The Plane Stress Problem

Transcription

1 . 14 The Plane Stress Problem 14 1

2 Chapter 14: THE PLANE STRESS PROBLEM 14 TABLE OF CONTENTS Page Introduction Plate in Plane Stress Mathematical Model Plane Stress Problem Description Given Problem Data Problem Unknowns Linear Elasticit Equations Governing Equations Boundar Conditions Weak Forms versus Strong Form Total Potential Energ Finite Element Equations Displacement Interpolation Element Energ Element Stiffness Equations Notes and Bibliograph References Eercises Solutions to. Eercises

3 INTRODUCTION Introduction We nowpass 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 1.7.1, continuum finite elements were invented in the aircraft industr (at Boeing, earl 1950s) to solve this kind of problem when it arose in the design and analsis of delta wing panels [164]. 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. 1 The distance between the plate faces is called the thickness and denoted b h. The midplane lies halfwa between the two faces. z The direction normal to the midplane is the transverse direction. Directions parallel to the midplane are called in-plane directions. The global ais z will be oriented along the transverse direction. Aes and are placed in the midplane, forming a right-handed Rectangular Cartesian Coordinate (RCC) sstem. Thus the midplane equation is z = 0. See Figure Figure A plate structure in plane stress. A plate loaded in its midplane is said to be in a state of plane stress, oramembrane state, if the following assumptions hold: 1. 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. In-plane displacements, strains and stresses can be taken to be uniform through the thickness. 4. The normal and shear 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 10% or less, than the shortest in-plane 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 through the thickness. Such plates are called transversel homogeneous or (in aerospace) monocoque plates. The last assumption ecludes wall constructions of importance in aerospace, in particular composite and honecomb sandwich plates. The development of mathematical 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. 1 If it is relativel thick, as in concrete pavements or Argentinian beefsteaks, the term slab is also used but not for plane stress conditions. 14 3

4 Chapter 14: THE PLANE STRESS PROBLEM 14 4 Midplane Mathematical idealization Plate Γ Ω Figure 14.. Mathematical model of plate in plane stress. Remark 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 are precluded. Although the construction of finite element models for those states has man common points with plane stress, we shall not 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 Remark 14.. 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). The BVP is posed over a plane domain with a boundar Ɣ, as illustrated in Figure 14.. 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 in-plane stresses through the thickness. See Figure Plane Stress Problem Description Given Problem Data Domain geometr. This is defined b the boundar Ɣ illustrated in Figure 14.. 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 knowwhether these forces are specified per unit of surface area or per unit length. 14 4

5 PLANE STRESS PROBLEM DESCRIPTION In-plane internal forces h p p p p p p z In-plane stresses h = Positive sign convention In-plane strains In-plane displacements h e e = e e h u u Figure Notation for in-plane stresses, strains, displacements and internal forces for a plate in plane stress. Displacement Boundar Conditions. These specif howthe 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 Problem Unknowns The 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. Thus 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(, ) = (14.1) u (, ) 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 allowstating the FE equations in matri form, these components are conventionall arranged to form a 3-component strain vector e(, ) = [ ] e (, ) e (, ) e (, ) 14 5 (14.)

6 Chapter 14: THE PLANE STRESS PROBLEM 14 6 Prescribed displacements u^ Displacement BCs u = u^ on Γ u Displacements u Bod forces b Γ Ω Kinematic e = D u in Ω D + b = 0 in Ω Equilibrium Strains e = E e in Ω Constitutive Stresses Force BCs T n = t^ or p T n = q^ on Γ t Prescribed tractions t or forces q Figure The Strong Form of the plane stress equations of linear elastostatics displaed as a Tonti diagram. Yellowboes identif prescribed fields whereas orange boes denote unknown fields. The distinction between Strong and Weak Forms is eplained in 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 s ratio effects. This strain does not enter the governing equations as unknown, however, because the associated stress zz is zero. This eliminates the contribution of zz e zz to the internal energ. Stresses. The in-plane stress field forms a tensor defined b three independent components:, and. As in the case of strains, to allowstating the FE equations in matri form, these components are conventionall arranged to form a 3-component stress vector (, ) = [ ] (, ) (, ) (, ) (14.3) The remaining three stress components: zz, z and 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 = h, p = h, p = h. (14.4) These p s also form a tensor. The are called membrane forces in the literature. See Figure Linear Elasticit 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

