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=
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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.

Plz tell the answers...... I require them too! :(

people do not want to share I guess..otherwise, the problems are too difficult?

only TWO days left for the deadline..please share some of the answers..

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

does anyone know the 8-4?

To obtain the answers to the questions, we need to apply the relevant equations and principles of mechanics. Here's how you can approach the problem:

For HW8_1_1:
1. Calculate the maximum moment in the cantilevered beam based on the applied load P.
2. Use the moment equation for a rectangular cross-section beam to find the maximum bending stress.
3. Substitute the appropriate values into the equations to obtain a symbolic expression for the maximum magnitude of stress, σmax.
4. To find the coordinate xmax where σmax occurs, consider the linearly varying side length of the square beam and the geometry of the cantilevered beam.

For HW8_1_2:
1. Calculate the maximum bending stress in the beam, σmax, using the obtained symbolic expression in HW8_1_1.
2. Calculate the stress on the beam cross-section adjacent to support B, σB, using the properties of the square cross-section.
3. Determine the ratio between σmax and σB. Note down the ratio.
4. Consider the scenario where the beam has a round cross-section with varying radius and determine if the ratio would be the same or different. Note down your answer.

For HW8_2_1:
1. Use the formula for the moment of inertia to find the moment of inertia of the steel pipe.
2. Use the formula for the moment of inertia and the given dimensions to find the moment of inertia of the plastic liner.
3. Use the modulus of elasticity and the moment of inertia values to calculate the stress in the steel and plastic.
4. Determine the maximum allowable magnitude of the distributed load by equating the stresses to the allowable stresses given in the problem.

For HW8_3:
1. Use the moment equation for a simply supported beam with a linearly varying distributed load to determine the beam slope at the mid-span (x=L/2) and the free end (x=L).
2. Use the double integration method to find the vertical displacement at the mid-span and the free end.

For HW8_4_1:
1. Calculate the moment of inertia of the cross-section of the beam.
- For a rectangular cross-section, use the formula for the moment of inertia for a rectangle.
2. Use the bending moment formula for a simply supported beam to find the maximum tensile stress in the beam.
3. Substitute the appropriate values into the equation to obtain a symbolic expression for the maximum tensile stress, σ+max.
4. To find the position ymax where σ+max occurs, consider the geometry and dimensions of the beam.

For HW8_4_2:
1. Use the bending moment equation for a simply supported beam with a concentrated moment to find the slope at the section x=2L where the moment is applied.
2. Substitute the appropriate values into the equation to obtain a symbolic expression for the beam slope at x=2L, ϑM.

Using these steps, you can solve each of the questions in the assignment by plugging in the relevant values and solving the equations.