The Plane Stress Problem
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1 . 4 The Plane Stress Problem 4
2 Chapter 4: THE PLANE STRESS PROBLEM 4 TABLE OF CONTENTS Page 4.. INTRODUCTION Plate in Plane Stress Mathematical Model PLANE STRESS PROBLEM DESCRIPTION Problem Data Problem Unknowns LINEAR ELASTICITY EQUATIONS Governing Equations Boundar Conditions Weak Forms versus Strong Form Total Potential Energ FINITE ELEMENT EQUATIONS Displacement Interpolation Element Energ Element Stiffness Equations EXERCISES
3 INTRODUCTION 4.. INTRODUCTION We now pass to the variational formulation of two-dimensional continuum finite elements. The problem of plane stress will serve as the vehicle for illustrating such formulations. As narrated in Appendi H, continuum finite elements were invented in the aircraft industr to solve this kind of problem when it arose in the design and analsis of delta wing panels. The problem is presented here within the framework of the linear theor of elasticit Plate in Plane Stress In structural mechanics, a flat thin sheet of material is called a plate. The distance between the plate faces is called the thickness and denoted b h. The midplane lies halfwa between the two faces. The direction normal to the midplane is called the transverse direction. Directions parallel to the midplane are called in-plane directions. The global ais z will be oriented along the transverse direction. Thus the midplane equation is z = 0. Ais and are placed in the midplane, forming a right-handed Rectangular Cartesian Coordinate (RCC) sstem. See Figure 4.. A plate loaded in its midplane is said to be in a state of plane stress, oramembrane state, if the following assumptions hold:. All loads applied to the plate act in the midplane direction, and are smmetric with respect to the midplane.. All support conditions are smmetric about the midplane. 3. All displacements, strains and stresses can be taken to be uniform through the thickness. 4. All stress components in the z direction are zero or negligible. The last two assumptions are not necessaril consequences of the first two. For the latter to hold, the thickness h should be small, tpicall 0% or less, than the shortest inplane dimension. If the plate thickness varies it should do so graduall. Finall, the plate fabrication must ehibit smmetr with respect to the midplane. To these four assumptions we add the following restriction: 5. The plate is fabricated of the same material though the thickness. Such plates are called transversel homogeneous or monocoque plates. The last assumption ecludes wall constructions of importance in aerospace, in particular composite and honecomb plates. The development of models for such configurations requires a more complicated integration over the thickness as well as the abilit to handle coupled bending and stretching effects, and will not be considered here. REMARK 4. Selective relaation from assumption 4 lead to the so-called generalized plane stress state, in which z stresses are accepted. The plane strain state is obtained if strains in the z direction is precluded. Although the construction of finite element models for those states has man common points with plane stress, we shall not At Boeing over the period If it is relativel thick, as in concrete pavements, or cheese slices, the term slab is also used but not for plane stress conditions. 4 3
4 Chapter 4: THE PLANE STRESS PROBLEM 4 4 z Figure 4.. A plate structure in plane stress. Midplane Plate Mathematical Idealization Γ Ω Figure 4.. Mathematical model of plate in plane stress. consider those models here. For isotropic materials the plane stress and plane strain problems can be mapped into each other through a fictitious-propert technique; see Eercise 4.. REMARK 4. Transverse loading on a plate produces plate bending, which is associated with a more comple configuration of internal forces and deformations. This subject is studied in more advanced courses Mathematical Model The mathematical model of the plate in plane stress is a two-dimensional boundar value problem (BVP). This problem is posed over a plane domain with a boundar Ɣ, as illustrated in Figure 4.. In this idealization the third dimension is represented as functions of and that are integrated through the plate thickness. Engineers often work with internal plate forces, which result from integrating the inplane stresses through the thickness. See Figure
