Simple Stress and Strain MCQ Quiz - Objective Question with Answer for Simple Stress and Strain - Download Free PDF

Concept:

For a tapered flat bar under axial load, the extension $\Delta L$ is given by:

where, P = axial load, L = length, E = modulus of elasticity, t = thickness,

b₁ b₂ = widths at ends

Calculation:

Given:

P = 80 kN

t = 10 mm

b₁ = 60 mm; b₂ = 40 mm

Explanation:

Pure Shear Stress

• A pure shear stress state occurs when material elements are subjected to equal and opposite forces that cause one set of planes to elongate while the perpendicular set contracts, without causing any change in volume. This happens when there is tension in one direction and an equal amount of compression in the perpendicular direction.

• In pure shear, there are no normal stresses along the principal axes — only shear stresses act. This state is often analyzed using Mohr’s Circle, which visually represents the relationship between normal and shear stresses.

• Pure shear stress leads to angular distortion of the material, meaning the shape changes (angles between planes are altered), but the volume of the material remains constant.

• Pure shear conditions are important in structural engineering and material science when analyzing situations such as torsion of circular shafts, punching shear in slabs, or material testing under shear load.

 Additional Information

• The concept of principal stresses is closely related to pure shear. In pure shear, the principal stresses are equal in magnitude but opposite in sign — one is tensile, one is compressive. There is no third stress component acting in the third direction (it is zero).

• Shear strain is the angular deformation resulting from pure shear. Under pure shear, the shape of the material element is distorted (angles change), but its volume remains unchanged.

• Mohr’s Circle is a graphical tool that is often used to represent the state of stress (including pure shear). In pure shear, the Mohr’s Circle is centered at zero normal stress, and its radius equals the shear stress. It helps visualize how shear and normal stresses transform on different planes.

Explanation:

Toughness and Stress-Strain Curves:

• Toughness is a measure of a material's ability to absorb energy up to fracture. It is represented by the area under the stress-strain curve of a material. The greater the area under the curve, the tougher the material. Toughness is influenced by several factors, including the material's strength, ductility, and grain size.
• In this scenario, we are comparing two materials, Material A (with finer grain structure) and Material B (with coarser grain structure), based on their stress-strain curves. The finer grain structure of Material A typically leads to enhanced strength and ductility due to the Hall-Petch effect, which states that finer grains increase a material's resistance to deformation.

Material A (finer grain size) has higher toughness than Material B (coarser grain size).

• This option is correct because finer grain structures, as in Material A, generally result in higher toughness. Finer grains improve both the strength and ductility of the material, contributing to a larger area under the stress-strain curve. This increased area directly correlates to higher toughness. In contrast, Material B, with its coarser grain structure, is likely to have lower strength and ductility, leading to a smaller area under its stress-strain curve and, hence, lower toughness.

Grain Structure and Mechanical Properties:

• Finer Grain Structure: Finer grains increase the number of grain boundaries, which act as obstacles to dislocation movement. This enhances strength (as per the Hall-Petch relationship) and can also improve ductility, resulting in higher toughness.
• Coarser Grain Structure: Coarser grains have fewer grain boundaries, making it easier for dislocations to move. This typically leads to lower strength and ductility, reducing the material's toughness.

Explanation:

Thermal Stress in Composite Bars

• Thermal stress refers to the stress induced in a material or structure due to a change in temperature. When a composite bar (a bar made up of two or more different materials) is subjected to a temperature change, each material tends to expand or contract based on its coefficient of thermal expansion. However, since the materials are bonded together, they restrict each other's free expansion or contraction, leading to thermal stress.
• In the context of composite bars, thermal stress depends on several factors, including the coefficient of thermal expansion, temperature change, modulus of elasticity, and the way materials are bonded. However, thermal stress DOES NOT depend on the area of the cross-section of the composite bar.

The formula for thermal stress is given as:

σ = E × α × ΔT

Where:

• σ: Thermal stress
• E: Modulus of elasticity
• α: Coefficient of thermal expansion
• ΔT: Temperature change

From the formula, it is clear that the thermal stress does not depend on the area of the cross-section. While the area of the cross-section may influence the total force generated due to the thermal stress (as force is given by stress × area), the stress itself remains independent of the cross-sectional area.

Explanation:

Modulus of Resilience

• The modulus of resilience is a property of materials that measures the maximum energy a material can absorb per unit volume without undergoing permanent deformation. It represents the area under the stress-strain curve up to the elastic limit (or proportional limit), where the material behavior remains elastic. This energy is recoverable after the removal of the applied load. The modulus of resilience is an important parameter in designing components subjected to dynamic and impact loads, as it provides insight into a material's ability to absorb energy without yielding.

