Stress Transformation

    General State of Stress

    The general state of stress at a point is characterized by three independent normal stress components and three independent shear stress components, and is represented by the stress tensor. The combination of the state of stress for every point in the domain is called the stress field.
    Stress tensor.
    T=[σxτxyτxzτxyσyτyzτxzτyzσz]
    Warning: Stress is a physical quantity and as such, it is independent of the chosen coordinate system. #str-ind

    Sign Convention

    Sign conventions on 2D elements
    • Positive normal stress acts outward from all faces.
    • Positive shear stress points towards the positive axis direction in a positive face.
    • Positive shear stress points towards the negative axis direction in a negative face.

    Plane Stress

    Plane stress occurs when two faces of the cube element are stress free.
    Plane stress.
    σz=τzx=τzy=0
    Example: Thin plates subject to forces acting in the mid-plane of the plate. #thn-plt

    Plane Stress Transformation

    We think of stresses acting on faces, so we often associate the state of stress with a coordinate system. However, the selection of a coordinate system is arbitrary (materials don't know about coordinates - it's a mathematical construct!) and we could choose to express the stress state acting on any set of faces aligned with any coordinate system axes. Furthermore, we can relate the states of stress in each coordinate system to one another through stress transformation equations.

    Heads up!

    Cauchys stress theorem builds on this content in later solid mechanics courses.

    For any surface that divides the body ( imaginary or real surface), the action of one part of the body on the other is equivalent to the system of distributed internal forces and moments and it is represented by the stress vector tn (also called traction), defined on the surface with normal unit vector n.

    The state of stress at a point in the body is defined by all the stress vectors tn associated with all planes (infinite in number) that pass through that point.

    Cauchys stress theorem.
    tn=Tn
    Cauchys stress theorem states that there exists a stress tensor T (which is independent of n), such that tn is a linear function of n.

    Transformation Sign Convention

    • Both the xy (original) and xy (transformed) systems follow the right-hand rule.
    • The orientation of an inclined plane (on which the normal and shear stress components are to be determined) will be defined using the angle θ. The angle θ is measured from the positive x to the positive x -axis. It is positive if it follows the curl of the right-hand fingers.
    Stress transformation equations. #str-trn
    σx=σx+σy2+σxσy2cos(2θ)+τxysin(2θ)σy=σx+σy2σxσy2cos(2θ)τxysin(2θ)τxy=σxσy2sin(2θ)+τxycos(2θ)
    Normal and shear stresses can be shown graphically as a function of θ to see how they overlap.
    Stresses based on the transformation angle chosen.

    2-D Mohr's Circle

    Mohr's circle is a graphical representation of stress transformations. The equations for stress transformations actually describe a circle if we consider the normal stress σ to be the x-coordinate and the shear stress π to be the y-coordinate.
    Circle centroid.
    C=σavg=σx+σy2=σ1+σ22 
    Cirlce radius.
    R=(σxσy2)2+τxy2 
    Two points of the circle.
     Point X:(σx,τxy) Point Y:(σy,τxy)
    The principal stresses are where the circle crosses the x-axis, and the maximum shear stress is the highest y-coordinate of the circle.

    Interactive Mohr's circle

    Example Problem: Interactive Mohr's circle #int-mrh

    σx: -80

    σy: 50

    τxy: -25

    σ1: 54.64

    σ2: -84.64

    τmax: 69.64

    σx: -80

    σy: 50

    τxy: -25

    Angle:

    θ= 0

    3-D Mohr's Circle

    Circle centroid.
    C=σavg=σ1+σ32 
    Cirlce radius.
    R=σ1σ32
    Points of the circle are plotted using each face.
    (σplane,τplane)
    The principal stresses are where the circle crosses the x-axis, and the maximum shear stress is the highest y-coordinate of the circle.

    Stresses on Inclined Planes

    Stresses on inclined planes. #str-inc
    σx=n tn=n Tn=σxcos2(θ)+2τxysin(θ)cos(θ)+σysin2(θ)τxy=s tn=s Tn=(σyσx)sin(θ)cos(θ)+τxy(cos2(θ)sin2(θ))

    Pure Shear

    A circular shaft under torsion develops pure shear on cross-sections between longitudinal planes (the faces of element a are parallel and perpendicular to the axis of the shaft).
    Torsion on inclined planes. #tor-inc
    σx=2τmaxsinθcosθ=τmaxsin(2θ)τxy=τmax(cos2θsin2θ)=τmaxcos(2θ)

    Principal Stresses

    σx can be maximized to find the principal stresses.
    Angle for maximum normal stress. #prn-str
    tan(2θp1)=2τxyσxσyθp2=θp1+90o
    The maximum/minimum normal stress values (the principal stresses) are associated with θp1 and θp2.
    Principal stresses
    σ1,2=σx+σy2±(σxσy2)2+τxy2 
    We use the convention that σ1>σ2.

    Alternative Approach: Eigenvalues

    The eigenvalues of the stress tensor are called the principal stresses, and the eigenvectors define the principal direction vectors.

    Because of symmetry, the stress tensor (T) has real eigenvalues (λ) and mutually perpendicular eigenvectors (v).

    Eigenvalues.
    Tv=λv(TλI)v=0 
    From linear algebra, we know that a system of linear equations Av=0 has a non-zero solution v if, and only if, the determinant of the matrix T is zero.
    Eigenvalues relate to stress. #egn-str
    λ1,2=σ1,2
    To find the eigenvectors, we plug our eigenvalues back into the equation (TλI)v=0.
    First eigenvector angle. #fst-egn
    θp1=tan1(σ1σxτxy)=tan1(τxyσ1σy) 
    We can repeat this procedure for the second eigenvalue, λ2=σ2.
    Second eigenvector angle.
    θp2=tan1(σ2σxτxy)=tan1(τxyσ2σy) 

    Maximum Shear Stress

    The orientations for the principal stress element and max shear stress element are 45o apart. τxy can also be maximized.
    Angle for maximum shear stress. #max-shr
    tan(2θs1)=(σxσy)2τxyθs2=θs1+90o
    The maximum/minimum in plane shear stress values are associated with θs1 and θs2.
    Maximum shear stress.
    |τmax|=(σxσy2)2+τxy2 
    A maximum shear stress element has an average normal stress.
    Average normal stress.
    σx=σy=σavg=σx+σy2 
    Warning: Unlike with the principal stress element, the normal stresses are not zero. #max-shr