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6 Plane Stress Transformations ASEN 3112 Lecture 6 Slide 1 ASEN - PDF document

Nov 14, 2022 •396 likes •642 views

ASEN 3112 - Structures 6 Plane Stress Transformations ASEN 3112 Lecture 6 Slide 1 ASEN 3112 - Structures Plane Stress State Recall that in a body in plane stress, the general 3D stress state with 9 components (6 independent) reduces to

  1. ASEN 3112 - Structures 6 Plane Stress Transformations ASEN 3112 Lecture 6 – Slide 1

  2. ASEN 3112 - Structures Plane Stress State Recall that in a body in plane stress, the general 3D stress state with 9 components (6 independent) reduces to 4 components (3 independent): τ σ τ xy xz plane stress σ xx τ xy xx 0 σ τ τ σ yy 0 τ yx yx yz yy τ τ σ 0 0 0 zx zy zz with = τ yx τ xy Plane stress occurs in thin plates and shells (e.g. aircraft & rocket skins, parachutes, balloon walls, boat sails, ...) as well as thin wall structural members in torsion. In this Lecture we will focus on thin flat plates and associated two-dimensional stress transformations ASEN 3112 Lecture 6 – Slide 2

  3. ASEN 3112 - Structures Flat Plate in Plane Stress or transverse dimension z Thickness dimension Top surface y x Inplane dimensions: in x,y plane ASEN 3112 Lecture 6 – Slide 3

  4. ASEN 3112 - Structures Mathematical Idealization as a Two Dimensional Problem y Midplane x Plate ASEN 3112 Lecture 6 – Slide 4

  5. ASEN 3112 - Structures Internal Forces, Stresses, Strains In-plane internal forces y + sign conventions for internal forces, dy dx stresses and strains p yy h y Thin plate in plane stress x dy p xx p z x xy dx dy In-plane stresses dx In-plane body forces dx dy y dx dy h y σ h y x yy τ = τ σ b y x xy yx xx b x x In-plane displacements In-plane strains dx dy dx dy h h y ε y u yy y ε γ = γ u x x x xx xy yx ASEN 3112 Lecture 6 – Slide 5

  6. ASEN 3112 - Structures Stress Transformation in 2D σ yy σ tt (a) τ tn (b) τ nt σ nn τ yx τ xy σ xx P P y t y n θ x x z z Local axes n,t Global axes rotate by θ with x,y stay fixed respect to x,y ASEN 3112 Lecture 6 – Slide 6

  7. ASEN 3112 - Structures Problem Statement σ tt τ tn τ nt σ nn P Plane stress transformation problem: given σ , σ , τ and angle θ xx yy xy t y express σ , σ and τ in terms of the data n tt nt nn θ x z This transformation has two major uses: Find stresses along a given skew direction Here angle θ is given as data Find max/min normal stresses, max in-plane shear and overall max shear Here finding angle θ is part of the problem ASEN 3112 Lecture 6 – Slide 7

  8. ASEN 3112 - Structures Analytical Solution This is also called method of equations in Mechanics of Materials books. A derivation using the wedge method gives σ = σ cos θ + σ sin θ + 2 τ sin θ cos θ 2 2 nn xx yy xy σ = σ sin θ + σ cos θ − 2 τ sin θ cos θ 2 2 tt yy xy xx τ = − ( σ − σ ) sin θ cos θ + τ (cos θ − sin θ ) 2 2 nt yy xy xx ο ο For quick checks when θ is 0 or 90 , see Notes. The sum of the two transformed normal stresses σ + σ = σ + σ nn tt xx yy is independent of the angle θ : it is called a stress invariant (mathematically, this is the trace of the stress tensor). A geometric interpretation using the Mohr's circle is immediate. ASEN 3112 Lecture 6 – Slide 8

  9. ASEN 3112 - Structures Double Angle Version Using double-angle trig relations such as cos 2 θ = cos θ - sin θ and 2 2 sin 2 θ = 2 sin θ cos θ , the transformation equations may be rewritten as σ + σ σ − σ yy σ = cos 2 θ + τ sin 2 θ yy xx + xx nn xy 2 2 σ − σ τ = − sin 2 θ + τ cos 2 θ yy xx xy nt 2 Here σ is omitted since it may be easily recovered as σ + σ − σ xx yy nn tt ASEN 3112 Lecture 6 – Slide 9

  10. ASEN 3112 - Structures Principal Stresses: Terminology The max and min values taken by the in-plane normal stress σ nn when viewed as a function of the angle θ are called principal stresses (more precisely, principal in-plane normal stresses , but qualifiers "in-plane" and "normal" are often omitted). The planes on which those stresses act are the principal planes . The normals to the principal planes are contained in the x,y plane. They are called the principal directions . The θ angles formed by the principal directions and the x axis are called the principal angles . ASEN 3112 Lecture 6 – Slide 10