The modulus of resilience can be mathematically expressed as:

U_r = (σ_y)² / 2E

Where:

• U_r = Modulus of resilience (energy per unit volume)
• σ_y = Yield strength of the material
• E = Modulus of elasticity of the material

The modulus of resilience is essentially the area of the triangle formed under the stress-strain curve up to the proportional limit (elastic region). The proportional limit is the point on the stress-strain curve up to which Hooke's Law is valid, and the relationship between stress and strain is linear.

Correct Option Analysis:

The correct option is:

Option 1: Proportional limit

This is the correct answer because the modulus of resilience is characterized by the area under the stress-strain curve up to the proportional limit. Within this region, the material behaves elastically, and all the energy absorbed during deformation can be recovered when the load is removed. Beyond the proportional limit, the material enters the plastic deformation region, where permanent deformation occurs, and the energy absorbed in this region is not recoverable. Hence, only the elastic region (up to the proportional limit) is used to calculate the modulus of resilience.

Explanation:

Ductile material fails along the principal shear plane as they are weak in shear and brittle material fails along with principal normal stress.

In Brittle materials under tension test undergoes brittle fracture i.e their failure plane is 90° to the axis of load and there is no elongation in the rod that’s why the diameter remains same before and after the load. Example: Cast Iron, concrete etc.

But in case of ductile materials, material first elongates and then fail, their failure plane is 45° to the axis of the load. After failure cup-cone failure is seen. Example Mild steel, high tensile steel etc.

Explanation:

Perfectly Plastic Material:

For this type of material, there will be only initial stress required and then the material will flow under constant stress.

The chart shows the relation between stress-strain in different materials. 

Explanation:

• Change in the temperature causes the body to expand or contract.
• Thermal stress is created when a change in size or volume is constrained due to a change in temperature.
• So an increase in temperature creates compressive stress and a decrease in temperature creates tensile stress.

Explanation:

ϵv = ϵx + ϵy + ϵz

Total strain or volumetric strain is given by 

There will be no change in volume if volumetric strain is zero.

ϵ_v = 0 ⇒  ν = 0.5

Explanation:

Since all the faces are free to expand the stresses due to temperature rise is equal to 0.

If the cube is constrained on all six faces, the stress produced in all three directions will be the same.

∴ thermal strain in x-direction = -α(ΔT) = 

σ_x = σ_y = σ_z = σ

Concept:

Let RA and RB be the reaction at support A and B respectively.

Free body diagram of the system is:

Calculation:

Given:

As per figure P = 10 kN, a = 1 m and b = 2 m.

Explanation:

Elastic recovery/strain: The strain recovered after the removal of the load is known as elastic strain.

Plastic strain: The permanent changes in dimension after the removal of load is known as plastic strain.

The load is removed when the stress was 200 MPa and the corresponding strain was 0.03

After the removal of load, the body recovered and the final strain found was 0.01.

∴ Elastic strain = 0.03 - 0.01 ⇒ 0.02 and Plastic strain = 0.01 respectively.

Concept:

Stress at any section of the bar is given by,

Calculation:

Given:

Load in section BC, P = 30 kN (compressive), 

cross-sectional area, A = 15 m² = 15 × 10⁶ mm²

Explanation:

Strain energy:

When a body is subjected to gradual, sudden, or impact load, the body deforms and work is done upon it. If the elastic limit is not exceeded, this work is stored in the body. This work-done or energy stored in the body is called strain energy.

Strain energy = Work done.

Case-I:

When a rigid body is slowly dropped on another body, it is a case of gradual loading:

Work-done on the bar = Area of load-deformation diagram 

Work stored in the bar = Area of resistance deformation diagram

We can write;

Case-II:

Work-done on the bar = Area of load-deformation diagram ⇒ P × δl

Work stored on the bar = Area of resistance deformation diagram

We can write;

∴ maximum stress intensity due to suddenly applied load is twice the stress intensity produced by the load of the same magnitude applied gradually.

Impact Loading:

When a load is dropped from a height before it commences to load the body, such loading is known as Impact loading.

The ratio of the stress or deflection produced due to impact loading to the stress or deflection produced due to static or gradual loading is known as the Impact factor.

∴ deflection due to sudden loading is twice that of gradual loading.

Explanation:

The stress-strain diagrams for different type of materials are given below: