Third Edition CHAPTER 6 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and ThinWalled Members © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. • Distribution of normal and shearing stresses satisfies Fx x dA 0 M x y xz z xy dA 0 Fz xz dA 0 M z y x M Fy xy dA V M y z x dA 0 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-2 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-3 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shear on the Horizontal Face of a Beam Element © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-4 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • Equilibrium of beam element F x 0 H D C dA A M D MC H y dA I A © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-5 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shear on the Horizontal Face of a Beam Element • Note, Q y dA A dM M D MC x V x dx • Substituting, VQ H x I H VQ q shear flow x I © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-6 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • Same result found for lower area H VQ q q x I Q Q 0 first moment wit h respect to neutral axis H H © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6-7 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6-8 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6-9 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 10 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Determination of the Shearing Stress in a Beam © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 11 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf 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 q x VQ x ave A A I t x VQ It © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 12 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 13 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • 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. © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 14 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 15 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf For A Narrow Rectangular Beam © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 16 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 17 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf • For American Standard (S-beam) and Wide-flange (Wbeam) beams VQ ave It V max Aweb © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 18 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 19 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 20 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 21 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 22 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 23 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 24 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 25 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 26 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 27 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 28 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problems 6.1, 6.3, 6.10, 6.11, 6.12, 6.13, 6.19, 6.20, 6.21, 6.22 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 29 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Longitudinal Shear on a Beam Element of Arbitrary Shape © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 30 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Longitudinal Shear on a Beam Element of Arbitrary Shape • Consider an element defined by the curved surface CDD’C’. Fx 0 H D C dA a © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 31 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Longitudinal Shear on a Beam Element of Arbitrary Shape • Except for the differences in integration areas, this is the same result obtained before which led to VQ H x I H VQ q x I © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 32 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 33 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 34 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 35 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 36 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses in Thin-Walled Members • The longitudinal shear force on the element is VQ H x I © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 37 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses in Thin-Walled Members • The corresponding shear stress is H VQ zx xz t x It © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 38 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses in Thin-Walled Members © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 39 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses in Thin-Walled Members © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 40 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 41 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 42 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 43 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 44 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 45 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 46 Third Edition MECHANICS OF MATERIALS © 2002 The McGraw-Hill Companies, Inc. All rights reserved. Beer • Johnston • DeWolf 6 - 47 Third Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Problems 6.30, 6.32, 6.33, 6.35, 6.36, 6.37, 6.39, 6.40, 6.41, 6.42 © 2002 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 48
