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
View Answer
Answer: c
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=∂u∂x
b) εX=∂x∂u
c) εX=dxdu
d) εX=x+u
View Answer
Answer: a
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=∂u∂x. 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=∂v∂y
b) εY=∂y∂v
c) εY=dydv
d) εY=v+y
View Answer
Answer: a
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=∂v∂y. 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=∂w∂z
b) εZ=∂z∂w
c) εZ=dzdw
d) εZ=z+w
View Answer
Answer: a
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=∂w∂z. Where, w is the displacement in z-direction.
5 - Question
The shearing strain component in X-Y plane is ________
a) Γxy=∂v∂x+∂u∂y
b) Γxy=∂v∂x−∂u∂y
c) Γxy=∂v∂y+∂u∂x
d) Γxy=∂v∂y−∂u∂x
View Answer
Answer: a
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=∂v∂x+∂u∂y.
6 - Question
The shearing strain component in Y-Z plane is ________
a) Γyz=∂w∂y+∂v∂z
b) Γyz=∂v∂x−∂w∂y
c) Γyz=∂v∂y+∂w∂x
d) Γyz=∂v∂y−∂w∂x
View Answer
Answer: a
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=∂w∂y+∂v∂z.
7 - Question
The shearing strain component in Z-X plane is ________
a) Γxz=∂w∂y+∂v∂z
b) Γxz=∂v∂x−∂w∂y
c) Γxz=∂u∂z+∂w∂x
d) Γxz=∂v∂y−∂w∂x
View Answer
Answer: c
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=∂u∂z+∂w∂x.
8 - Question
The strain tensor is given by ____________
a) ⎛⎝⎜εxzεyxεzxεxyεyzεzyεxxεyyεzz⎞⎠⎟
b) ⎛⎝⎜εxyεyxεzyεxxεyyεzzεxzεyzεzz⎞⎠⎟
c) ⎛⎝⎜εxxεyxεzxεxyεyyεzyεxzεyzεzz⎞⎠⎟
d) ⎛⎝⎜εxxεyyεzzεxyεyxεzyεxzεyzεzx⎞⎠⎟
View Answer
Answer: c
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⎞⎠⎟
View Answer
Answer: c
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
View Answer
Answer: a
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
View Answer
Answer: b
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) ∂2εy∂y2+∂2εy∂x2=∂2Γxy∂x∂y
b) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y
c) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y
d) ∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y
View Answer
Answer: b
Explanation: The strain equations in terms of displacements is given by,
εX=∂u∂x—————−(1)
εY=∂v∂y—————−(2)
Γxy=∂v∂x+∂u∂y———(3)
Differentiating (1) twice with respect to y, (2) twice with respect to x and (3) once with respect to x and then y,
∂2εx∂y2=∂3udx∂y2————−(4)
∂2εy∂x2=∂3vdy∂x2————−(5)
∂2Γxy∂x∂y=∂3udx∂y2+∂3vdy∂x2——−(6)
∴ from (4), (5) and (6), we get,
∂2εx∂y2+∂2εy∂x2=∂2Γxy∂x∂y which is a compatibility equation.