Semiloof Curved Thin Shell Elements

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1 Semiloof Curved Thin Shell Elements General Element Name Y,v,θy X,u,θx Z,w,θz Element Group Element Subgroup Element Description Number Of Nodes Freedoms Node Coordinates TSL 1 2 Semiloof 3 QSL A family of shell elements for the analysis of arbitrarily curved shell geometries, including multiple branched junctions. The elements can accommodate generally curved geometry with varying thickness and anisotropic and composite material properties. The element formulation takes account of both membrane and flexural deformations. As required by thin shell theory, transverse shearing deformations are excluded. or 8 numbered anticlockwise. U, V, W: at corner nodes. U, V, W, θ 1, θ 2 : (loof rotations) at mid-side nodes (see Notes). X, Y, Z: at each node. Geometric Properties t 1... t n Thickness at each node. Also see Composite Geometry data chapter. Material Properties Linear Isotropic: MATERIAL PROPERTIES (Elastic: Isotropic) Orthotropic: MATERIAL PROPERTIES ORTHOTROPIC (Elastic: Orthotropic Plane Stress) MATERIAL PROPERTIES ORTHOTROPIC SOLID (Elastic: Orthotropic Solid) Anisotropic: MATERIAL PROPERTIES ANISOTROPIC 3 (Elastic: Anisotropic Thin Plate) Rigidities. RIGIDITIES (Rigidities: Shell) Matrix 21

2 Semiloof Curved Thin Shell Elements Joint Concrete Biaxial: MATERIAL PROPERTIES NONLINEAR 2 (Concrete: Biaxial) Elasto-Plastic Stress resultant: MATERIAL PROPERTIES NONLINEAR 29 (Elastic: Isotropic, Plastic: Resultant) (ifcode not required) Tresca: MATERIAL PROPERTIES NONLINEAR 1 (Elastic: Isotropic, Plastic: Tresca, Hardening: Isotropic Hardening Gradient, Isotropic Plastic Strain or Isotropic Total Strain) Mohr- Coulomb: Drucker- Prager: Volumetric Crushing: Rubber Composite Composite shell: Field Stress Potential Creep Damage Viscosity MATERIAL PROPERTIES NONLINEAR 3 (Elastic: Isotropic, Plastic: Mohr-Coulomb, Hardening: Granular) MATERIAL PROPERTIES NONLINEAR (Elastic: Isotropic, Plastic: Drucker-Prager, Hardening: Granular) COMPOSITE PROPERTIES STRESS POTENTIAL VON_MISES, HILL, HOFFMAN CREEP PROPERTIES (Creep) DAMAGE PROPERTIES SIMO, OLIVER (Damage) Loading Prescribed PDSP, TPDSP Value Concentrated CL Loads Element Loads Distributed UDL Loads FLD Body Forces CBF Prescribed variable. U, V, W: at corner nodes. U, V, W, θ 1, θ 2 : at mid-side nodes. Concentrated loads. Px, Py, Pz: at corner nodes. Px, Py, Pz, M 1, M 2 : at mid-side nodes. Uniformly distributed loads. Wx, Wy, Wz: midsurface local pressures for element. Constant body forces for element. Xcbf, Ycbf, Zcbf, Ωx, Ωy, Ωz, αx, αy, αz 21

3 BFP, BFPE Body force potentials at nodes/for element. ϕ 1, ϕ 2, ϕ 3, 0, Xcbf, Ycbf, Zcbf, where ϕ 1, ϕ 2, ϕ 3 are the face loads in the local coordinate system. Velocities VELO Velocities. Vx, Vy, Vz: at nodes. Accelerations ACCE Initial SSI, SSIE Stress/Strains Residual Stresses SSIG SSR, SSRE SSRG Accelerations. Ax, Ay, Az: at nodes. Initial stresses/strains at Gauss points. (1) Resultants (for model 29 and RIGIDITIES) Nx, Ny, Nxy, Mx, My, Mxy, εx, εy, γxy, ψx, ψy, ψxy: forces, moments/unit width and membrane/flexural strains in local directions. (2) Components (in all cases except for nonlinear model 29 and RIGIDITIES). 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (σx, σy, σxy, εx, εy, γxy) Bracketed terms repeated for each layer. Residual stresses at Gauss points. (1) Resultants (for model 29) Nx, Ny, Nxy, Mx, My, Mxy: forces, moments/unit width in local directions. (2) Components (for all nonlinear material models except model 29). 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, (σx, σy, σxy) Bracketed terms repeated for each layer. Temperatures TEMP, TMPE Temperatures at nodes/for element. T, 0, 0, dt/dz, To, 0, 0, dto/dz Field Loads Temp Dependent Loads Output LUSAS Solver Stress resultant: Nx, Ny, Nxy, Mx, My, Mxy: forces, moments/unit width in local directions. Stress (default): σx, σy, σxy, σmax, σmin, β, σe: in local directions (see Notes). Strain: εx, εy, γxy, ψx, ψy, ψxy: membrane, flexural strains in local directions. 21

