Engineering Questions with Answers - Multiple Choice Questions

# Strain Components and Compatibility Equations

1 - Question

There are __________ strain components for a three dimensional case.
a) 2
b) 3
c) 6
d) 8

Explanation: There are six strain components for a three dimensional case and there are three linear strain components and three shearing strain components.

2 - Question

If u is the displacement in x-direction, the linear strain component in is defined by __________
a) εX=ux
b) εX=xu
c) εX=dxdu
d) εX=x+u

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,
εX=ux. Where, u is the displacement in x-direction.

3 - Question

If v is the displacement in y-direction, the linear strain component in is defined by __________
a) εY=vy
b) εY=yv
c) εY=dydv
d) εY=v+y

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,
εY=vy. Where, v is the displacement in y-direction.

4 - Question

If w is the displacement in z-direction, the linear strain component in is defined by __________
a) εZ=wz
b) εZ=zw
c) εZ=dzdw
d) εZ=z+w

Explanation: The strain is the ratio of change in dimension of an object to its original dimension. Therefore, the linear strain component in is defined by,
εZ=wz. Where, w is the displacement in z-direction.

5 - Question

The shearing strain component in X-Y plane is ________
a) Γxy=vx+uy
b) Γxy=vxuy
c) Γxy=vy+ux
d) Γxy=vyux

Explanation: In order to find the shearing strain component in X-Y plane, let us consider a plane lamina of size dx, dy in X-Y plane. If u is the displacement in x-direction and v is the displacement in y-direction, the shearing strain component is,
Γxy=vx+uy.

6 - Question

The shearing strain component in Y-Z plane is ________
a) Γyz=wy+vz
b) Γyz=vxwy
c) Γyz=vy+wx
d) Γyz=vywx

Explanation: In order to find the shearing strain component in Y-Z plane, let us consider a plane lamina of size dy, dz in Y-Z plane. If w is the displacement in z-direction and v is the displacement in y-direction, the shearing strain component is,
Γyz=wy+vz.

7 - Question

The shearing strain component in Z-X plane is ________
a) Γxz=wy+vz
b) Γxz=vxwy
c) Γxz=uz+wx
d) Γxz=vywx

Explanation: In order to find the shearing strain component in Z-X plane, let us consider a plane lamina of size dx, dz in Z-X plane. If w is the displacement in z-direction and u is the displacement in x-direction, the shearing strain component is,
Γxz=uz+wx.

8 - Question

The strain tensor is given by ____________
a) ⎛⎝⎜εxzεyxεzxεxyεyzεzyεxxεyyεzz⎞⎠⎟
b) ⎛⎝⎜εxyεyxεzyεxxεyyεzzεxzεyzεzz⎞⎠⎟
c) ⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟
d) ⎛⎝⎜εxxεyyεzzεxyεyxεzyεxzεyzεzx⎞⎠⎟

Explanation: The strain tensor is given by,
⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟
where, the diagonal elements represent the linear strain components and the off-diagonal elements represent the shearing strain components.

9 - Question

The strain tensor is given by ____________
a) ⎛⎝⎜1/2Γxz1/2Γyx1/2Γzx1/2Γxy1/2Γyz1/2Γzyεxxεyyεzz⎞⎠⎟
b)⎛⎝⎜1/2Γxy1/2Γyx1/2Γzyεxxεyyεzz1/2Γxz1/2Γyz1/2Γzz⎞⎠⎟
c) ⎛⎝⎜εxx1/2Γyx1/2Γzx1/2Γxyεyy1/2Γzy1/2Γxz1/2Γyzεzz⎞⎠⎟
d) ⎛⎝⎜εxxεyyεzz1/2Γxy1/2Γyx1/2Γzy1/2Γxz1/2Γyz1/2Γzx⎞⎠⎟

Explanation: The strain tensor is given by,
⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟
where, the diagonal elements represent the linear strain components and the off-diagonal elements represent the shearing strain components. The linear strain components of a diagonal is equal to half the sum shearing strain components.
∴ εyx=1/2Γyx
εxz=1/2Γxz
εzy=1/2Γzy.
∴ the strain tensor is given by,
⎛⎝⎜εxx1/2Γyx1/2Γzx1/2Γxyεyy1/2Γzy1/2Γxz1/2Γyzεzz⎞⎠⎟.

10 - Question

The linear strain of a diagonal is equal to half the sum shearing strain components.
a) True
b) False

Explanation: The linear strain of a diagonal is equal to half the sum shearing strain components. Thus if εyx, εxz and εzy represent the linear strains of the diagonal of a plane lamina,
∴ εyx=1/2Γyx
εxz=1/2Γxz
εzy=1/2Γzy.

11 - Question

The compatibility equations are results of application of stress equations.
a) True
b) False

Explanation: The equations resulting from the applications of the strain equations are known as the compatibility equations or the Saint-Venant’s equations.

12 - Question

___________ is a compatibility equation.
a) 2εyy2+2εyx2=2Γxyxy
b) 2εxy2+2εyx2=2Γxyxy
c) 2εxy2+2εyx2=2Γxyxy
d) 2εxy2+2εyx2=2Γxyxy