STIFFNESS CHARACTERISTICS OF GEOMETRICALLY NON-LINEAR BEAM FINITE ELEMENT/GEOMETRIŠKAI NETIESINIO LENKIAMO STRYPO BAIGTINIO ELEMENTO STANDUMO RODIKLIŲ NUSTATYMAS
Authors: Mikoliūnas Audrius , Kačianauskas Rimantas 
Published: March 2019

Two-dimensional geometrically non-linear beam element is considered in this paper. The explicit expressions of stiffness characteristics of element with three nodes are derived and tested.

Among models of the geometrically non-linear beams, the elements with 2 nodes dominate [1–8]. Such elements produce constant axial force. The idea of more complex elements with tree nodes was suggested in [3]. In this paper geometrically non-linear flat bending beam element with 3 nodes for evaluating of axial force is investigated and nonlinear stiffness characteristics are derived.

Basic relations of element e are derived using virtual displacement method. On the level of element e, the principle of virtual displacements is expressed by equalities (1–3). Using displacement approach, displacement functions are prescribed in the bounds of one finite element. Generalised deformations are obtained by introducing displacements approximation (4) and inserting them into non-linear geometric equations (5–6). Variation of deformation energy (3) is expressed in (7). Putting equality (7) into (1), it is possible to write equality of virtual works in terms of non-linear algebraic equations (8). Non-linear stiffness matrix is presented as the sum of 3 matrices (9). The first matrix [K 0e ] (linear matrix) is the matrix of small deflections, which is independent on deformed shape. The second matrix [K Ne ] is the matrix of large deflections. The third matrix [K Ge ] is a geometrical stiffness matrix. It reflects the second member of equality (7). Expressions of geometrically non-linear stiffness matrices are greatly dependent on the introduced assumptions and appropriate elements.

Shallow beam finite element is shown in Fig 1. This finite element has 3 nodes. In the initial configuration a beam can be straight (Fig 1a), or curved (Fig 1b). The initial configuration of a beam is described by a vector z = {z1 αx1, Z2, αx2}T of a beam final nodal co-ordinates, where z i means nodes co-ordinates, αxi—initial rotations (Fig lb). However, the initial configuration is a relative statement, and is generally described by vector z. If in initial configuration the element is straight, vector z=0. Physical properties of the element are denoted with capital EA (tensional rigidity) and EI (flexural rigidity). The finite element has 7 degrees of freedom: 3 of them are defined at each end of the element (2 linear and 1 rotation) and 1 in the middle of the element Vector Ue of nodal deflections for this element is split into two parts: Ue= {u, w}T , u = {u1, u2,u3}T, w'=z+w, w={w1,Θx1, w2, Θx2}T. Deflection u3 shows the deflection of the middle beam node, which is not proportional to the final nodal deflections. To be more strict, u3 is straightened by linear law. So the linear element in the direction of longitudinal deformation expression is (11). The deflection of a point which is moved from the centre of plane surface in distance z1, deflection u (in direction x) is expressed in (12). Deformation is expressed by summarised deformations (13). So the deformed element only longitudinal deformation Δ is assigned, which is shown in (16). Evaluating earlier received expressions, it is possible to make equality of virtual work (1), where generalised vectors Θ e (x)={Δx, κ}T and Q e (x)={N, M}T. Generalised deformations Δ and κ expressed by deflections approximating expressions (4). For convenience, vertical and horizontal deflections are separated (17). By analogy with deflections, vector z and its derivatives are approximated by (24–25). Beam's curvature (15) is also expressed by nodal deflections: (26–27). Putting (20, 21, 26) into (16) and (15, 16) into (13) and expression (28) is got. Evaluating that the element work in elastic stage, expression (10) can be rewritten (29). The final stiffness matrix expression (9) is given in the form of block matrices (30). Expressions of block matrices are presented by (31–33). Having completed operations in expressions (31–33), final stiffness matrix is (34–36). After integrating, linear matrix is (37). Analogous operations are performed with matrices (35)–(38). Elements of this matrix are calculated using computer algebra. Matrix (36) consists of three parts. The first integral (39) is stiffness matrix of bending beam. If we assume that axial force in beam's length is invariable, the third integral is equal to (41). Assuming that axial force in the length of element is not constant (the axial force is calculated according to forms (28,29)), the expressions of geometrical stiffness matrix become very complicated.

Analysis of geometrically non-linear system of finite elements is described by algebraic equation (42). Usually expression of non-linear deformation is investigated as a process varying in time t, where outer load F and deflections U are functions of time: F≡6F(t) and U≡6U(t). Load in the moment of time t i+1 = t i + Δt is added in portions (43). Deflections are expressed by analogy with (44). Non-linear model (42) is expressed by increments (45). Vector of residuals γ reflects solution of equation (42) inadequacy of state variables. Nowadays there exists many algorithms of different complexity for solution of non-linear problems [2–4,9,10]. The majority of methods that have already become standard uses different Newton-type variety of algorithms. Classical Newton-type algorithms are adapted to non-linear process with so called “softening” curve to model (Fig 2a). In the work there was done and realised a combined algorithm for non-linear process with “hardening” or “softening” curve to model. The illustration of algorithm is given in Fig 3. Using the algorithm in every load step, tangent stiffness matrix is counted twice. The first matrix corresponds to tangent of load step at the beginning (tangent 1), and the second one to the step at the end (tangent 2). Algorithm is implemented in the program created by the authors.

A simple cantilever beam (Fig 4) is taken for the test. History of deformation was investigated. The results are given in non-dimensional quantities (Fig 5). Euler's method is realised as a particular case of implemented algorithm. The same example was also solved using program ANSYS, where beam elements are used and described only by two nodes. The results presented show obviously the advantages of three-node element and validity of proposed assumptions.