European Congress on Computational Methods in Applied Sciences and Engineering (ECCOMAS 212) J. Eberhardsteiner et.al. (eds.) Vienna, Austria, September 1-14, 212 A NOTE ON RELATIONSHIP BETWEEN FIXED-POLE AND MOVING-POLE APPROACHES IN STATIC AND DYNAMIC ANALYSIS OF NON-LINEAR SPATIAL BEAM STRUCTURES Gordan Jelenić 1, Maja Gaćeša 1, and Miran Saje 2 1 University of Rijeka, Faculty of Civil Engineering Radmile Matejčić 3, 51 Rijeka, Croatia e-mail: {gordan.jelenic,maja.gacesa@gradri.hr 2 University of Ljubljana, Faculty of Civil and Geodetic Engineering Jamova cesta 2, 11 Ljubljana, Slovenia e-mail: msaje@fgg.uni-lj.si Keywords: 3D beams, non-linear analysis, configuration-dependent interpolation, fixed-pole approach Abstract. Fixed-pole approach in the non-linear analysis of beam structures offers a different and promising framework for development of new element types with particular potential in the area of conservative time integrators. The finite elements formulated using this approach, however, utilise non-standard degrees of freedom which makes them difficult to combine with standard finite elements in existing codes. A modified approach which combines the benefits of the fixed-pole approach with the use of standard kinematic unknowns from the existing (movingpole) approaches is presented in this work along with a family of solution procedures for static analysis of geometrically non-linear spatial beam structures. Such a hybrid formulation opens a number of questions which are specifically addressed.
1 INTRODUCTION 1.1 Notation L, x initial length of the beam, co-ordinate of a point along the centroid axis of the beam E i orthonormal base vectors of the material co-ordinate system (i 1, 2, 3) I, 3 3 unit and null matrix, respectively r, Λ position vector of a point along the beam, rotation matrix of a triad rigidly attached to the cross-section u displacement vector N, M vectors of stress and stress-couple resultants in material co-ordinate system n, m vectors of stress and stress-couple resultants in spatial co-ordinate system n, m fixed-pole vectors of stress and stress-couple resultants in spatial co-ordinate system Γ, K translational and rotational strain measures in material co-ordinate system γ, κ translational and rotational strain measures in spatial co-ordinate system γ, κ fixed-pole translational and rotational strain measures in spatial co-ordinate system n e, m e vectors of applied distributed forces and torques n e, m e fixed-pole vectors of applied distributed forces and torques F, F L,M, M L applied concentrated forces at ends of the beam F, F L, M, M L fixed pole applied concentrated forces at ends of the beam C N, C M translational and rotational constitutive matrices E, G Young s (elastic) modulus, shear modulus A 1, A 2, A 3 cross-sectional and shear areas J, I 2, I 3 torsional and cross-sectional moments of inertia A diag [A 1, A 2, A 3 ] J ρ diag [J, I 2, I 3 ] ρ material density k, π vectors of specific translational momentum and angular momentum δr, δϑ variation of the position vector, spin variable N number of interpolation points (nodes) of a finite element I i (x) i-th Lagrangian interpolation polynomial of order N 1 G (if not in C N ) weak form g i, q i i, qi m, q i e residual and nodal vectors of internal, inertial and external forces at node i K ij stiffness matrix relating the trial functions at node j to the residual at node i ( ), ( ) ( ), ( ) t ( ) i, ( ) j δ( ), ( ) derivatives of ( ) with respect to x, and time t cross-product operator ( ), transposed quantity ( ) ( ) at nodes i and j variation of ( ), incremental change of ( ) The geometrically exact beam theory provided by Simo in 1985 [1] makes one of the milestones in the development of the non-linear finite-element method in introducing a non-linear manifold as the solution space and the related differential geometry tools in order to implement a robust solution procedure [2]. This approach, however, turns out to be incompatible with algorithmic preservation of strain invariance with respect to a rigid-body motion, or simultaneous conservation of energy and the momentum vectors during a free motion of an unloaded beam. Borri and Bottasso [3, 4] propose the fixed-pole approach in which they introduce new stresscouple resultant and angular momentum - defined with respect to a unique point for the whole structure - the fixed pole. Different implementations of this general concept result in the algo- 2
