Copyright © 2011 Pearson Education South Asia Pte Ltd Chapter Objectives To determine the shear stress in straight beams subjected to transverse loading. To calculate the shear flow in beams composed of several members. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd In-class Activities Reading Quiz Applications Shear in a straight beam Shear formula Shear stresses in beams Shear flow in built-up beams Concept Quiz Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ 1) Which one of the following statements is not true? Shear stresses cause warping of cross section Warping effect is negligible for slender beams “Plane section remains plane” is valid for bending of deep beam Shear forces in beams cause non-linear shear-strain distributions over the cross section Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd READING QUIZ (cont) Which one of the following statements is not true? The shear formula should not be used to determine the shear stress… …on cross sections that are short or flat …at points of sudden cross-sectional changes …at a point on an inclined boundary None of the above 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

SHEAR IN A STRAIGHT BEAM Transverse shear stress always has its associated longitudinal shear stress acting along longitudinal planes of the beam. Copyright © 2011 Pearson Education South Asia Pte Ltd

SHEAR IN A STRAIGHT BEAM (cont) Effects of Shear Stresses: Warping of cross section Copyright © 2011 Pearson Education South Asia Pte Ltd

SHEAR IN A STRAIGHT BEAM (cont) Note: Warping” violates the assumptions of “plane section remains plane” in flexure and torsion formulae “Warping” is negligible in “slender beam” Copyright © 2011 Pearson Education South Asia Pte Ltd

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

Copyright © 2011 Pearson Education South Asia Pte Ltd SHEAR IN BEAMS Rectangular cross section Shear –stress distribution is parabolic Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd SHEAR IN BEAMS (cont) Wide-flange beam Shear-stress distribution is parabolic but has a jump at the flange-to-web junctions. Limitations on the use of shear formula Not on cross sections that are short or flat Not at points of sudden cross sectional changes (e.g. flange-to-web junction in wide flange beam) Not at a joint on an inclined boundary Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 A steel wide-flange beam has the dimensions shown in Fig. 7–11a. If it is subjected to a shear of V = 80 kN, plot the shear-stress distribution acting over the beam’s cross-sectional area. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 (cont) Solution The moment of inertia of the cross-sectional area about the neutral axis is For point B, tB’ = 0.3m, and A’ is the dark shaded area shown in Fig. 7–11c Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 1 (cont) Solution For point B, tB = 0.015m, and QB = QB’, For point C, tC = 0.015m, and A’ is the dark shaded area in Fig. 7–11d. Considering this area to be composed of two rectangles, Thus, Copyright © 2011 Pearson Education South Asia Pte Ltd

SHEAR FLOW IN BUILT-UP BEAM Shear flow ≡ shear force per unit length along longitudinal axis of a beam. q = shear flow V = internal resultant shear I = moment of inertia of the entire cross-sectional area Copyright © 2011 Pearson Education South Asia Pte Ltd

SHEAR FLOW IN BUILT-UP BEAM (cont) Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 Nails having a total shear strength of 40 N are used in a beam that can be constructed either as in Case I or as in Case II, Fig. 7–18. If the nails are spaced at 90 mm, determine the largest vertical shear that can be supported in each case so that the fasteners will not fail. Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 (cont) Solution Since the cross section is the same in both cases, the moment of inertia about the neutral axis is Case I For this design a single row of nails holds the top or bottom flange onto the web. For one of these flanges, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd EXAMPLE 2 (cont) Solutions Case II Here a single row of nails holds one of the side boards onto the web. Thus, Copyright © 2011 Pearson Education South Asia Pte Ltd

Copyright © 2011 Pearson Education South Asia Pte Ltd CONCEPT QUIZ 1) Which one of the following statements is not true? The shear center always lies on an axis of symmetry of the cross section. A crack along the member at a distance a/3 above/below the neutral axis will first start to appear due to shear. A crack along the member at the neutral axis will first start to appear due to shear. The centroid of the cross section coincides with the neutral axis. Copyright © 2011 Pearson Education South Asia Pte Ltd