4 Semiloof Curved Thin Shell Elements LUSAS Modeller See Results Tables (Appendix K). Local Axes Local y axis The local element y-axis at a point coincides with a curvilinear line ξ = constant in the natural coordinate system which lies in the shell mid-surface. Local x axis The local x-axis at a point is perpendicular to the local y-axis in the positive η direction and is tangential to the shell mid-surface. Local z axis The local z-axis forms a right-hand set with the x and y axes and the direction is given by the ordering of the element nodes according to a right-hand screw rule. The local z-axis +ve direction defines the element top surface. TSL z x y QSL8 η y z x ξ

5 Sign Convention Thin shell element (see Notes). Formulation Geometric Nonlinearity Total Lagrangian For large displacements, rotations up to 1 radian and small strains. Updated Lagrangian For large displacements, rotation increments up to 1 radian and small strains. Eulerian Co-rotational Integration Schemes Stiffness Default. 3-point (TSL), -point (QSL8). Fine (see Options). 3x3 (QSL8) Coarse (see Options). 2x2 (QSL8) Mass Default. 3-point (TSL), -point (QSL8). Fine (see Options). 3x3 (QSL8) Mass Modelling Consistent mass (default). Lumped mass. Options 18 Invokes fine integration rule. 19 Invokes coarse integration rule. 32 Suppresses stress output but not resultants. 3 Outputs element stress resultants. Updated Lagrangian geometric nonlinearity. Outputs strains as well as stresses. 9 Outputs local direction cosines at nodes and Gauss points. 87 Total Lagrangian geometric nonlinearity. 102 Switch off load correction stiffness due to centripetal acceleration. 10 Lumped mass matrix. 138 Output yield flags only. 139 Output yielded Gauss points only. 19 Suppress extrapolation of stresses to nodes. 170 Suppress transfer of shape function arrays to disk. 218

6 Semiloof Curved Thin Shell Elements Notes on Use 1. The element formulations are based on an isoparametric approach with constraints to invoke the Kirchhoff hypothesis for thin shells. 2. The variation of stresses within the elements may be regarded as linear. 3. The loof rotations refer to rotations about the element edge at the loof points. The positive direction of a loof rotation is defined by a right-hand screw rule applied to a vector running in the direction of the lower to higher numbered corner nodes. It should be noted that this direction is enforced on a global level which means that the loof rotations along the adjoining edge of several elements will be consistent in terms of direction and ordering. The ordering is such that loof point 1 is located between the lower numbered node and the appropriate mid-side node. Similarly loof point 2 lies between the mid-side node and the higher numbered node along an element edge. The loof rotations are actually specified at the element mid-side nodes.. The elements pass the patch test for convergence for mixed triangular and quadrilateral element geometry.. Stress output to the LUSAS output file is on lines: Stresses due to membrane action. Top surface stresses due to bending action. Top surface stresses due to membrane and bending action. Bottom surface stresses due to membrane and bending action.. Stresses will not be output when using RIGIDITIES or material model 29. Averaged stresses will not be processed when using RIGIDITIES. 7. The through-thickness integration is performed explicitly for linear analyses and a -point Newton-Cotes rule is utilised for materially nonlinear analyses with continuum material models. The through-thickness integration rules are as follows: Linear models: 3-layers. Nonlinear models: -layers. Composite model: Variable. Restrictions Ensure mid-side node centrality Avoid excessive element curvature Avoid excessive aspect ratio Recommendation on Usage These elements may be utilised for analysing flat and curved 3D shell structures where the transverse shear effects do not influence the solution. The configuration of 219

7 the nodal freedoms provides an element suitable for modelling intersecting shells, e.g. tubular joints and also for use with solid elements (HX20). The elements may be combined with the Semiloof beam (BSL3,BSL,BXL) for analysing ribbed plates and shells. The quadrature points of the 3-point rule are non-standard. The coarse 2*2 quadrature rule provides the most effective element if the mesh is highly constrained. However, the element possesses two mechanisms, the usual in-plane hourglass mechanism encountered when reduced integration is utilised with 8-noded elements and an out of plane mechanism. The in-plane mechanism is rarely activated but the out-of-plane mechanism may be more troublesome, particularly where elements are regular and have one zero principal curvature, e.g. a cylinder subject to internal pressure. Provided the mechanisms are not activated the element with 2*2 provides the best results. The -point quadrature rule provides an element with a performance below that of the element with 2*2 quadrature, but considerably better than the element with 3*3 quadrature. However, the element possesses a 'near' mechanism which may be activated for lightly constrained meshes, particularly if out of plane loads are present. The middle integration point of the point rule is only implemented as a method of reducing the excitation of spurious modes (or mechanisms) which are present with the 2*2 integration rule. The th integration point is actually weighted with an arbitrarily small value which has the effect of stabilising the results. For these reasons, values from the middle integration point are not taken into account for the nodal extrapolation. The 3*3 quadrature rule provides an element that has no mechanisms but tends to provide over-stiff solutions. Therefore, a finer discretisation is required than if the -point quadrature rule is used. 220

3D Semiloof Thin Beam Elements

3D Semiloof Thin Beam Elements General Element Name Z,w,θz Y,v,θy Element Group X,u,θx Element Subgroup Element Description Number Of Nodes BSL3, BSL4 y 1 4 x z 2 Semiloof 3 Curved beam elements in 3D