3.18 Exercises
95
v = 2x
3
+ 6x
2
+ y
2
+ z + 5
w = x
3
+ 3y
3
+ 8x y + 4
5. Derive the compatibility equation in terms of strain and displacements.
6. At a point in a stressed material, the stresses acting are: σ x = 300 N/mm
2 ,
σ y = 250 N/mm
2 and σ z = 220 N/mm
2 . If γ = 0.3, calculate the volumetric
strain.
7. In a steel bar subjected to three-dimensional stress system, the elongations
measured in the three principal directions over a length of 100 cm were found
to be 1.8 mm, 1.2 mm and 0.6 mm, respectively, along the x, y and z axes.
Calculate the volumetric strain and new volume of the material.
8. The displacement components in a strained body are:
u = 0.02x y + 0.03y
2
v = 0.03x
2
+ 0.02z
3 y
w = 0.02x y
2
+ 0.06z
2
Determine the strain matrix at the point (3, 2, −5).
9. The strain components at a point with respect to xyz co-ordinate system are:
ε x = 0.01
ε y = 0.02
ε z = 0.03
γ xy = γ yz = γ xz = 0.016
If the co-ordinate axes are rotated about z-axis through 45° in the anticlockwise
direction, determine the new strain component.
10. The components of strain tensor at a given point are given by the following
array of terms:
ε i j =
⎡
⎣
0.01 0.02 0.05
0.02 0.03 0.04
0.05 0.04 0.05
⎤
⎦
Determine
(a) Octahedral normal and shearing strains.
(b) Deviator and spherical strain tensors.
11. The displacement field components at a point are given by
u = −0.01y
2
+ 0.15x yz
v = 0.02x
2 y + 0.03x
2 z
w = 0.15x yz − 0.01x
2 yz
Determine the strain tensor at the point (2, −1, 3).
12. At a point in a body, the components of strain are