Finite elements for plates and shells Sstem LEC 008/11/04 1
Plates and shells A Midplane The mid-surface of a plate is plane, a curved geometr make it a shell The thickness is constant. Ma be isotropic, anisotropic, composite and laered. The isotropic case is considered. Thin and thick cases are take into account. A Sstem LEC 008/11/04
Comparison plate-beam A plate can be regarded as the two-dimensional analogue of a beam: - both carr transverse loads b bending action; - but the have significant differences; - a beam tpicall has a single bending moment; a plate has two bending moments (and two twisting moments) - a deflection of a beam need not strain its ais; deflection of a plate will strain its midsurface Bending stiffness EI 3 Et 1 Flessione di una trave Sstem LEC 008/11/04 Bending of a beam 3
t<1/10 the span of the plate ww(,) Thin plates (Kirchhoff) Transverse shear deformation is neglected Arbitrar point P displacement γ z γ 0 γ 0 z w w u z v z Sstem LEC 008/11/04 4
Sstem LEC 008/11/04 5 The strain displacement relations give the first of these equation is the onl one used in the beam theor. In a thin plate the stress normal to the midplane is considered negligible Accordingl, the plane stress equation gives 0 + z w z v u w z v w z u σ γ ε ε σ Dε D [ ] E 1 ν 1 ν 0 ν 1 0 0 0 1 ν ( )
Stress distribution σ E z σ 1 ν 1 ν w τ ν 1 w w zg Like in a beam, stresses var linearl with distance from the midsurface Transverse shear stresses τ z and τ z are also present, even though transverse deformation is neglected Transverse shear stresses var quadraticall through the thickness Sstem LEC 008/11/04 6
Moments The stresses give rise to bending moment M and M and twisting moment M Moments are function of and and are computed for unit length in the plane of the plate M M M M M M (, (, (, ) ) ) dm dm dm z( σ da) z( σ da) z( σ da) da 1 dz da 1 dz da 1 dz M t / t / σ zdz M t / t / σ zdz M t / τ t / zdz Sstem LEC 008/11/04 7
Sstem LEC 008/11/04 8 6 /) ( 6 /) ( 6 /) ( t M t z t t M t z t t M t z t τ τ τ τ τ σ σ σ σ σ σ σ σ σ σ Maimum magnitude of stresses Formula for σ can be regarded as the fleure formula σ M c/i applied to a unit width of the plate with ct/
Sstem LEC 008/11/04 9 ) 1(1 ) (1 Bending stiffness 0 0 0 3 ν ν νσ σ Et EJ D w w w M Bending of a thin plate
A plate has a wide cross section Bending of a plate - top and bottom edge of a cross section remain straight -parallel ais when M is applied When a plate is bent to a clindrical surface, onl M acts: M w 0 w 0 w 0 σ νσ The fleural stress σ is accompanied b the stress σ Stress σ constrains the plate against the deformation ε, thereb stiffening the plate The amount of stiffening is proportional to 1/(1-ν ), so that a unit width of the plate has Bending stiffness 3 EI Et D (1 ν ) 1(1 ν ) Sstem LEC 008/11/04 10
Thick plates (Mindlin) Sstem LEC 008/11/04 γ z 0 γ 0 γ 0 Accounts for transverse shear deformation z The assumption that right angle in a cross section are preserved must be abandoned The planes initiall normal to the midsurface eperience rotations different from rotations of the midsurface itself (, ) 11
Sstem LEC 008/11/04 1 z ε z v z z u ε z z w w z γ γ γ + If w w Transverse shear deformations vanish and the equations reduce to the equations for thin plates. Strain-displacements relations
Loads Loads in the z direction, either distribuited or concentrated,to the lateral surface z±t/, or to the edge of the plate Bending moment whose vector is tangent to the edge At the point where concentrated lateral load (z direction) is applied: - Kirchhoff theor predicts infinite bending; - Mindlin theor predicts infinite bending and infinite displacement The infinities disappear if the concentrated load is applied over a small area instead. Sstem LEC 008/11/04 13
Membrane forces Internal force resultant in the plane of the plate (membrane forces) can develop as a consequence of the deflection ma be present because of loads component tangent to the midsurface can significantl influence the response of the plate to load Membrane forces have a stress stiffening effect: - if tensile the effectivel increase the fleural rigidit - if compressive the decrease it Compressive membrane forces ma become large enough to produce buckling Sstem LEC 008/11/04 14
