MECHANICS OF MATERIALS

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1 2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 6 MECHANICS OF MATERIALS 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

2 Shear on the Horizontal Face of a Beam Element Consider prismatic beam For equilibrium of beam element F 0 H da H M x Note, Q D M M D yda A C M I C A A D yda dm x V x dx C Substituting, VQ H x I H VQ q shear x I flow 6-2

3 Shear on the Horizontal Face of a Beam Element Shear flow, H VQ q shear x I where Q yda I first A y AA' moment of 2 da flow areaabovey secondmoment of fullcrosssection Same result found for lower area H VQ q q x I QQ 0 first moment wit h respect to neutralaxis H H 1 6-3

4 Concept Application 6.1 A beam is made of three planks, nailed together. Knowing that the spacing between nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail. SOLUTION: 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. 6-4

5 Concept Application

Determination of the Shearing Stress in a

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. H qx VQ x ave A A I tx VQ It On the upper and lower surfaces of the beam, yx = 0. It follows that xy = 0 on the upper and lower edges of the transverse sections. b<=h/4,c1 及 C2 點之剪應力值不會超過沿中性軸之應力平均值 0.8% (p377) 6-6

7 Shearing Stresses xy in Common Types of Beams For a narrow rectangular beam, xy max VQ 3V y 1 Ib 2 A c 3V 2 A 2 2 For American Standard (S-beam) and wide-flange (W-beam) beams ave max VQ It V A web 6-7

8 Concept Application 6.2 τ all =1.75MPa, Check that the design is acceptable from the point of view of the shear stresses 6-8

9 Sample Problem

10 Sample Problem 6.2 SOLUTION: Develop shear and bending moment diagrams. Identify the maximums. A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used, all 12MPa 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

11 Sample Problem 6.2 SOLUTION: Develop shear and bending moment diagrams. Identify the maximums. V M max max 14.5kN 10.95kNm 6-11

12 Sample Problem I 12 1 bd I 1 S 6 bd c md 0.015m d

The longitudinal shear force on the element is VQ H x I The corresponding shear stress

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 VQ H x I The corresponding shear stress is H VQ zx xz tx It Previously found a similar expression for the shearing stress in the web VQ xy It xy xz NOTE: 0 0 in the flanges in the web 6-13

14 Shearing Stresses in Thin-Walled Members The variation of shear flow across the section depends only on the variation of the first moment. VQ q t I 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. 6-14

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. 6-15

16 Sample Problem 6.3 Knowing that the vertical shear is 200 kn in a W250x101 rolled-steel beam, determine the horizontal shearing stress in the top flange at the point a. 6-16

17 Sample Problem 6.5 ( 自行練習 ) 6-17

18 Problems 6.37 Knowing that a given vertical shear V causes a maximum shearing stress of 75 MPa in an extruded beam having the cross section shown, determine the shearing stress at the three points indicated. 6-18

19 Problems

20 Problems

21 Problems

22 Problems

23 Unsymmetric Loading of Thin-Walled Members Beam loaded in a vertical plane of symmetry deforms in the symmetry plane without twisting. My x I ave VQ It Beam without a vertical plane of symmetry bends and twists under loading. My x I ave VQ It 6-23

application of a torque as well as the shear load.

24 Unsymmetric Loading of Thin-Walled Members If the shear load is applied such that the beam does not twist, then the shear stress distribution satisfies VQ D B E ave V q ds F q ds q ds F It B F and F indicate a couple Fh and the need for the application of a torque as well as the shear load. Fh 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. The point O is referred to as the shear center of the beam section. A D 6-24

25 Concept Application 6.5 Determine the location for the shear center of the channel section with b = 100 mm, h = 150 mm, and t = 4 mm F h Vthb e V 4I where b bvq F q ds ds I I I web th 2I 2 Combining, 2 Vthb 4I 6b flange h h V 1 th 12 b 100mm e h 150mm 2 2 3b 3100mm 2 V I th b 4I b h st ds bt 12 3 h bt 2 2 e 40mm 6-25

26 Concept Application 6.6 Determine the shear stress distribution for V = 10 kn q VQ t It Shearing stresses in the flanges, Shearing stress in the web, VQ It B max 1 V 8 ht4 bh 3V 4b h th 6b ht 2th6 bh N 4 0.1m 0.15m 0.15m 60.1m 0.15m VQ It V It Vhb st th 6b h th6 bh N 0.1m 0.004m 0.15m 60.1m 0.15m m h Vh s 2 2I 6Vb 13.3MPa 18.3MPa 6-26

27 Concept Application

28 Problems

29 Problems

30 Problems

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 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