7 LINEAR ELASTICITY EQUATIONS Governing Equations The three internal fields: displacements, strains and stresses (14.1) (14.3) are connected b three field equations: kinematic, constitutive and internal-equilibrium equations. If initial strain effects are ignored, these equations read [ e e e [ ] [ ] / 0 [ u = 0 / u / / ] [ E11 E 1 E 13 = E 1 E E 3 E 13 E 3 E 33 ] [ ] + [ / 0 / 0 / / ], ][ e e e [ b ], b ] = [ ] 0. 0 (14.5) 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 compact matri version of (14.5) is e = Du, = Ee, D T + b = 0, (14.6) 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 11 = E = E/(1 ν ), E 33 = 1 E/(1 + ν) = G, E 1 = ν E 11 and E 13 = E 3 = 0. See also Eercise Boundar Conditions Boundar conditions prescribed on Ɣ ma be of two tpes: displacement BC or force BC (the latter is 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 Displacement boundar conditions are prescribed on Ɣ u in the form u = û. (14.7) Here û are prescribed displacements. Often û = 0. This happens in fied portions of the boundar, as the ones illustrated in Figure Force boundar conditions (also called stress BCs and traction BCs in the literature) are specified on Ɣ t. The take the form n = ˆt. (14.8) Here ˆt are prescribed surface tractions specified as a force per unit area (that is, not integrated through the thickness), and n is the stress vector shown in Figure The dependence on (, ) has been omitted to reduce clutter. 14 7

8 Chapter 14: THE PLANE STRESS PROBLEM 14 8 t n (unit eterior normal) ;; ;; Γ u ; ^ + u = 0 Boundar displacements u ^ are prescribed on Γu (figure depicts fiit condition) Γ t t^ Boundar tractions ^t or boundar forces q^ are prescribed on Γ t n nt ^t n ^t t ^t nn Stress BC details (decomposition of forces q ^ would be similar) Figure Displacement and force (stress, traction) boundar conditions for the plane stress problem. An alternative form of (14.8) uses the internal plate forces: p n = ˆq. (14.9) Here p n = n h and ˆq = ˆt h. This form is used more often than (14.8) in structural design, particularl when the plate wall construction is inhomogeneous. The components of n in Cartesian coordinates followfrom Cauch s stress transformation formula [ ] [ n n = + n n 0 n = n + n 0 n n ] [ ], (14.10) 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 (14.8) splits into two scalar conditions: ˆt = n and ˆt = n. The derivation of (14.10) is the subject of Eercise It is sometimes convenient to write the condition (14.8) in terms of normal n and tangential t directions: nn = ˆt n, nt = ˆt t (14.11) in which nn = n n + n n and nt = n n + n n. Remark 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. 14 8

9 LINEAR ELASTICITY EQUATIONS Prescribed displacements u^ Displacement BCs u = u^ on Γ u Displacements u Bod forces b Γ Ω Kinematic e = D u in Ω δπ= 0 in Ω Equilibrium (weak) Strains e = E e in Ω Constitutive Stresses Force BCs (weak) δπ = 0 on Γt Prescribed tractions t or forces q Figure The TPE-based Weak Form of the plane stress equations of linear elastostatics. Weak links are marked with gre lines Weak Forms versus Strong Form We introduce nowsome further terminolog from variational calculus. The Tonti diagram of Figure 14.4 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 The internal equilibrium equations and stress BC become weak links, which are drawn 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 belowis based on this particular Weak Form 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 = 1 h T e d = 1 = U W. (14.1) h e T Eed. (14.13) The derivation details are relegated to Eercise E14.5. 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. (14.14) Ɣ t 14 9

10 Chapter 14: THE PLANE STRESS PROBLEM 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 to the plane stress problem are collected here and epressed in matri form. The domain of Figure 14.7(a) is discretized b a finite element mesh as illustrated in Figure 14.7(b). From this mesh we etract a generic element labeled e with n 3 node points. In the subsequent derivations the number n is kept arbitrar. Therefore, the formulation is applicable to arbitrar two-dimensional elements, for eample those sketched in Figure Departing from previous practice in 1D elements, the element node points will be labelled 1 through n. These are called local node numbers. The element domain and boundar are denoted b e and Ɣ e, respectivel. (a) Γ Ω (c) (b) Ω Figure Finite element discretization and etraction of generic element. The element has n degrees of freedom. These are collected in the element node displacement vector u e = [ u 1 u 1 u... u n u n ] T. (14.15) Displacement Interpolation 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=1 N e i (, ) u i, u (, ) = where Ni e (, ) are the element shape functions. In matri form: u(, ) = n i=1 e Γ N e i (, ) u i, (14.16) [ ] [ ] u (, ) N e = 1 0 N e 0... Nn e 0 u (, ) 0 N1 e 0 N e... 0 Nn e u e = Nu e. (14.17) This N (with superscript e omitted to reduce clutter) is called the shape function matri. It has dimensions n. For eample, if the element has 4 nodes, N is 8. The interpolation condition on the element shape function Ni e (, ) states that it must take the value one at the i th node and zero at all others. This ensures that the interpolation (14.17) is correct at the nodes. Additional requirements on the shape functions are stated in later Chapters. e 3 This is the so called element isotrop condition, which is studied and justified in advanced FEM courses

