Dr. Konstantinos A. Sierros

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 hollow with an inside diameter of 1.0 in.?

Problem 3.5-7
The normal strain in the 45 ° direction 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 d
2 of the tube is 0.8 in., what is the inside diameter d
1
?

4.4: Relationships between loads, shear forces and bending moments
• 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
FIG. 4-10
Element of a beam used in deriving the relationships between loads, shear forces, and bending moments. (All loads and stress resultants are shown in their positive directions.)
Copyright 2005 by Nelson, a division of Thomson Canada Limited

4.4: Distributed loads
• Consider a distributed load of intensity q and its relationship to the shear 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

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

4.4: Loads in the form of couples
• The last case to be considered is a load in the form of a couple M o
• From equilibrium of the element in the vertical direction we obtain V
1
= 0 which shows that the shear force does not change at the point of application of a couple
• If we take equilibrium of moments we obtain M
1
= -M o
. This equation shows that the bending moment decreases by M o as we move from left to right through the point of load application. Thus, the bending moment changes abruptly at the point of application of a couple

4.5: Shear force and bending-moment diagrams
•
When designing a beam, we need to know how the shear forces and bending moments 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
FIG. 4-11
Shear-force and bending-moment diagrams for a simple beam with a concentrated load
Copyright 2005 by Nelson, a division of Thomson Canada Limited

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
•
Cut through the beam at a cross-section to the left of the load P and at distance x from the support at A and draw FBD (fig 4-11b)
(0 < x < α)
FIG. 4-11
Shear-force and bending-moment diagrams for a simple beam with a concentrated load
Copyright 2005 by Nelson, a division of Thomson Canada Limited

4.5: Shear force and bending-moment diagrams
Concentrated load
• Next cut through the beam to the right of the load P (α < x < L) and draw a
FBD (fig 4-11c)
…and
(α < x < L)
FIG. 4-11
Shear-force and bending-moment diagrams for a simple beam with a concentrated load
Copyright 2005 by Nelson, a division of Thomson Canada Limited