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

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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To obtain the symbolic expressions for the given quantities, we need to understand the given parameters and their relationship.

For HW6_1A:
The torsional stiffness of the shaft, KT, is given by the equation: KT = Q/Φ.
To express KT in terms of t, L, G, and dA, we need to determine the values of Q and Φ.

For HW6_1B:
The maximum shear strain on the generic x-section along the shaft, γmax(x), needs to be expressed in terms of t, L, G, Q, x, and dA.
To obtain this expression, we need to consider the given parameters and their relationship.

For HW6_1CX:
The challenge question asks to determine the torque Q0 required to obtain a maximum strain of 2% for given values of L, t, dA, and G.
After finding Q0, we need to plot γmax(x) using MATLAB code.

For HW6_2:
In this problem, a solid composite shaft is considered, which consists of an inner core and an outer sleeve.
We need to find symbolic expressions for different quantities, such as the axial torque resultant, rotation field, angle of twist, and maximum magnitude of shear stress, in terms of the given parameters.

For HW6_3:
In this problem, a round shaft is fixed at both ends and loaded by a constant distributed torque t0.
We need to determine the maximum value of the distributed load, t0,m, that can be applied to the shaft if we want to limit the rotation of the midsection to a maximum value φm.

To obtain the symbolic expressions for each of these problems, we need to analyze the given parameters and equations, apply relevant formulas, and manipulate the variables to get the desired expressions. It may be helpful to refer to relevant textbooks, lecture notes, or online resources to understand the concepts and equations involved in each problem.