ME 323: Mechanics of MaterialsFall 2020

Homework 5Due Wednesday, October 7

Problem 5.1 (10 points)

Beam ABCD is loaded as shown below. The beam is simply supported (pin support at A and roller at B).

(a) Determine the support reactions at A and B.

(b) Sketch the shear force and bending moment diagrams. Label the shear force and bending moment val-

ues at cross-sections A, B, C, and D, and at the locations of maximum and minimum values. Mention

the order of the curve in each section (first order/linear,second order/quadratic, third order/cubic,

etc) as well.

(c) Indicate the coordinate x, within segment AB, at which the bending moment is maximum.

Parameters: L=6 ft, p0=15 kips/ft, PC=40 kips, MC=30 kip.ft and MD=60 kip.ft

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ME 323: Mechanics of MaterialsFall 2020

Homework 5Due Wednesday, October 7

Problem 5.2(10 points)

A cantilever beam with a circular cross section (diameter d) is loaded as shown below. A uniformly

distributed load p0 is applied over AB, a concentrated couple MC is applied at C, and a concentrated force

PD is applied at the end of the beam at D.

(a) Construct the shear force and bending moment diagrams for the beam.

(b) Determine the maximum flexural stress, in magnitude, of the beam.

(c) Identify the positions on the beam where the maximum tensile flexural stress and maximum com-

pressive flexural stress occur, i.e., determine the coordinate x of the cross-section and the coordinates

(y, z) within such cross-section.

Parameters: d = 20 mm, p0 = 1 kN/m, MC = 10 kN.m, and PD = 3 kN

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ME 323: Mechanics of MaterialsFall 2020

Homework 5Due Wednesday, October 7

Problem 5.3(10 points)Consider the shear force and bending moment diagrams from Problem 5.1.

Part 1: The beam has the solid circular cross section in Fig. 1 with diameter d = 12 in.

(a) Determine the maximum flexural stress, in magnitude, and the location (x,y) where it occurs.

(b) Sketch the flexural stress distribution on the cross section identified in (a).

(c) Determine the maximum transverse shear stress and the location (x,y) where it occurs.

Part 2: The beam has the hollow circular cross section, with outer diameter d0 = 12 in. and inner

diameter di = 10 in.

(a) Design the cross section so that the hollow beam has the same maximum flexural stress as the

solid beam, found in Part 1.(a). Let ratio of the outer and inner diameter be 1.2 (do = 1.2di).Determine both the outer and inner diameters required to achieve this design goal.

(b) Compare the areas of the new hollow cross section and the solid cross section of Part 1 and

indicate which one is larger. As a designer, would you choose the hollow or the solid cross

section?

Part 3: The beam has the triangular cross section with b = 12 in.

(a) Determine the transverse shear stress at the centroid of the cross section located just to the left

of point B on the beam.

Figure 1: Cross section of beam for Part� 1 Cross section of beam for Part� 2 Cross section of beam for Part� 3.

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ME 323: Mechanics of MaterialsFall 2020

Homework 5Due Wednesday, October 7

Problem 5.4 Conceptual (5 points)

A shear force V and bending moment M act at a cross section of a trapezoidal cross-sectioned

beam. Consider the five points (i), (ii), (iii), (iv) and (v) on the beam cross section, as shown above.

Match up the state of stress at each of these five points with the stress elements (a) through (o)

shown below. If you choose “(o) NONE of the above”, provide a sketch of the correct state of stress

for your answer.

The state of stress at point (i) is

The state of stress at point (ii) is

The state of stress at point (iii) is

The state of stress at point (iv) is

The state of stress at point (v) is

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