.
- Hence, the bending moment at that end must be zero.
Step 3: Analyze each option.
(A) Is always zero if it does not carry couple at the end: Correct, because supports of a simply supported beam cannot resist bending moment unless a couple is applied externally.
(B) Is zero if the beam has uniformly distributed load only: Incorrect reasoning, because the bending moment at the supports is zero for any loading type (UDL, point load, etc.) as long as no end couple is applied.
(C) Is zero if the beam has concentrated loads only: Incorrect for the same reason as (B). The type of load does not change the end moment condition.
(D) May or may not be zero: Incorrect for a simply supported beam, because it is zero unless an end couple is applied (which is exactly covered in option A).
Step 4: Conclusion.
Therefore, the correct statement is (A) Is always zero if it does not carry couple at the end.

Step 1: Recall standard elastic relations.
In the theory of elasticity, Young’s modulus, bulk modulus, and Poisson’s ratio are interrelated material properties for isotropic and homogeneous materials.
Step 2: Write the known relation.
The standard formula connecting Young’s modulus ( ), bulk modulus ( ), and Poisson’s ratio ( ) is given by:

Step 3: Match with the options.
Comparing the derived relation with the given options, it directly matches option (B).
Step 4: Conclusion.
Hence, the correct relationship is .

Step 1: Understand Mohr’s circle.
Mohr’s circle is a graphical representation of the state of stress at a point. It helps visualize normal and shear stresses acting on different planes.
Step 2: Identify what Mohr’s circle provides.
Using Mohr’s circle, we can determine:
- Principal stresses
- Maximum shear stress
- Orientation of principal planes
Step 3: Evaluate the options.
(A) Bending stresses: These are obtained using bending theory, not Mohr’s circle.
(B) Principal stresses: Correct, Mohr’s circle is primarily used to determine principal stresses graphically.
(C) Torsional shear stresses: These are calculated using torsion theory, not Mohr’s circle directly.
(D) None: Incorrect, since Mohr’s circle has clear applications.
Step 4: Conclusion.
Therefore, Mohr’s circle is a graphical method to find principal stresses.

Step 1: Define the point of contra-flexure.
The point of contra-flexure is the point on a beam where the bending moment changes its sign.
This means the curvature of the beam changes from sagging to hogging (or vice-versa).
Step 2: Connect sign change with bending moment value.
For bending moment to change sign, it must pass through zero at some location.
So, at the point of contra-flexure:
- Bending moment is zero, and
- It changes sign across that point (positive to negative or negative to positive).
Step 3: Analyze the options.
(A) Point where Shear force is maximum: Incorrect, because contra-flexure is related to bending moment sign change, not maximum shear force.
(B) Point where Bending moment is maximum: Incorrect, because contra-flexure occurs at zero bending moment, not maximum.
(C) Point where Bending moment is zero: Incomplete, because bending moment can be zero at supports in simply supported beams, but that does not necessarily indicate a sign change.
(D) Point where Bending moment is zero but also changes sign from positive to negative: Correct, this captures both necessary conditions for contra-flexure.
Step 4: Conclusion.
Hence, the point of contra-flexure is correctly described by option (D).