Module 3 - Exercise
1. 2. 3.
Explain strain tensor. Derive the strain displacement relation at any point in an elastic body. The displacement at a point ( x , y) are as given below
u = 5 x + 3 x y + x + y 4
2
2
v = y + 2 xy + 4 Compute the values of normal and shearing strains at a point (3, -2) and verify whether compatibility exists or not? Determine the strain components at point (1,2,3) for the following displacement field. 3
4.
u = 8 x + 2 y + 6 z + 10 2
v = 2 x + 6 x + y + z + 5 3
2
w = x + 3 y + 8 xy + 4 Derive the compatibility compatibilit y equation in terms of strain and displacements At a point in a stressed material, material, the stresses stresses acting are: are: 3
5. 6.
2
s y
3
= 250 N / mm2 and s z = 220 N / mm 2 . If
g
s x
= 300 N / mm2 ,
= 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 1000 mm were found to be 1.8mm, 1.2mm and 0.6mm 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: 2 u = 0.02 xy + 0.03 y
v = 0.03 x + 0.02 z y 2
3
w = 0.02 xy + 0.06 z Determine the strain matrix at the point (3, 2, -5) 9. The strain components at a point with respect to xyz coordinate system are: 2
2
e y = 0.02 = 0.01 g xy = g yz = g xz = 0.016 e x
g xy
= 0.03
If the coordinate axes are rotated about z-axis through 45 0 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:
e ij
é 0.01 0.02 0.05ù = êê0.02 0.03 0.04úú êë0.05 0.04 0.05úû
Determine (a) Octahedral normal and Shearing strains (b) Deviator and Spherical strain tensors 11. The displacement field components at a point are are given by
u = -0.01 y + 0.15 xyz 2
v = 0.02 x y + 0.03 x z 2
2
w = 0.15 xyz - 0.01 x yz 2
1
Determine the strain tensor at the point (2, -1, 3) 12. At a point in a body the components of strain are
= -0.000832
e x
=0
g xy
g yz
e y
= -0.000832
= 0.00145
g xz
e z
= 0.001664
=0
Find the principal strains 13. The components of strain at a point in a body are
= 0.01
e x
e y
= 0.03
g xy
= -0.05
e z
= 0.01
g yz
g xz
= 0.05 = 0.008
Find the principal strains. 14. At a point in a material the state of strain is represented by e x
= 0.00233
e xy
= -0.00152
e y
= 0.00091
e yz
= 0.00085
e zx
= 0.00110
= 0.00125
e z
Find the direction cosines of the principal strains. 15. The principal strains at a point are given by e 1
= 2 ´ 10 -3
= -3 ´ 10 -3
e 2
e 3
= -4 ´ 10 -3
Calculate the octahedral normal and shearing strains. 16. The strain components at a point are given by e x
= 10 xy + 12 z;
g xy
= 4 xy 2
e y
= 6 xy 2 + 2 yz;
g yz
= 2 yz 2
= 2 x 2 z + 2 y;
e z
g xz
= 2 xz 2
Verify whether the compatibility equations are satisfied or not at the point (1, -1, 2) 17. For the given displacement field
(
)
(
)
u = c x + 2 z , v = c 4 x + 2 y + z , w = 4cz Where c is a very small constant, determine the strain at (2, 1, 3) in the direction
0, -
2
1 2
,
2
2
1 2
18. A state of plane strain in a steel plate is defined by the following data e x
= 0.00050
e y
= 0.00014
e z
= 0.00036
Construct a Mohr’s circle and find the magnitudes and directions of principal strains. 19. The following strains were measured in a structure during the test by means of strain gauges e 0
= 650 ´ 10 -6
e 60
= -200 ´ 10 -6
e 120
= 250 ´ 10 -6
Determine the following (a) Magnitude of principal strains (b) Orientation of principal planes
2
20. Data taken from a 45 0 strain rosette reads as follows: e 0 = 750 micrometers/m e 45 e 90
= -110 micrometers/m = 210 micrometers/m
Find the magnitudes and the directions of principal strains. 21. Using an equiangular strain rosette, the following strains were measured at a point in a material. e 0 = 600 micrometres/ms, e 60 = -200 micrometres/m, e 120 = 300 micrometers/m Calculate the magnitudes and directions of principal strains.
3
