BENDING STRESSES IN BEAMS

1 RAGHU ENGINEERING COLLEGE(AUTONOMOUS)DEPARTMENT OF MECHANICAL ENGINEERINGM echanics of solidsUNIT-3 Flexural and Shear STRESSES in BeamsPrepared bySinguru RajeshAssistant Professor1 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC1 CONTENTS: BENDING in BEAMS STRESSES due to BENDING Simple BENDING or Pure BENDING Assumptions of theory of Simple BENDING Neutral Axis Moment of Resistance BENDING of FlitchedBeams Shear STRESSES in beamsSinguru Rajesh Mechanics of Solids Mechanical Engineering REC2 FLEXURALSTRESSES:Theoryofsimplebending Assumptions Derivationofbendingequation:M/I=f/y=E/RN eutralaxis Determinationbendingstresses sectionmodulusofrectangularandcircularse ctions(SolidandHollow),I,T,AngleandChann elsections.

2 Derivationofformula Shearstressdistributionacrossvariousbeam ssectionslikerectangular,circular,triang ular,I, in beamsSinguru Rajesh Mechanics of Solids Mechanical Engineering REC3 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC STRESSES due to bending412/39eBendingeeeCompressionTensi onShear5 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC WHAT THE LOADS DO????Singuru Rajesh Mechanics of Solids Mechanical Engineering REC6 LCBW KNSFDBMD-Base LineSFD and BMD for OVER HANGING BEAM POINT LOAD with Point LoadA Base Linea CBACBDD+-KNDW KN KNmKNmAaPURE BENDING or SIMPLE BENDINGS imple BENDING or Pure BENDING Whensomeexternalforceactsonabeam,theshea rforceandbendingmomentsaresetupatallthes ectionsofthebeam Duetoshearforceandbendingmoment, Stressesintroducedbybendingmomentareknow nasbendingstresses.

3 Rajesh Mechanics of Solids Mechanical Engineering REC7 Simple BENDING or Pure BENDING Whensomeexternalforceactsonabeam,theshea rforceandbendingmomentsaresetupatallthes ectionsofthebeam Duetoshearforceandbendingmoment, Stressesintroducedbybendingmomentareknow nasbendingstresses. Rajesh Mechanics of Solids Mechanical Engineering REC8 , ,independentlyofthelayer, Rajesh Mechanics of Solids Mechanical Engineering REC9 Theory of simple , ,independentlyofthelayer, Rajesh Mechanics of Solids Mechanical Engineering REC10 Theory of simple bendingTheory of simple BENDING Rajesh Mechanics of Solids Mechanical Engineering REC11 Theory of simple BENDING DuetothebendingactiontheelementABCD isdeformedtoA B C D (concavecurve).

4 Rajesh Mechanics of Solids Mechanical Engineering REC12 ThelayerACisshortenedtoA C .Henceitissubjectedtocompressivestress. ThelayerBDiselongatedtoB D .Henceitissubjectedtotensilestresses. Hencetheamountofshorteningdecreasefromth etoplayertowardsbottomandtheamountofelon gationdecreasesfromthebottomlayertowards top. Rajesh Mechanics of Solids Mechanical Engineering RECN eutral Axis Forabeamsubjectedtoapurebendingmoment,th estressesgeneratedontheneutrallayeriszer o. Neutralaxisisthelineofintersectionofneut rallayerwiththetransversesection /y=E/RSinguru Rajesh Mechanics of Solids Mechanical Engineering REC14 Neutral Axis Forabeamsubjectedtoapurebendingmoment,th estressesgeneratedontheneutrallayeriszer o.

5 Neutralaxisisthelineofintersectionofneut rallayerwiththetransversesection /y=E/RSinguru Rajesh Mechanics of Solids Mechanical Engineering REC15 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC16 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC17 REMEMBER POINTS inguru Rajesh Mechanics of Solids Mechanical Engineering REC18 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC19 Moment of Resistance Duetothetensileandcompressivestresses,fo rcesareexertedonthelayersofabeamsubjecte dtosimplebending.