Beam with immovable hinge supports Hinge support remains eactl a distance L apart, regardless of how much load is applied. Beam develop the usual fleural stresses and also membrane force N that support part of the applied load b spring effect. If the deflected shape is a parabola with center deflection w c the uniform distributed load supported b spring action is Sstem LEC 008/11/04 q s 64 3 Ebt 4 L 4 w t c 3 15
For a simple supported beam w c 5 384 qbl EI 4 q b The total load supported b beam and spring action occurring simultaneousl is 3 3 Ebt wc wc q qs + qb 4 1.3 + 6. 4 L t t For w c /t0.5 (HP: t<<l) spring and beam action each support about half of the total load This argument is of little value for beams because immovible supports are not found in practice. The value of the argument is its implication for problems of thin plates Sstem LEC 008/11/04 16
Observation The counterpart of the spring action in a beam is strain of the midsurface in a plate. Deflection of a plate ww(, ) produce no strain of the midsurface onl if w describes a developable surface, i.e. clinder or cone. In general load produces a deflected shape that is not developable. Accordingl, in general there are strains at the midsurface, and membrane forces appear that carr out part of the load. Sstem LEC 008/11/04 17
Finite elements for plates A plate is a thin solid and might be modeled b 3D solid elements A solid element is wasteful of d.o.f., as it computes transverse normal stress and transverse shear stresses, all of which are considered negligible in a thin plate. Also thin 3D elements invite numerical troubles because stiffness associated with ε z is ver much larger than other stiffnesses. Sstem LEC 008/11/04 18
The plate element has half as man d.o.f. as the comparable 3D element and omit ε z from its formulation Thickness appear to be zero, but the correct value is used in its formulation Circular plates can be modeled b shell of revolution elements, simpl b making shell elements flat rather than clindrical or conic. Each of such element is thus a flat annular ring, joined to adjacent annular elements at its inner and outer radii. Sstem LEC 008/11/04 19
Patch tests for plate elements A plate element must be able to displa states of constant σ, σ and τ if it is to pass patch test. These states must be displaed in each zconstant laer. This means that a valid plate element must pass patch tests for states of constant M, M, and M.. w cost cost Kirchoff theor Mindlin theor Patch test for constant M Sstem LEC 008/11/04 0
Kirchhoff plate elements 1 d.o.f. Nodal d.o.f.: w w(, ) w w The assumed w field is a polnomial in and The stiffness matri is k B T DBdV where D is the matri of the fleural rigidities B is contrived to produce curvature when it operates on nodal d.o.f. that describe a lateral displacement field ww(,) Sstem LEC 008/11/04 1
This element is incompatible: that is, if n is a direction normal to an element edge, w n is not continuous between elements for some loading conditions Accordingl, the element cannot guarantee a lower bound on computed displacements, so results ma converge from above rather from below. A compatible rectangular element with corner nodes onl require that twist w/ also be used as a nodal d.o.f. which is undesiderable. Eperience in formulation of plate elements has shown that the Midlin formulation is more productive and Midlin plate elements are in common use also for thin plates. Sstem LEC 008/11/04
Sstem LEC 008/11/04 3 ), ( ), ( ), ( w w Mindlin plate elements A Mindlin element is based on three fields each interpolated from nodal values. If all interpolations use the same polnomial, then for an element of n nodes: Nd i i i n i i i i w N N N w 1 0 0 0 0 0 0
In the stiffness matri k { DBdV 55 D include the 3b 3 matri for plane stress and also shear moduli associated with the two transverse shear strain. Integration with respect ot z is done eplicitl. B T Integration in the plane of the element is done numericall if the element is isoparametric. Four nodes and eight nodes quadrilater elements are popular based on the same N i used for a plane elements. In an laer zconstant, the behaviour of a Mindlin plate can be deduced from the behaviour of the corresponding element provided that the integrand are integrated b the same quadrature rule. Sstem LEC 008/11/04 4
