Reference no: EM13897533

1. The components of the stress tensor in a rectangular cartesian coordinate system x₁, x₂, x₃ at a point P are given in appropriate units by

Find:

(a) The traction at P on the plane normal to the x₁-axis;

(b) The traction at P on the plane whose normal has direction ratios 1: -3:2;

(c) The traction at P on a plane through P parallel to the plane x₁+2x₂+3x₃ = 1;

(d) The principal stress components at P;

(e) The directions of the principal axes of stress at P. Verify that the principal axes of stress are mutually orthogonal.

The coordinates in x^-₁, x^-₂, x^-₃ are related to x₁, x₂, x₃ by

x^-₁ = 1/3(x₁ - 2x₂ + 2x₃) , x^-₂ = 1/3(-2x₁ + x₂ + 2x₃) , x^-₃ = 1/3(-2x₁ - 2x₂ - x₃)

Verify that this transformation is orthogonal, and find the components of the stress tensor defined above in the new coordinate system. Use the answer to check the answers to (d) and (e) above.

2. In plane stress (T₁₃ = T₂₃ = T₃₃ = 0) show that if the x^-₁ and x^-₂-axes are obtained by rotating the x₁ and x₂-axes through an angle a about the x₃-axis, then

T^-₁₁ = ½(T₁₁+T₂₂) + 1/2(T₁₁-T₂₂) cos2α + T₁₂ sin2α

T^-₂₂ = ½(T₁₁+T₂₂) - 1/2(T₁₁-T₂₂) cos2α - T₁₂ sin2α

T^-₁₂ = -½(T₁₁-T₂₂) + sin2α + T₁₂cos2α

3. If, in appropriate units

find the principal components of stress, and show that the principal directions which correspond to the greatest and least principal components are both perpendicular to the x₂-axis.

4. A cantilever beam with rectangular cross-section occupies the region -a ≤x₁ ≤a, -h ≤ x₂ ≤ h, 0 ≤ x₃ ≤1. The end x₃ = I is built-in and the beam is bent by a force P applied at the free end x₃=0 and acting in the x₂ direction. The stress tensor has components where A, B and C are constants.

(a) Show that this stress satisfies the equations of equilibrium with no body forces provided 2B+ C= 0;

(b) Determine the relation between A and B if no traction acts on the sides x₂ = ±h;

(c) Express the resultant force on the free end x₃ = 0 in terms of A, B and C and hence, with (a) and (b), show that C= -3P/4ah³.

5. The stress in the cantilever beam of Problem 4 is now given by

Where C and D are constants.

(a) Show that this stress satisfies the equations of equilibrium with no body forces;

(b) Show that the traction on the surface x₂ = -h is zero;

(c) Find the magnitude and direction of the traction on the surface x₂= h, and hence the total force on this surface;

(d) Find the resultant force on the surface x₃ = l. Prove that the traction on this surface exerts zero bending couple on it provided that C(5l² -2h²)+ 5D =0.

6. The stress components in a thin plate bounded by x₁ = ±L and x₂ = ±h are given by

T₁₁ = Wm²cos(1/2πx₁/L)sinh mx₂,

T₂₂ = -1/4Wπ²L^-2cos(1/2πx₁/L)sinh mx₂

 T₁₂ = 1/2WπmL^-1 sin(1/2πx₁/L)cosh mx₂,

T₁₃ = T₂₃ = T₃₃ = 0

where W and in are constants.

 (a) Verify that this stress satisfies the equations of equilibrium, with no body forces;

(b) Find the tractions on the edges x₂ = h and x₁ = -L;

(c) Find the principal stress components and the principal axes of stress at (0, h, 0) and at (L, 0, 0).

7. A solid circular cylinder has radius a and length L, its axis coincides with the x₃-axis, and its ends lie in the planes x₃= -L and x₃=0. The cylinder is subjected to axial tension, bending and torsion, such that the stress tensor is given by

where α, β, γ and δ are constants.

(a) Verify that these stress components satisfy the equations of equilibrium with no body forces;

(b) Verify that no traction acts on the curved surface of the cylinder;

(c) Find the traction on the end x₃=0, and hence show that the resultant force on this end is an axial force of magnitude πα²β, and that the resultant couple on this end has components (1/4πα⁴ δ, -1/4πα⁴γ, 1/2 πα⁴α) about the x₁, x₂ and x₃-axes;

(d) For the case in which bending is absent (γ = 0, δ = 0) find the principal stress components. Verify that two of these components are equal on the axis of the cylinder, but that elsewhere they are all different provided that α ≠ 0. Find the principal stress direction which corresponds to the intermediate principal stress component.

8. A cylinder whose axis is parallel to the x₃-axis and whose normal cross-section is the square -a ≤x₁ ≤a, -a ≤ x₂ ≤ a, is subjected to torsion by couples acting over its ends x₃=0 and x₃= L. The stress components are given by T₁₃ = ∂Ψ/∂x₂, T₂₃ = -∂Ψ/∂x₁, T₁₁ = T₁₂ = T₂₂ = T₃₃ = 0, where Ψ = Ψ (x₁, x₂).

(a) Show that these stress components satisfy the equations of equilibrium with no body forces;

(b) Show that the difference between the maximum and minimum principal stress components is 2{(∂Ψ/∂x₁)² + (∂Ψ/∂x₂)²}^1/2, and find the principal axis which corresponds to the zero principle stress component;

(c) For the special case Ψ = (x₁²- a²)(x₂²- a²) show that the lateral surfaces are free from traction and that the couple acting on each end face is 32a⁶/9.

9. Let n be a unit vector, t⁽ⁿ⁾ the traction on the surface normal to n, and S the magnitude of the shear stress on this surface, so that S is the component of t⁽ⁿ⁾ perpendicular to n. Prove that as n varies, S has stationary values when n is perpendicular to one of the principal axes of stress, and bisects the angle between the other two. Prove also that the maximum and minimum values of S are ± 1/2(T₁- T₃).