6 Wehaveseenthatforceonalayerofcrosssectio nalareadAatadistanceyfromtheneutralaxis, dF= (E x y x dA)/RMoment of force dF about the neutral axis= dF x y= (E x y x dA)/R x y= E/R x (y dA).Singuru Rajesh Mechanics of Solids Mechanical Engineering REC20 Hencethetotalmomentofforceabouttheneutra laxis=Bendingmomentapplied= E/Rx(y dA)=E/RxIxx M = E/R x IxxOr M/Ixx=E/RHence M/Ixx=E/R = b/y; bis also known as flexural stress (Fb).HenceM/Ixx=E/R=Fb/y Rajesh Mechanics of Solids Mechanical Engineering REC21 Condition of Simple BENDING Howeverinpracticalapplications,thebendin gmomentvariesfromsectiontosectionandthes hearforceisnotzero.

7 Butinthesectionwherebendingmomentismaxim um,shearforce(derivativeofbendingmoment) iszero. Rajesh Mechanics of Solids Mechanical Engineering REC22 BENDING of fletched BEAMS Abeammadeupoftwoormoredifferentmaterials assumedtoberigidlyconnectedtogetherandbe havinglikeasinglepieceiscalledaflitchedb eamoracompositebeam. Rajesh Mechanics of Solids Mechanical Engineering REC23 Thebendingstressescanbecalculatedusingtw oconditions. Straindevelopedonalayerataparticulardist ancefromtheneutralaxisisthesameforbothth ematerials Momentofresistanceofcompositebeamisequal tothesumofindividualmomentofresistanceof themembersSinguru Rajesh Mechanics of Solids Mechanical Engineering REC24 Using condition-1: 1/E1= 2/E2; 1= 2 x (E1/E2) or 1 = 2 x m; where m= E1/E2 is the modular ratio between steel and woodUsing condition-2:M=M1+ M2.

8 M1= 1x I1/yM1= 2x I2/ySinguru Rajesh Mechanics of Solids Mechanical Engineering REC25 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC26 or Composite BeamsSinguru Rajesh Mechanics of Solids Mechanical Engineering REC27 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC28 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC29 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC30 Problem 1 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC31 Problem 1 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC32 Problem 1 Rajesh Mechanics of Solids Mechanical Engineering REC33 Section Modulus for Various Shapes or Beam SectionsSinguru Rajesh Mechanics of Solids Mechanical Engineering REC34 Section Modulus for Various Shapes or Beam Sectionsd/2dSinguru Rajesh Mechanics of Solids Mechanical Engineering REC35120 Problem 2 Singuru Rajesh Mechanics of Solids

9 Mechanical Engineering REC36 Problem 3 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC37A steel plate of width 200mm and of thickness 50mm is bent into a circular arc of radius 20m. Determine the max stress induced and the BENDING moment which will produce the max stress. Take E= 2 x 10^5 N/mm^ Rajesh Mechanics of Solids Mechanical Engineering REC38 Problem 4 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC39 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC40 Problem 4: Calculate the maximum stress induced in a cast iron pipe of external diameter 40mm, of internal diameter 20mm and of length 4m then the pipe is supported at its ends and carry a point load of 80N at its.

10 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC415 Singuru Rajesh Mechanics of Solids Mechanical Engineering REC42 Problem 4: Calculate the maximum stress induced in a cast iron pipe of external diameter 40mm, of internal diameter 20mm and of length 4m then the pipe is supported at its ends and carry a point load of 80N at its :Singuru Rajesh Mechanics of Solids Mechanical Engineering REC435 Singuru Rajesh Mechanics of Solids Mechanical Engineering Rajesh Mechanics of Solids Mechanical Engineering REC455 Singuru Rajesh Mechanics of Solids Mechanical Engineering Rajesh Mechanics of Solids Mechanical Engineering Rajesh Mechanics of Solids Mechanical Engineering Rajesh Mechanics of Solids Mechanical Engineering Rajesh Mechanics of Solids Mechanical Engineering