4.10 Exercises
123
4.10 Exercises
1. State and explain generalised Hooke’s law
2. State and explain the following:
(a) Saint-Venant’s principle (b) Principle of superposition (c) Uniqueness
theorem
3. The displacement field in a body is specified as u = x
3
+ 3, v = 3y
2 z, w =
x +3z. Determine the strain and stress components at a point whose coordinates
are (1, 2, 3). Take E = 2 × 10
5 N/mm
2 and ν = 0.32.
4. A homogeneous and isotropic elastic solid with Young’s modulus 200 GPa and
Poisson’s ratio 0.3 is subjected to a displacement field given by
u =
x
3 y
2 z − 4x y
2 z
2
10
−4
v =
x
2 y
2 z
3
+ 3x yz
2
10
−4
w =
2x
4
− 3y
2 z
2
+ x yz
4
10
−4
Evaluate the stress components at a point whose coordinates are (2.5, 4.0, −
6.0). Express the above state of stress as the sum of its spherical and deviator
components.
5. Calculate the principal strains and then the principal stresses at the point (2,
4) and the directions of principal stresses with respect to the x-axis for the
following displacement field.
u = 4x
4
+ 2x
2 y
2
+ x + 3
v = y
4
+ 3x
2 y + 1
6. Prove the following relationships:
(i) G =
E
2(1+v)
(ii) λ =
2Gυ
(1−2υ)
(iii) K =
EG
3(3G−E)
7. A triaxial state of principal stress acts on the faces of a unit cube. Show that
these stresses will not produce a volume change if ν = 0.5. The material is a
linearly elastic isotropic material. If ν = 0.5, show that the condition necessary
for the volume to remain unchanged is σ 1 + σ 2 + σ 3 = 0.
8. A member made of isotropic bronze is subjected to a state of plane strain.
Determine σ z , ε x , ε y and γ xy , if σ x = 90 MPa, σ y = −50 MPa, and τ xy =
70MPa.
Take E = 82.6 GPa and ν = 0.35.
9. A square plate with 800 mm sides parallel to the x and y-axes has a uniform
thickness of 10 mm and is made of an isotropic steel (E = 82.6 GPa and ν =
0.29). The plate is subjected to plane strain state. If σ x = σ 1 = 500 MPa and
ε x = 2ε y , determine the magnitude of σ y = σ 2 and σ z = σ 3 , assuming linearly
elastic conditions.
10. The components of a strain tensor at a point are given as:
123
4.10 Exercises
1. State and explain generalised Hooke’s law
2. State and explain the following:
(a) Saint-Venant’s principle (b) Principle of superposition (c) Uniqueness
theorem
3. The displacement field in a body is specified as u = x
3
+ 3, v = 3y
2 z, w =
x +3z. Determine the strain and stress components at a point whose coordinates
are (1, 2, 3). Take E = 2 × 10
5 N/mm
2 and ν = 0.32.
4. A homogeneous and isotropic elastic solid with Young’s modulus 200 GPa and
Poisson’s ratio 0.3 is subjected to a displacement field given by
u =
x
3 y
2 z − 4x y
2 z
2
10
−4
v =
x
2 y
2 z
3
+ 3x yz
2
10
−4
w =
2x
4
− 3y
2 z
2
+ x yz
4
10
−4
Evaluate the stress components at a point whose coordinates are (2.5, 4.0, −
6.0). Express the above state of stress as the sum of its spherical and deviator
components.
5. Calculate the principal strains and then the principal stresses at the point (2,
4) and the directions of principal stresses with respect to the x-axis for the
following displacement field.
u = 4x
4
+ 2x
2 y
2
+ x + 3
v = y
4
+ 3x
2 y + 1
6. Prove the following relationships:
(i) G =
E
2(1+v)
(ii) λ =
2Gυ
(1−2υ)
(iii) K =
EG
3(3G−E)
7. A triaxial state of principal stress acts on the faces of a unit cube. Show that
these stresses will not produce a volume change if ν = 0.5. The material is a
linearly elastic isotropic material. If ν = 0.5, show that the condition necessary
for the volume to remain unchanged is σ 1 + σ 2 + σ 3 = 0.
8. A member made of isotropic bronze is subjected to a state of plane strain.
Determine σ z , ε x , ε y and γ xy , if σ x = 90 MPa, σ y = −50 MPa, and τ xy =
70MPa.
Take E = 82.6 GPa and ν = 0.35.
9. A square plate with 800 mm sides parallel to the x and y-axes has a uniform
thickness of 10 mm and is made of an isotropic steel (E = 82.6 GPa and ν =
0.29). The plate is subjected to plane strain state. If σ x = σ 1 = 500 MPa and
ε x = 2ε y , determine the magnitude of σ y = σ 2 and σ z = σ 3 , assuming linearly
elastic conditions.
10. The components of a strain tensor at a point are given as: