Help with the following ones:
2-2-1, 2-2-3, 2-2-4
Q2_1: QUIZ 2, PROBLEM #1
The shaft ABC is a solid circular cylinder of constant outer diameter 2R and length 3L. The shaft is fixed between walls at A and C and it is composed of two segments made of different materials. The left third of the shaft (AB) is composed of a linear isotropic elastic material of shear modulus G0, while the right two-thirds of the shaft (BC) is composed of a different linear elastic material of shear modulus 2G0. The right segment, BC, is subjected to a uniform distributed torque per unit length t0[N⋅m/m].
Obtain symbolic expressions in terms of R, G0, L, t0, and x for the quantities below. In your answers, leave rationals as fractions and enter G0, t0, and π as G_0, t_0 and pi, respectively.
Q2_1_1 : 100.0 POINTS
The x-component of the reaction torque at C:
TCx= unanswered
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Q2_1_2 : 60.0 POINTS
The twist rate dφdx(x), and the position x0 along the shaft where the twist rate goes to zero (dφdx(x0)=0):
for0≤x<L,dφdx(x)= unanswered
forL<x≤3L,dφdx(x)= unanswered
dφdx(x0)=0atx0= unanswered
You have used 0 of 4 submissions
Q2_1_3 : 60.0 POINTS
The maximum absolute value of the shear stress in the shaft (τmax) and its location (rτmax, xτmax):
τmax= unanswered
rτmax= unanswered
xτmax= unanswered
You have used 0 of 4 submissions
Q2_1_4 : 100.0 POINTS
The maximum value of the rotation field φ(x) along the shaft (φmax), and the position along the shaft where the maximum rotation occurs (xφmax):
φmax= unanswered
xφmax= unanswered
Q2_2: QUIZ 2, PROBLEM #2
The composite beam AB, of length L=2m, is free at A (x=0) and fixed at B (x=L) and is composed of a round cylindrical core of constant radius R0=1cm bonded inside a sleeve of thickness R0 (outer radius 2R0=2cm). The beam is loaded, as indicated, by a downward linearly varying distributed load per unit length of magnitude
q(x)=q0xL,withq0=2.76kN/m.
The material moduli are:
For the core, EC=70GPa=E0
For the sleeve, ES=210GPa=3E0
Q2_2_1 : 60.0 POINTS
Obtain a symbolic expression for the internal bending moment resultant in terms of L, q0 (enter as q_0), and x:
M(x)= unanswered
You have used 0 of 4 submissions
Q2_2_2 : 60.0 POINTS
Obtain a symbolic expression for the effective section stiffness of the beam (EI)eff in terms of R0 and E0 (enter these as R_0 and E_0, leave rationals as fractions, and enter π as pi):
(EI)eff= unanswered
You have used 0 of 4 submissions
Q2_2_3 : 60.0 POINTS
Obtain symbolic expressions for the curvature at the neutral axis 1ρ(x) and the slope ϑ(x) of the beam in terms of L, q0, R0, E0, and x (again, leave rationals as fractions and enter π as pi):
1ρ(x)= unanswered
ϑ(x)= unanswered
You have used 0 of 4 submissions
Q2_2_4 : 70.0 POINTS
Obtain the numerical value (in cm) for the displacement at the free end, vA=v(x=0):
vA= cm unanswered
You have used 0 of 4 submissions
Q2_2_5 : 70.0 POINTS
Obtain the numerical values in MPa for the maximum tensile stresses in the core (σmax,C) and in the sleeve (σmax,S):
σmax,C= MPa unanswered
σmax,S= MPa unanswered
You have used 0 of 4 submissions
No one has answered this question yet.
To solve the given problems, you need to apply the relevant equations and formulas related to mechanics and material properties. Here is a step-by-step guide on how to approach each problem:
Q2_1_1:
To find the x-component of the reaction torque at C (TCx), first find the total torque at C due to the applied torque (t0) on segment BC. Then, since the shaft is in equilibrium, the reaction torque at C will be equal in magnitude but opposite in direction. Use the equation:
TCx = -t0 * L * (2/3)
Q2_1_2:
This problem requires finding the twist rate (dφdx) at different regions of the shaft and the position (x0) where the twist rate goes to zero (dφdx(x0)=0). Use the equations:
For 0 ≤ x < L:
dφdx = (-t0 * L * x)/(2G0 * R^2)
For L ≤ x ≤ 3L:
dφdx = (-t0 * L)/(4G0 * R^2)
To find x0, set the twist rate equation equal to zero and solve for x0.
Q2_1_3:
To find the maximum absolute value of the shear stress in the shaft (τmax) and its location (rτmax, xτmax), use the equation:
τmax = (t0 * L)/(3πR^3G0)
To find the location where τmax occurs, substitute τmax into the twist rate equation and solve for x.
Q2_1_4:
To find the maximum value of the rotation field (φmax) along the shaft and the position (xφmax) where the maximum rotation occurs, use the equations:
For 0 ≤ x < L:
φmax = (-t0 * L^2 * x)/(4G0 * R^2)
For L ≤ x ≤ 3L:
φmax = (-t0 * L^2)/(16G0 * R^2)
To find xφmax, substitute φmax into the twist rate equation and solve for x.
Q2_2_1:
To find the internal bending moment resultant (M(x)), you need to consider the different bending moments created by the distributed load (q(x)) and the material properties of the core and sleeve. Use the equations:
For 0 ≤ x ≤ R0:
M(x) = q0 * x * (3E0 * π * R0^3)/(4L)
For R0 < x ≤ L:
M(x) = q0 * x * (15E0 * π * R0^4)/(32L)
Q2_2_2:
To find the effective section stiffness of the beam (EI)eff, you need to consider the properties of the core and sleeve. Use the equations:
For 0 ≤ x ≤ R0:
(EI)eff = (E0 * π * R0^4)/4
For R0 < x ≤ L:
(EI)eff = (15E0 * π * R0^4)/32
Q2_2_3:
To find the curvature at the neutral axis (1/ρ(x)) and the slope (θ(x)) of the beam, you need to consider the bending moments and material properties. Use the equations:
For 0 ≤ x ≤ R0:
1/ρ(x) = (q0 * x)/(2E0)
θ(x) = (q0 * x^2)/(2E0)
For R0 < x ≤ L:
1/ρ(x) = (3q0 * x)/(8E0)
θ(x) = (3q0 * x^2)/(4E0)
Q2_2_4:
To find the displacement at the free end (vA), you need to consider the applied load, beam properties, and boundary conditions. Use the equation:
vA = (5q0 * L^4)/(16E0 * R0^3)
Q2_2_5:
To find the maximum tensile stresses in the core (σmax, C) and in the sleeve (σmax, S), you need to consider the bending moments and material properties. Use the equations:
For the core:
σmax, C = (3q0 * R0)/(2π)
For the sleeve:
σmax, S = (15q0 * R0)/(4π)
Remember to substitute the given values for q0, E0, R0, L, and x in the respective equations to obtain numerical values.
Please note that these are general guidelines for solving the problems. Make sure to double-check the equations and assumptions before submitting your answers.