Mechanics of Materials

Mechanics of Materials – MAE 243 (Section 002) Spring 2008 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 d2 of the tube is 0.8 in., what is the inside diameter d1?

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

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 Discarding products of differentials because they are negligible compared to the other terms Moments from left hand side Counterclockwise +ve

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 Mo • From equilibrium of the element in the vertical direction we obtain V1 = 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 M1 = -Mo. This equation shows that the bending moment decreases by Mo 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

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)

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) (α < x < L) …and

Mechanics of Materials – MAE 243 (Section 002) Spring 2008