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).