Copyright © 2011 Pearson Education South Asia Pte Ltd Chapter Objectives To show how to use the method of sections for determining the internal loadings in a member. To introduce the concepts of normal and shear stress. To analyze and design members subject to axial load and direct shear. To define normal and shear strain, and show how they can be determined for various types of problems. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd In-Class Activities Reading Quiz Applications Normal Stress Shear Stress Normal Strain Shear Strain Cartesian Strain Components Concept Quiz Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ What is the normal stress in the bar if P=10 kN and 500 mm²? 0.02 kPa 20 Pa 20 kPa 200 N/mm² 20 MPa Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ (cont) What is the average shear stress in the internal vertical surface AB (or CD), if F=20 kN, and AAB=ACD=1000 mm²? 20 N/mm² 10 N/mm² 10 kPa 200 kN/m² 20 MPa Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ (cont) The center portion of the rubber balloon has a diameter of d = 100 mm. If the air pressure within it causes the balloon’s diameter to become d = 125 mm, determine the average normal strain in the rubber. 0.2 0.25 π 0.25 1.25 Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ (cont) 4. What is the unit of strain? mm mm/m Micron no unit Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd APPLICATIONS Where does the average normal stress occur? Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd APPLICATIONS (cont) Will the total shear force over the anchor length be equal to the total tensile force σtensile A in the bar? Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd APPLICATIONS Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd AVERAGE NORMAL STRESS Will the total shear force over the anchor length be equal to the total tensile force σtensile A in the bar? Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd NORMAL SHEAR STRESS Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 The bar in Fig. 7-14a has a constant width of 35 mm and a thickness of 10 mm. Determine the maximum average normal stress in the bar when it is subjected to the loading shown. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 (cont) Solution By inspection, different sections have different internal forces. Graphically, the normal force diagram is as shown. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 (cont) Solution By inspection, the largest loading is in region BC, Since the cross-sectional area of the bar is constant, the largest average normal stress is Copyright © 2011 Pearson Education South Asia Pte Ltd

DESIGN OF SIMPLE CONNECTION For normal force requirement For shear force requirement Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 The rigid bar AB shown in Fig. 1–29a is supported by a steel rod AC having a diameter of 20 mm and an aluminum block having a cross sectional area of 1800 mm2. The 18-mm-diameter pins at A and C are subjected to single shear. If the failure stress for the steel and aluminum is and , respectively, and the failure shear stress for each pin is , determine the largest load P that can be applied to the bar. Apply a factor of safety of FS=2. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 (cont) Solution The allowable stresses are There are three unknowns and we apply the equations of equilibrium, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 (cont) Solution We will now determine each value of P that creates the allowable stress in the rod, block, and pins, respectively. For rod AC, Using Eq. 1, For block B, Using Eq. 2, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 (cont) Solution For pin A or C, Using Eq. 1, When P reaches its smallest value (168 kN), it develops the allowable normal stress in the aluminium block. Hence, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd NORMAL STRAIN Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd SHEAR STRAIN Copyright © 2011 Pearson Education South Asia Pte Ltd

CARTESIAN STRAIN COMPONENTS The approximate lengths of the sides of the parallelepiped are The approximate angles between sides, again originally defined by the sides Δ x, Δ y and Δ z are Notice that the normal strains cause a change in volume of rectangular element, whereas the shear strain cause a change in shape Copyright © 2011 Pearson Education South Asia Pte Ltd

CARTESIAN STRAIN COMPONENTS (cont) Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 3 The slender rod creates a normal strain in the rod of where z is in meters. Determine (a) displacement of end B due to the temperature increase, and (b) the average normal strain in the rod. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 3 (cont) Solution Part (a) Since the normal strain is reported at each point along the rod, it has a deformed length of The sum along the axis yields the deformed length of the rod is The displacement of the end of the rod is therefore Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 3 (cont) Solution Part (b) Assumes the rod has an original length of 200 mm and a change in length of 2.39 mm. Hence, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 4 Due to a loading, the plate is deformed into the dashed shape shown in Fig. 2–6a. Determine (a) the average normal strain along the side AB, and (b) the average shear strain in the plate at A relative to the x and y axes. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 4 (cont) Solution Part (a) Line AB, coincident with the y axis, becomes line AB’ after deformation, thus the length of this line is The average normal strain for AB is therefore The negative sign indicates the strain causes a contraction of AB. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 4 (cont) Solution Part (b) As noted, the once 90° angle BAC between the sides of the plate, referenced from the x, y axes, changes to θ’ due to the displacement of B to B’. Since then is the angle shown in the figure. Thus, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd CONCEPT QUIZ 1) The thrust bearing is subjected to the loads as shown. Determine the order of average normal stress developed on cross section through B, C and D. C > B > D C > D > B B > C > D D > B > C Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd CONCEPT QUIZ (cont) 2) The rectangular membrane has an un-stretched length L1 and width L2. If the sides are increased by small amounts ΔL1 and ΔL2, determine the normal strain along the diagonal AB. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd CONCEPT QUIZ (cont) 3) The rectangular plate is subjected to the deformation shown by the dashed line. Determine the average shear strain γxy of the plate. Copyright © 2011 Pearson Education South Asia Pte Ltd