rithms which naturally inherit the strain-invariance of the underlying formulation with respect to a rigid body motion [3] or are capable of simultaneous conservation of both energy and momentum vectors [4]. The main problem with the fixed-pole finite element implementation is that it uses nonstandard problem unknowns, and therefore is not easily combinable with existing finite element codes. For this reason, we derive and present a formulation which uses standard kinematic unknowns but is inherently based on the fixed-pole approach, along with with the algorithmic benefits of this approach. 2 KINEMATIC AND CONSTITUTIVE EQUATIONS Kinematic equations follow from the Reissner-Simo theory [1]. The material translational and rotational strain measures are given as follows: Γ Λ t γ Λ t r E 1 ˆK Λ t κλ Λ t Λ, (1) while the spatial translational and angular velocity vectors, v and w, are defined as v ṙ and ŵ Λ t Λ. Varying (1) leads to δγ δr + r δϑ δκ δϑ (2). Let us consider a linear elastic material with the material cross-sectional stress and stresscouple resultants defined as [ ] N CN Γ, (3) M C M K with EA GI t C N GA 1, C M EI 1. (4) GA 2 EI 2 3 THE STANDARD APPROACH The standard approach [2] given by Simo and Vu-Quoc follows naturally from the principle of virtual work: V i + V m V e, (5) where V i, V m and V e are the virtual works of internal, inertial and external forces, respectively, which are defined as follows: n L [ ] I V i δγ t δκ t dx δr m t δϑ t d dx n r I d dx, (6) m dx k V m δr t δϑ t dx, (7) π V e δr t δϑ t ne F dx + δr m t δϑ t F + δr e M t L L δϑt L. (8) M L 3
A certain irregularity in the integrals should be noted while the virtual quantities simply dot-multiply the specific vectors of inertial and external forces, an additional operator occurs between the virtual quantities and the stress resultants: [ ] I d dx r I d dx. (9) After interpolating virtual quantities δr, δϑ using Lagrangian polynomials I i (x): δr I i (x)δr i, δϑ i1 I i (x)δϑ i, (1) i1 we have the approximate weak form: G h t δr i δϑ t i g i, (11) i1 from which, because of the arbitrariness of δr i and δϑ i, follows the vector of nodal dynamic residuals: g i q i i + q i m q i e, (12) with q i i, q i m and q i e as the nodal vectors of internal, inertial and external forces, respectively, given as follows: [ I q i i ] I n i I i r I i dx, (13) I m k q i m I i dx, (14) π q i e I i ne F dx + δ1 i F + δn i L. (15) m e 4 THE FIXED-POLE APPROACH In [3, 4] Borri and Bottasso thoroughly investigated the idea of replacing the stress-couple resultant m and the specific angular momentum π, which are defined with respect to the beam reference axis at a cross-section, with another stress-couple resultant m and specific angular momentum π, which are to be defined with respect to a unique point for the whole structure the fixed pole naturally taken to be the origin of the spatial frame, ie: M M L m r n + m, (16) π r k + π. (17) We also note that n n and k k. In order to obtain a relationship between standard and fixed-pole strain measures, we request the strain energy density to remain invariant to the change of virtual quantities: 1 2 ( γ n + κ m) 1 (γ n + κ m) (18) 2 4
By substituting the relationship (16) between standard and fixed pole stress resultants, we obtain the relationship between standard and fixed-pole spatial strain measures: γ γ + r κ, (19) κ κ. (2) It is also useful to note the relationship between the time derivatives of standard and fixedpole specific momenta: k k, π ṙ {{ k + r k + π, (21) ṙ Aρṙ and considering virtual work of the inertial forces (7) to be an objective quantity gives us the fixed-pole kinematics which is virtual-work conjugate to the time-derivatives of the specific fixed-pole momenta as δ r δr + r δϑ, δ ϑ (22) δϑ, from where their respective derivatives follow as δ r δr + r δϑ + r δϑ, δ ϑ δϑ. (23) Analogously to (16), the fixed-pole distributed moment is introduced: m e m e + r n e, (24) noting that n e n e. We also introduce the relationship between standard and fixed-pole concentrated moments at ends of the beam: M M + r F, M L M L + r L F L, (25) noting that F F and F L F L. 4.1 Original formulation We rewrite the expression for the virtual work of internal forces (6) and substitute (23) and (16) into it: n L V i (δr + r δϑ) t δϑ t dx ((δr + r δϑ) n + δϑ m) dx m ((δ r r δϑ r δϑ + r δϑ) n + δϑ ( m r n)) dx ( δ r n + δ ϑ ) m dx, (26) 5