5 PLANE STRESS PROBLEM DESCRIPTION Inplane stresses z h σ σ = σ σ p p p h p p p Inplane internal forces Figure 4.3. Internal stresses and plate forces. 4.. PLANE STRESS PROBLEM DESCRIPTION 4... Problem Data The given data includes: Domain geometr. This is defined b the boundar Ɣ illustrated in Figure 4.. Thickness. Most plates used as structural components have constant thickness. If the thickness does var, in which case h = h(, ), it should do so graduall to maintain the plane stress state. Material data. This is defined b the constitutive equations. Here we shall assume that the plate material is linearl elastic but not necessaril isotropic. Specified Interior Forces. These are known forces that act in the interior of the plate. There are of two tpes. Bod forces or volume forces are forces specified per unit of plate volume; for eample the plate weight. Face forces act tangentiall to the plate faces and are transported to the midplane. For eample, the friction or drag force on an airplane skin is of this tpe if the skin is modeled to be in plane stress. Specified Surface Forces. These are known forces that act on the boundar Ɣ of the plate. In elasticit these are called surface tractions. In actual applications it is important to know whether these forces are specified per unit of surface area or per unit length. Displacement Boundar Conditions. These specif how the plate is supported. Points on the plate boundar ma be fied, allowed to move in one direction, or subject to multipoint constraints. In addition smmetr and antismmetr lines ma be identified as discussed in Chapter 8. If no displacement boundar conditions are imposed, the plate structure is said to be free-free. 4 5
6 Chapter 4: THE PLANE STRESS PROBLEM Problem Unknowns The three unknown fields are displacements, strains and stresses. Because of the assumed wall fabrication homogeneit the in-plane components are assumed to be uniform through the plate thickness. Consequentl the dependence on z disappears and all such components become functions of and onl. Displacements. The in-plane displacement field is defined b two components: u(, ) = [ ] u (, ) u (, ) (4.) The transverse displacement component u z (,, z) component is generall nonzero because of Poisson s ratio effects, and depends on z. However, this displacement does not appear in the governing equations. Strains. The in-plane strain field forms a tensor defined b three independent components: e, e and e. To allow stating the FE equations in matri form, these components are conventionall arranged to form a 3-component strain vector e(, ) = [ ] e (, ) e (, ) e (, ) (4.) The factor of in e shortens strain energ epressions. The shear strain components e z and e z vanish. The transverse normal strain e zz is generall nonzero because of Poisson effects. This strain does not enter the governing equations as unknown because the associated stress σ zz is zero, which eliminates the contribution of σ zz e zz to the internal energ. Stresses. The in-plane stress field forms a tensor defined b three independent components: σ, σ and σ. As in the case of strains, to allow stating the FE equations in matri form, these components are conventionall arranged to form a 3-component stress vector σ(, ) = [ ] σ (, ) σ (, ) σ (, ) (4.3) The remaining three stress components: σ zz, σ z and σ z, are assumed to vanish. The plate internal forces are obtained on integrating the stresses through the thickness. Under the assumption of uniform stress distribution, p = σ h, p = σ h, p = σ h. (4.4) These p s also form a tensor. The are often called membrane forces in the literature. See Figure
7 LINEAR ELASTICITY EQUATIONS Displacement BC Prescribed displacements u^ u = u^ on Γ u Displacements u Bod forces b Γ Ω Kinematic Strains e e = D u in Ω Constitutive σ = E e in Ω Equilibrium Stresses σ D T σ+ b = 0 in Ω Force BC σ T n = t^ or p T n = q^ on Γ t Prescribed ^ tractions t or forces q^ Figure 4.4. The Strong Form of the plane stress equations of linear elastostatics displaed as a Tonti diagram. Yellow boes identif prescribed fields whereas orange boes denote unknown fields. The distinction between Strong and Weak Forms is eplained in LINEAR ELASTICITY EQUATIONS We shall develop plane stress finite elements in the framework of classical linear elasticit. The necessar governing equations are presented below. The are graphicall represented in the Strong Form Tonti diagram of Figure Governing Equations The three internal fields: displacements, strains and stresses (4.)