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MECHANICS OF MATERIALSFourth Beer Johnston DeWolfSample Problem 4 2 Based on the cross section geometry calculate the location of the section.

centroid and moment of inertia I x I A d 2 Apply the elastic flexural formula tofind the maximum tensile andcompressive stresses .

A cast iron machine part is acted uponby a 3 kN m couple Knowing E Calculate the curvature165 GPa and neglecting the effects of 1 Mfillets determine a the maximum EItensile and compressive stresses b .

the radius of curvature 2006 The McGraw Hill Companies Inc All rights reserved 4 1 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfSample Problem 4 2.

Based on the cross section geometry calculatethe location of the section centroid andmoment of inertia Area mm2 y mm yA mm31 20 90 1800 50 90 103.

2 40 30 1200 20 24 103 A 3000 yA 114 10 yA 114 10Y 38 mmI x I A d 2 121 bh3 A d 2 .

1 90 203 1800 122 1 30 403 1200 182 I 868 103 mm 4 868 10 9 m 4 2006 The McGraw Hill Companies Inc All rights reserved 4 2 MECHANICS OF MATERIALSFourth Beer Johnston DeWolf.

Sample Problem 4 2 Apply the elastic flexural formula to find themaximum tensile and compressive stresses M c A 3 kN m 0 022 m A 76 0 MPa A .

I 868 10 9 m 4M cB 3 kN m 0 038 m B 131 3 MPa B I 868 10 9 m 4 Calculate the curvature.

3 kN m 1 20 95 10 3 m 1 165 GPa 868 10 9 m 4 47 7 m 2006 The McGraw Hill Companies Inc All rights reserved 4 3.

MECHANICS OF MATERIALSFourth Beer Johnston DeWolfBending of Members Made of Several Materials Consider a composite beam formed fromtwo materials with E1 and E2 .

Normal strain varies linearly x Piecewise linear normal stress variation 1 E1 x 2 E2 x Neutral axis does not pass through.

section centroid of composite section Elemental forces on the section aredF1 1dA 1 dA dF2 2 dA 2 dA x Define a transformed section such that 1 x 2 n x dF2 .

nE1 y dA E1 y n dA E E1 2006 The McGraw Hill Companies Inc All rights reserved 4 4 MECHANICS OF MATERIALS.

Fourth Beer Johnston DeWolfExample 4 03 Transform the bar to an equivalent crosssection made entirely of brass Evaluate the cross sectional properties.

of the transformed section Calculate the maximum stress in thetransformed section This is the correctmaximum stress for the brass pieces ofBar is made from bonded pieces of.

steel Es 29x106 psi and brass Determine the maximum stress in the Eb 15x106 psi Determine the steel portion of the bar by multiplyingmaximum stress in the steel and the maximum stress for the transformedbrass when a moment of 40 kip in section by the ratio of the moduli ofis applied elasticity .

2006 The McGraw Hill Companies Inc All rights reserved 4 5 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfExample 4 03 Transform the bar to an equivalent cross section.

made entirely of brass Es 29 106 psin 1 933Eb 15 106 psibT 0 4 in 1 933 0 75 in 0 4 in 2 25 in.

Evaluate the transformed cross sectional properties1 b h3 1 2 25 in 3 in 3I 12 T 12 5 063 in 4 Calculate the maximum stresses.

Mc 40 kip in 1 5 in m 4 11 85 ksiI 5 063 in b max m b max 11 85 ksi.

s max n m 1 933 11 85 ksi s max 22 9 ksi 2006 The McGraw Hill Companies Inc All rights reserved 4 6 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfReinforced Concrete Beams.

Concrete beams subjected to bending moments arereinforced by steel rods The steel rods carry the entire tensile load belowthe neutral surface The upper part of theconcrete beam carries the compressive load .

In the transformed section the cross sectional areaof the steel As is replaced by the equivalent areanAs where n Es Ec To determine the location of the neutral axis bx x n As d x 0.

1 b x2 n As x n As d 0 The normal stress in the concrete and steel x c x s n x 2006 The McGraw Hill Companies Inc All rights reserved 4 7.

MECHANICS OF MATERIALSFourth Beer Johnston DeWolfSample Problem 4 4 Transform to a section made entirelyof concrete .

Evaluate geometric properties oftransformed section Calculate the maximum stressesin the concrete and steel A concrete floor slab is reinforced with.

5 8 in diameter steel rods The modulusof elasticity is 29x106psi for steel and3 6x106psi for concrete With an appliedbending moment of 40 kip in for 1 ftwidth of the slab determine the maximum.

stress in the concrete and steel 2006 The McGraw Hill Companies Inc All rights reserved 4 8 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfSample Problem 4 4.

Transform to a section made entirely of concrete Es 29 106 psin 8 06Ec 3 6 106 psinAs 8 06 2 4 85 in 4 95 in 2.

Evaluate the geometric properties of thetransformed section 12 x 4 95 4 x 0 x 1 450 inI 13 12 in 1 45 in 3 4 95 in 2 2 55 in 2 44 4 in 4 Calculate the maximum stresses .

Mc1 40 kip in 1 45 in c c 1 306 ksiI 44 4 in 4Mc2 40 kip in 2 55 in s 18 52 ksi s n 8 06.

I 44 4 in 4 2006 The McGraw Hill Companies Inc All rights reserved 4 9 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfStress Concentrations.

Stress concentrations may occur Mc m K in the vicinity of points where theloads are applied in the vicinity of abrupt changes.

in cross section 2006 The McGraw Hill Companies Inc All rights reserved 4 10 MECHANICS OF MATERIALSFourth Beer Johnston DeWolfPlastic Deformations.

For any member subjected to pure bending x mstrain varies linearly across the If the member is made of a linearly elastic material the neutral axis passes through the section centroidand x .

For a material with a nonlinear stress strain curve the neutral axis location is found by satisfyingFx x dA 0 M y x dA For a member with vertical and horizontal planes ofsymmetry and a material with the same tensile and.

compressive stress strain relationship the neutralaxis is located at the section centroid and the stress strain relationship may be used to map the straindistribution from the stress distribution 2006 The McGraw Hill Companies Inc All rights reserved 4 11.

MECHANICS OF MATERIALSFourth Beer Johnston DeWolfPlastic Deformations When the maximum stress is equal to the ultimatestrength of the material failure occurs and the.

corresponding moment MU is referred to as theultimate bending moment The modulus of rupture in bending RB is foundfrom an experimentally determined value of MUand a fictitious linear stress distribution .

RB may be used to determine MU of any membermade of the same material and with the samecross sectional shape but different dimensions 2006 The McGraw Hill Companies Inc All rights reserved 4 12 MECHANICS OF MATERIALS.

Fourth Beer Johnston DeWolfMembers Made of an Elastoplastic Material Rectangular beam made of an elastoplastic material x Y m m Y M Y Y maximum elastic moment.

If the moment is increased beyond the maximumelastic moment plastic zones develop around anelastic core 1 yY2 M 2 MY 1 3 2 yY elastic core half thickness.

c In the limit as the moment is increased further theelastic core thickness goes to zero corresponding to afully plastic deformation M p 32 M Y plastic moment.

k shape factor depends only on cross section shape 2006 The McGraw Hill Companies Inc All rights reserved 4 13The upper part of the concrete beam carries the compressive load. A 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.

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