New Formula for Geometric Stiffness Matrix Calculation

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1 Journal of Applied Mathematics and Phsics, 016, 4, Published Online April 016 in SciRes. ew Formula for Geometric Stiffness Matri Calculation I. ěmec 1*, M. rcala, I. Ševčík, H. Štekbauer 1 1 Facult of Civil Engineering, Brno Universit of echnolog, Brno, Czech Republic FEM Consulting, S.R.O., Brno, Czech Republic Received 4 June 015; accepted 4 April 016; published 7 April 016 Copright 016 b authors and Scientific Research Publishing Inc. his work is licensed under the Creative Commons Attribution International License (CC BY). Abstract he standard formula for geometric stiffness matri calculation, which is convenient for most engineering applications, is seen to be unsatisfactor for large strains because of poor accurac, low convergence rate, and stabilit. For ver large compressions, the tangent stiffness in the direction of the compression can even become negative, which can be regarded as phsical nonsense. So in man cases rubber materials eposed to great compression cannot be analzed, or the analsis could lead to ver poor convergence. Problems with the standard geometric stiffness matri can even occur with a small strain in the case of plastic ielding, which eventuates even greater practical problems. he authors demonstrate that amore precisional approach would not lead to such strange and theoreticall unjustified results. An improved formula that would eliminate the disadvantages mentioned above and leads to higher convergence rate and more robust computations is suggested in this paper. he new formula can be derived from the principle of virtual work using a modified Green-Lagrange strain tensor, or from equilibrium conditions where in the choice of a specific strain measure is not needed for the geometric stiffness derivation (which can also be used for derivation of geometric stiffness of a rigid truss member). he new formula has been verified in practice with man calculations and implemented in the RFEM and SCIA Engineer programs. he advantages of the new formula in comparison with the standard formula are shown using several eamples. Kewords Geometric Stiffness, Stress Stiffness, Initial Stress Stiffness, angent Stiffness Matri, Finite Element Method, Principle of Virtual Work, Strain Measure * Corresponding author. How to cite this paper: ěmec, I., rcala, M., Ševčík, I. and Štekbauer, H. (016) ew Formula for Geometric Stiffness Matri Calculation. Journal of Applied Mathematics and Phsics, 4,

2 1. Introduction Stress stiffening is an important source of stiffness and must be taken into account when analzing structures. he standard formula for geometric stiffness matrices is introduced b a number of authors, such as Zienkiewicz, Bathe, Cook, Beltschko, Simo, Hughes, Bonet, de Souza eto and others [1]-[10]. he standard formula has been shown to be satisfactor in a large amount of cases, though certain difficulties such as low accurac, poor convergence rate and poor solution stabilit were discovered when solving problems that included the evaluation of etreme stress and strain states. Some authors, e.g. Cook [4], have suggested an improvement for bars and some authors dealt with nonlinear models describing large (finite) deformation (strain) behavior of materials and structures [11]-[1]. However, as far as the authors know, no general solution to the problem has been suggested for a D or 3D continuum. Upon this ascertainment, thoughts arose concerning the phsical essence of geometric (or stress) stiffness and the formula for evaluating geometric stiffness matrices. As a result, a new formula for geometric stiffness matri calculation is suggested. he presentation of this new formula, which should substantiall improve analsis of structures eposed to large strain, is the subject matter of this paper. In Section, the standard formula for geometric stiffness matrices is presented. Section 3 shows the phsical background of geometric stiffness based on equilibrium. In Section 4, the new, improved formula for geometric matrices is introduced. he advantages of the new formula, including a substantiall improved rate of convergence and stabilit, are demonstrated b eamples in Section5. Conclusions are presented in Section 6.. he Standard Formula for Stress-Stiffness Matrices Let us show the general calculation algorithm for the geometric stiffness matri (sometimes also called the stress stiffness matri or initial stress matri) of an element in an updated Lagrangian formulation. Let the following hold for each component u of displacement vector u : where u ia is the value of displacement Let us define matri as follows: i u n = u (1) i a ia a= 1 u i in node a and n is the number of element nodes. [ ] = I, I,, ni () 1 where I is the unit diagonal matri of the order 3 3, where 3 is the dimension of the problem. hen, the following relation can be written for the displacement vector: u= d (3) where d is the vector of deformation parameters of the element containing all the components u ia in such an arrangement that for each node a all components u i are listed. Let us define matri g a containing the first derivatives of base functions for node a with respect to spatial coordinates and matri G, which is formed b sub-matrices I a, a a= = a, g I I az, I g a G = = [ g1, g,, ga,, gn] (5) he operator denotes the tensor (Kronecker) matri product. Further, let us define matri Σ b multipling each component of the Cauch stress tensor b the unit diagonal matri: I I z I Σ = I = I z I (6) sm. zz I (4) 734