-(4.4) are connected b three field equations: kinematic, constitutive and internal-equilibrium equations. If initial strain effects are ignored, these equations read [ e e e [ σ σ σ ] [ ] / 0 [ u = 0 / u / / ] [ E E E 3 = E E E 3 E 3 E 3 E 33 ] [ ] σ σ + σ [ / 0 / 0 / / ], ][ e e e [ b ], b ] = [ ] 0. 0 (4.5) In these equations, b and b are the components of the interior bod force b, E is the 3 3 stress-strain matri of plane stress elastic moduli, D is the 3 smmetric-gradient operator and its transpose the 3 tensor-divergence operator. The dependence on (, ) has been omitted to reduce clutter. 4 7
8 Chapter 4: THE PLANE STRESS PROBLEM 4 8 t ; ;; Γ u ;; ;; u ^ = 0 + Γ t t^ σ n σ σ nt nn ^t n ^t t ^t Stress BC details (decomposition of forces ^q would be similar) n (unit eterior normal) Boundar displacements u ^ are prescribed on Γu Boundar tractions ^t or boundar forces ^q are prescribed on Γt The compact matri version of (4.5) is Figure 4.5. Displacement and force (stress, traction) boundar conditions for the plane stress problem. e = Du, σ = Ee, D T σ + b = 0, (4.6) in which E = E T. If the plate material is isotropic with elastic modulus E and Poisson s ratio ν, the moduli in the constitutive matri E reduce to E = E = E/( ν ), E 33 = E/( + ν) = G, E = ν E and E 3 = E 3 = 0. See also Eercise Boundar Conditions Boundar conditions prescribed on Ɣ ma be of two tpes: displacement BC or force BC (also called stress BC or traction BC). To write down those conditions it is conceptuall convenient to break up Ɣ into two subsets: Ɣ u and Ɣ t, over which displacements and force or stresses, respectivel, are specified. See Figure 4.5. Displacement boundar conditions are prescribed on Ɣ u in the form u = û. (4.7) Here û are prescribed displacements. Often û = 0. This happens in fied portions of the boundar, as the ones illustrated in Figure 4.5. Force boundar conditions (also called stress BCs and traction BCs in the literature) are specified on Ɣ t. The take the form σ n = ˆt. (4.8) Here ˆt are prescribed surface tractions specified as a force per unit area (that is, not integrated through the thickness), and σ n is the stress vector shown in Figure
9 LINEAR ELASTICITY EQUATIONS An alternative form of (4.8) uses the internal plate forces: p n = ˆq. (4.9) Here p n = σ n h and ˆq = ˆt h. This form is used more often than (4.8) in structural design, particularl when the plate wall construction is nonhomogeneous. The components of σ n in Cartesian coordinates follow from Cauch s stress transformation formula [ ] [ σ n σ n = + σ n n 0 n = σ n + σ n 0 n n ] [ ] σ σ, (4.0) σ in which n and n denote the Cartesian components of the unit normal vector n (e) (also called the direction cosines of the normal). Thus (4.8) splits into two scalar conditions: ˆt = σ n and ˆt = σ n. It is sometimes convenient to write the condition (4.8) in terms of normal n and tangential t directions: σ nn = ˆt n, σ nt = ˆt t (4.) in which σ nn = σ n n + σ n n and σ nt = σ n n + σ n n. REMARK 4.3 The separation of Ɣ into Ɣ u and Ɣ t is useful for conciseness in the mathematical formulation, such as the energ integrals presented below. It does not ehaust, however, all BC possibilities. Frequentl at points of Ɣ one specifies a displacement in one direction and a force (or stress) in the other. An eample of these are roller and sliding conditions as well as lines of smmetr and antismmetr. To cover these situations one needs either a generalization of the split, in which Ɣ u and Ɣ t are permitted to overlap, or to define another portion Ɣ m for mied conditions. Such generalizations will not be presented here, as the become unimportant once the FE discretization is done Weak Forms versus Strong Form We introduce now some further terminolog from variational calculus. The Tonti diagram of Figure 4.4 is said to displa the Strong Form of the governing equations because all relations are verified point b point. These relations, called strong links, are shown in the diagram with black lines. A Weak Form is obtained b relaing one or more strong links. Those are replaced b weak links, which enforce relations in an average or integral sense rather than point b point. The weak links are then provided b the variational formulation chosen for the problem. Because in general man variational forms of the same problem are possible, there are man possible Weak Forms. On the other hand the Strong Form is unique. The Weak Form associated with the Total Potential Energ (TPE) variational form is illustrated in Figure 4.6. The internal equilibrium equations and stress BC become weak links, which are indicated b gra lines. These equations are given b the variational statement δ = 0, where the TPE functional is given in the net subsection. The FEM displacement formulation discussed below is based on this particular Weak Form. 4 9
