Strength of materials - Chapter 4: Stress transformation - Nguyễn Sỹ Lâm

Nội dung text: Strength of materials - Chapter 4: Stress transformation - Nguyễn Sỹ Lâm

1. CHAPTER 4: STRESS TRANSFORMATION 4.1 Introduction 4.2 Plane Stress State 4.3 Transformation of plane stress 4.4 Morh’s circle for plane stress 4.5 Hooke’s Laws 4.6 Transformation of plane strain 4.7 Morh’s circle for plane strain
2. 4.1 INTRODUCTION SIGN CONVENTION: Normal stress: Tension is positive Compression is Negative Shear stress: two subscripts + First subscript denotes the face on which the stress acts + Second gives the direction on the stress vector Positive face (+): normal axis follows the positive direction of the original axis Negative face (-): normal axis follows the negative direction of the original axis
3. 4.2 PLANE STRESS STATE • Plane Stress - state of stress in which two faces of the cubic element are free of stress. For the illustrated example, the state of stress is defined by  x,  y, xy and  z  zx  zy 0.
4. 4.2 PLANE STRESS STATE • State of plane stress occurs in a thin plate subjected to forces acting in the midplane of the plate.
5. 4.3 TRANSFORMATION OF PLANE STRESS θ is positive if the rotation is counter clockwise from x to x’ • The equations may be rewritten to yield      x y x y cos 2   sin 2  x 22 xy      x y x y cos 2   sin 2  y 22 xy   xysin 2   cos 2  x y2 xy
6. 4.3 TRANSFORMATION OF PLANE STRESS Maximum Shearing Stresses 2  x  y 2  max R  xy 2  x  y tan2s 2 xy Note : defines two angles separated by 90o and o offset from  p by 45     x y ave 2
7. 4.3 TRANSFORMATION OF PLANE STRESS EXAMPLE 4.01 SOLUTION: • Find the element orientation for the principal stresses from 2 xy tan 2 p  x  y • Determine the principal stresses from 2  x  y  x  y 2  max,min  xy 2 2 • Calculate the maximum shearing stress with For the state of plane stress shown, 2 determine (a) the principal panes,  x  y 2 max  xy (b) the principal stresses, (c) the 2 maximum shearing stress and the   corresponding normal stress.  x y 2
8. 4.3 TRANSFORMATION OF PLANE STRESS EXAMPLE 4.01 • Calculate the maximum shearing stress with 2  x  y 2 max  xy 2 30 2 40 2 max 50 MPa  x 50 MPa  xy 40 MPa s  p 45  10 MPa x s 18.4, 71.6 • The corresponding normal stress is   50 10   x y ave 2 2  20 MPa
9. 4.4 MORH’S CIRCLE FOR PLANE STRESS • With Mohr’s circle uniquely defined, the state of stress at other axes orientations may be depicted. • For the state of stress at an angle  with respect to the xy axes, construct a new diameter X’Y’ at an angle 2 with respect to XY. • Normal and shear stresses are obtained from the coordinates X’Y’.
10. 4.4 MORH’S CIRCLE FOR PLANE STRESS EXAMPLE 4.02 For the state of plane stress shown, (a) construct Mohr’s circle, determine (b) the principal planes, (c) the SOLUTION: principal stresses, (d) the maximum • Construction of Mohr’s circle shearing stress and the corresponding  x  y 50 10 normal stress.  20 MPa ave 2 2 CF 50 20 30 MPa FX 40 MPa R CX 30 2 40 2 50 MPa
11. 4.4 MORH’S CIRCLE FOR PLANE STRESS EXAMPLE 4.02 • Maximum shear stress s  p 45 max R   ave s 71.6 max 50 MPa  20 MPa
12. 4.4 MORH’S CIRCLE FOR PLANE STRESS EXAMPLE 4.03 • Principal planes and stresses XF 48  max OA OC CA  max OA OC BC tan 2 p 2.4 CF 20 80 52 80 52 2 p 67.4  max 132 MPa  min 28 MPa  p 33.7 clockwise
13. 4.4 MORH’S CIRCLE FOR PLANE STRESS
14. 4.4 MORH’S CIRCLE FOR PLANE STRESS Application of Morh’s circle to the Three-Dimensional Analysis of Stress • Transformation of stress for an element • The three circles represent the rotated around a principal axis may be normal and shearing stresses for represented by Mohr’s circle. rotation around each principal axis. • Points A, B, and C represent the • Radius of the largest circle yields the principal stresses on the principal planes maximum shearing stress. (shearing stress is zero) 1    max 2 max min
15. 4.4 MORH’S CIRCLE FOR PLANE STRESS Application of Morh’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 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
16. 4.5 HOOKE’S LAW DILATATION: BULK MODULUS • Relative to the unstressed state, the change in volume is e 1  1  x 1  y 1  z  1 1  x  y  z   x  y  z 1 2    E x y z dilatation (change in volume per unit volume) • For element subjected to uniform hydrostatic pressure, 3 1 2 p e p E k E k bulk modulus 3 1 2 • Subjected to uniform pressure, dilatation must be negative, therefore 0  1 2
17. 4.5 HOOKE’S LAW Relation Among E, , and G • An axially loaded slender bar will elongate in the axial direction and contract in the transverse directions. • An initially cubic element oriented as in top figure will deform into a rectangular parallelepiped. The axial load produces a normal strain. • If the cubic element is oriented as in the bottom figure, it will deform into a rhombus. Axial load also results in a shear strain. • Components of normal and shear strain are related, E 1  2G
18. 4.6 TRANSFORMATION FOR PLANE STRAIN • State of strain at the point Q results in different strain components with respect to the xy and x’y’ reference frames. 2 2   x cos   y sin   xy sin cos   45 1    OB 2 x y xy  xy 2OB x  y • Applying the trigonometric relations used for the transformation of stress, x  y x  y  xy  cos2 sin 2 x 2 2 2 x  y x  y  xy  cos2 sin 2 y 2 2 2     x y x y sin 2 xy cos2 2 2 2
19. 4.7 MORH’S CIRCLE FOR PLANE STRAIN Three-Dimensional Analysis of Strain • Previously demonstrated that three principal 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 principal planes of stress are also the principal planes of strain. • Rotation about the principal axes may be represented by Mohr’s circles.
20. 4.7 MORH’S CIRCLE FOR PLANE STRAIN Three-Dimensional Analysis of Strain • Consider the case of plane stress,  x  a  y b  z 0 • Corresponding normal strains,    a b a E E    a b b E E        c E a b 1  a b • 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.