Sample Problem 4.2Slide 2Slide 3Bending of Members Made of Several MaterialsExample 4.03Slide 6Reinforced Concrete BeamsSample Problem 4.4Slide 9Stress ConcentrationsPlastic DeformationsSlide 12Members Made of an Elastoplastic Material© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 1Sample Problem 4.2A cast-iron machine part is acted upon by a 3 kN-m couple. Knowing E = 165 GPa and neglecting the effects of fillets, determine (a) the maximum tensile and compressive stresses, (b) the radius of curvature.SOLUTION:•Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. 2dAIIAAyYx•Apply the elastic flexural formula to find the maximum tensile and compressive stresses.IMcm•Calculate the curvatureEIM1© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 2Sample Problem 4.2SOLUTION:Based on the cross section geometry, calculate the location of the section centroid and moment of inertia.mm 383000101143AAyY33332101143000104220120030402109050180090201mm ,mm ,mm Area,AyAAyy      49-432312123121231212m10868 mm1086818120040301218002090IdAbhdAIIx© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 3Sample Problem 4.2•Apply the elastic flexural formula to find the maximum tensile and compressive stresses.4949m10868m038.0mkN 3m10868m022.0mkN 3IcMIcMIMcBBAAmMPa 0.76AMPa 3.131B•Calculate the curvature  49-m10868GPa 165mkN 31EIMm 7.47m1095.2011-3© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 4Bending of Members Made of Several Materials•Consider a composite beam formed from two materials with E1 and E2.•Normal strain varies linearly.yx•Piecewise linear normal stress variation.yEEyEExx222111Neutral axis does not pass through section centroid of composite section.•Elemental forces on the section aredAyEdAdFdAyEdAdF222111  12112EEndAnyEdAynEdF •Define a transformed section such thatxxxnIMy21© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 5Example 4.03Bar is made from bonded pieces of steel (Es = 29x106 psi) and brass (Eb = 15x106 psi). Determine the maximum stress in the steel and brass when a moment of 40 kip*in is applied.SOLUTION:•Transform the bar to an equivalent cross section made entirely of brass•Evaluate the cross sectional properties of the transformed section•Calculate the maximum stress in the transformed section. This is the correct maximum stress for the brass pieces of the bar.•Determine the maximum stress in the steel portion of the bar by multiplying the maximum stress for the transformed section by the ratio of the moduli of elasticity.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 6Example 4.03•Evaluate the transformed cross sectional properties  431213121in. 063.5in. 3in. 25.2 hbITSOLUTION:•Transform the bar to an equivalent cross section made entirely of brass.in 25.2in 4.0in 75.0933.1in 4.0933.1psi1015psi102966TbsbEEn•Calculate the maximum stresses  ksi 85.11in. 5.063in. 5.1in.kip 404IMcm  ksi 85.11933.1maxmaxmsmbn  ksi 22.9ksi 85.11maxmaxsb© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 7Reinforced Concrete Beams•Concrete beams subjected to bending moments are reinforced by steel rods.•In the transformed section, the cross sectional area of the steel, As, is replaced by the equivalent areanAs where n = Es/Ec.•To determine the location of the neutral axis,   002221dAnxAnxbxdAnxbxsss•The normal stress in the concrete and steelxsxcxnIMy•The steel rods carry the entire tensile load below the neutral surface. The upper part of the concrete beam carries the compressive load.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 8Sample Problem 4.4A concrete floor slab is reinforced with 5/8-in-diameter steel rods. The modulus of elasticity is 29x106psi for steel and 3.6x106psi for concrete. With an applied bending moment of 40 kip*in for 1-ft width of the slab, determine the maximum stress in the concrete and steel.SOLUTION:•Transform to a section made entirely of concrete.•Evaluate geometric properties of transformed section.•Calculate the maximum stresses in the concrete and steel.© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 9Sample Problem 4.4SOLUTION:•Transform to a section made entirely of concrete. 2285466in95.4in 206.806.8psi 106.3psi 1029scsnAEEn•Evaluate the geometric properties of the transformed section.     422331in4.44in55.2in95.4in45.1in12in450.10495.4212Ixxxx•Calculate the maximum stresses.4241in44.4in55.2inkip4006.8in44.4in1.45inkip40IMcnIMcscksi306.1cksi52.18s© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 10Stress ConcentrationsStress concentrations may occur:•in the vicinity of points where the loads are appliedIMcKm•in the vicinity of abrupt changes in cross section© 2006 The McGraw-Hill Companies, Inc. All rights reserved. MECHANICS OF MATERIALSFourthEditionBeer • Johnston • DeWolf4 - 11Plastic Deformations•For any member subjected to pure