10 Chapter 4: THE PLANE STRESS PROBLEM 4 0 Displacement BC Prescribed u = u^ Displacements displacements u^ on Γ u u Bod forces b Γ Ω Kinematic e = D u in Ω Equilibrium δπ= 0 in Ω Strains e Constitutive σ = E e in Ω Stresses σ Force BC δπ = 0 on Γ t Prescribed ^ tractions t or forces q^ Figure 4.6. The TPE-based Weak Form of the plane stress equations of linear elastostatics. Weak links are marked with gre lines Total Potential Energ As usual the Total Potential Energ functional for the plane stress problem is given b The internal energ is the elastic strain energ: U = = U W. (4.) h σ T e d = h e T Ee d. (4.3) The derivation details are relegated to Eercise E4.5. The eternal energ is the sum of contributions from known interior and boundar forces: W = h u T b d + h u T ˆt dɣ. Ɣ t (4.4) Note that the boundar integral over Ɣ is taken onl over Ɣ t. That is, the portion of the boundar over which tractions or forces are specified FINITE ELEMENT EQUATIONS The necessar equations to appl the finite element method are collected here and epressed in matri form. The domain of Figure 4.7(a) is discretized b a finite element mesh as illustrated in Figure 4.7(b). From this mesh we etract a generic element labeled (e) with n 3 node points. In the subsequent derivation the number n is kept arbitrar. Therefore, the formulation is applicable to arbitrar two-dimensional elements, for eample those sketched in Figure
11 4 4.4 FINITE ELEMENT EQUATIONS (a) (b) (c) Γ Ω (e) Γ (e) Ω Figure 4.7. Finite element discretization and etraction of generic element. Departing from previous practice in D elements, the element node points will be labelled through n. These are called local node numbers. The element domain and boundar are denoted b (e) and Ɣ (e), respectivel. The element has n degrees of freedom. These are collected in the element node displacement vector Displacement Interpolation u (e) = [ u u u... u n u n ] T. (4.5) The displacement field u (e) (, ) over the element is interpolated from the node displacements. We shall assume that the same interpolation functions are used for both displacement components. 3 Thus u (, ) = n i= N (e) i (, ) u i, u (, ) = where N (e) i (, ) are the element shape functions. In matri form: u(, ) = n i= [ ] [ u (, ) N (e) 0 N (e) 0... N (e) = u (, ) N (e) i (, ) u i, (4.6) n 0 0 N (e) 0 N (e)... 0 N n (e) ] u (e) = N (e) u (e). (4.7) The minimum conditions on N (e) i (, ) is that it must take the value one at the i th node and zero at all others, so that the interpolation (4.7) is correct at the nodes. Additional requirements on the shape functions are stated in later Chapters. Differentiating the finite element displacement field ields the strain-displacement relations: e(, ) = n n 0 N n (e) N n (e) u (e) = Bu (e). (4.8) 3 This is the so called element isotrop condition, which is studied and justified in advanced FEM courses. 4
12 Chapter 4: THE PLANE STRESS PROBLEM n = 3 n = 4 n = 6 n = Figure 4.8. Eample two-dimensional finite elements, characterized b their number of nodes n. The strain-displacement matri B = DN (e) is 3 n. The superscript (e) is omitted from it to reduce clutter. The stresses are given in terms of strains and displacements b σ = Ee= EBu (e), which is assumed to hold at all points of the element Element Energ To obtain finite element stiffness equations, the variation of the TPE functional is decomposed into contributions from individual elements: where and δ (e) = δu (e) δw (e) = 0. (4.9) U (e) = h σ T e d (e) = h e T Ee d (e) (4.0) (e) (e) W (e) = h u T b d (e) + h u T ˆt dɣ (e) (4.) (e) Ɣ (e) Note that in (4.) Ɣ t (e) has been taken equal to the complete boundar Ɣ (e) of the element. This is a consequence of the fact that displacement boundar conditions are applied after assembl, to a free-free structure. Consequentl it does not harm to assume that all boundar conditions are of stress tpe insofar as forming the element equations Element Stiffness Equations Inserting the relations u = Nu (e), e = Bu (e) and σ = Ee into (e) ields the quadratic form in the nodal displacements (e) = u(e)t K (e) u (e) u (e)t f (e), (4.) where the element stiffness matri is K (e) = h B T EB d (e), (e) (4.3) 4