11 FINITE ELEMENT EQUATIONS n = 3 n = 4 n = 6 n = 1 Figure Eample plane stress finite elements, characterized b their number of nodes n Differentiating the finite element displacement field ields the strain-displacement relations: N1 e N 0 e N e 0... n 0 e(, ) = N e 0 1 N e 0 N e... 0 n N1 e N1 e N e N e N e... n Nn e ue = Bu e. (14.18) This B = DNis called the strain-displacement matri. It is dimensioned 3 n. For eample, if the element has 6 nodes, B is 3 1. The stresses are given in terms of strains and displacements b = 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: δ e = δu e δw e = 0. (14.19) where U e = 1 h T e d e = 1 h e T Ee d e (14.0) e e and W e = h u T b d e + h u T ˆt dɣ e (14.1) e Ɣ e Note that in (14.1) Ɣt 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 = Ee into e ields the quadratic form in the nodal displacements e = 1 uet K e u e u et f e. (14.) 14 11

12 Chapter 14: THE PLANE STRESS PROBLEM 14 1 Here the element stiffness matri is K e = h B T EB d e, e (14.3) and the consistent element nodal force vector is f e = h N T b d e + e h N T ˆt dɣ e. Ɣ e (14.4) In the second integral of (14.4) 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. Notes and Bibliograph The plane stress problem is well suited for introducing continuum finite elements, from both historical and technical standpoints. Some books use the Poisson equation for this purpose, but problems such as heat conduction cannot illustrate features such as vector-mied boundar conditions and shear effects. The first continuum structural finite elements were developed at Boeing in the earl 1950s to model delta-wing skin panels [3,164]. A plane stress model was naturall chosen for the panels. The paper that gave the method its name [5] used the plane stress problem as application driver. The technical aspects of plane stress can be found in an book on elasticit. A particularl readable one is the ecellent tetbook b Fung [69], which is unfortunatel out of print. References Referenced items have been moved to Appendi R. 14 1

13 14 13 Eercises Homework Eercises for Chapter 14 The Plane Stress Problem EXERCISE 14.1 [A+C: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 ( zz = z = z = 0) and plane strain (e zz = e z = e z = 0) as given in an tetbook on elasticit, are [ ] [ ][ ] plane stress: = E 1 ν 0 e ν 1 0 e 1 ν, ν e ] 1 ν 0 ] (E14.1) plane strain: [ = E(1 ν) (1 + ν)(1 ν) 1 ν ν 1 ν ν (1 ν) [ e Showthat 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. You ma also chose to answer this eercise b doing the inverse process: go from plane strain to plain stress b replacing a fictitious modulus and Poisson s ratio in the plane strain constitutive matri. This device permits reusing a plane stress FEM program to do plane strain, or vice-versa, as long as the material is isotropic. Partial answer to go from plane stress to plane strain: ν = ν/(1 ν). EXERCISE 14. [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 ν E λ = (1 + ν)(1 ν), µ = G = E (1 + ν). (E14.) Conversel µ(3λ + µ) E = λ + µ, ν = λ (λ + µ). (E14.3) Substitute (E14.3) into (E14.1) 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 (E14.4) 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. e e. 4 Equation (E14.4) 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 [143, pp. 54ff]

14 Chapter 14: THE PLANE STRESS PROBLEM EXERCISE 14.3 [A:0] Include thermoelastic effects in the plane stress constitutive field equations, assuming a thermall isotropic material with coefficient of linear epansion α. Hint: start from the two-dimensional Hooke s lawincluding temperature: e = 1 E ( ν ) + α T, e = 1 E ( ν ) + α T, e = /G, (E14.5) in which T = T (, ) and G = 1 E/(1 + ν). Solve for stresses and collect effects of T in one vector of thermal stresses. EXERCISE 14.4 [A:15] Derive the Cauch stressto-traction equations (14.10) using force equilibrium along and and the geometric relations shown in Figure E14.1. (This is the wedge method in Mechanics of Materials.) Hint: t ds = d + d, etc. = t d ds d n(n =d/ds, n =d/ds) Figure E14.1. Geometr for deriving (14.10). t EXERCISE 14.5 [A:5=5+5+15] 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 = 1 ( e + e + e + e ) = 1 ( e + e + e ). (E14.6) (a) Showthat (E14.6) can be written in terms of strains onl as U = 1 et Ee, (E14.7) (b) and hence justif (14.13). Showthat (E14.6) can be written in terms of stresses onl as U = 1 T C, (E14.8) (c) where C = E 1 is the elastic compliance (strain-stress) matri. Suppose ou want to write (E14.6) in terms of the etensional strains {e, e } and of the shear stress =. This is known as a mied representation. Show that U = 1 [ e e ] T [ A11 A 1 A 13 A 1 A A 3 A 13 A 3 A 33 ][ e e ], (E14.9) and eplain howthe 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