3 If state of the stress is not negligible, the potential energ of the internal forces should be completed b the following term: 1 u u ( ) d 1 d 1 = I = d Σ d = d K d (7) hen, the following formula for the geometric matri of the element can be written: d d = = K Σ G Σ G (8) Integration is carried out on the deformed bod (in the current configuration) and the derivatives are performed with respect to the spatial coordinates. he component of the matri K relating the element node a to the element node b can also be written simpl in matri notation: or in indicial notation: ( ) d K = I (9) ab a b a b K abij kl ijd i, j 1,, 3 = δ = k l Similar formulae also hold for a total Lagrangian formulation, but the second Piola-Kirchhoff stress tensor is then used instead of the Cauch stress, and integration is carried out on the undeformed bod 0 (in the original configuration) while the derivatives are performed with respect to the material coordinates. 3. he Source of Geometric Stiffness he Phsical Background Let us consider the truss member shown in Figure 1. ode is loaded b the force F parallel to the ais and sliding in the same direction. he equilibrium equation in the direction in node can be written as follows (10) R( ) = ( ) F = 0 (11) α =. ( ) where ( ) = cosα is the horizontal component of the internal force at node and cos l R is the residual or out-of-balance force. he horizontal stiffness K at node is defined simpl b the relation d d d d d d d K = = = = + = + = + d d d l d l l dl l d l dl l l l R l d d = + 1 cos sin K = α + α = M + K dl l l l dl l his formula is independent of an strain measure or pertinent constitutive relations. It can be seen that stiffness K consists of two parts. he first part, K M, represents the material stiffness and depends on the strain measure and constitutive relations. he second part, K, which does not depend on the material or the strain and stress measures chosen, but onl on the geometr and the normal force, represents so-called geometric stiffness. It can be seen that if the angle α is zero, no geometric stiffness will occur regardless of the normal force value. Let us show a derivation of a formula for geometric stiffness matri of a truss member (see Figure ) in a finite element formulation and let us start with a simple derivation based on equilibrium conditions. Let d be a vector of the nodal displacements of an element, and let r be a vector of residual forces; the stiffness matri of the element can then be defined as follows: r K = d (1) (13) 735

4 Figure 1. russ member in an arbitrar position in D. Figure. russ member: the ais is the ais of the rod in its original position. A geometric (stress) stiffness matri can be obtained b an equilibrium condition when onl the initial stress state and pertinent infinitesimal nodal displacement for each row of the matri is taken into account. Such a definition of a geometric stiffness matri is independent of the strain tensor chosen. o simplif the following derivations let s introduce both, the coordinates with the ais aligned with the ais of the rod and corresponding displacement vector u and let s restrict the deformation to the plane. Let the vector of the nodal displacements of the element be [ u, v, u, v ] d = (14) 1 1 where u and v are the displacement components in the direction of the and ais, respectivel. he well known material stiffness matri of the truss element in D is then defined b the following relation: EA K M = (15) l ote that the truss element has no lateral material stiffness. In general, arbitrar term of a stiffness matri K ij is defined as the derivative of an unbalanced force r i with respect to the deformation parameter d j as is defined b (13). Based on this definition, the geometric stiffness matri of the truss element subjected to tensile force can be easil derived. he moment equilibrium condition for the truss member in the configuration with the lateral displacement dv in node 1 is sufficient to obtain the transversal diagonal stiffness term K : he moment equilibrium condition can be written as follows: dv dl cos dα = 0 (16) 736