13 FINITE ELEMENT EQUATIONS and the consistent element nodal force vector is f (e) = h N T b d (e) + h N T ˆt dɣ (e). (4.4) (e) Ɣ (e) In the second integral the matri N is evaluated on the element boundar onl. The calculation of the entries of K (e) and f (e) for several elements of historical or practical interest is described in subsequent Chapters. 4 3
14 Chapter 4: THE PLANE STRESS PROBLEM 4 4 Homework Eercises for Chapter 4 The Plane Stress Problem EXERCISE 4. [A:5] Suppose that the structural material is isotropic, with elastic modulus E and Poisson s ratio ν. The in-plane stress-strain relations for plane stress (σ zz = σ z = σ z = 0) and plane strain (e zz = e z = e z = 0) as given in an tetbook on elasticit, are [ ] [ ][ ] σ plane stress: σ = E ν 0 e ν 0 e σ ν, 0 0 ν e ] ν 0 ] (E4.) plane strain: [ σ σ σ = E( ν) ( + ν)( ν) ν ν ν ν ( ν) [ e Show that the constitutive matri of plane strain can be formall obtained b replacing E b a fictitious modulus E and ν b a fictitious Poisson s ratio ν in the plane stress constitutive matri and suppressing the stars. Find the epression of E and ν in terms of E and ν. This device permits reusing a plane stress FEM program to do plane strain, as long as the material is isotropic. EXERCISE 4. [A:5] In the finite element formulation of near incompressible isotropic materials (as well as plasticit and viscoelasticit) it is convenient to use the so-called Lamé constants λ and µ instead of E and ν in the constitutive equations. Both λ and µ have the phsical dimension of stress and are related to E and ν b λ = e e ν E ( + ν)( ν), µ = G = E ( + ν). (E4.) Conversel µ(3λ + µ) E = λ + µ, ν = λ (λ + µ). (E4.3) Substitute (E4.3) into (E4.) to epress the two stress-strain matrices in terms of λ and µ. Then split the stress-strain matri E of plane strain as E = E µ + E λ (E4.4) in which E µ and E λ contain onl µ and λ, respectivel, with E µ diagonal and E λ33 = 0. This is the Lamé or {λ, µ} splitting of the plane strain constitutive equations, which leads to the so-called B-bar formulation of near-incompressible finite elements. 4 Epress E µ and E λ also in terms of E and ν. For the plane stress case perform a similar splitting in which where E λ contains onl λ = λµ/(λ + µ) with E λ33 = 0, and E µ is a diagonal matri function of µ and λ. 5 Epress E µ and E λ also in terms of E and ν.. 4 Equation (E4.4) is sometimes referred to as the deviatoric+volumetric splitting of the stress-strain law, on account of its phsical meaning in plane strain. That meaning is lost, however, for plane stress. 5 For the phsical significance of λ see I. Sokolnikoff, The Mathematical Theor of Elasticit, McGraw-Hill, nd ed., 956, p. 54ff. 4 4
15 4 5 Eercises σ σ = σ t d ds d σ n(n =d/ds, n =d/ds) t Figure E4.. Geometr for deriving (4.0). EXERCISE 4.3 [A:0] Derive the Cauch stress-to-traction equations (4.0) using force equilibrium along and and the geometric relations shown in Figure E4.. Hint: t ds = σ d + σ d, etc. EXERCISE 4.4 [A:5] Include thermoelastic effects in the plane stress field equations, assuming a thermall isotropic material with coefficient of linear epansion α. EXERCISE 4.5 [A:5=5+5+5] A plate is in linearl elastic plane stress. It is shown in courses in elasticit that the internal strain energ densit stored per unit volume is U = (σ e + σ e + σ e + σ e ) = (σ e + σ e + σ e ). (E4.5) (a) Show that (E4.5) can be written in terms of strains onl as U = et Ee, (E4.6) (b) and hence justif (4.3). Show that (E4.5) can be written in terms of stresses onl as U = σt C σ, (E4.7) (c) where C = E is the elastic compliance (strain-stress) matri. Suppose ou want to write (E4.5) in terms of the etensional strains {e, e } and of the shear stress σ = σ. This is known as a mied representation. Show that U = [ e e σ ] T [ A A A 3 A A A 3 A 3 A 3 A 33 ][ e e σ ], (E4.8) and eplain how the entries A ij can be calculated 6 in terms of the elastic moduli E ij. 6 The process of computing A is an instance of partial inversion of matri E. See Remark
The Plane Stress Problem
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