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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)=

unanswered

v(x=L2)=

unanswered

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=

unanswered

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