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# Mechanics of Deformable Solids ME 222

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This 16 page Class Notes was uploaded by Princess Rolfson on Saturday September 19, 2015. The Class Notes belongs to ME 222 at Michigan State University taught by Dahsin Liu in Fall. Since its upload, it has received 49 views. For similar materials see /class/207530/me-222-michigan-state-university in Mechanical Engineering at Michigan State University.

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Date Created: 09/19/15

IME 222 Mechanics of Deformable Solids Lecture 16 Shearing Stresses in Beams Department of Mechanical Engineering A Loos 161 IME 222 Mechanics of Deformable Solids Introduction 0 Transverse loading applied to a beam results in normal and shearing stresses in any given transverse section of the beam Normal stress created by the bending couple M Shearing stresses created by shear V 0 Internal normal and shearing forces exerted on a given transverse section of a prismatic beam With a vertical plane of symmetry are equivalent to the bending couple M and the shear force V y Ux Department of Mechanical Engineering A Loos 16392 IME 222 Mechanics of Deformable Solids Introduction Distribution of normal and shearing stresses satis es the siX equilibrium equations zFx 10di 0 ZMX 4er erydA 0 2F 2 jrxydA V 2M 2 27di 2 O m erszo ZMZ J yaxM Department of Mechanical Engineering A Loos 16393 IME 222 Mechanics of Deformable Solids Introduction Department of Mechanical Engineering A Loos Consider a small cubic element located in the vertical plane of symmetry of the beam sz 0 Examine the stresses on the faces of the cube On each of the two faces perpendicular to the x axis Normal stress 0x Shearing stress rxy When shearing stresses rxy are exerted on the vertical faces of an element equal stresses ryx must be exerted on the horizontal faces Longitudinal shearing stresses must exist in any member subjected to transverse loading 164 IME 222 Mechanics of Deformable Solids Shear on the Horizontal Face of a Beam Element Consider prismatic beam AB with a vertical plane of symmetry that supports various concentrated and distributed loads At a distance x from end A detach an element CDD C of length Ax extending across the Width of the beam from the upper surface of the beam to a horizontal plane located at a distance y I from the neutral axis Department of Mechanical Engineering A Loos 16395 ME 222 Mechanics of Deformable Solids Shear on the Horizontal Face of a Beam Element lL For equilibrium of beam element ZFx O AHIlt0C 0DdA 0 where the integral extends over the shaded area a of the section located above the line y yl a 2 I AH M IydA The forces exerted on the element Note a I Vertical shearing forces VC and VD Q 2 J ydA I Horizontal shearing force AH exerted on the lower face of element a dM I Horizontal normal forces 5ch and M D MC 2 Ax VAx chA dx 39 Possible load wa Substituting AH QM Department of Mechanical Engineering A Loos I 16396 I ME 222 Mechanics of Deformable Solids Shear on the Horizontal Face of a Beam Element 0 Shear ow horizontal shear per unit length qshear ow 0 Where Qme 173 a F the first moment of with respect to gal c39 b7 g c 171quot NA the neutral axis of the portion of the I Z I cross section located either above of below the point at which q is being computed I I 2sz 10 2 second moment of the full cross section centroidal moment of inertia of the entire cross sectional area Department of Mechanical Engineering A Loos 16397 IME 222 Mechanics of Deformable Solids Shear on the Horizontal Face of a Beam Element 2 quot Em 0 Same result found for lower area leE b quotgit 1 7 F M C D D C 3 quot AH VQ39 Ax I Q Q39 0 first moment with respect to neutral axis AH39 2 AH I y 1 Cquot D Department of Mechanical Engineering A Loos 16398 IME 222 Mechanics of Deformable Solids Example 601 100 mm 1 i SOLUTION v 20 mm 0 Determine the horizontal force per 100 unit length or shear ow q on the 20 mm mm lower surface of the upper plank 20 mm 0 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 Department of Mechanical Engineering A Loos 16399 IME 222 Mechanics of Deformable Solids Example 601 r0100 m L A l SOLUTION 0320 1 9 0060 m 39 T 0 Determlne the h0r1z0nta1 force per M l unit length or shear ow q 0n the lower surface of the upper plank VQ 500N120 X 10 m q 2 T Z 64 1620 X 10 m Q Ay 37041V 0020m gtlt 0100 m0060m m 120 X 10 6m3 0 Calculate the corresponding shear force in each nail for a nail spacing of 2 0100m0020m3 25 mm 0020m gtlt 0100m0060m2 F 2 0025mm 2 O Ozsmx3704Nm 21620gtlt106m4 1 3 E0020m0100m Department of Mechanical Engineering A Loos 163910 ME 222 Mechanics of Deformable Solids Determination of the Shearing Stress in a Beam al rm 0 The average shearing stress on the horizontal O D A C face of the element is obtained by dividing the if shearing force on the element by the area of Dquot thefazzre T qAx Q yxavf ave AA AA 1 tAx VQ TM 2 7 If the Width of the beam is comparable or large relative to its depth the shearing stresses at D 1 and D 2 are signi cantly higher than at D 0 Shear stress distribution along D lD 2 39 TXy exerted on the transverse plane 39 TyX exerted on the horizontal plane I TXy Tyx I Tave lt Txymax Tyxmax Di39 Department of Mechanical Engineering A Loos 1611 ME 222 Mechanics of Deformable Solids Determination of the Shearing Stress in a Beam On the upper and lower surfaces of the beam ryx 0 It follows that er O on the upper and lower edges of the transverse sections Department of Mechanical Engineering A Loos 1639 IME 222 Mechanics of Deformable Solids Shearing Stresses 23W in Common Types of Beams 0 For a narrow rectangular beam VQ 3V y2 rxyz z 1 2 1b 2A 0 3V For American Standard Sbeam Ar I jB I 1 and W1de ange Wbeam beams V 2live 2 1 T V max Aweb Department of Mechanical Engineering A Loos 1639 ME 222 Mechanics of Deformable Solids Further Discussion of the Distribution of Stresses in a Narrow Rectangular Beam E 0 Consider a narrow rectangular cantilever beam subjected to load P at its free end 3P y2 ny T 1 a 2 2A CZ x 1 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 SaintVenant 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 for typical beam sections of interest Department of Mechanical Engineering A Loos 1639 ME 222 Mechanics of Deformable Solids Sample Problem 62 251xips l kip 25 kips r2 ftlt 3 him 2ampT I 10 ft 0 Determine the beam depth based on allowable normal stress SOLUTION 7 0 Develop shear and bending moment diagrams Identify the maximums A timber beam is to support the three concentrated loads shown Knowing that for the grade of timber used 0 Determine the beam depth based on allowable shear stress 0a 2 1800 psi Ta 2 120 psi 0 Requlred beam depth 1s equal to the larger of the two depths found determine the minimum required depth d of the beam Department of Mechanical Engineering A Loos 1639 IME 222 Mechanics of Deformable Solids Sample Problem 62 27 kipx l kip 25 kips A C D E B Develop shear andbend1ng mement t dlagrams Identlfy the mammums 3 kips 3 kips L2 f tlt 3 ft gtlt 3 39t ilt2 ftgti Vmax 3klps v 1 Mmax 75kipft90kipin Z 39ips 6 115 5 kip x e 5 kip 15 6 e 3 kips M f k1 t Skipft 0 p t 6 kip ft Department of Mechanical Engineering A Loos 163916

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