009 The McGraw-Hill Companies, nc. All rights reserved. Fifth S E CHAPTER 6 MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members
Shear on the Horizontal Face of a Beam Element Consider prismatic beam For equilibrium of beam element Fx 0 H D C da H Note, Q M D M M D y da A C M C dm dx A A y da x V x Substituting, H x H q x shear flow 6-3
Shear on the Horizontal Face of a Beam Element Shear flow, q where Q H x first moment of y da A y A A' da area above second moment of full crosssection Same result found for lower area H q x Q Q 0 H H shear q first moment with respect to neutral axis flow y 1 6-4
Example 6.01 A beam is made of three planks, nailed together. Knowing that the spacing between nails is 5 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail. SOLUTON: Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. Calculate the corresponding shear force in each nail. 6-5
Example 6.01 6-6
Determination of the Shearing Stress in a Beam The average shearing stress on the horizontal face of the element is obtained by dividing the shearing force on the element by the area of the face. ave H A t q x A x t x On the upper and lower surfaces of the beam, yx = 0. t follows that xy = 0 on the upper and lower edges of the transverse sections. b<=h/4,c1 及 C 點之剪應力值 不會超過沿中性軸之應力 平均值 0.8% (p377) 6-7
Shearing Stresses xy in Common Types of Beams For a narrow rectangular beam, xy max b 3V A 3V 1 A y c For American Standard (S-beam) and wide-flange (W-beam) beams ave max t V A web 6-8
Sample Problem 6.1 6-9
Sample Problem 6. SOLUTON: Develop shear and bending moment diagrams. dentify the maximums. A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used, all 1MPa 0.8MPa all determine the minimum required depth d of the beam. Determine the beam depth based on allowable normal stress. Determine the beam depth based on allowable shear stress. Required beam depth is equal to the larger of the two depths found. 6-10
Sample Problem 6. SOLUTON: Develop shear and bending moment diagrams. dentify the maximums. V M max max 14.5 kn 10.95 knm 6-11
Sample Problem 6. S 1 1 b d c 1 6 3 b d 1 6 0.09m d 0.015 md
Problems 6.9, 6.1 6-13
Shearing Stresses in Thin-Walled Members Consider a segment of a wide-flange beam subjected to the vertical shear V. The longitudinal shear force on the element is H zx xz x The corresponding shear stress is NOTE: xy 0 xz 0 H t x t Previously found a similar expression for the shearing stress in the web xy t in the flanges in the web 6-14
Shearing Stresses in Thin-Walled Members The variation of shear flow across the section depends only on the variation of the first moment. q t For a box beam, q grows smoothly from zero at A to a maximum at C and C and then decreases back to zero at E. The sense of q in the horizontal portions of the section may be deduced from the sense in the vertical portions or the sense of the shear V. 6-15
Shearing Stresses in Thin-Walled Members For a wide-flange beam, the shear flow increases symmetrically from zero at A and A, reaches a maximum at C and then decreases to zero at E and E. The continuity of the variation in q and the merging of q from section branches suggests an analogy to fluid flow. 6-16
Sample Problem 6.3 Knowing that the vertical shear is 00 kn in a W50x101 rolled-steel beam, determine the horizontal shearing stress in the top flange at the point a. 6-17
Sample Problem 6.4 6-18
Sample Problem 6.5 6-19
Problems 6.35 6-0
Unsymmetric Loading of Thin-Walled Members Beam loaded in a vertical plane of symmetry deforms in the symmetry plane without twisting. x My ave t Beam without a vertical plane of symmetry bends and twists under loading. My x ave t 6-1
Unsymmetric Loading of Thin-Walled Members f the shear load is applied such that the beam does not twist, then the shear stress distribution satisfies ave t V D q ds B q ds q ds F F and F indicate a couple Fh and the need for the application of a torque as well as the shear load. Fh The point O is referred to as the shear center of the beam section. F Ve When the force P is applied at a distance e to the left of the web centerline, the member bends in a vertical plane without twisting. B A E D 6-
Example 6.05 Determine the location for the shear center of the channel section with b = 100 mm, h = 150 mm, and t = 4 mm where e b F q 0 1 1 web th Combining, F h V ds Vthb 4 Vthb 4 b 6b flange h h V 1 1 b 100mm e h 150mm 3b 3 100mm ds V th 3 th b 4 b 0 0 h st ds 1 bt 1 3 h bt e 40mm 6-3
Example 6.06 Determine the shear stress distribution for V = 10 kn q t t Shearing stresses in the flanges, Shearing stress in the web, t B max 1 V 8 ht4b h 3V 4b h 1 1th 6b ht th6b h 310000 N4 0.1m 0.15m 0.004 m0.15m6 0.1m 0.15m t V t Vhb st h Vh s 6Vb 1 1th 6 b h th6b h 610000 N0.1m 0.004 m0.15m6 0.1m 0.15m 13.3M Pa 18.3M Pa 6-4
Example 6.07 6-5
Problems 6.66 6-6