  11. ASEN 3112 - Structures Principal Angles To find the principal angles , set the derivative of σ with respect nn to θ to zero. Using the double-angle version, d σ = (σ − σ ) sin 2θ + 2τ cos 2θ = 0 nn yy xx d θ This is satisfied for θ = θ if p 2 τ tan 2 θ = xy (*) σ − σ p yy xx It can be shown that (*) provides two principal double angles, 2 θ and 2 θ , within the range of interest, which is [0, 360 ] or o p 1 p 2 [ − 180 ,180 ] (range conventions vary between textbooks). o o o The two values differ by 180 . On dividing by 2 we get the principal angles θ and θ that differ by 90 . Consequently the o p 1 p 2 two principal directions are orthogonal . ASEN 3112 Lecture 6 – Slide 11

  12. ASEN 3112 - Structures Principal Stress Values Replacing the principal angles given by (*) of the previous slide into the expression for σ and using trig identities, we get nn σ − σ σ + σ 2 yy σ = + τ yy xx xx 2 xy 1,2 2 2 in which σ denote the principal normal stresses. Subscripts 1 and 2 1,2 correspond to taking the + and − signs, respectively, of the square root. A staged procedure to compute these values is described in the next slide. ASEN 3112 Lecture 6 – Slide 12

  13. ASEN 3112 - Structures Staged Procedure To Get Principal Stresses 1. Compute σ − σ σ + σ 2 yy yy xx σ = , R = + + τ xy xx 2 2 av 2 Meaning: σ is the average normal stress (recall that σ + σ is an xx yy av invariant and so is σ ), whereas R is the radius of Mohr's circle av described later. This R also represents the maximum in-plane shear value, as discussed in the Lecture notes. 2. The principal stresses are σ = σ + R , σ = σ − R av av 1 2 3. The above procedure bypasses the computation of principal angles . Should these be required to find principal directions, use equation (*) of the Principal Angles slide. ASEN 3112 Lecture 6 – Slide 13

  14. ASEN 3112 - Structures Additional Properties 1. The in-plane shear stresses on the principal planes vanish 2. The maximum and minimum in-plane shears are + R and − R, respectively 3. The max/min in-plane shears act on planes located at +45 and -45 from the principal planes. These are the principal shear planes 4. A principal stress element (used in some textbooks) is obtained by drawing a triangle with two sides parallel to the principal planes and one side parallel to a principal shear plane For further details, see Lecture notes. Some of these properties can be visualized more easily using the Mohr's circle , which provides a graphical solution to the plane stress transformation problem ASEN 3112 Lecture 6 – Slide 14

  15. ASEN 3112 - Structures Numeric Example principal directions σ =10 psi |τ | = R =50 psi 2 max σ = 20 psi θ = 108.44 yy 2 (b) (a) τ = τ =30 psi (c) 18.44 +45 σ =110 psi xy yx 1 = 63.44 θ =18.44 σ =100 psi P P 1 P xx x x principal principal planes planes y y t principal planes n θ (d) principal 45 x x stress element plane of max 45 P inplane shear For computation details see Lecture notes ASEN 3112 Lecture 6 – Slide 15

  16. ASEN 3112 - Structures Graphical Solution of Example Using Mohr's Circle 2 θ = 36.88 +180 = 216.88 τ = shear 2 τ = 50 stress (a) Point in plane stress max 50 σ = 20 psi yy 40 H (a) τ = τ =30 psi 30 Radius R = 50 xy yx 20 σ = normal σ =100 psi 10 P stress 0 20 40 60 80 100 xx 0 σ = 10 C −10 2 σ = 110 −20 y 1 −30 V −40 2 θ = 36.88 1 −50 τ = −50 x min (b) Mohr's circle coordinates of blue points are H: (20,30), V:(100,-30), C:(60,0) ASEN 3112 Lecture 6 – Slide 16

  17. ASEN 3112 - Structures What Happens in 3D? This topic be briefly covered in class if time allows, using the following slides. If not enough time, ask students to read Lecture notes (Sec 7.3), with particular emphasis on the computation of the overall maximum shear ASEN 3112 Lecture 6 – Slide 17

  18. ASEN 3112 - Structures General 3D Stress State σ τ τ xy xz xx σ τ τ yx yz yy σ τ τ zy zx zz There are three (3) principal stresses, identified as σ , σ , σ 2 1 3 ASEN 3112 Lecture 6 – Slide 18

  19. ASEN 3112 - Structures Principal Stresses in 3D (2) The σ turn out to be the eigenvalues of the stress matrix. i They are the roots of a cubic polynomial (the so-called characteristic polynomial) τ σ −σ τ xy xz xx C (σ) = det τ σ −σ τ yx yz yy τ τ σ −σ zx zy zz = −σ + I σ − I σ + I = 0 3 2 1 2 3 The principal directions are given by the eigenvectors of the stress matrix. Both eigenvalues and eigenvectors can be numerically computed by the Matlab function eig(.) ASEN 3112 Lecture 6 – Slide 19

  20. ASEN 3112 - Structures 3D Mohr Circles (Yes, There Is More Than One) Overall τ = shear + max shear stress All possible stress states at the material point lie on the grey shaded area between the outer Inner Mohr's and inner circles circles σ = normal stress σ Principal stress σ Principal stress σ 2 3 1 Outer Mohr's circle The overall maximum shear, which is the radius of the outer Mohr's circle, is important for assessing strength safety of ductile materials ASEN 3112 Lecture 6 – Slide 20

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