and obtain the virtual work of internal forces using the fixed-pole virtual quantities and stress and stress-couple resultants: ( ) { d n m V i δ r dx t δ ϑ t dx. (27) Note that there is no irregularity (9) in this term! In a like manner, we rewrite the expression for virtual work of the inertial forces (7) and substitute (22) and (21) into it: V m ( (δ r r δϑ) k + δϑ π r k ) (δ r r δϑ) t δϑ t to get the fixed-pole virtual work of inertial forces as V m δ r t δ ϑ t dx { k π r k dx ( δ r k ) + δ ϑ π dx, (28) k dx. (29) π Finally, the virtual work of external forces (8) may be treated in the same way, so that after substituting (25) and (22) into it: { V e (δ r r δϑ) t δϑ t n e dx+ m e r n e { + (δ r r δϑ ) t δϑ t F + M r F { + (δ r L r L δϑ L ) t δϑ t L F L, (3) M L r L F L we arrive at the following fixed-pole virtual work of external forces: V e δ r t δ ϑ t ne F dx + δ r m t δ ϑ t F + δ r e M t L δ ϑ t L L. M L Deciding to interpolate the fixed-pole virtual quantities δ r, δ ϑ using Lagrangian polynomials I i (x), δ r I i (x)δ r i, δ ϑ i1 we obtain the approximate weak form: I i (x)δ ϑ i, (31) i1 G h δ r t i δ ϑ t i ḡi, (32) i1 from which follows, because of the arbitrariness of δ r i and δ ϑ i, ḡ i q i i + q i m q i e, (33) 6
with ḡ i as the fixed-pole vector of nodal dynamic residuals and q i i, q i m and q i e as the fixed-pole nodal vectors of internal, inertial and external forces, respectively, given as q i i I i dx, (34) n m k q i m I i dx, (35) π q i e 4.2 Modified formulation I i { ne m e dx + δ i 1 F F + δn M i L. (36) M L As shown in (32), the fixed-pole nodal dynamic residual ḡ i is virtual-work conjugate to the fixed-pole nodal virtual displacements and rotations which are non-standard and prevent the elements from being combined with standard elements in a finite-element mesh. This problem, however, is easily overcome by noting that from (22) the nodal fixed-pole virtual quantities are related to the standard nodal virtual quantities via: [ ] δ ri δri δ ϑ + r i δϑ i I ri δri, (37) i δϑ i I δϑ i so that the fixed-pole weak form (32) becomes with G h [ ] I δr t i δϑ t i ḡ r i i I i1 δr t i δϑ t i g i, (38) i1 g i q i i + q i m q i e, (39) and q i i q i m q i e [ ] I n I i r dx, (4) r i I m [ ] I k I i r dx, (41) r i I π [ ] I I i ne F r dx + δ i F 1 + δ i L N. (42) r i I m e M M L After substituting (4), (41) and (42) in g i and then back to (32), it can be shown that, using a non-linear interpolation of the type: δr δϑ I i [δr i + (r i r) δϑ i ] i1 (43) I i δϑ i i1 is equivalent to the standard (Lagrangian) interpolation of the fixed-pole virtual quantities (31). 7
5 SOLUTION PROCEDURE The non-linear equation (39) may be solved using the Newton-Raphson solution procedure. In order to concentrate on the issues that arise when attempting to utilise a fixed-pole approach in the hybrid manner presented here, in which the standard displacement and rotation quantities have been kept as the problem unknowns, we proceed by considering only the static case. When linearising (39), account has to be taken of the fact that Λ ϑλ and Γ Λ t ( r + r ϑ) as well as [1] ( ( ) ) t ( ( ) ) K new Λ t newλ new exp ϑ Λ old exp ϑ Λ old ( ( )) t ( ( )) ( ( )) t ( ) Λ t old exp ϑ exp ϑ Λold + Λ t old exp ϑ exp ϑ Λ old {{ I K + K old, (44) where and ( ) exp ϑ I + sin ϑ ϑ ϑ 1 cos ϑ + ϑ 2, (45) ϑ 2 with H ϑ sin ϑ ϑ ϑ t + ϑ 3 K Λ t oldh ϑ, (46) sin ϑ ϑ The strain measure updates for K and Γ then follow as 1 cos ϑ I ϑ. (47) ϑ 2 K new K old + K, (48) Γ new : Λ t newr new E 1. (49) After expanding equation (39) into a Taylor series around a known configuration and omitting higher-order terms we have where g i + g i, (5) g i q i i q i e. (51) After the linearisation of the modified nodal internal force vector (4) we have q i i [ ] I i ri r L [ ] [ ] I n r dx + I n i dx+ ϑ i ϑ r r i I m ϑ [ ] [ ] [ ] [ ] I Λ + I i CN Λ t r + r ϑ r r i I Λ C M Λ t ϑ dx. (52) 8
Linearizing the modified nodal external force vector (42) we have: [ ] q i e I i ri r dx. (53) n e ϑ i ϑ An interesting phenomenon occurs when linearising the external force vector, which is a direct consequence of the use of the fixed-pole virtual quantities and their conversion back to standard virtual quantities: even though the external loading is configuration-independent, the modified nodal external loading vector (42) is not. As a result of this, the stiffness matrix will have an additional part as a result of the linearisation of q i e. In order to complete the solution procedure, the unknown functions and/or their iterative changes must be interpolated in some way. Within this paper we consider three different interpolation options, which arise as a consequence of the fact that the virtual position vector has been interpolated in a non-linear manner, dependent on the actual position vector not only at the nodal points, but also at x (i.e. the integration point). 