Bending deformation Elements strains ε are independent of, an order o quadrature will report the same strain energ of pure bending However this element displas spurious shear strain. If a/t is large, transverse shear strain γ z becomes large and the element is too stiff in bending, unless γ z is evaluated at 0, where it vanishes. But one-point quadrature for all strains will introduce four instabilit modes. Sstem LEC 008/11/04 5
This observation suggests selective integration in which one-point quadrature is applied to transverse shear terms and four-point quadrature is applied to bending terms. Two instabilit modes remain that are controlled b stabilization matrices. Eight nodes Mindlin elements have the analogous shortcomings and ma also be treated b selective integration. Calculated stresses are usuall more accurate at the Gauss points. Sstem LEC 008/11/04 6
Support conditions Support conditions are classed as clamps, simple, or free, in direct analog to the possible support conditions of a beam. Nodal d.o.f. that must be prescribed for these support conditions are the following Clamped wϑ n ϑ s 0 ------ Simpl supported w0 M n 0 Free ------- QM n M ns 0 Sstem LEC 008/11/04 7
Bending of square plate D 3 Et 1(1 ν ) Maimum deflection Distributed load q f 4 αql D { α0.00406 simpl supported edges α0.0016 clamped edges Concentrated load P f αpl D { α0.0116 simpl supported edges α0.00560 clamped edges Sstem LEC 008/11/04 8
Twisting test cae E10 7 ν0.3 t0.05 W 3 0.009 L Sstem LEC 008/11/04 9
Shells and shell theor The geometr of a shell is defined b its thickness and its midsurface, which is a curved surface in space. Load is carried b a combination of membrane action and bending action. A thin shell can be ver strong if membrane action dominates, in the same wa that a wire can carr great load in tension but onl small load in bending. However, no shell is completel free of bending stresses. The appear or near point load, line loads, reinforcements, junctures, change of curvature and supports. An concentration of load or geometrical discontinuit can be epected to produce bending stresses, often much larger, of membrane stresses, but quite localized.
Eample of load and geometr that produce bending Aial force G must be transferred through the structure to the support. The simple support around the base AA applies aiall direct line load, which has a shell normal component that causes bending. Around BB the clindrical and conical part eert shell-normal load component on one other. Shell-normal load is transferred across FF because the clindrical and spherical shells tr to epand different amounts under internal pressure Line load EE is obviousl shell-normal, as is the restrain provided b reinforcing ring DD
Internal forces and moments associated with bending at a discontinuit such as CC are the following Fleural stress and bending moment in a shell are related in the same wa as for a plate Membrane stresses, constant through the thickness, would be superposed on the fleural stress t M M V V t M M νσ σ σ σ νσ σ σ K K 0 0 0 0 6 0 0 0 6 0
How much is the boundar in which bending ma be important? A simple approimation can be obtained from the theor of a shell of revolution. Analtical solution for radial displacement and bending moment as function of aial distance from the end is w e λ 31 λ R ( C sinλ + C cosλ) ( ν ) t 1 Where, given R the radius and t the thickness of the shell, 1/ 4 For λ 3 0.5, e -λ 0.07 (ν0) Rt The end displacement and the bending moment declines to 7%. This is the estimate value of the boundar laer. In a FE analsis al least two element must be used to span the boundar laer. Sstem LEC 008/11/04 33
Shell tangent edge loads Shell tangent edge loads produce actions that are not confined to a boundar laer. Aial loads act on end A of the unsupported clindrical shell, but the largest displacement appear at the end B in apparent contradiction of Saint-Venant principle. Saint-Venant principle is applicable to massive isotropic boides. Thin-walled structure and high anisotropic structures ma behave quite differentl.
