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=
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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=
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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)=
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v(x=L)=
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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=
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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=
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No one has answered this question yet.

Anyone....... plz post answers I require them too! Thanks in Advance!

8-2,8-2-1, 8-3, 8-3-2X, 8-4-1a,8-4-2???

does anyone know the 8-4?

To solve these engineering homework problems, you will need to apply principles of mechanics and structural analysis. Here are step-by-step instructions on how to solve each problem:

HW8_1_1:
To obtain a symbolic expression for the maximum magnitude of stress, σmax, in the beam and the coordinate of the cross section, xmax, where σmax occurs, you need to consider the bending moment and the moment of inertia of the tapered square cross-section. The maximum stress occurs at the section with the maximum bending moment. Use the following steps:
1. Determine the moment of inertia of the tapered square cross-section as a function of x, using the area moment of inertia equation for a square cross-section.
2. Determine the bending moment as a function of x using the equation for a cantilever beam subjected to a concentrated load.
3. Substitute the expression for the moment of inertia and bending moment into the formula for stress to obtain a symbolic expression for σmax.
4. To find xmax, solve the equation for the bending moment at the section where σmax occurs and substitute the expression for x from the previous step.

HW8_1_2:
To obtain the 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, you need to compare the stress formula in HW8_1_1 to the stress at section B. Determine the bending moment at section B and substitute it into the formula for stress. Compare the expressions for σmax and σB to calculate the ratio.
For the round cross-section case, you would need to calculate the moment of inertia of the round cross-section using the equation for moment of inertia of a circular cross-section and then repeat the steps mentioned above to calculate σmax and compare it to the stress adjacent to support B, σB.

HW8_2_1:
To determine the maximum allowable magnitude of the distributed load on the beam, you need to ensure that the stress in both the steel and plastic remain below their respective allowable limits.
1. Calculate the moments of inertia for the steel, plastic, and the total cross-sectional area.
2. Determine the stress in the steel and plastic using the formulas for stress in a beam subjected to a distributed load.
3. Set the stress in the steel equal to the allowable stress and solve for the distributed load.
4. Substitute the values of the dimensions and allowable stresses to find the numerical value of the maximum allowable magnitude of the distributed load on the beam.

HW8_3:
To obtain symbolic expressions for the beam slope and vertical displacement at the free end (x=L) and at the beam mid-span (x=L/2), you need to solve the differential equations of the beam deflection.
1. Set up the differential equation of beam deflection using the flexure formula.
2. Integrate the differential equation to obtain a general solution.
3. Use the boundary conditions of the beam to determine the constants of integration.
4. Substitute the values of q0, L, and EI into the resulting expressions to find the slope and vertical displacement at the desired locations.

HW8_4_1:
To obtain symbolic expressions for the maximum tensile stress in the beam, σ+max, and for the position on the cross-section, ymax, where it occurs, you need to consider the bending moment and the moment of inertia of the statically indeterminate beam.
1. Apply the principle of superposition to determine the moment at the section where the concentrated moment M0 is applied. Consider the beams on either side of the applied moment separately and solve for the reactions and bending moments at the support.
2. Determine the moment of inertia of the given cross-section.
3. Apply the flexure formula and solve for the maximum tensile stress and the position on the cross-section where it occurs.

HW8_4_2:
To obtain a symbolic expression for the beam slope at the section x=2L, where the moment is applied (ϑM=ϑ(x=2L)), you need to solve the differential equation of beam deflection considering the applied moment at x=2L.
1. Set up the differential equation of beam deflection using the flexure formula.
2. Apply the principle of superposition and consider the beams on either side of the applied moment separately.
3. Integrate the differential equation to obtain a general solution.
4. Use the boundary conditions of the beam to determine the constants of integration.
5. Substitute the values of E, L, h, b, and M0 into the resulting expression to find the slope at the desired location.

Remember to carefully follow these steps and apply the appropriate formulas to accurately solve each problem.