MECHANICS 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.Stress and Strain – Axial Loading© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 2ContentsStress & Strain: Axial LoadingNormal StrainStress-Strain TestStress-Strain Diagram: Ductile MaterialsStress-Strain Diagram: Brittle Materials Hooke’s Law: Modulus of ElasticityElastic vs. Plastic BehaviorFatigueDeformations Under Axial LoadingExample 2.01Sample Problem 2.1Static IndeterminacyExample 2.04Thermal StressesPoisson’s RatioGeneralized Hooke’s LawDilatation: Bulk ModulusShearing StrainExample 2.10Relation Among E, ν, and GSample Problem 2.5Composite MaterialsSaint-Venant’s PrincipleStress Concentration: HoleStress Concentration: FilletExample 2.12Elastoplastic MaterialsPlastic DeformationsResidual StressesExample 2.14, 2.15, 2.16© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 3Stress & Strain: Axial Loading• Suitability of a structure or machine may depend on the deformations in the structure as well as the stresses induced under loading. Statics analyses alone are not sufficient.• Considering structures as deformable allows determination of member forces and reactions which are statically indeterminate.• Determination of the stress distribution within a member also requires consideration of deformations in the member.• Chapter 2 is concerned with deformation of a structural member under axial loading. Later chapters will deal with torsional and pure bending loads.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 4Normal Strainstrain normalstress====LAPδεσLAPAPδεσ===22LLAPδδεσ===22© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 5Stress-Strain Test© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 6Stress-Strain Diagram: Ductile Materials© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 7Stress-Strain Diagram: Brittle Materials© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 8Hooke’s Law: Modulus of Elasticity• Below the yield stressElasticity of Modulus or Modulus Youngs==EEεσ• Strength is affected by alloying, heat treating, and manufacturing process but stiffness (Modulus of Elasticity) is not.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 9Elastic vs. Plastic Behavior• If the strain disappears when the stress is removed, the material is said to behave elastically. • When the strain does not return to zero after the stress is removed, the material is said to behave plastically.• The largest stress for which this occurs is called the elastic limit.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 10Fatigue• Fatigue properties are shown on S-N diagrams.• When the stress is reduced below the endurance limit, fatigue failures do not occur for any number of cycles.• A member may fail due to fatigueat stress levels significantly below the ultimate strength if subjected to many loading cycles.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 11Deformations Under Axial LoadingAEPEE ===σεεσ• From Hooke’s Law:• From the definition of strain:Lδε=• Equating and solving for the deformation,AEPL=δ• With variations in loading, cross-section or material properties,∑=iiiiiEALPδ© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 12Example 2.01in.618.0 in. 07.1psi10296==×=−dDESOLUTION:• Divide the rod into components at the load application points.• Apply a free-body analysis on each component to determine the internal force• Evaluate the total of the component deflections.Determine the deformation of the steel rod shown under the given loads.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 13SOLUTION:• Divide the rod into three components:• Apply free-body analysis to each component to determine internal forces,lb1030lb1015lb1060333231×=×−=×=PPP• Evaluate total deflection,()()()in.109.753.01610309.01210159.012106010291133336333222111−×=⎥⎥⎦⎤⎢⎢⎣⎡×+×−+××=⎟⎟⎠⎞⎜⎜⎝⎛++=∑=ALPALPALPEEALPiiiiiδin.109.753−×=δ22121in 9.0in. 12====AALL233in 3.0in. 16==AL© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 14Sample Problem 2.1The rigid bar BDE is supported by two links AB and CD. Link AB is made of aluminum (E = 70 GPa) and has a cross-sectional area of 500 mm2. Link CD is made of steel (E = 200 GPa) and has a cross-sectional area of (600 mm2). For the 30-kN force shown, determine the deflection a) of B, b) of D, and c) of E.SOLUTION:• Apply a free-body analysis to the bar BDE to find the forces exerted by links AB and DC.• Evaluate the deformation of links ABand DC or the displacements of Band D.• Work out the geometry to find the deflection at E given the deflections at B and D.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 15Sample Problem 2.1Displacement of B:()()()()m10514Pa1070m10500m3.0N10606926-3−×−=×××−==AEPLBδ↑= mm 514.0BδDisplacement of D:()()()()m10300Pa10200m10600m4.0N10906926-3−×=×××==AEPLDδ↓= mm 300.0DδFree body: Bar BDE()()ncompressioFFtensionFFMABABCDCDB kN60m2.0m4.0kN3000M kN90m2.0m6.0kN3000D−=×−×−==+=×+×−==∑∑SOLUTION:© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf2 - 16Sample Problem 2.1Displacement of D:()mm