5.1 Interpolation option 1 Within this option r and ϑ are interpolated in the same way as δr and δϑ (43): r ϑ I j [ r j + (r j r) ϑ j ] j1 I j ϑ j, while the unknown displacement functions are interpolated as: j1 r (54) I k r k. (55) k1 It should be noted that this interpolation is in contradiction with the interpolation for r in (54), which obviously does not follow as a linearisation of (55). The above contradiction necessarily results in loss of quadratic convergence of the Newton-Raphson solution process, and has been introduced as a simplest means of providing r(x) in (54). After introducing (54) into (52) and (53) we have: q i i K ij rj int, (56) ϑ j q i e K ij rj ext, (57) ϑ j where [ ] int I i L [ ] n (δ ij I j ) n I j n( r j dx + I i I j r) ( r dx r i ) n m [ ] [ ] [ ] [ ] [ ] I Λ + I i I j CN Λ t I rj r r r i I Λ C M Λ t dx, I (58) K ij 9
and and finally K ij ext [ ] I i (I j δ ij ) n e I j n e ( r j dx, (59) r) K ij K ij int Kij ext. (6) 5.2 Interpolation option 2 Within this option, r is interpolated using (55). Interpolation of the displacement increments follows consistently by linearising r and ϑ: r I j r j, ϑ j1 I j ϑ j. (61) After introducing (61) into (52) and (53) we have q i i K ij rj int, (62) ϑ j q i e K ij rj ext, (63) ϑ j where and K ij int + [ ] I i (δ n ij I j ) dx + ] [ Λ [ I I i r r i I K ij ext j1 [ ] [ ] I n I i I j dx+ r r i I m ] [ ] [ ] [ CN Λ t I j ] I I j r Λ C M Λ t I j dx, I I i (I j δ ij ) [ ] dx, (64) n e and finally K ij K ij int Kij ext. (65) In this interpolation option, different interpolations have been used for the test functions and the trial functions, which is bound to make the tangent stiffness matrix strongly non-symmetric. 5.3 Interpolation option 3 Within this option r is not interpolated, but only updated as follows: r new (x) r old (x) + r(x), (66) where r old (x) is the last known value for r(x), not necessarily associated with an equilibrium state and r(x) following interpolation of the Newton-Raphson changes given in (54). This is a consistent choice which yields a tangent stiffness matrix. In this case the values of r at both integration and nodal points must be saved. The earlier results (56) (6) can still be used, but now noting that: r r g r(x g ). (67) 1
6 CONCLUDING REMARKS The fixed-pole approach, first introduced by Borri and Bottasso [3] has been analysed and discussed. The approach uses a new set of stress-couple resultants and specific angular momentum, defined with respect to the fixed-pole rather than the reference line at the cross-section, which in turn implies that r(x), the standard position vector, ceases to be the problem unknown. The presented modified fixed-pole approach makes it easy to combine the standard choice of the problem unknowns with the benefits of the original fixed-pole approach. The solution procedure for a spatial beam under static loading has been presented and the complexities of the modified approach discussed. In particular, the approach is based on a non-linear interpolation of the virtual displacements, which is dependent on the position vector field itself, which in turn opens a number of combinations in interpolating the position vector field and its iterative change. The solution procedure for a spatial beam under dynamic loading will be derived and the numerical examples shown in future works. ACKNOWLEDGEMENT Research resulting with this paper was made within the scientific project No 114-- 325: Improved accuracy in non-linear beam elements with finite 3D rotations financially supported by the Ministry of Science, Education and Sports of the Republic of Croatia. The second author additionally acknowledges the support of the Croatian Science Foundation which funded the bilateral project No 3.1/129: Preservation of mechanical constants in numerical time integration of non-linear beam equations of motion. REFERENCES [1] J.C. Simo. A finite strain beam formulation. The three-dimensional dynamic problem. Part I. Computer Methods in Applied Mechanics and Engineering, 49:55 7, 1985. [2] J.C. Simo and L. Vu-Quoc. On the dynamics in space of rods undergoing large motions A geometrically exact approach. Computer Methods in Applied Mechanics and Engineering, 66:125 161, 1986. [3] M. Borri and C. Bottasso. An intrinsic beam model based on a helicoidal approximation Part I: Formulation. International Journal for Numerical Methods in Engineering, 37(13):2267 2289, 1994. [4] C. Bottasso and M. Borri. Integrating finite rotations. Computer Methods in Applied Mechanics and Engineering, 164(3-4):37 331, 1998. 11