The given figure shows the Mohr's circle of stress for two unequal and like principal stresses (σx and σy) acting at a body across two mutually perpendicular planes. The normal stress on an oblique section making an angle θ with the minor principle plane is given by <img src="/images/question-image/mechanical-engineering/strength-of-materials/1528354684-24.JPG" title="Strength of Materials mcq question image" alt="Strength of Materials mcq question image">

The given figure shows the Mohr's circle of stress for two unequal and like principal stresses (σ_x and σ_y) acting at a body across two mutually perpendicular planes. The resultant stress is given by

If p₁ and p₂ are mutually perpendicular principal stresses acting on a soil mass, the normal stress on any plane inclined at angle $${\theta ^ \circ }$$ to the principal plane carrying the principal stress p₁, is:

$$\frac{{{{\text{p}}_1} - {{\text{p}}_2}}}{2} + \frac{{{{\text{p}}_1} + {{\text{p}}_2}}}{2}\sin \,2\theta $$

$$\frac{{{{\text{p}}_1} - {{\text{p}}_2}}}{2} + \frac{{{{\text{p}}_1} + {{\text{p}}_2}}}{2}\cos\,2\theta $$

$$\frac{{{{\text{p}}_1} + {{\text{p}}_2}}}{2} + \frac{{{{\text{p}}_1} - {{\text{p}}_2}}}{2}\cos\,2\theta $$

$$\frac{{{{\text{p}}_1} + {{\text{p}}_2}}}{2} + \frac{{{{\text{p}}_1} - {{\text{p}}_2}}}{2}\sin \,2\theta $$

Consider the following statements: X. Two-dimensional stresses applied to a thin plater in its own plane represent the plane stress condition. Y. Normal and shear stresses may occur simultaneously on a plane. Z. Under plane stress condition, the strain in the direction perpendicular to the plane is zero. Which of the above statements are correct?

2 only

1 and 2

2 and 3

1 and 3

The 3-dimensional state of stress consists of 3 unequal principal stresses acting on a point, this is called the triaxial state of stress. If two of the three principal stresses are equal, the state of stress is known as __________

cylindrical

uniaxial

spherical/hydrostatic

triaxial

The three-dimensional state of stress consists of 3 unequal principal stresses acting on a point, this is called the triaxial state of stress. If all the three principal stresses are equal, the state of stress is known as __________

cylindrical

uniaxial

spherical/hydrostatic

triaxial

When a body is subjected to biaxial stress i.e. direct stresses $$\left( {{\sigma _{\text{x}}}} \right)$$ and $$\left( {{\sigma _{\text{y}}}} \right)$$ in two mutually perpendicular planes accompanied by a simple shear stress $$\left( {{\tau _{{\text{xy}}}}} \right),$$ then maximum normal stress is

$$\frac{{{\sigma _{\text{x}}} + {\sigma _{\text{y}}}}}{2} + \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} - {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} + {\sigma _{\text{y}}}}}{2} - \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} - {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} - {\sigma _{\text{y}}}}}{2} + \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} + {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} - {\sigma _{\text{y}}}}}{2} - \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} + {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

When a body is subjected to biaxial stress i.e. direct stresses $$\left( {{\sigma _{\text{x}}}} \right)$$ and $$\left( {{\sigma _{\text{y}}}} \right)$$ in two mutually perpendicular planes accompanied by a simple shear stress $$\left( {{\tau _{{\text{xy}}}}} \right),$$ then minimum normal stress is

$$\frac{{{\sigma _{\text{x}}} + {\sigma _{\text{y}}}}}{2} + \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} - {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} + {\sigma _{\text{y}}}}}{2} - \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} - {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} - {\sigma _{\text{y}}}}}{2} + \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} + {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

$$\frac{{{\sigma _{\text{x}}} - {\sigma _{\text{y}}}}}{2} - \frac{1}{2}\sqrt {{{\left( {{\sigma _{\text{x}}} + {\sigma _{\text{y}}}} \right)}^2} + 4\tau _{{\text{xy}}}^2} $$

A vertical ore body having a consistent width of 100 m has been excavated by the open pit method up to a depth of 100 m at a pit slope angle of 45°. The figure shows the cross section of the open pit. If the ore body is further excavated by a depth of 10 m, maintaining the same pit slope angle, the ratio of waste rock to ore is

1.2

2.1

3.0

3.9

The figure shows a shape ABC and its mirror image A₁B₁C₁ across the horizontal axis (X-axis). The coordinate transformation matrix that maps ABC to A₁B₁C₁ is

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Body with material properties that are different in three mutually perpendicular directions at a point in the body and further has three mutually perpendicular planes of material property symmetry are called as ________

Orthotropic materials

Anisotropic materials

Non-homogeneous materials

Isotropic materials