MECHANICS OF MATERIALS

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1 CHAPTER MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members 006 The McGraw-Hill Companies, nc. All rights reserved.

2 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 τ xy in Common Types of Beams Further Discussion of the Distribution of Stresses in a... Sample Problem 6. 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 The McGraw-Hill Companies, nc. All rights reserved. 6 -

ntroduction Transverse loading applied to a beam results in normal

Distribution of normal and shearing stresses satisfies ( yτ zτ )

z x ( yσ x ) M 0 When shearing stresses are exerted on the

the horizontal faces Longitudinal shearing stresses must exist in

3 ntroduction Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. Distribution of normal and shearing stresses satisfies ( yτ zτ ) Fx σ xda 0 M x xz xy da F τ da V M zσ da 0 F y z τ xy xz da 0 M y z x ( yσ x ) M 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. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-3

4 Shear on the Horizontal Face of a Beam Element Consider prismatic beam For equilibrium of beam element F 0 H + σ σ da ( ) H Note, M x Q D M D y da A M C M C dm dx A A D y da C x V x Substituting, VQ H x H VQ q xx shear flow 006 The McGraw-Hill Companies, nc. All rights reserved. 6-4

Shear on the Horizontal Face of a Beam Element Shear flow, where H VQ q xx Q y da shear first moment of A y A+ A' da flow area above second moment of full cross

5 Shear on the Horizontal Face of a Beam Element Shear flow, where H VQ q xx Q y da shear first moment of A y A+ A' da flow area above second moment of full cross section Same result ltfound dfor lower area H VQ q q x Q + Q 0 first moment with respect to neutral axis H H 006 The McGraw-Hill Companies, nc. All rights reserved. 6-5 y 1

upper plank. Calculate the corresponding shear force in each nail.

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 5 mm and that the vertical shear in the beam is V 500 N, determine the shear force in each nail. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-6

Example 6.01 SOLUTON: Q Ay ( 0.00 m 0.100 m)( 0.060 ) m 6 3 10 10 m 1 1 + [ + ( 0.00 m)( 0.100 m) 1 1 3 ( 0.100m )( 0.

0 10 6 m 4 ] Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank.

7 Example 6.01 SOLUTON: Q Ay ( 0.00 m m)( ) m m [ + ( 0.00 m)( m) ( 0.100m )( 0.00m ) 3 ( 0.00 m m)( m) m 4 ] Determine the horizontal force per unit length or shear flow q on the lower surface of the upper plank. 6 VQ (500N)(10 10 q 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 006 The McGraw-Hill Companies, nc. All rights reserved. 6-7

Determination of the Shearing Stress in a Beam The average shearing

the shearing force on the element by the area of the face.

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. H q x VQ x τ ave A A t x VQ t 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 transverse 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. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-8

Shearing Stresses τ xy in Common Types of

9 Shearing Stresses τ xy in Common Types of Beams For a narrow rectangular beam, VQ 3V V τ xy 1 b A 3V τ max A y c For American Standard (S-beam) and wide-flange (W-beam) beams τ ave τ max VQ t V A web 006 The McGraw-Hill Companies, nc. All rights reserved. 6-9

Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam Consider a narrow rectangular

independent of the distance from the point of application of the load.

From Saint-Venant s principle, effects of the load application mode are negligible except in immediate vicinity

10 Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam Consider a narrow rectangular cantilever beam subjected to load P at tits free end: τ xy 3 P y 1 A c σ x + Pxy 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 ibl for typical beam sections of interest. t 006 The McGraw-Hill Companies, nc. All rights reserved. 6-10

Sample Problem 6. SOLUTON: Develop shear and bending moment diagrams. dentify the maximums. A timber beam is to support the three concentrated loads shown.

11 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 dt determine the minimum ii required ddepth 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. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-11

Sample Problem 6. 1 3 bd 1 Determine the beam depth based on allowable normal stress.

) Determine the beam depth hbased on allowable shear stress. 1 S bd c 6 3 τ V max all 1 A ( 3.5in.

13 Sample Problem bd 1 Determine the beam depth based on allowable normal stress. M σ max all S 1800 psi d 9.6in lb in. ( in. ) Determine the beam depth hbased on allowable shear stress. 1 S bd c 6 3 τ V max all 1 A ( 3.5in. ) d lb in. d 10 psi ( ) ( 3.5in. ) d d 10.71in. d Required beam depth is equal to the larger of the two. d 10.71in. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-13

Longitudinal Shear on a Beam Element of fabit Arbitrary Shape We

14 Longitudinal Shear on a Beam Element of fabit 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 C da a Except for the differences in integration areas, this is the same result obtained before which led to VQ H VQ H x q x 006 The McGraw-Hill Companies, nc. All rights reserved. 6-14

A square box beam is constructed dfrom four planks as shown. Knowing that the spacing between nails is 1.5 in.

15 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 dfrom 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. 006 The McGraw-Hill Companies, nc. All rights reserved. 6-15

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 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