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# Elasticity – State of Stress at a Point

The force systems acting on an elastic body in equilibrium are _______

a) body forces only

b) surface forces only

c) both body and surface forces

d) body stresses

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Answer: c

Explanation: The force systems acting on an elastic body in equilibrium are of two kinds and they are body forces and surface forces. The surfaces forces acts on the surface of the body while the body forces are distributed over the volume of the body.

___________ is an example of surface force.

a) inertia force

b) gravitational force

c) magnetic force

d) hydrostatic force

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Answer: d

Explanation: The hydrostatic force acts on the surface of the object in all directions with a given magnitude and hence is an example of a surface force. The inertia force, gravitational force and magnetic force are the examples of the body forces.

Surfaces forces are applied _________

a) externally at boundaries of body

b) internally at boundaries of body

c) only on one side in internal of the body

d) throughout the volume of the body

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Answer: a

Explanation: The forces distributed over the surface of the body, like pressure of one body over another body, are called surface forces. The surface forces are applied externally at boundaries of body.

Dimensionally a surface force is defined as _____________

a) force at a point

b) pressure per unit area

c) force per unit area

d) force per unit volume

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Answer: c

Explanation: Surface forces are the forces distributed over the surface of the body. The surface forces are applied externally at boundaries of body. Hence, dimensionally a surface force is defined as force per unit area.

Body forces are _________

a) external forces at boundaries of body

b) internal forces at boundaries of body

c) forces only on one side in internal of the body

d) forces distributed throughout the volume of the body

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Answer: d

Explanation: The forces distributed throughout the volume of the body are called body forces. Such forces are gravitational forces, inertia forces, magnetic forces and seepage forces, etc.

Dimensionally a body force is defined as _____________

a) force at a point

b) pressure per unit area

c) force per unit area

d) force per unit volume

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Answer: d

Explanation: Body forces are the forces distributed over the entire volume of a body. Such forces are gravitational forces, inertia forces, magnetic forces and seepage forces, etc. Dimensionally a body force is defined as force per unit volume.

The stress tensor is given by ___________

a) ⎡⎣⎢σxxτyxτzxτxyσyyτzyτxzτyzσzz⎤⎦⎥

b) ⎡⎣⎢σzzτyxτzxτxyσyyτzyτxzτyzσxx⎤⎦⎥

c) ⎡⎣⎢σxxτyxτzxτzzσyyτzyτxzτyzσzz⎤⎦⎥

d) ⎡⎣⎢σxxτyxτzxτyyσyyτyyτxzτyzσzz⎤⎦⎥

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Answer: a

Explanation: The stress tensor is given by,

⎡⎣⎢σxxτyxτzxτxyσyyτzyτxzτyzσzz⎤⎦⎥

The main diagonal elements are the normal stresses which are compressive. They are represented by “σ”. The off-diagonal elements are shear stress. They are represented by “τ”.

In a stress tensor, each stress component in it is represented by__________

a) magnitude only

b) direction only

c) both magnitude and direction

d) opposite direction

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Answer: c

Explanation: In a stress tensor, each stress component in it is represented by both magnitude and direction. For example, σ_{xx} signifies the normal stress acting on the face of element perpendicular to x-axis.

The component τ_{xy} denotes ___________

a) normal stress in x-direction

b) normal stress perpendicular to y-axis

c) shear stress acting perpendicular to x-axis

d) shear stress acting perpendicular to y-axis

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Answer: c

Explanation: The shear stresses in the stress tensor are represented by τ. The shear stress τ_{xy} denotes a stress acting in the face of the element perpendicular to x-axis and acting in the direction of the y-axis.

The component σ_{z} denotes __________

a) normal stress in x-direction

b) normal stress acting perpendicular to z-axis

c) shear stress acting perpendicular to z-axis

d) shear stress acting perpendicular to y-axis

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Answer: b

Explanation: The normal stresses in the stress tensor are represented by σ. The normal stress σ_{z} denotes the normal stress acting on the face of the element that is perpendicular to the z-axis.

The component τ_{yz} denotes ________

a) normal stress in x-direction

b) normal stress perpendicular to y-axis

c) normal acting perpendicular to x-axis

d) shear stress acting perpendicular to y-axis

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Answer: d

Explanation: The shear stresses in the stress tensor are represented by τ. The shear stress τ_{yz} denotes a stress acting in the face of the element perpendicular to y-axis and acting in the direction of the z-axis.

At a point, there are/is _________ normal stress.

a) 1

b) 2

c) 3

d) 4

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Answer: c

Explanation: At a point, there are three normal stresses. The normal stresses are represented by σ. The three normal stresses at a point are σ_{x}, σ_{y} and σ_{z}. The x, y and z represent the signifies that stress acts on the face of the element that is perpendicular to x,y and z directions respectively.

At a point there are ______ shear stresses.

a) 2

b) 4

c) 6

d) 8

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Answer: c

Explanation: At a point, there are six shear stresses. The shear stresses in the stress tensor are represented by τ. The six shear stresses are τ_{XY}, τ_{YX}; τ_{YZ}, τ_{ZY}; τ_{ZX}, τ_{XZ}. The first letter of the subscript represents the direction of stress acting on the face of element perpendicular to that axis. The second letter of the subscript represents the direction of stress along which it acts.

There are _______ independent shearing stresses.

a) 2

b) 3

c) 6

d) 8

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Answer: b

Explanation: At a point there are six shearing stresses. At equilibrium of the elemental volume, moment about z-axis ∑M_{z} = 0.

∴ (τ_{YX}.dx.dz)dy = (τ_{XY} .dy.dz)dx

∴ τ_{YX} = τ_{XY} .

Similarly,

τ_{YZ} = τ_{ZY} and τ_{ZX} = τ_{XZ}.

Hence, out of six shearing stresses, there are only three independent shearing stresses.

The total independent stresses at a point are _________

a) 3

b) 6

c) 9

d) 12

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Answer: b

Explanation: At a point there are three normal stresses and six shear stresses. The three normal stresses at a point are σ_{x}, σ_{y} and σ_{z}. But out of six shearing stresses, there are only three independent shearing stresses. Therefore, the total independent stresses at a point are six stresses.