Determine the torque that initiates yielding in the box

Question

Question 1. A 60^o strain rosette is mounted on a aluminum alloy component (E = 10,200 ksi, v = 0.30) that is in a state of plane stress (σ_zz = σ_xz = σ_yz = 0).

The following readings are obtained from each gauge:

ε_a = 1,200(10^-6)
ε_b = -600(10^-6)
ε_c = 600(10^-6)

a. Determine ε_xx, ε_yy, and γ_xy.
b. Using Generalized Hooke's Law, determine the remaining three stress component (σ_xx, σ_yy, and τ_xy ) and the remaining three strain components (ε_zz, γ_xz, and γ_yz).

Question 2. Rigid bar ABC is supported by three bars (AD, BE, and CF) with the σ-ε behavior shown. Each support bar has an area of A = 0.05 in².

a. Determine the load, P_y, when the first bar yields and the downward displacement, δ_y, of rigid bar ABC with this load applied.

b. Determine the load, P_P, when all bars have yielded and the downward displacement, δP, of the rigid bar with this load applied.

c. Determine the load, P_U, when the first bar reaches its ultimate stress and the downward displacement, δU, of the rigid bar with this load applied.

Question 3. A certain isotropic material is subjected to a state of plane stress where σ_xx = 78 ksi, σ_yy = 24 ksi, and τ_xy = 36 ksi.

Determine the minimum yield stress in uniaxial tension, σY, this material must have if it has not yet yielded, using:

a. the maximum shear stress (Tresca) criterion.
b. the maximum distortional energy (Von Mises) criterion.

Question 4. Using Castigliano's Theorem, determine the vertical displacement at A for the beam shown. Take E = 12 GPa and G = 1.0 GPa.

Question 5. Due to a manufacturing error a box beam was fabricated with the four different wall thickness shown. The material used to make the box beam has a allowable shear stress of tallow = 140 MPa, and a shear modulus of G = 77.5 GPa.

a. Determine the torque that initiates yielding in the box beam.

b. Determine the total angle of twist if the box beam is 1 m long, and subjected to a torque of T = 2.0 kN.m.