4Pure BendingSlide 3Other Loading TypesSymmetric Member in Pure BendingBending DeformationsStrain Due to BendingStress Due to BendingBeam Section PropertiesProperties of American Standard ShapesDeformations in a Transverse Cross SectionMECHANICS OF MATERIALSFourth EditionFerdinand P. BeerE. Russell Johnston, Jr.John T. DeWolfLecture Notes:J. Walt OlerTexas Tech UniversityCHAPTER© 2006 The McGraw-Hill Companies, Inc. All rights reserved. 4Pure Bending© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 2Pure BendingPure BendingOther Loading TypesSymmetric Member in Pure BendingBending DeformationsStrain Due to BendingBeam Section PropertiesProperties of American Standard ShapesDeformations in a Transverse Cross SectionSample Problem 4.2Bending of Members Made of Several MaterialsExample 4.03Reinforced Concrete BeamsSample Problem 4.4Stress ConcentrationsPlastic DeformationsMembers Made of an Elastoplastic MaterialExample 4.03Reinforced Concrete BeamsSample Problem 4.4Stress ConcentrationsPlastic DeformationsMembers Made of an Elastoplastic MaterialPlastic Deformations of Members With a Single Plane of S...Residual StressesExample 4.05, 4.06Eccentric Axial Loading in a Plane of SymmetryExample 4.07Sample Problem 4.8Unsymmetric BendingExample 4.08General Case of Eccentric Axial Loading© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 3Pure BendingPure Bending: Prismatic members subjected to equal and opposite couples acting in the same longitudinal plane© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 4Other Loading Types•Principle of Superposition: The normal stress due to pure bending may be combined with the normal stress due to axial loading and shear stress due to shear loading to find the complete state of stress.•Eccentric Loading: Axial loading which does not pass through section centroid produces internal forces equivalent to an axial force and a couple•Transverse Loading: Concentrated or distributed transverse load produces internal forces equivalent to a shear force and a couple© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 5Symmetric Member in Pure BendingMdAyMdAzMdAFxzxyxx00•These requirements may be applied to the sums of the components and moments of the statically indeterminate elementary internal forces.•Internal forces in any cross section are equivalent to a couple. The moment of the couple is the section bending moment.•From statics, a couple M consists of two equal and opposite forces.•The sum of the components of the forces in any direction is zero.•The moment is the same about any axis perpendicular to the plane of the couple and zero about any axis contained in the plane.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 6Bending DeformationsBeam with a plane of symmetry in pure bending:•member remains symmetric•bends uniformly to form a circular arc•cross-sectional plane passes through arc center and remains planar•length of top decreases and length of bottom increases•a neutral surface must exist that is parallel to the upper and lower surfaces and for which the length does not change•stresses and strains are negative (compressive) above the neutral plane and positive (tension) below it© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 7Strain Due to BendingConsider a beam segment of length L.After deformation, the length of the neutral surface remains L. At other sections,  mxmmxcycρcyyLyyLLyLor linearly) ries(strain va© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 8Stress Due to Bending•For a linearly elastic material,linearly) varies(stressmmxxcyEcyE•For static equilibrium,dAycdAcydAFmmxx00First moment with respect to neutral plane is zero. Therefore, the neutral surface must pass through the section centroid.•For static equilibrium,   IMycySMIMccIdAycMdAcyydAyMxmxmmmmx ngSubstituti2© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 9Beam Section Properties•The maximum normal stress due to bending,modulussection inertia ofmoment section cISISMIMcmA beam section with a larger section modulus will have a lower maximum stress•Consider a rectangular beam cross section,AhbhhbhcIS6136131212Between two beams with the same cross sectional area, the beam with the greater depth will be more effective in resisting bending.•Structural steel beams are designed to have a large section modulus.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 10Properties of American Standard Shapes© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 11Deformations in a Transverse Cross Section•Deformation due to bending moment M is quantified by the curvature of the neutral surfaceEIMIMcEcEccmm11•Although cross sectional planes remain planar when subjected to bending moments, in-plane deformations are nonzero,yyxzxy•Expansion above the neutral surface and contraction below it cause an in-plane curvature,curvature canticlasti