CONICAL SHAFT WITH DISTRIBUTED TORSIONAL LOAD
The conical shaft AB is made of steel, with shear modulus G0. The shaft has length L and is free at A (where x=0) and fixed at B (where x=L). The shaft is solid, with a linear taper so that the radius of the generic (circular) cross section is defined by the equation R(x)=R0(xL), where R0 is the radius at the wall, B. The shaft is subjected to an applied distributed torque per unit length tx(x)=t0(xL)4, where t0 is the magnitude of the applied distributed torque at B (x=L).
The KNOWN quantities in this problem are:
L=2m
G0=70GPa
R0=2cm
t0=2kN⋅m/m
In symbolic expressions, do NOT substitute ANY of the given numerical values of the known quantities, enter L, G0, R0, and t0 as L, G_0, R_0, and t_0, respectively, and enter π as pi.
1)Obtain a symbolic expression for the axial torque resultant in the bar T(x) in terms of x, t0, and L.
T(x)=
2)Compute the numerical value of the reaction TBx at support B, in units of kN·m. (Note: You will need to provide your answer to the second decimal digit.)
TBx=
3)Obtain a symbolic expression for the twist rate along the shaft, dφdx, in terms of x, G0, t0, and R0.
dφdx(x)=
4)Compute the numerical magnitude (absolute value) of the maximum shear stress in the shaft, τmax, in units of MPa:
τmax=
Enter symbolic expressions for the location (r,x) at which this maximum shear stress occurs, in terms of R0 and L:
r=
x=
5)Obtain a symbolic expression for the angle of rotation of the free end A of the shaft in terms of t0, G0, R0, and L:
φ(x=0)=
Compute the numerical value of the angle of rotation of the free end, in units of radians. (Note: You will need to provide your answer to the third decimal digit.):
φ(x=0)=
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