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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
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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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4.5: Shear force and bending-moment diagrams
Concentrated load
Simply supported beam AB and concentrated load P (fig 4-11a). We can
determine the reactions of the beam
Copyright 2005 by Nelson, a division of Thomson Canada Limited
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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Copyright 2005 by Nelson, a division of Thomson Canada Limited
FIG. 4-11
Shear-force andbending-momentdiagrams for a simplebeam with aconcentrated load
4.5: Shear force and bending-moment diagrams
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