MECHANICS OF MATERIALS Sample Problem 4.2

Transcription

1 Sample Problem 4. SOLUTON: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. ya ( + Y Ad ) A A cast-iron machine part is acted upon by a kn-m couple. Knowing 65 GPa and neglecting the effects of fillets, determine (a) the maimum tensile and compressive stresses, (b) the radius of curvature. Apply the elastic fleural formula to find the maimum tensile and compressive stresses. Mc m Calculate the curvature M 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -

2 Sample Problem 4. SOLUTON: Based on the cross section geometry, calculate the location of the section centroid and moment of inertia. Area, mm y, mm 50 0 y A, mm A 000 ya 4 0 Y ya 4 0 A mm ( ) ( + Ad bh + Ad ) ( 90 ) ( ) mm m 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -

3 Sample Problem 4. Apply the elastic fleural formula to find the maimum tensile and compressive stresses. Mc m M c kn m 0.0 m A A A m M c kn m 0.08m B B B m 0 MPa. MPa Calculate the curvature M kn m ( 65 GPa 9 4 )( m ) m m 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -

4 Bending of Members Made of Several Materials Consider a composite beam formed from two materials with and. Normal strain varies linearly. ε y Piecewise linear normal stress variation. y ε ε y Neutral ais does not pass through section centroid of composite section. My n n lemental forces on the section are df y da da df da Define a transformed section such that ( n ) y y df da ( n da ) n y da 006 The McGraw-Hill Companies, nc. All rights reserved. 4-4

5 ample 4.0 SOLUTON: Transform the bar to an equivalent cross section made entirely of brass valuate the cross sectional properties of the transformed section Bar is made from bonded pieces of steel ( 90 6 s psi) and brass ( b 50 6 psi). Determine the maimum stress in the steel and brass when a moment of 40 kip*inin is applied. Cl Calculate lt the maimum stress in the transformed section. This is the correct maimum stress for the brass pieces of the bar. Determine the maimum stress in the steel portion of the bar by multiplying the maimum stress for the transformed section by the ratio of the moduli of elasticity. 006 The McGraw-Hill Companies, nc. All rights reserved. 4-5

6 ample 4.0 SOLUTON: Transform the bar to an equivalent cross section made entirely of brass. n b T s b psi.9 psi 0.4 in in in.5 in valuate the transformed cross sectional properties b h T 5.06 in. 4 (.5 in. )( in. ) Calculate the maimum stresses m Mc ( 40 kip in. )(.5 in. ) 5.06 in ksi ( b ) ma m ( b ) ma ( ) n.9.85 ki ksi ( ) 9 ki ksi s ma m s.85 ksi ma The McGraw-Hill Companies, nc. All rights reserved. 4-6

7 Reinforced Concrete Beams Concrete beams subjected to bending moments are reinforced by steel rods. The steel rods carry the entire tensile load below the neutral surface. The upper part of the concrete beam carries the compressive load. n the transformed section, the cross sectional area of the steel, A s, is replaced by the equivalent area na s where n s / c. To determine the location of the neutral ais, ( b ) n A ( d ) 0 s b + n A n A d s s The normal stress in the concrete and steel My n c 006 The McGraw-Hill Companies, nc. All rights reserved. 4-7 s 0

8 Sample Problem 4.4 A concrete floor slab is reinforced with 5/8-in-diameter steel rods. The modulus of elasticity is 906psi for steel and.606psi for concrete. With an applied bending moment of 40 kip*in for -ft width of the slab, determine the maimum stress in the concrete and steel. SOLUTON: Transform to a section made entirely of concrete. valuate geometric properties of transformed section. Calculate the maimum stresses in the concrete and steel. 006 The McGraw-Hill Companies, nc. All rights reserved. 4-8

9 Sample Problem 4.4 SOLUTON: Transform to a section made entirely of concrete. n na s s c psi 8.06 psi ( 5 in ) 4.95in 8.06 π 4 8 π valuate the geometric properties of the transformed section ( 4 ) 0.450in 4 ( in)(.45in) + ( 4.95in )(.55in) 44.4 in Calculate the maimum stresses. c s Mc Mc n 40kip in.45in in 40kip in.55in in c.06ksi 8.5ksi 006 The McGraw-Hill Companies, nc. All rights reserved. 4-9 s

10 Stress Concentrations Stress concentrations may occur: in the vicinity of points where the loads are applied in the vicinity of abrupt changes in cross section m K Mc 006 The McGraw-Hill Companies, nc. All rights reserved. 4-0

11 Plastic Deformations For any member subjected to pure bending ε y ε c strain varies linearly across the section m fthe member is made of a linearly elastic material, the neutral ais passes through the section centroid and My For a material with a nonlinear stress-strain strain curve, the neutral ais location is found by satisfying F da 0 M y da For a member with vertical and horizontal planes of symmetry yand a material with the same tensile and compressive stress-strain relationship, the neutral ais is located at the section centroid and the stress- strain relationship may be used to map the strain distribution from the stress distribution. 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -

12 Plastic Deformations When the maimum stress is equal to the ultimate strength of the material, failure occurs and the corresponding moment M U is referred to as the ultimate bending moment. The modulus of rupture in bending, R B, is found from an eperimentally determined value of M U and a fictitious linear stress distribution. R B M U c R B may be used to determine M U of any member made of the same material and with the same cross sectional shape but different dimensions. 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -

13 Members Made of an lastoplastic Material Rectangular beam made of an elastoplastic material m Y m Mc Y MY Y c maimum elastic moment f the moment is increased beyond the maimum elastic moment, plastic zones develop around an elastic core. M Y y M Y Y y c elastic core half - thickness n the limit as the moment is increased further, the elastic core thickness goes to zero, corresponding to a fully plastic deformation. M p M k M M Y p Y plastic moment shape factor (depends only on cross section shape) 006 The McGraw-Hill Companies, nc. All rights reserved. 4 -