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Lecture Strength of Materials I: Chapter 4 - PhD. Tran Minh Tu

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Chapter 4 - State of stress and strength hypothese. The following will be discussed in this chapter: State of stress at a point, plane stress, mohr’s circle, special cases of plane stress, stress – strain relations, strength hypotheses.

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Nội dung Text: Lecture Strength of Materials I: Chapter 4 - PhD. Tran Minh Tu

  1. STRENGTH OF MATERIALS 1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, Vietnam
  2. CHAPTER 4 State of Stress and Strength Hypothese 1/10/2013
  3. Contents 4.1. State of stress at a point 4.2. Plane Stress 4.3. Mohr’s Circle 4.4. Special cases of plane stress 4.5. Stress – Strain relations 4.6. Strength Hypotheses 1/10/2013 3
  4. 4.1. State of stress at a point • External loads applied to the body => The body is deformed =>The stress is occurred • At a point K on the arbitrary section, there n  are 2 types of stress: normal stress s and shearing stress t y K  • The state of stress at a point K is a set of all stresses components acting on all sections, which go through this point z x • The most general state of stress at a point may be represented by 6 components, s x ,s y ,s z normal stresses t xy , t yz , t zx shearing stresses (Note: t xy  t yx , t yz  t zy , t zx  t xz ) 1/10/2013 4
  5. 4.1. State of stress at a point • Principal planes: no shear stress acts on • Principal directions: the direction of the principal planes • Principal stresses: the normal stress act on the principal plane • There are three principal planes , which are perpendicular to each other and go through a point • Three principal stresses: s1, s2, s3 with: s1 ≥ s2 ≥ s3 • Types of state of stress: - Simple state of stress: 2 of 3 principal stresses equal to zeros - Plane state of stress: 1 of 3 principal stresses equal to zeros - General state of stress: all 3 principal stresses differ from zeros 1/10/2013 5
  6. 4.2. Plane Stress • Plane Stress – the 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 s x , s y , t xy and s z  t zx  t zy  0. • State of plane stress occurs in a thin plate subjected to the forces acting in the mid-plane of the plate. y sy O sy tyx x tyx y sx sx txy txy x 6 z
  7. 4.2. Plane Stress Sign Convention： • Normal Stress: positive: tension; negative: compression • Shear Stress: positive: the direction associated with its subscripts are plus-plus or minus-minus; negative: the directions are plus-minus or minus-plus 4.2.1. Complementary shear stresses： • The shear stresses with the same subscripts in two orthogonal planes (e.g. txy and tyx) are equal y 1/10/2013 7
  8. 4.2. Plane Stress sy 4.2.2. Stresses on Inclined Planes： u Sign Convention：    >0 - counterclockwise； sx txy  su >0 – pull out u  t uv - clockwise O x su Fu 0  v s u A  s x A cos2   t xy A cos  sin  y sx  s y A sin 2   t yx A sin  cos   0 txy tuv tyx F v 0 sy τuv A - τ xy Acos 2 α - σ x Acosαsinα Asin  A Acos  +τ yx Asin2 α +σ y Asinαcosα = 0 1/10/2013 8
  9. 4.2. Plane Stress 4.2.2. Stresses on Inclined Planes： su u s x s y s x s y su   cos 2  t xy sin 2 txy  2 2 x s x s y tuv t uv  sin 2  t xy cos 2 sx sy 2 y v tyx -  > 0: counterclockwise from the x axis to u axis sy 1/10/2013 9
  10. 4.2. Plane Stress 4.2.3. Principal stresses are maximum and minimum stresses ： By taking the derivative of su to  and setting it equal to zero: ds u 2t xy  0 => tg2 p =- d sx sy  p  p1, p 2    p  90 0 sx s y  s x s y  2 s max, min  s 1,2(3)      t 2 2  2  xy 1/10/2013 10
  11. 4.2. Plane Stress 4.2.4. Maximum in-plane shear stresses dt sx sy  0 => tg2 s = =>  s = p  450 d 2t xy => An element subjected to maximum shear stresses will be 450 from the position of an element that is subjected to the principal stress  s x s y  2 t max,min     t 2  2  xy 4.2.5. The first invariant of plane stress The sum of the normal stresses has the same value in each coordinate system s x  s y  s u  s v  const 1/10/2013 11
  12. 4.3. Mohr’s Circle Using the transformation relations: sx  s y sx  s y ( su - )2  ( cos 2  t xy sin 2 )2 2 2 sx  s y ( tuv )2  ( sin 2  t xy cos 2 )2 2 sx  s y   sx  s y  2 2  s  u -   t 2 uv     t 2 xy Mohr’ Circle  2   2   sx  s y   sx  s y  2 Center: I  ,0  Radius: R    t xy 2  2   2  1/10/2013 12
  13. 4.3. Mohr’s Circle Plane Stress t uv sy tyx t u t K max u uv sx  M  sx t xy txy txy R tyx 02 sy O ` s u B I s u A 01  sx  s y  I ,0  s2 u1  2  sy  sx  s y  2 R    t xy 2  2  sx t min Điểm cực  M s y , t xy  u2 s1 1/10/2013 13
  14. 4.4. Special Cases of Plane Stress 4.4.1. Uniaxial tension 4.4.2. Pure shear 1/10/2013 14
  15. 4.4. Special Cases of Plane Stress 4.4.3. Special plane stress t t t s I s s s t t smin smax t s s 2 s1  s 3 s 2 s max,min  s1,3      t 2 t max      t2 2 2 2 2 1/10/2013 15
  16. 4.5. Stress – Strain relations 1. Uniaxial stress y sx  sx x  y   s x z    s x E E E 2. Pure shear z x t xy y  xy   yz   zx  0 G txy 1/10/2013 z x16
  17. 4.5. Stress – Strain relations 3. General state of stress - Assumption: The normal strain sy causes only the normal stress. The Shear strain causes only the shear y stress sx - The Principle of superposition sz txy sx sy sz x    E E E z x  1 E  s x   s y sz   1/10/2013 17
  18. 4.5. Stress – Strain relations a. Normal stress – normal strain relation Generalized Hooke’s law  x  s x   s y  s z  1    1  s 1   s 2  s 3  1 E 1 E  y  s y   s x  s z  E  2  1 s 2   s 3  s 1  E  z  s z   s x  s y  1  3  s 3   s 2  s 1  1 E E b. Shear stress – shear strain relation t xy t xz t yz  xy   xz   yz  G G G E with E, , G are Young modulus, Poisson ratio, G shear modulus, which the relation among them: 2 1    1/10/2013 18
  19. 4.5. Stress – Strain relations 1 1 - Plane stress  x  s x  s y  1  s 1  s 2  E E t xy  xy  1  y  s y  s x  1  2  s 2  s 1  G E E 4. Normal stress – unit volume change relation V  a1a2a3 V1  a1 ( 1   1 )a2 ( 1   2 )a3 ( 1   3 ) V1  V   1   2   3 V 1  2 1  2  (s1 s 2 s 3 )  (s x s y s z ) E E 1/10/2013 19
  20. 4.5. Stress – Strain relations 5. Strain energy s1 1 1 u  us  ut  s  t 2 2 s2 Principal element: t = 0 => s3 1 1 1 u  s 1 1  s 2 2  s 3 3 2 2 2  1 2E   s 12  s 22  s 32  2 s 1s 2  s 3s 2  s 1s 3  1/10/2013 20
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