need help for HW

hello Y’ll,I am taking advance mechanics of materials and I need help solving this Homework. the Homework file uploaded below.

HW7: CE-4440 Advanced Strength and Applied

Elasticity

Fall 2020

Due, Tues. Dec. 1st

S.Moorthy, 3255F PFT

Phone: 578-4846,Email:[email protected]

1) If the torsional solution for a cylinder of equilateral triangular cross-section shown in figure, is

to be determined using the Prandtl stress function Φ = k(x−√3y− 2 3h)(x+√3y− 2 3h)(x+ 1 3h),

(a) Verify that the stress function Φ satisfies Poisson′s equation

(b) Calculate the shear stress τzx(x, y) and τzy(x, y) for given torque T acting on the cross-section

of the beam

(c) Calculate the angle of twist per unit length θ for given torque T and calculate the effective

polar moment of inertia Je of the cross-section

(d) Calculate the warping function w(x, y) for the cross-section (Grad students only)

.

.

3

3

3

h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). . h/ 2h −−−−− h −−−− 3 3 h 2h x=− y+−−−− 2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in figure. Calculate the shear stress distribution and the angle of twist per unit length for each cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last cross-section). .

h/

2h

−−−−−

h

−−−−

3

3

h

2h

x=− y+−−−−

2) An internal torque T = 2.5 mN · m acts on each of the three thin cross-sections shown in

figure. Calculate the shear stress distribution and the angle of twist per unit length for each

cross-section. (Assume G = 40 GPa and t1 = t2 = t4 = 2 mm and t3 = t5 = 3 mm for the last

cross-section).

.

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