MECHANICS OF CHAPTER 6MATERIALS - IIT Bombay

1 MECHANICS OF MATERIALSCHAPTER6 Shearing Stresses in Beams and Thin-Walled MembersMECHANICS OF MATERIALS4 -26-2 Introduction MyMdAFdAzMVdAFdAzyMdAFxzxzzxyxyyxyxzxxx 0000 Distribution of normal and shearing stresses satisfies ( from equilibrium)Transverse loading applied to beam results in normal and shearing stresses in transverse shearing stresses must exist in any member subjected to transverse shearing stresses are exerted on vertical faces of an element, equal stresses exerted on horizontal facesMECHANICS OF MATERIALSV ertical and Horizontal Shear Stresses4 -3 MECHANICS OF MATERIALSS hear Stress in BeamsTwo beams glued together along horizontal surfaceWhen loaded, horizontal shear stress must develop along glued surface in order to prevent sliding betweenthe OF MATERIALS6-5 Shear on Horizontal Face of Beam ElementConsider prismatic beamEquilibrium of element CDC D ACDACDxdAyIMMHdAHF 0xVxdxdMMMdAyQCDA Let.

2 Flowshear IVQxHqxIVQHMECHANICS OF MATERIALS6-6 Shear on Horizontal Face of Beam Elementwheresection cross full ofmoment second above area ofmoment first '21 AAAdAyIydAyQSame result found for lower areaHHqIQVIQVxHqdAyQQAA )(zero) isNA wrt area ofmoment first ( 021 flowshear IVQxHqShear flow, MECHANICS OF MATERIALS6-7 Example made of three planks, nailed together. Spacing between nails is 25 mm. Vertical shear in beam is V= 500 N. Find shear force in each OF MATERIALS6-8 Example ] [ IyAQSOLUTION:Find horizontal force per unit length or shear flow qon lower surface of upper )m10120)(N500(46-36 IVQqCalculate corresponding shear force in each nail for nail spacing of 25 )( () ( FMECHANICS OF MATERIALS6-9 Determination of Shearing StressAverageshearing stress on horizontal face of element is shearing force on horizontal face divided by area of If width of beam is comparable or large relative to depth, the shearing stresses at D 1and D 2are significantly higher than at D, , the above averaging is not averaging is across dimension t(width) which is assumed much less than the depth, so this averaging is allowed.)

3 On upper and lower surfaces of beam, tyx= 0. It follows that txy= 0 on upper and lower edges of transverse ; MECHANICS OF MATERIALS6-10 Shearing Stresses txyin Common Types of BeamsFor a narrow rectangular beam,AVcyAVIbVQxy23123max22 For I beamswebaveAVItVQ max MECHANICS OF MATERIALS6-11 Example beam supports three concentrated loads. allall Find minimum required depth dof OF MATERIALS6-12 Example MVMECHANICS OF MATERIALS6-13 Example Determine depth based on allowable normal stress. 1012236max ddSMall Determine depth based on allowable shear stress. ddAVall Required depthmm322 dMECHANICS OF MATERIALS6-14 Longitudinal Shear Element of Arbitrary ShapeHave examined distribution of vertical components txyon transverse section. Now consider horizontal components only the integration area is different, hence result same as before, , Will use this for thin walled members alsoIVQxHqxIVQH Consider element defined by curved surface CDD C.

