State briefly the upper and lower bound theorems of

Question

Question 1 -

A cantilever beam of length 2L, height 2h and depth t is in a state of plane stress. On its upper surface, it is subjected to a uniform pressure of magnitude w, as shown in Figure 1. The beam is supported at each end by a shearing force/unit depth of magnitude wL.

a) To solve this problem using the Theory of Elasticity, the material must satisfy five conditions (continuity, homogeneity, isotropy, elasticity, linearity), the deformation must be small, and the load must be conservative.

Describe these assumptions with one sentence (each), and for each one provide an example of an exception (e.g. material, deformation, or load type that violates the condition).

b) Provide all the conditions that an Airy stress function Φ(x, y) must satisfy in order to describe this loading condition correctly.

c) Show that the function Φ(x, y) = wx²/8 {(y/h)³ - 3(y/h) - 2} + Ay³ - wy⁵/40h³ satisfies the conditions above, calculating the parameter A where required.

d) Write down complete relationships for the stress field in terms of the beam's dimensions, applied load magnitude, and position (x, y), and describe how you would calculate the strain and displacement fields. Provide the relationships that are necessary for this procedure, but do not carry out the calculations.

Question 2 -

a) The Love-Kirchoff hypotheses are approximations that reduce the analysis of bending of plates from three to two dimensions. Describe briefly (in two sentences max) each of the hypotheses. State for what plate geometries they are reasonable.

b) A plate of dimensions a x b is simply supported on all four sides. It is subjected to a line load of magnitude q₀ at position y = y₀, see Figure 2. The Navier double series solution requires that this load be transformed into the form:

q(x, y) = ∑_m∑_n q_mn sin (mπx/a)sin(nπy/b).

Find the coefficients q_mn of this series for the given setup.

c) A long hollow torsional member has the trapezoidal cross-section shown in Figure 3a, with all walls having uniform thickness t throughout. Find an expression of the torsional stiffness of the member in terms of the dimensions a , t, and the shear stiffness G of the material.

d) The member of Figure 3a) is stiffened by adding two internal webs of length a, as shown in Figure 3b. Calculate the factor of increase of the torsional stiffness.

Question 3 -

a) (i) Explain why there is no volumetric change during the plastic deformation of metals, referring to the mechanism of plastic flow in these materials.

(ii) Can volumetric plastic straining occur in a porous metal? Provide your reasons.

b) A thin metal wire with a square cross section is stretched in a tensile testing machine. The machine loads the wire by applying a stress which increases linearly with time by σ_zz = at, where a is a proportionality constant. Between the grips of the machine, the wire passes through a sealed pressure vessel, which can apply any constant compressive pressure p. The wire material obeys the Swift hardening law σ^- = C(ε0 + ε^-)^N, where C, ε₀,  N are material constant, and ε- is the effective strain ε^- = √(2/3(ε'ijε'ij)). Yielding does not occur until σ^- > C(ε₀)N.

(i) Write the full stress tensor and deviatoric stress tensor for this setup, and show that the von Mises equivalent stress is given by the expression σ^- = p + σ_zz.

(ii) Using the Lévy-Mises flow rule, write out expressions for the strain rates ε^·_xx = dε_xx/dt, ε^·_yy, ε^·_zz, ε^-· in terms of p, σ_zz, λ^·.

(iii) Using the Swift hardening law, show that the equivalent strain follows the relationship: Nε^-·/ε₀+ε^- = a/p+at.

(iv) Show that the axial strain in the wire is given by the relationship: ε_zz = ε₀[(p+at/p+at₁)^1/N - 1], where t₁ is the time at which yielding first occurs.

Question 4 -

a) (i) State briefly the upper and lower bound theorems of plasticity.

(ii) Describe the differences between the upper bound and slip line field methods. When do they coincide?

b) A ductile metal bar of initial thickness 4h is extruded in a die to a final thickness 2h, as shown in Figure 4. The material has a shear yield strength k, and friction between the metal bar and the die walls can be assumed negligible.

(i) Use the upper bound technique, with planes of tangential velocity discontinuity as shown by the dashed lines in Figure 4, to calculate the dependence of the extrusion force F per unit depth upon the variable x.

(ii) From the above, obtain the minimum extrusion force per unit depth required to perform this operation.

(iii) Explain briefly why the above solution is not exact.