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[[Category: Users:General FEM]] == General Description == === Element Type === * This beam element is a 2 node non-linear 3D-beam for large rotations and small deformations (Green-Lagrange-strains) taking into account shear deformation (Timoshenko-beam element). * This beam element has 6 DOFs per node (three translations and three rotations). * The stiffness matrix is obtained using a one-point Gaussian quadrature, which results in an under-integration of the shear terms. This prevents shear-locking. === Degrees of Freedom === For the BeamNL element use the 3 translatoric degrees of freedom ''DISP_X, DISP_Y, DISP_Z'' and the 3 rotatoric degrees of freedom ''ROT_X, ROT_Y, ROT_Z''. === Orientation of the local coordinate system === [[File:BeamNL_Orientation.png|250px|frame|right|Orientation of the local coordinate system for BeamNL]] The BeamNL element uses the following definition for the determination of the local coordinate system in each node (needed for the orientation of I_22 and I_33,...), based on the idea of sets of members (dt.; ''Stabzug''). These coordinate systems also have the role of a local Frenet-frame that is rotated with the element during a non-linear analysis. The difference between the current and the reference Frenet-frames is used to determine the curvatures of the element: * the local x-axis ('''x'''_loc) is defined as the tangent to the element in the considered point. This has important consequences: ** need for a neighbour-element search (inside one set of members, other elements have to be linked using [[Users:General FEM Analysis/BCs Reference/Dirichlet|''MPC-COUPLING'']]), ** for the definition of kinks, two different sets of members have to be used (else kinks would be smoothed out), ** for starting members (i.e. with only 1 adjacent elemetn), the connection of node 1 to node 2 is the local x-axis. * the local y-axis ('''z'''_loc) is defined with the help of an orientation point (''ORIENTATION_POINT_BEAMNL'', external to the beam element itself and common for one set of members; to be defined in the element property (see below)): an auxiliary vector ('''vec'''_aux) pointing from node 1 to this external orientation point is set up. The local axis y is the result of the cross-product of this auxiliary vector and the local x-axis (mathematically spoken: '''y'''_loc = '''vec'''_aux x '''x'''_loc). * the local z-axis ('''z'''_loc) is perpendicular to the other two local axes, following the right-hand-rule for x-y-z (i.e. once again using the cross-product). This makes z point in the half-space of the orientation point. * '''addition:''' If the local x-axes at both ends (i.e. nodes) of the element don't point in the same direction (due to the situation of the orientation point), the x-axis at the second node is switched by 180 degrees. == Input Parameters == === Parameter Description === {| border="1" cellpadding="3" cellspacing="0" |colspan="3" style="background:#efefef;"| Compulsory Parameters |- !Parameter !Values, Default(*) !Description |- !MAT |EL-MAT ''int'' |Number for the used Material e.g. MAT=EL-MAT 1 |- !AREA | |Definition of the cross-sectional area of the beam |- !SHEAR_SECTION_Y, SHEAR_SECTION_Z | |Definition of the shear-cross-sectional area of the beam in y- and z-direction |- !I_22, I_33 | |Definition of the moments of inertia around the local y- and z-axis |- !I_23 | |Definition of the mixed moment of inertia |- !I_T | |Definition of the torsional resistance I_T |- !ORIENTATION_POINT_BEAMNL | |External orientation point for the set of beam members. |- |} === Example of a Complete Input Block === <pre> EL-PROP 1 : BEAMNL MAT= EL-MAT 1 AREA=0.015 SHEAR_SECTION_Y = 0.0075 SHEAR_SECTION_Z = 0.015 I_22=1.125 I_33=0.03125 I_23=1.23 I_T=1.15625 ORIENTATION_POINT_BEAMNL = 0.0, 10.0, 0.0 </pre> <pre> BC-DIRICHLET 2: MPC-COUPLING DOFS = DISP_X NODES = 21, 121 DOFS = DISP_Y NODES = 21, 121 DOFS = ROT_Z NODES = 21, 121 </pre> <pre> EL-TOP 1 ! ID PART PROP NODE NUMBERS NEL 1 1 1 1 2 </pre> === Use of the shear sections SHEAR_SECTION_Y and SHEAR_SECTION_Z === The shear sections ''SHEAR_SECTION_Y'' and ''SHEAR_SECTION_Z'' depend on the cross section of the beam and can be obtained by multiplying the area of the beam ''AREA'' with the shear correction factor α which is dependent on the shape of the sections. For a rectangle, α=5/6. === Use of the rotation parameter THETA === The rotation parameter ''THETA'' and its use is explained in the section concerning the coordinate system above. === The mixed moment of inertia I_23 === The mixed moment of inertia ''I_23'' is equal to 0 if one of the two local axes y and z is an axis of symmetry of the section. === The external point of orientation ORIENTATION_POINT_BEAMNL === The external point of orientation ''ORIENTATION_POINT_BEAMNL'' has to be entered with its coordinates in the global XYZ-frame in order to define the orientation of the local coordinate system of the beam. == Element Loading == For the moment, only nodal forces in the three global directions can be applied (i.e. ''Fx, Fy, Fz''). === Pressure === * not defined yet === Dead Load === * not defined yet === Snow Load === * not defined yet == Tests and Benchmarks == === Static linear analysis === [[File:linBeamTests2.jpg|300px|frame|right|Static linear test for element BeamNL]] For the moment, the element BeamNL has successfully been tested in 3D in all its linear static features, including * bending, axial deformation, torsion, * shear deformation (separately definable for both local axes), * rotation around the local axis. As an example, the structure on the right was part of the final tests. == Theory == The element implementation mainly follows the implementation of a non-linear 3D-beam element in the PhD-thesis of Roland Sauer <ref name="Sauer1998"> Sauer, R.: Eine einheitliche Finite-Element-Formulierung für Stab- und Schalentragwerke mit endlichen Rotationen, Institut für Baustatik, Universität Karlsruhe, 1998</ref>, and some papers of F. Gruttmann et al. As it is based on linear ansatz-functions, the BeamNL-element needs sufficient fine discretization in order to converge to the true solution. == References == <references/>

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