4 AdAHFCDx 0 MECHANICS OF MATERIALS6-15 Example box beam constructed from four planks. Spacing between nails is 44 mm. Vertical shear force V= kN. Find shearing force in each OF MATERIALS6-16 Example :Determine the shear force per unit length along each edge of the upper plank. lengthunit per force edge qfIVQqBased on the spacing between nails, determine the shear force in each nail. FFor the upper plank, 3mm64296mm47mm768mm1 yAQFor the overall beam cross-section, 441214121mm10332mm76mm112 IMECHANICS OF MATERIALS6-17 Shearing Stresses in Thin-Walled MembersConsider I-beam with vertical shear shear force on element isxIVQH ItVQxtHxzzx Corresponding shear stress isNOTE:0 xy 0 xz in the flangesin the webPreviously had similar expression for shearing stress webItVQxy Shear stress assumed constant through thickness t, , due to thinnnessour averaging is now OF MATERIALS6-18 Shearing Stresses in Thin-Walled MembersThe variation of shear flow across the section depends only on the variation of the first For a box beam, qgrows smoothly from zero at A to a maximum at Cand C and then decreases back to zero at sense of qin the horizontal portions of the section may be deduced from the sense in the vertical portions or the sense of the shear OF MATERIALS6-19 Shearing Stresses in Thin-Walled MembersFor wide-flange beam, shear flow qincreases symmetrically from zero at Aand A , reaches a maximum at Cand then decreases to zero at Eand E.

5 The continuity of the variation in qand the merging of qfrom section branches suggests an analogy to fluid OF MATERIALS6-20 Example shear is 200 kNin a W250x101 rolled-steel beam. Find horizontal shearing stress at a. QShear stress at a, ItVQ MECHANICS OF MATERIALSWork out this example of a wide flange beam (Doubly symmetric) MECHANICS OF MATERIALS6-22 Unsymmetric Loading of Thin-Walled MembersBeam loaded in vertical plane of symmetry, deforms in symmetry plane without Beam without vertical plane of symmetry bends and twists under MECHANICS OF MATERIALSB ending+TorsioneffectBending+Torsioneffec tPure BendingMECHANICS OF MATERIALS6-24 Unsymmetric Loading of Thin-Walled MembersPoint O is shear centerof the beam shear load applied such that beam does not twist, then shear stress distribution satisfiesFdsqdsqFdsqVItVQEDBADBave Fand F form a couple Fh.

6 Thus we have a torque as well as shear load. Static equivalence yields, VehF Thus if force P applied at distance eto left of web centerline, the member bends in vertical plane without twisting. Net torsional moment is Fh-Ve= 0, so shear stresses due to bending shear only, and not due to torsional load not applied thru shear center then net torsional moment exists, so total shear stress due to bending shear & torsional shear (ref. open thin walled torsion) MECHANICS OF MATERIALSF acts about Shear CenterWhen force applied at shear center, it causes pure bending & no location depends on cross-sectional geometry cross-section has axis of symmetry, then shear center lies on the axis of symmetry (but it may not be at centroid itself). If cross section has two axes of symmetry, then shear center is located at their intersection.

7 This is the only case where shear center and centroid coincide. MECHANICS OF MATERIALSWant to find shear flow and shear center of thin-walled open I and Z -sections L and T -sections intersection of the two straight limbs, , where bending shear stresses cause zero torsional moment. Thin-walled cross sections are very weak in torsion, therefore load must be applied through shear center to avoid excessive twistingMECHANICS OF MATERIALS6-27 Example 100615010036322 bthtbtefwfmm40 eDetermine location of shear center of channel section with b= 100 mm, h= 150 mm, and t= 4 mmVhFe IhbVtdshstIVdsIVQdsqdstFfbbfbbfxz4220000 sMECHANICS OF MATERIALS6-28 Example shear stress distribution for V= 10 kNItVQtq Shearing stress in the web, hsxyxywfwwfwxythbthtshtshbtVItVQ Shearing stresses in the flanges, )(22 hbthtVbbIVhsIVhhstItVItVQfwBxzfffxz qss1fwBxyBxztt becauseonly)()( MECHANICS OF MATERIALSS hear center of a thin walled semicircular cross-section(a) Find shear stress ( xq)

8 At an angle , , at section bbFind the first moment of the cross-sectional area between point aand section bb sincos20trrdtrydAQ trVttrtrVItVQx sin22/sin32 sVVy MECHANICS OF MATERIALS(b) Find the shear center (S)Moment about geometric center of circle O, due to the shear force is VeShear stress acting on element dACorresponding force is xqdAand moment due to this force is )(dArdMxo rVtrdtrVrdMMoo4sin20 trddA reVeMo4 trVx sin2