tapered thin-wall circular shaft has constant wall thickness, t, length L, and diameters linearly varying between dA at the support A(x=0) and dB at its free end B(x=L). The shaft is homogeneous with shear modulus G

Question

HW6_1A : 20.0 POINTS

Obtain a symbolic expression for the torsional stiffness of the shaft KT=Q/Φ, in terms of t, L, G, dA (you will have factors of π in your answers: enter π as "pi" ):
KT=
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HW6_1B : 20.0 POINTS

Obtain a symbolic expression for the maximum shear strain on the generic x-section along the shaft, γmax(x), in terms of t, L, G, Q, x, dA (you will have factors of π in your answers: enter π as "pi" ):

γmax(x)=
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HW6_1CX : 0.0 POINTS

CHALLENGE QUESTION! (no points, just for fun!)
This challenge question is just for fun: it gives you no points, so you do not NEED to get the right solution. Indeed it is not even graded.

For L=0.5 m, t=2 mm, dA=4 cm, and G=70 GPa, obtain the torque Q0 that you need to apply to the shaft if you want to obtain a maximum value of 2% strain.

Then, use these values to plot γmax(x) by writing MATLAB code in the blank command window below. If you succeed, take a screenshot of your plot (NOT THE CODE) and post it in the discussion forum under the "Gamma- Challenge!" thread.

Note: be careful when you write your expression for γmax(x) in MATLAB. Remember that element-wise division needs the period, so if you need to define a vector y = 1/x where you want to obtain each element of y as the inverse of the corresponding element of x, you need to define y as: y = 1./x

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HW6_2: SOLID COMPOSITE SHAFT SUBJECTED TO DISTRIBUTED TORQUE

A composite shaft of length L is constructed from an inner core of radius R and modulus Gc=5G0, and a sleeve of outer radius 2√R and modulus Gs=G0, bonded together. One end of the shaft, B, is fixed and the other, A, is free to rotate as shown in the figure. A uniform distributed torque, tx(x)=t0 (t0 = constant with units of N⋅m/m), is applied to the shaft in the direction shown in the figure.

Obtain symbolic expressions in terms of R0, G0, L, t0, x for the following quantities.

(NOTE: you will have factors of π in your answers: enter π as "pi".)

HW6_2A : 10.0 POINTS

The axial torque resultant:
T(x)=
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HW6_2B : 10.0 POINTS

The rotation field φ(x) along the shaft:
φ(x)=
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HW6_2C : 10.0 POINTS

The angle of twist:
ΦAB=
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HW6_2D : 10.0 POINTS

The maximum magnitude of shear stress, τmax, in the shaft:
τmax=
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HW6_3: STATICALLY INDETERMINATE SHAFT UNDER DISTRIBUTED LOADING

The round shaft in the figure has length L and is fixed at both ends. The shaft is loaded by a constant distributed torque t0. The modulus of the material, G, and the polar moment of inertia of the cross section, Ip, are known.

HW6_3 : 40.0 POINTS

If we want to limit the rotation of the midsection of the shaft to a maximum value, φ(L2)=φm, what is the maximum value of the distributed load, t0,m, that can be applied to the shaft?

Provide your answer as a symbolic expression in terms of L, G, Ip, φm (write as "I_p" and "phi_m"):

t0,m=
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