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

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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Lecture Strength of Materials I: Chapter 4 - PhD. Tran Minh Tu STRENGTH OF MATERIALS1/10/2013 TRAN MINH TU - University of Civil Engineering, 1 Giai Phong Str. 55, Hai Ba Trung Dist. Hanoi, VietnamCHAPTER 4 State of Stress and Strength Hypothese 1/10/2013 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 Hypotheses1/10/2013 3 4.1. State of stress at a point• External loads applied to the body =>The body is deformed =>The stress isoccurred • 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 ofall stresses components acting on allsections, 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 44.1. 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 otherand 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 54.2. 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 z4.2. Plane StressSign 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-plus4.2.1. 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 74.2. Plane Stress sy 4.2.2. Stresses on Inclined Planes： u Sign Convention：    >0 - counterclockwise； sx txy  su >0 – pull out u  t uv - clockwiseO 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 = 01/10/2013 84.2. Plane Stress4.2.2. 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 94.2. Plane Stress4.2.3. 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 ...