This set of Soil Mechanics Multiple Choice Questions & Answers (MCQs) focuses on “Elasticity Elements – Stress Functions”.

1. The stress function was introduced by __________

a) G.B Airy

b) Terzaghi

c) Darcy

d) Meyerhof

View Answer

Explanation: The stress function was introduced by G.B Airy in 1862. The stress function is denoted by Φ and is called Airy stress function. Darcy gave the law of flow of water through soils.

2. The σ_{x} in terms of stress function is given by __________

a) \(\frac{∂Φ}{∂z}\)

b) \(\frac{∂^2 Φ}{∂z^2}\)

c) \(\frac{∂^2 Φ}{∂x2}\)

d) \(\frac{∂Φ}{∂x}\)

View Answer

Explanation: It is convenient to reduce the compatibility and equilibrium equations into a single equation in terms of stress function Φ.

The stress function for stress in the x-direction is given by,

\(\frac{∂^2 Φ}{∂z^2}.\)

3. The σ_{z} in terms of stress function is given by __________

a) \(\frac{∂Φ}{∂z}\)

b) \(\frac{∂^2 Φ}{∂z^2}\)

c) \(\frac{∂^2 Φ}{∂x^2}\)

d) \(\frac{∂Φ}{∂x}\)

View Answer

Explanation: It is convenient to reduce the compatibility and equilibrium equations into a single equation in terms of stress function Φ.

The stress function for stress in the z-direction is given by,

\(σ_z=\frac{∂^2 Φ}{∂x^2}.\)

4. The shear stress τ_{xz} in terms of stress function is given by __________

a) \(\frac{∂Φ}{∂z}-γx\)

b) \(-\frac{∂^2Φ}{∂x∂z}-γx\)

c) \(\frac{∂^2Φ}{∂x^2} -γx\)

d) \(\frac{∂Φ}{∂x}-γx\)

View Answer

Explanation: It is convenient to reduce the compatibility and equilibrium equations into a single equation in terms of stress function Φ.

The shear stress function τ

_{xz}is given by,

\(τ_{xz}=-\frac{∂^2Φ}{∂x∂z}-γx\)

5. For both plane stress as well as plain strain case the equilibrium equation in x-direction is _______

a) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}=0\)

b) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}+X=1\)

c) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+X=0\)

d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)

View Answer

Explanation: The equilibrium equation in x-direction is \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+\frac{∂τ_{zx}}{∂z}+X=0\)

In the plain strain case, one dimension (y) is very large in comparison to the other two directions. So, the strain components in this direction are zero. Also in plain stress condition, the stresses in y-direction are considered as zero.

∴ The equation reduces to \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)

6. For both plane stress as well as plain strain case the equilibrium equation in z-direction is _______

a) \(\frac{∂τ_{xz}}{∂x}+\frac{∂σ_z}{∂z}+γ=0\)

b) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}+γ=1\)

c) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{yx}}{∂y}+γ=0\)

d) \(\frac{∂σ_x}{∂x}+\frac{∂τ_{zx}}{∂z}=0\)

View Answer

Explanation: The equilibrium equation in x-direction is \(\frac{∂τ_{xz}}{∂x} + \frac{∂τ_{yz}}{∂y} +\frac{∂σ_z}{∂z}+γ=0.\)

In the plain strain case, one dimension (y) is very large in comparison to the other two directions. So, the strain components in this direction are zero. Also in plain stress condition, the stresses in y-direction are considered as zero.

∴ The equation reduces to \(\frac{∂τ_{xz}}{∂x}+\frac{∂σ_z}{∂z}+γ=0\)

7. For two dimensional case, for both plane stress as well as plain strain case the compatibility equation is _______

a) \(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} =\frac{∂^2 Γ_{xz}}{∂x∂z}\)

b) \(\frac{∂^2 ε_z}{∂z^2} +\frac{∂^2 ε_y}{∂x^2} =\frac{∂^2 Γ_{zy}}{∂z∂y}\)

c) \(\frac{∂^2 ε_x}{∂y^2} +\frac{∂^2 ε_y}{∂x^2} =\frac{∂^2 Γ_{xy}}{∂x∂y}\)

d) \(\frac{∂^2 ε_z}{∂z^2} +\frac{∂^2 ε_y}{∂x^2} =0\)

View Answer

Explanation: For two dimensional case, the six compatibility equations are evidently reduced to one single equation;

\(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} =\frac{∂^2 Γ_{xz}}{∂x∂z}.\)

This is because, in the plain strain case, one dimension (y) is very large in comparison to the other two directions. So, the strain components in this direction are zero. Also in plain stress condition, the stresses in y-direction are considered as zero.

8. The compatibility equation in terms of plane stress case is given by ________

a) \((\frac{∂^2}{∂x^2} +\frac{∂^2}{∂z^2})=0\)

b) \((\frac{∂^2}{∂y^2} +\frac{∂^2}{∂z^2})(σ_y+σ_z )=0\)

c) \((\frac{∂^2}{∂x^2} +\frac{∂^2}{∂y^2})(σ_x+σ_y )=0\)

d) \((\frac{∂^2}{∂x^2} +\frac{∂^2}{∂z^2})(σ_x+σ_z )=0\)

View Answer

Explanation: For two dimensional case, the six compatibility equations are evidently reduced to one single equation;

\(\frac{∂^2 ε_x}{∂z^2} +\frac{∂^2 ε_z}{∂x^2} = \frac{∂^2 Γ_{xz}}{∂x∂z} ———————(1)\)

From the Hooke’s law equation,

\(ε_x=\frac{1}{E} (σ_x-μσ_z ),\)

\(ε_z=\frac{1}{E} (σ_z-μσ_x ) \) and

\(Γ_{xz}=\frac{2(1+μ)}{E} τ_{zx}\)

Substituting these values in (1) and simplifying further we get,

\((\frac{∂^2}{∂x^2} + \frac{∂^2}{∂z^2})(σ_x+σ_z)=0.\)

**Sanfoundry Global Education & Learning Series – Soil Mechanics.**

To practice all areas of Soil Mechanics, __here is complete set of 1000+ Multiple Choice Questions and Answers__.

**Next Steps:**

- Get Free Certificate of Merit in Geotechnical Engineering
- Participate in Geotechnical Engineering Certification Contest
- Become a Top Ranker in Geotechnical Engineering
- Take Geotechnical Engineering Tests
- Chapterwise Practice Tests: Chapter 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
- Chapterwise Mock Tests: Chapter 1, 2, 3, 4, 5, 6, 7, 8, 9, 10

**Related Posts:**

- Buy Geotechnical Engineering I Books
- Apply for Civil Engineering Internship
- Practice Civil Engineering MCQs
- Apply for Geotechnical Engineering I Internship
- Practice Geotechnical Engineering II MCQs