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Mechanics of MaterialsMAE 243 (Section 002)

Spring 2008

Dr. Konstantinos A. Sierros

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Problem 3.4-7

Four gears are attached to a circular shaft and transmit the

torques shown in the figure. The allowable shear stress in the

shaft is 10,000 psi.

(a) What is the required diameter d of the shaft if it has a solid

cross section?

(b) What is the required outside diameter d if the shaft is hollowwith an inside diameter of 1.0 in.?

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Problem 3.5-7

The normal strain in the 45direction on the surface of a circular

tube (see figure) is 880 x (10^-6) when the torque T = 750 lb-in.

The tube is made of copper alloy with G = 6.2 x 910^6) psi.

If the outside diameter d2of the tube is 0.8 in., what is the inside

diameter d1?

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4.4: Relationships between loads, shear forces and bending

moments

Copyright 2005 by Nelson, a division of Thomson Canada Limited

FIG. 4-10

Element of a beam

used in deriving therelationshipsbetween loads, shearforces, and bendingmoments. (All loadsand stress resultants areshown in their positivedirections.)

Distributed loads and concentrated loads are positive when they act downward

on the beam and negative when they act upward

A couple acting as a load on a beam is positive when it is counterclockwise

and negative when it is clockwise

Shear forces V and bending moments M acting on the sides of the element are

shown in their positive directions

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4.4: Distributed loads

Consider a distributed load of intensity q and its relationship to theshear force V

Consider the moment equilibrium of the beam element we can relate

the shear force V with the bending moment M

Moments from left hand side

Counterclockwise +ve

Discarding products of

differentials because they are

negligible compared to the

other terms

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4.4: Concentrated loads

Consider a concentrated load P acting on the beam element

It can be shown that the bending moment M does not change as we pass

through the point of application of a concentrated load

At the point of application of a concentrated load P, the rate of change dM/dx

of the bending moment decreases abruptly by an amount equal to P

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4.5: Shear force and bending-moment diagrams


FIG. 4-11

Shear-force andbending-momentdiagrams for a simplebeam with a

concentrated load

When designing a beam, we need to know how the shear forces and bendingmoments vary throughout the length of the beam. Minimum and maximum

values are of special importance

Information of this kind is provided by graphs in which the shear force and

bending moment are plotted as ordinates (y coordinate) and the distance x along

the axis of the beam is plotted as the abscissa (x coordinate)

Shear force and

bending moment

diagrams

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Concentrated load

Simply supported beam AB and concentrated load P (fig 4-11a). We can

determine the reactions of the beam


FIG. 4-11

Shear-force andbending-momentdiagrams for a simplebeam with aconcentrated load

Cut through the beam at a cross-section to

the left of the load P and at distance x fromthe support at A and draw FBD (fig 4-11b)

(0 < x < )

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FIG. 4-11

Shear-force andbending-momentdiagrams for a simplebeam with aconcentrated load


Concentrated load

Next cut through the beam to the right of the load P ( < x < L) and draw a

FBD (fig 4-11c)

( < x < L)

and

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