Finite elements for shells The most direct mode to obtain a shell element is to combine a membrane element and a bending element. A simple triangular element can be obtained b combining the plane stress triangle with the plane bending triangle. The resulting element is flat and has five d.o.f. or si d.o.f. per node, depending on whether or not the shell-normal rotation zi at node i is present in the plane stress element. A quadrilater element can be produced in similar fashion, b combining quadrilater plane and plate elements. A four node quadrilater element is in general a warped element because its nodes are not all coplanar. A modest amount of warping can seriousl degrade the performance of an element.
Flat elements Advantages of a flat element include: - simplicit of formulation, - simplicit in the description of the geometr, - element s abilit to represent rigid-bod motion without strain. Disadvantage include: - the representation of a smoothl curved shell surface b flat or slightl warped facets. There is a discretization error associated with the lack of coupling between membrane and bending action within individual elements. Common advice is that flat element should span no more than roughl 10 of the arc of the actual shell.
Curved elements based on shell theor avoid some shortcomings of flat elements but introduce other difficulties: - more data are needed to describe the geometr of a curved element, - formulation, based on shell theor, is complicated, - membrane and bending action are coupled within the element
Isoparametric shell elements Isoparametric shell elements occup a middle ground between flat elements and curved elements based on shell theor. One begins with a 3d solid element The element can model a shell if thickness t is small in comparison with other dimensions. However, such an element has the defect noted for plate bending: invite numerical troubles because the stiffness associated with ε z is ver much larger than the other stiffness.
b c b b c b b a b a b b c b b a w w t v v t v v w w t u u t u u + + Accordingl, the element is transformed, the number of nodes is reduced from 0 to 8, b epressing translational d.o.f. of the 0-node element in terms of translational and rotational d.o.f. of the 8-node element. Node A and C appear in the solid element, but not in the shell elements. Displacement at A and C are
With relations like these for all thickness direction lines of nodes, shape function of the 0 node element are transformed so as to operate on the three translations and two rotations at each node of the 8-node element. A Mindlin shell element is obtained. Thickness direction normal stress is taken as zero, and stress-strain relations reflect this assumption. The element matri is integrated numericall. A reduced or selective integration scheme ma be used to avoid transverse shear locking and membrane locking. Sstem LEC 008/11/04 40
Test cases a) Shell roof with q 90 for unit area: membrane action dominates b) Pinched clinder, F1: mambrane and bending action are active c) Hemisphere, F: bending action dominates d) Twisted strip: F110-6: sensitivit of the elements to warping can be tested
Problem R o b L t E ν A Roof 5 50 0.5 43 10 6 0.3 0.304 FPeso Clinder F1 Hemisphere F Strip F110-6 F10-6 300 600 3.00 3 10 6 0.3 0.185 10-4 10 -- 0.04 68.5 10 6 0.3 0.094 1.1 1 0.003 9 10 6 0. 556 10-6 194 10-6
Shell of revolution A Aaa A Aaa A Aaa A Aaa Aaa In cross section, an element for a shell of revolution resembles a beam element. Like an element for a solid of revolution, an element for a shell of revolution has nodal circle, rather than nodal points. Tpicall there are two nodal circles for element.
A shell of revolution element ma be flat (conical) or curved. The simplest formulation resembles the D beam element, in that is used: - a cubic lateral displacement field, - a linear meridianal displacement field, - each nodal circle has two translational (radial and aial) and one rotational as d.o.f. Conical shell elements have advantages and disadvantages like those of other flat element Spherical shell loaded b internal pressure and modeled b a coarse mesh of conical shell elements displa spurious bending moments. Sstem LEC 008/11/04 44
Eamples
From the beam behaviour 3 PL 6 v 37310 mm 3EI PLR σ z 0. 0713MPa I Shear center (-R 0) Torque about the shear center T -RP Rotation of the loaded end ϑ-11. 10-6 rad Total deflection v-37310-6 +R ϑ-933 10-6 mm Sstem LEC 008/11/04 46