HW8_1: CANTILEVERED BEAM WITH TAPERED SQUARE CROSS SECTION
A cantilevered beam of length L has a square cross section of side length linearly varying from hA at the free end to hB=3hA, at the fixed support. A concentrated load, P, is applied at the free end A as indicated.
Take the x axis with origin at A, oriented from A to B as indicated.
HW8_1_1 : 20.0 POINTS
Obtain a symbolic expression in terms of P, L, hA (enter as h_A) for the maximum magnitude of stress, σmax, in the beam, and for the coordinate of the cross section, xmax, where σmax occurs:
σmax=
unanswered
xmax=
unanswered
HW8_1_2 : 10.0 POINTS
Obtain the (dimensionless) ratio between the maximum magnitude of bending stress in the beam, σmax and the largest magnitude of stress on the beam cross section immediately adjacent to support B, σmax(x=L)=σB. Would this ratio change if instead of having a square cross section the beam had a round cross section of outer radius R(x) linearly varying from RA to RB=3RA?
σmaxσB=
unanswered
σmaxσB ratio for round cross section would be:
same different
HW8_2: SIMPLY-SUPPORTED LINED STEEL PIPE WITH DISTRIBUTED LOAD
A plastic lined steel pipe supports the constant distributed load q [N/m] over the central portion of beam AB, with a=4 m. The steel pipe has outer diameter d3=100 mm, and inner diameter d2=94 mm. the plastic liner has inner diameter d1=82 mm. The modulus of elasticity of the steel is 75 times the modulus of the plastic.
HW8_2_1 : 30.0 POINTS
Given that the allowable stress in the steel is 350 MPa, and the allowable stress in the plastic is 6 MPa, determine the numerical value, in Nm, of the maximum allowable magnitude of distributed load on the beam:
N/m unanswered
HW8_3: DEFLECTION OF A CANTILEVER BEAM WITH VARYING DISTRIBUTED LOAD
The cantilevered beam AB is fixed at the wall at A and subjected to a (downward) distributed load linearly varying from zero at the free end B to a maximum magnitude q0 [N/m] at the wall, A. The beam has length L and uniform section stiffness, EI.
Obtain symbolic expressions, in terms of q0 (enter as q_0), L, and EI ( enter as EI without the multiplication symbol) for the beam slope and vertical displacement at the free end (x=L) and at the beam mid-span (x=L/2)
HW8_3 : 30.0 POINTS
ϑ(x=L2)=
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v(x=L2)=
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ϑ(x=L)=
unanswered
v(x=L)=
unanswered
STATICALLY INDETERMINATE BEAM WITH CONCENTRATED MOMENT
Beam AB is homogeneous, with modulus E. The beam has known length 3L, height h, and width b. The beam is fixed at A, and simply supported at B. Take the x axis on the neutral axis, oriented from left to right, with origin at the wall, A. The beam is subjected to a concentrated moment of magnitude M0 applied at x=2L as indicated.
HW8_4_1 : 20.0 POINTS
Obtain symbolic expressions, in terms of h, b, M0, (enter as M_0) for the maximum tensile stress in the beam, σ+max, and for the position on the cross section, ymax, where it occurs.
σ+max=
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ymax=
unanswered
HW8_4_2 : 10.0 POINTS
Obtain a symbolic expression, in terms of E, L, h, b, M0, (enter as M_0) for the beam slope at the section x=2L, where the moment is applied, ϑM=ϑ(x=2L):
ϑM=
unanswered
By the Oath of David Thou shalt not Cheat.
8-2,8-2-1, 8-3, 8-3-2X, 8-4-1a,8-4-2???
To obtain the symbolic expressions for the given questions, we need to apply principles of mechanics and engineering analysis. Here are the explanations on how to approach each question:
HW8_1_1: To find the maximum magnitude of stress (σmax) in the cantilevered beam and the coordinate (xmax) where it occurs, you can use the equation for stress in bending:
σ = My/I
where σ is the stress, M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the cross-section.
For a square cross-section, the moment of inertia can be expressed as I = (h^4)/12.
The bending moment can be calculated as M = P(L-x), where P is the concentrated load applied at the free end.
To find σmax, you need to maximize the moment of the cross-section, which occurs at the fixed support (x = 0).
Thus, σmax = P(hA^3/12) / (hA^2/6) = 2P/hA.
The coordinate where σmax occurs (xmax) is at the fixed support (x = 0).
HW8_1_2: The ratio between the maximum magnitude of bending stress (σmax) and the largest magnitude of stress on the adjacent support (σmax(x = L) = σB) can be calculated as σmax/σB.
For a square cross-section, σB occurs at the fixed support and can be calculated as σB = 6P/ (hA^2).
To determine if this ratio would change for a round cross-section with a linearly varying outer radius, you would need to perform a similar analysis for the round cross-section.
In this case, the moment of inertia for a round cross-section changes along the length of the beam. You need to integrate the moment of inertia using the varying outer radius function. The bending moment and stress calculations would also change accordingly.
Comparing the ratios for the square cross-section and round cross-section would indicate if the ratio changes or remains the same.
HW8_2_1: To determine the maximum allowable magnitude of the distributed load on the beam, you need to consider the stress limits for both the steel and the plastic.
The stress in the steel can be calculated using the formula: stress_steel = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the steel pipe cross-section.
Similarly, the stress in the plastic can be calculated using the same formula, but with the appropriate values for M, y, and I for the plastic liner.
Equate the calculated stresses to their respective allowable stress limits and solve for the maximum distributed load (q) on the beam.
HW8_3: To obtain the symbolic expressions for beam slope (ϑ) and vertical displacement (v) at different locations on the cantilever beam, you need to integrate the equations of beam deflection.
For a cantilever beam with a linearly varying distributed load, you can use the standard differential equations for beam deflection in terms of q0, L, and EI.
By integrating these equations and applying the appropriate boundary conditions, you can find the symbolic expressions for ϑ(x=L/2), v(x=L/2), ϑ(x=L), and v(x=L).
HW8_4_1: To find the maximum tensile stress (σ+max) in the statically indeterminate beam and the position (ymax) where it occurs, you need to consider the effects of bending and shear stresses.
The bending stress can be calculated using the formula: σ_bending = My/I, where M is the bending moment, y is the distance from the neutral axis, and I is the moment of inertia of the cross-section.
The shear stress can be calculated using the formula: τ_shear = VQ/It, where V is the shear force, Q is the first moment of area, I is the moment of inertia, and t is the thickness of the cross-section.
To find the maximum tensile stress (σ+max), consider the combination of bending and shear stresses.
The position (ymax) where it occurs can be determined by calculating the values of y that result in the maximum tensile stress.
Consider the cross-section of the beam and evaluate the stresses at different locations to find the maximum tensile stress and its position.
HW8_4_2: The beam slope (ϑM) at the section where the concentrated moment is applied can be calculated using the equation:
ϑ = d^2v/dx^2,
where v is the vertical displacement and x is the distance along the beam.
To obtain the symbolic expression for ϑM, consider the properties of the beam (h, b) along with E, L, and M0 in the equation.
Remember, these explanations provide the general approach to solving the questions. The actual calculations may involve additional equations, integration, and substitution.