Strength of Materials - 3 - Mechanical Engineering MCQ

• The force of resistance per unit area, offered by a body against deformation is known as stress.
• The external force acting on the body is called the load or force.
• When a body is subjected to some external force, there is some change of dimension of the body. The ratio of change of dimension of the body to the original dimension is know as strain.

Point load is that load which acts over a small distance. Because of concentration over small distance this load can may be considered as acting on a point. Point load is denoted by P and symbol of point load is arrow heading downward (↓).

Uniformly distributed load is that whose magnitude remains uniform throughout the length.

Uniformly Varying Load (Non – Uniformly Distributed Load): It is that load whose magnitude varies along the loading length with a constant rate.

Uniformly varying load is further divided into two types:

• Triangular Load: whose magnitude is zero at one end of span and increases constantly till the 2nd end of the span.
• Trapezoidal Load: which is acting on the span length in the form of trapezoid. Trapezoid is generally form with the combination of uniformly distributed load (UDL) and triangular load.

Shear stress on a beam section is:

(i) Zero at extreme fibres

(ii) Maximum at neutral axis

. Maximum Principal stress theory was postulated by Rankine. It is suitable for brittle materials.

2. Maximum Principal strain theory was postulated by St Venant. This theory is not accurate for brittle and ductile materials both.

3. Maximum Shear stress theory was postulated by Tresca. This theory is suitable for ductile materials. Its results are the safest.

4. Maximum shear strain energy theory was postulated by Von-mises. Its results in case of pure shear are the accurate for ductile materials.

The maximum deflection will occur at

Bending Moment Equation

For the constant bending moment: σ ∝ 1/Z

So, bending stress is inversely proportional to the Section Modulus 'YH

The general relationship between the shear force diagram, bending moment diagram and loading diagram will be:

1. Shear force diagram is 1^o higher than loading diagram.

2. Bending moment diagram is 1^o higher than shear force diagram.

3. For the uniformly varying load on the beam, the shear force diagram is parabolic in nature.

4. For the uniformly varying load on the beam, the bending moment diagram is also parabolic but is 1^o higher than shear force diagram.

Mohr circle is a graphical representation of plane stress which helps in determining the relationships between normal and shear stresses acting on any inclined plane at a point in a stressed body.

Mohr’s circle of stresses is a graphical method of finding normal, tangential and resultant stresses on an oblique plane.

Moment of inertia (I₁) of circular plate of diameter “D” is given by:

.

Moment of inertia (I₂) of square plate of side equal to diameter “D” of circular plate is given by:

Elastic Rubber: 0.5

Wood: Because wood is orthotropic, 12 constants are required to describe elastic behaviour: 3 moduli of elasticity, 3 moduli of rigidity, and 6 Poisson’s ratios (vary from 0.02 to 0.47). These elastic constants vary within and among species and with moisture content and specific gravity.

Copper: 0.33

Steel: 0.27 - 0.30

Maximum shear stress (τ) is given by:

Effective length of columns under different end conditions are:

1. For both ends hinged: L_e = L

2. One end fixed and another end free: L_e = 2L

3. Both ends fixed = Le=

4. One end fixed and other is hinges: Le=

Radius of Mohr’s circle is given by:

r=

At principal plane, σ_x = σ₁, σ_y = σ₂

τ_xy = 0

r=

σ₁ = p , σ₂ = -p

r=

r = p

Section modulus is defined as the ratio of moment of inertia of a beam about its C.G to the maximum distance of extreme x-section of the beam (Y_max).

Stress on x face, A(200, 0) and Stress on y face, B(0, 0)

Maximum shear stress = Radius of Mohr circle

τ_max= σ/2 = 100 MPa

Relation between Modulus of elasticity (E), Shear Modulus (G) and Bulk modulus (K) is given by:

Now, rewriting the equation as follows:

3KE + GE = 9KG

3KE – 9KG = - GE

K(3E – 9G) = - GE

Strength is defined as the ability of the material to resist, without rupture, external forces causing various types of stresses. Breaking strength is the ability of a material to withstand a pulling or tensile force.

Toughness is defined as the ability of the material to absorb energy before fracture takes place. In other words, toughness is the energy for failure by fracture. Toughness is measured by a quantity called modulus of toughness. Modulus of toughness is the total area under a stress-strain curve in tension test, which also represents the work done to fracture the specimen.

Hardness is defined as the resistance of a material to penetration or permanent deformation. It usually indicates resistance to abrasion, scratching, cutting or shaping.

Stiffness or rigidity is defined as the ability of the material to resist deformation under the action of external load. Modulus of elasticity is the measure of stiffness.

Limit of poisons ratio varies between -1 to 0.5.

μ = 0.5 for rubber.

Slenderness ratio ( λ ) of long column is defined as the ratio of Effective length of column to its least radius of gyration.

The neutral axis is an axis in the cross-section of a beam (a member resisting bending) or shaft along which there are no longitudinal stresses or strains. If the section is symmetric, isotropic and is not curved before a bend occurs, then the neutral axis is at the geometric centroid. All fibers on one side of the neutral axis are in a state of tension, while those on the opposite side are in compression.