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 01 The McGraw-Hill Companies, nc. All rights reserved.
ntroduction Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. Distribution of normal and shearing stresses satisfies F F F x y z ( yτ zτ ) σ xda 0 M x xz τ da V M z σ da 0 τ xy xz da 0 M y z xy x ( yσ x ) M da 0 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. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-
Shear on the Longitudinal Surface of a Beam Element Consider prismatic beam For equilibrium of beam element F 0 H + ( σ σ ) da H Note, Q M D a' x y da M C M D dm dx M C a' a' D y da x V x C Substituting, H x H q x shear flow 01 The McGraw-Hill Companies, nc. All rights reserved. 3-3
Shear on the Longitudinal Surface of a Beam Element Shear flow, where H q x Q a' yda first A y moment of da shear flow areaabovey secondmoment of fullcrosssection Same result found for lower area H q x Q + Q 0 H H first moment with respect to neutral axis 1 01 The McGraw-Hill Companies, nc. All rights reserved. 3-4
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 5 mm and that the vertical shear in the beam is V 500 N, determine the shear force in each nail. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-5
Example 6.01 Q Ay ( 0.00 m 0.100m)( 0.060 m) 10 10 1 1 + [ + ( 0.00 m)( 0.100 m) 1 1 6 ( 0.100m)( 0.00m) ( 0.00m 0.100 m)( 0.060m) 16.0 10 m 6 3 m 4 3 3 ] SOLUTON: Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. 6 (500N)(10 10 q -6 16.0 10 m 3704 N m Calculate the corresponding shear force in each nail for a nail spacing of 5 mm. 4 m F ( 0.05m) q (0.05m)( 3704 N F 9.6 N 3 ) m 01 The McGraw-Hill Companies, nc. All rights reserved. 3-6
Determination of the Shearing Stress in a Beam The average shearing stress on the longitudinal surface of the element is obtained by dividing the shearing force on the element by the area of the face. τ ave H A t q x A x t x On the upper and lower surfaces of the beam, τ yx 0. t follows that τ xy 0 on the upper and lower edges of the cross-sections. f the width of the beam is comparable or large relative to its depth, the shearing stresses at D 1 and D are significantly higher than at D. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-7
Shearing Stresses τ xy in Common Types of Beams For a narrow rectangular beam, τ τ xy max b 3V A 3V 1 A y c For American Standard (S-beam) and wide-flange (W-beam) beams τ ave τ max t V A web 01 The McGraw-Hill Companies, nc. All rights reserved. 3-8
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, σ all 1800 psi τ 10 psi 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. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-9
Sample Problem 6. SOLUTON: Develop shear and bending moment diagrams. dentify the maximums. V M max max 3kips 7.5kip ft 90kip in 01 The McGraw-Hill Companies, nc. All rights reserved. 3-10
Sample Problem 6. S 1 3 bd 1 Determine the beam depth based on allowable normal stress. σ all M S 1800 psi max d 9.6in. 90 10 3 lb in. ( 0.5833in. ) Determine the beam depth based on allowable shear stress. c 1 6 1 6 bd ( 3.5in. ) d ( 0.5833in. ) d 3V τ max all A 3 3000lb 10 psi d d 10.71in. ( 3.5in. ) d Required beam depth is equal to the larger of the two. d 10.71in. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-11
Longitudinal Shear on a Beam Element of Arbitrary Shape We have examined the distribution of the vertical components τ xy on a transverse section of a beam. We now wish to consider the horizontal components τ xz of the stresses. Consider prismatic beam with an element defined by the curved surface CDD C. ( σ σ ) Fx 0 H + D a Except for the differences in integration areas, this is the same result obtained before which led to H H x q x C da 01 The McGraw-Hill Companies, nc. All rights reserved. 3-1
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. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-13
Example 6.04 SOLUTON: For the upper plank, Q A y 3 4.in ( 0.75in. )( 3in. )( 1.875in. ) For the overall beam cross-section, 1 1 4 ( 4.5in) 1 ( 3in) 7.4in 4 1 4 Determine the shear force per unit length along each edge of the upper plank. q f ( )( 3) 600lb 4.in 7.4in q lb 46.15 in edge force per unit length 4 lb 9.3 in Based on the spacing between nails, determine the shear force in each nail. F F f l 80.8lb lb 46.15 in ( 1.75in) 01 The McGraw-Hill Companies, nc. All rights reserved. 3-14
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 x The corresponding shear stress is τ H t x zx τ xz NOTE: τ xy 0 τ xz 0 t Previously found a similar expression for the shearing stress in the web τ xy t in the flanges in the web 01 The McGraw-Hill Companies, nc. All rights reserved. 3-15
Shearing Stresses in Thin-Walled Members The variation of shear flow across the section depends only on the variation of the first moment. q τt 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. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-16
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. 01 The McGraw-Hill Companies, nc. All rights reserved. 3-17
Plastic Deformations Recall: M Y c σy maximum elastic moment For M PL < M Y, the normal stress does not exceed the yield stress anywhere along the beam. For PL > M Y, yield is initiated at B and B. For an elastoplastic material, the half-thickness of the elastic core is found from Px Maximum load which the beam can support is P 3 y M Y Y 01 The McGraw-Hill Companies, nc. All rights reserved. 3-18 max M 1 1 3 c The section becomes fully plastic (y Y 0) at the wall when 3 PL M Y M p L p
Plastic Deformations Preceding discussion was based on normal stresses only Consider horizontal shear force on an element within the plastic zone, H ( σ σ ) da ( σ σ ) da 0 C D Therefore, the shear stress is zero in the plastic zone. Shear load is carried by the elastic core, 3 P y 1 A y 3 P A As A decreases, τ max increases and may exceed τ Y 01 The McGraw-Hill Companies, nc. All rights reserved. 3-19 τ τ xy max Y Y where Y A by Y
Sample Problem 6.3 SOLUTON: For the shaded area, Q ( 4.31in)( 0.770in)( 4.815in) 15.98in 3 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, τ t τ.63ksi ( )( 3) 50 kips 15.98in ( 4 394in )( 0.770in) 01 The McGraw-Hill Companies, nc. All rights reserved. 3-0