Mechanics of Solids I. Transverse Loading

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1 Mechanics of Solids I Transverse Loading

Introduction o Transverse loading applied

stresses satisfies F da 0 M y z da 0 x x x

2 Introduction o Transverse loading applied to a beam results in normal and shearing stresses in transverse sections. o Distribution of normal and shearing stresses satisfies F da 0 M y z da 0 x x x xz xy F da V M z da 0 y xy y x F da 0 M y M z xz z x Shear V is the result of a transverse shearstress distribution that acts over the beam s x-section.

3 Introduction o o When shearing stresses are exerted on the vertical faces of an element, equal stresses must be exerted on the horizontal faces. Longitudinal shear stress ( yx ) must exist o Shear does not occur in a beam subjected to pure bending

4 Shear on the Horizontal Face of a Beam Element o Consider prismatic beam o For equilibrium of beam element F 0 H da x D C A MD MC H I o Note, Q y da A A y da dm MD MC x V x dx o Substituting, VQ H x I H VQ q shear flow x I

5 Shear on the Horizontal Face of a Beam Element Shear flow, VQ q I shear flow where Q A y da first moment of area above y I AA' 2 y da second moment of full cross section 1 Same result found for lower area H H

6 Example 6.1 A beam is made of three planks, nailed together. Knowing that the spacing between the nails is 25 mm and that the vertical shear in the beam is V = 500 N, determine the shear force in each nail.

7 Shearing Stress in a Beam o 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: Shear formula. ave H qx VQ x A A I t x ave VQ It

8 Midterm 2/49 y z B A C 80 mm V z = 30 kn A beam with rectangular cross section subject to a vertical shear V y and a horizontal shear V z as shown. Determine shear stress at point A, B, C and D on the beam cross section. 10 mm x D V y = 20 kn 10 mm 50 mm

o If the width of the beam is comparable or large relative to its depth, the shearing stresses

9 Shearing Stress in a Beam xy and yx exerted on a transverse and a horizontal plane through D are equal. o If 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. o 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.

10 Shearing Stresses xy in Common Types of Beams For a narrow rectangular beam, parabola xy max VQ 3 V y 1 Ib 2 A c 3 V 2 A 2 2 By comparison, max is 50% greater than the average shear stress determined from avg = V/A.

Shearing Stresses xy in Common Types of Beams Wide-flange beam (W-beam) and

web. Using analysis similar to a rectangular x-section, the shear stress

jump in shear stress at the flange-web junction since x- sectional thickness

11 Shearing Stresses xy in Common Types of Beams Wide-flange beam (W-beam) and Standard beam (S-beam) A wide-flange beam consists of two (wide) flanges and a web. Using analysis similar to a rectangular x-section, the shear stress distribution acting over x-section is shown ave max VQ It V A web o There is a jump in shear stress at the flange-web junction since x- sectional thickness changes at this point o The web carries significantly more shear force than the flanges

12 Example 6.2 A timber beam is to support the three concentrated loads shown. Knowing that for the grade of timber used, all 12 MPa, 0.8 MPa all determine the minimum required depth d of the beam.

Longitudinal Shear on an Arbitrary Shape Beam

F 0 H da x D C A H VQ q x I Shear flow is

13 Longitudinal Shear on an Arbitrary Shape Beam Earlier, we learn how to calculate shear flow along horizontal surfaces How to calculate q along vertical surfaces? F 0 H da x D C A H VQ q x I Shear flow is calculated by using the same equation. But by cutting through the vertical surface!

14 Example 6.3 A square box beam is constructed from four planks as shown. Knowing that the spacing between nails is 44 mm. and the beam is subjected to a vertical shear of magnitude V = 2.5 kn, determine the shearing force in each nail.

o The longitudinal shear force on the element is VQ H x I o The corresponding shear

15 Shearing Stresses in Thin-Walled Members o Consider a segment of a wide-flange beam subjected to the vertical shear V. o The longitudinal shear force on the element is VQ H x I o The corresponding shear stress is zx o NOTE: 0 xy xz H VQ xz t x It o Previously found a similar expression for the shearing stress in the web VQ xy It 0 in the flanges in the web

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 VQ I 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.

Shearing Stresses in Thin-Walled Members o For a wide-flange beam, the shear flow increases symmetrically from zero at A and A, reaches a maximum at C

17 Shearing Stresses in Thin-Walled Members o 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. o The continuity of the variation in q and the merging of q from section branches suggests an analogy to fluid flow.

18 Shearing Stresses in Thin-Walled Members Directional sense of q is such that shear appears to flow through the x-section inward at beam s top flange combining and then flowing downward through the web then separating and flowing outward at the bottom flange

19 Shearing Stresses in Thin-Walled Members Important notes If a member is made from segments having thin walls, only the shear flow parallel to the walls of member is important Shear flow varies linearly along segments that are perpendicular to direction of shear V Shear flow varies parabolically along segments that are inclined or parallel to direction of shear V On x-section, shear flows along segments so that it contributes to shear V yet satisfies horizontal and vertical force equilibrium

20 Example 6.4 Knowing that a given vertical shear V causes a maximum shearing stress of 75 Mpa in the hat-shaped extrusion shown, determine the corresponding shearing stress at (a) point a, (b) point b.

MECHANICS OF MATERIALS

MECHANICS OF MATERIALS 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