MECHANICS OF MATERIALS

Transcription

1 Application of Mohr s Circle to the Three- Dimensional Analysis of Stress If A and B are on the same side of the origin (i.e., have the same sign), then a) the circle defining max, min, and τ max for the element is not the circle corresponding to transformations within the plane of stress b) maximum shearing stress for the element is equal to half of the maximum stress c) planes of maximum shearing stress are at 45 degrees to the plane of stress 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-1 Yield Criteria for Ductile Materials Under Plane Stress Failure of a machine component subjected to uniaxial stress is directly edicted from an equivalent tensile test Failure of a machine component subjected to plane stress cannot be directly edicted from the uniaxial state of stress in a tensile test specimen It is convenient to determine the incipal stresses and to base the failure criteria on the corresponding biaxial stress state Failure criteria are based on the mechanism of failure. Allows comparison of the failure conditions for a uniaxial stress test and biaxial component loading 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-1

2 Yield Criteria for Ductile Materials Under Plane Stress Maximum shearing stress criteria: Structural component is safe as long as the maximum shearing stress is less than the maximum shearing stress in a tensile test specimen at yield, i.e., τ Y max < τy = For a and b with the same sign, a b τ max = or < Y For a and b with opposite signs, a b τ Y max = < 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-3 Yield Criteria for Ductile Materials Under Plane Stress Maximum distortion energy criteria: Structural component is safe as long as the distortion energy per unit volume is less than that occurring in a tensile test specimen at yield. 1 6G 1 ( + ) < ( ) a a a b a b + ud < uy b 6G b < Y Y Y 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-4

3 Fracture Criteria for Brittle Materials Under Plane Stress Brittle materials fail suddenly through rupture or fracture in a tensile test. The failure condition is characterized by the ultimate strength U. Maximum normal stress criteria: Structural component is safe as long as the maximum normal stress is less than the ultimate strength of a tensile test specimen. a < b < U U 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-5 Stresses in Thin-Walled Pressure Vessels Cylindrical vessel with incipal stresses 1 = hoop stress = longitudinal stress Hoop stress: = 0 = ( t x) p( r x) F z 1 = t F x = 0 = = t 1 = 1 Longitudinal stress: ( π rt) p( π r ) 006 The McGraw-Hill Companies, Inc. All rights reserved

4 Stresses in Thin-Walled Pressure Vessels Points A and B correspond to hoop stress, 1, and longitudinal stress, Maximum in-plane shearing stress: 1 τmax( in plane) = = 4t Maximum out-of-plane shearing stress corresponds to a 45 o rotation of the plane stress element around a longitudinal axis τmax = = t 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-7 Stresses in Thin-Walled Pressure Vessels Spherical essure vessel: 1 = = t Mohr s circle for in-plane transformations reduces to a point = 1 = = constant τmax(in-plane) = 0 Maximum out-of-plane shearing stress τ 1 max = 1 = 4t 006 The McGraw-Hill Companies, Inc. All rights reserved

5 Transformation of Plane Strain Plane strain - deformations of the material take place in parallel planes and are the same in each of those planes. Plane strain occurs in a plate subjected along its edges to a uniformly distributed load and restrained from expanding or contracting laterally by smooth, rigid and fixed supports components of strain : ( γ = γ 0) x y γ xy z = zx zy = Example: Consider a long bar subjected to uniformly distributed transverse loads. State of plane stress exists in any transverse section not located too close to the ends of the bar. 006 The McGraw-Hill Companies, Inc. All rights reserved. 7-9 Transformation of Plane Strain State of strain at the point Q results in different strain components with respect to the xy and x y reference frames. ( θ ) = cos θ + sin θ + γ sinθ cosθ γ OB = x xy = OB ( 45 ) = 1 ( x + y + γ xy ) ( + ) x Applying the trigonometric relations used for the transformation of stress, x y γ = sin θ + y y xy x + y x y γ xy x = + cosθ + x + y x y γ xy y = cosθ γ x y xy cosθ sin θ sin θ 006 The McGraw-Hill Companies, Inc. All rights reserved

6 Mohr s Circle for Plane Strain The equations for the transformation of plane strain are of the same form as the equations for the transformation of plane stress - Mohr s circle techniques apply. Abscissa for the center C and radius R, x + y x y γ xy ave = R = + Principal axes of strain and incipal strains, γ xy tan θ p = x max = ave + R y min = ave R Maximum in-plane shearing strain, ( ) γ max = R = x y xy γ The McGraw-Hill Companies, Inc. All rights reserved Three-Dimensional Analysis of Strain Previously demonstrated that three incipal axes exist such that the perpendicular element faces are free of shearing stresses. By Hooke s Law, it follows that the shearing strains are zero as well and that the incipal planes of stress are also the incipal planes of strain. Rotation about the incipal axes may be reesented by Mohr s circles. 006 The McGraw-Hill Companies, Inc. All rights reserved

7 Three-Dimensional Analysis of Strain For the case of plane strain where the x and y axes are in the plane of strain, - the z axis is also a incipal axis - the corresponding incipal normal strain is reesented by the point Z = 0 or the origin. If the points A and B lie on opposite sides of the origin, the maximum shearing strain is the maximum in-plane shearing strain, D and E. If the points A and B lie on the same side of the origin, the maximum shearing strain is out of the plane of strain and is reesented by the points D and E. 006 The McGraw-Hill Companies, Inc. All rights reserved Three-Dimensional Analysis of Strain Consider the case of plane stress, = = 0 x = a y b z Corresponding normal strains, a ν b a = E E b = c = ν a + E b E ν ν ( a + b ) = ( a + b ) E ν Strain perpendicular to the plane of stress is not zero. If B is located between A and C on the Mohr-circle diagram, the maximum shearing strain is equal to the diameter CA The McGraw-Hill Companies, Inc. All rights reserved

8 Measurements of Strain: Strain Rosette Strain gages indicate normal strain through changes in resistance. With a 45 o rosette, x and y are measured directly. γ xy is obtained indirectly with, xy = OB ( ) γ + x y Normal and shearing strains may be obtained from normal strains in any three directions, 1 = x cos θ1 + y sin θ1 + γ xy sinθ1 cosθ1 = x cos θ + y sin θ + γ xy sinθ cosθ 3 = x cos θ3 + y sin θ3 + γ xy sinθ3 cosθ3 006 The McGraw-Hill Companies, Inc. All rights reserved