Third E CHAPTER 6 Shearing MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Stresses in Beams and Thin- Walled Members
Shearing Stresses in Beams and Thin-Walled Members ntroduction Shear on the Horizontal Face of a Beam Element Example 6.01 Determination of the Shearing Stress in a Beam Shearing Stresses t xy 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. Unsymmetric Loading of Thin-Walled Members Example 6.05 Example 6.06 6-2
ntroduction 6 -
ntroduction 6-4
Shear on the Horizontal Face of a Beam Element Consider prismatic beam For equilibrium of beam element F 0 H da H Note, Q M D x M M D y da A C M C dm dx A y da A D D x V x Substituting, H x H q x shear flow 6-5
Shear on the Horizontal Face of a Beam Element Shear flow, q where Q H x shear first moment of y da A y A A' 2 da flow area above second moment of full crosssection Same result found for lower area H q x Q Q 0 H H q first moment with respect to neutral axis y 1 6-6
Example 6.01 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. 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. 6-7
Example 6.01 Q 0.020 m 0.100 m 0.060 m 12010 1 12 2[ Ay 0.020 m 0.100 m 1 12 6 0.100 m 0.020 m 0.020 m 0.100 m 0.060 m 16.2010 m 6 m 4 2 ] SOLUTON: Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. 6 (500N)(12010 q -6 16.2010 m 704 N m Calculate the corresponding shear force in each nail for a nail spacing of 25 mm. 4 m F ( 0.025m) q (0.025m)( 704 N F 92.6N ) m 6-8
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 t q x A x t x On the upper and lower surfaces of the beam, t yx = 0. t follows that t xy = 0 on the upper and lower edges of the transverse sections. f the width of the beam is comparable or large relative to its depth, the shearing stresses at D 1 and D 2 are significantly higher than at D. 6-9
Shearing Stresses t xy in Common Types of Beams For a narrow rectangular beam, t t xy max b V 2 A V 1 2 A y c 2 2 For American Standard (S-beam) and wide-flange (W-beam) beams t ave t max t V A web 6-10
Sample Problem 6.2 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 1800psi t 120psi 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. 6-11
Sample Problem 6.2 SOLUTON: Develop shear and bending moment diagrams. dentify the maximums. V M max max kips 7.5kip ft 90kip in 6-12
Sample Problem 6.2 S 1 12 c 1 6 bd 1 6 bd.5in. 2 d 2 0.58in. d 2 Determine the beam depth based on allowable normal stress. all M S 1800 psi max 9010 d 9.26in. 0.58in. lb in. Determine the beam depth based on allowable shear stress. V t max all 2 A 000lb 120 psi 2.5in. d d 10.71in. d 2 Required beam depth is equal to the larger of the two. d 10.71in. 6-1
Example 6.04 SOLUTON: 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. 6-14
Example 6.04 SOLUTON: For the upper plank, Q Ay 4.22in 0.75in. in. 1.875in. For the overall beam cross-section, 1 12 4.5in 1 in 27.42in 4 12 Determine the shear force per unit length along each edge of the upper plank. q f 600lb 4.22in 27.42in q lb 46.15 2 in edge force per unit length 4 lb 92. in Based on the spacing between nails, determine the shear force in each nail. F lb f 46.15 1.75in in F 80.8lb 6-15
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 t zx t xz x The corresponding shear stress is NOTE: t xy 0 t xz 0 H t x t Previously found a similar expression for the shearing stress in the web t xy t in the flanges in the web 6-16
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. 6-17
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 the 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. 6-18
Sample Problem 6. SOLUTON: For the shaded area, Q 4.1in 0.770in 4.815in 15.98in Knowing that the vertical shear is 50 kips in a W10x68 rolled-steel beam, determine the horizontal shearing stress in the top flange at the point a. The shear stress at a, t t t 2.6ksi 50kips 15.98in 4 94in 0.770in 6-19