Fourth 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 © 2006 The McGraw-Hill Companies, Inc. All rights reserved. Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses in Beams and Thin-Walled Members Introduction Shear on the Horizontal Face of a Beam Element Example 6.01 Determination of the Shearing Stress in a Beam Shearing Stresses txy in Common Types of Beams Further Discussion of the Distribution of Stresses in a ... Sample Problem 6.2 Longitudinal Shear on a Beam Element of Arbitrary Shape Example 6.04 Shearing Stresses in Thin-Walled Members Plastic Deformations Sample Problem 6.3 Unsymmetric Loading of Thin-Walled Members Example 6.05 Example 6.06 © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6-2 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Introduction • Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. • Distribution of normal and shearing stresses satisfies F x x dA 0 F y t xy dA V F z t xz dA 0 y t xz z t xy dA 0 M y z x dA 0 M z y x M M x • When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces • Longitudinal shearing stresses must exist in any member subjected to transverse loading. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6-3 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shear on the Horizontal Face of a Beam Element • Consider prismatic beam • For equilibrium of beam element F x 0 H D C dA A M H D MC I y dA A • Note, Q y dA A dM MD MC x V x dx • Substituting, H VQ x I q © 2006 The McGraw-Hill Companies, Inc. All rights reserved. H x VQ shear flow I 6-4 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shear on the Horizontal Face of a Beam Element • Shear flow, q H x VQ shear flow I • where Q y dA A first moment of area above y1 I 2 y dA A A' second moment of full cross section • Same result found for lower area q H x Q Q 0 VQ q I first moment wit h respect to neutral axis H H © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6-5 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Example 6.01 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. 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. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6-6 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Example 6.01 SOLUTION: • Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. q VQ ( 500 N )(120 10 16.20 10 I Q Ay 3704 N 0 . 020 m 0 . 100 m 0 . 060 m 120 10 I 1 12 6 0 . 020 m 4 m • Calculate the corresponding shear force in each nail for a nail spacing of 25 mm. m 0 . 100 m 3 2 [ 1 0 . 100 m 0 . 020 m 12 0 . 020 m 0 . 100 m 0 . 060 m ] 2 16 . 20 10 m 3 m ) 3 3 6 -6 6 m 4 © 2006 The McGraw-Hill Companies, Inc. All rights reserved. F ( 0 . 025 m ) q ( 0 . 025 m )( 3704 N m F 92 . 6 N 6-7 Fourth 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. t ave H A q x A VQ x I t x VQ It • On the upper and lower surfaces of the beam, tyx= 0. It follows that txy= 0 on the upper and lower edges of the transverse sections. • If the width of the beam is comparable or large relative to its depth, the shearing stresses at D1 and D2 are significantly higher than at D. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6-8 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Shearing Stresses txy in Common Types of Beams • For a narrow rectangular beam, 2 3V y 1 t xy 2 Ib 2 A c VQ t max 3V 2 A • For American Standard (S-beam) and wide-flange (W-beam) beams t ave VQ t max © 2006 The McGraw-Hill Companies, Inc. All rights reserved. It V A web 6-9 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam • Consider a narrow rectangular cantilever beam subjected to load P at its free end: 2 3P y 1 t xy 2 2 A c x Pxy I • Shearing stresses are independent of the distance from the point of application of the load. • Normal strains and normal stresses are unaffected by the shearing stresses. • From Saint-Venant’s principle, effects of the load application mode are negligible except in immediate vicinity of load application points. • Stress/strain deviations for distributed loads are negligible for typical beam sections of interest. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 10 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 6.2 SOLUTION: • Develop shear and bending moment diagrams. Identify the maximums. • Determine the beam depth based on allowable normal stress. A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used, all 1800 psi t all 120 psi • Determine the beam depth based on allowable shear stress. • Required beam depth is equal to the larger of the two depths found. determine the minimum required depth d of the beam. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 11 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 6.2 SOLUTION: Develop shear and bending moment diagrams. Identify the maximums. V max 3 kips M max 7 . 5 kip ft 90 kip in © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 12 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Sample Problem 6.2 • Determine the beam depth based on allowable normal stress. all M max S 3 90 10 lb in. 1800 psi 0 . 5833 in. d 2 d 9 . 26 in. 3 I 1 bd • Determine the beam depth based on allowable shear stress. 12 S I c 2 1bd 6 t all 2 1 3 . 5 in. d 6 0 . 5833 in. d 2 3 V max 2 120 psi A 3 3000 lb 2 3.5 in. d d 10 . 71 in. • Required beam depth is equal to the larger of the two. d 10 . 71 in. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 13 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Longitudinal Shear on a Beam Element of Arbitrary Shape • We have examined the distribution of the vertical components txy on a transverse section of a beam. We now wish to consider the horizontal components txz of the stresses. • Consider prismatic beam with an element defined by the curved surface CDD’C’. F x 0 H D C dA a • Except for the differences in integration areas, this is the same result obtained before which led to H VQ I © 2006 The McGraw-Hill Companies, Inc. All rights reserved. x q H x VQ I 6 - 14 Fourth Edition MECHANICS OF MATERIALS Beer • Johnston • DeWolf Example 6.04 SOLUTION: • Determine the shear force per unit length along each edge of the upper plank. • Based on the spacing between nails, determine the shear force in each nail. A square box beam is constructed from four planks as shown. Knowing that the spacing between nails is 1.5 in. and the beam is subjected to a vertical shear of magnitude V = 600 lb, determine the shearing force in each nail. © 2006 The McGraw-Hill Companies, Inc. All rights reserved. 6 - 15

Descargar
# 6 - WordPress.com