5 For the infinitesimal angle dα it can be assumed that cos dα = 1, and the following term for the stiffness term K can be derived: K d dv l = = (17) When introducing a displacement du in the direction of the ais of the member, the end forces are in equilibrium and no additional force and therefore no geometric stiffness will occur in this direction. From equilibrium equations and smmetr of the stiffness matri it is eas to determine the other coefficients of the geometric stiffness matri, particularl K 4, K 4 and K 44. he remaining coefficients of the matri are zeros. he geometric stiffness matri then has the following form: K = (18) l he same formula corresponds with Formula (1) and is presented also b Cook in [4], the same as man other authors. he geometric stiffness matri for a truss member can also be derived from the principle of virtual work, which will be described later. hen a strain measure and constitutive law must be introduced, which is not applicable for a rigid truss, where geometric stiffness also eists. he resulting tangent stiffness matri K is defined as the sum of the material and geometric stiffness matri: K = KM + K (19) When appling the general standard algorithm for geometric stiffness matrices to the truss element in question, we obtain: u u = = v d (0) [, ] = I I (1) 1 where I is the identit matri of order and the base functions where 1 1 =, = l l i are defined as follows: () u = d = Gd (3) G = = l (4) Substituting in the formulae the formula for the geometric stiffness matri reads: Σ = = (5) A = Al (6) K = d dl G G = G G = l l (7)

6 his geometric stiffness matri differs from that in Formula (18) and introduces also an aial stiffening. But no reason was found b the authors for concluding that normal force had led to a change in the aial stiffness of the element. So let us derive the geometric stiffness matri of a truss element in a more undisputable wa based on the principle of virtual work. With deformation restricted to the plane, the Green-Lagrange strain tensor is defined where For truss the principle of virtual work becomes u 1 u v ε = + + = e + η e u 1, u η v = = + (8) (9) Sδε d= W (30) et where S is the nd Piola-Kirchhoff stress in the aes at the following calculated time step t + t. Assuming t+ t t equalit S S = S + S = + S we obtain the incremental epression of the (30) Sδε d+ δηd= Wet δ ed (31) and the linearized equation of the principle of virtual work (virtual displacement) simplifies to: Assuming (5) we obtain e where Eeδed+ δη d= Wet δ ed (3) δ e EAeδ el+ δη l= Fδu δ el (33) δ u δu u δv v = and δη = + e u 1 = = l [ ] δe δ δ (34) d (35) = d [ ] d = d d (36) l l l δu u δv v δη = + = δ d G Gd = δ d d (37) l hen the equation of the principle of virtual work can be written as follows: where δd K d + δd K d = δd f δd f (38) M et int EA KM =, f int = (39) l

7 After transformation into global coordinate sstem K = d G Σ G = (40) l = R, where ( α) sin ( α) ( α) cos ( α) cos C S R = = sin S C (41) d = d, where R 0 = 0 R and after elimination of the vector of virtual displacements we get: (4) KM = KM (43) C CS C CS EA CS S CS S K M = (44) l C CS C CS CS S CS S K = K = (45) l [ ] f f (46) int = in = C S C S ( ) K + K d = f f (47) M et int he geometric stiffness matri (45) is the same as that obtained b use the standard Formula (7) and the first row of the matri does not correspond with Formula (1). Let us tr to derive the geometric stiffness matri of a truss element using a more accurate strain measure. he approimate nature of the linear relation between the deformation and displacement can be shown on a fibre of initial length ds. Without an loss of generalization, let us introduce a sstem of coordinates with the origin at the starting point of the fibre and with the ais oriented in the original direction of the fibre. Let us denote b ds the length of the fibre in the deformed bod (Figure 3). Let us denote b the vector of displacement of the starting point of the fibre. he end-point of the fibre will be displaced b vector u+ du. Using the formula for the bod-diagonal of a cuboid with dimensions ds + du, dv, dw, we can epress the new length of the fibre using the following relation: Introducing stretch λ = ds ds and considering we obtain the following relation for stretch of the fibre: ( ) ds = ds + du + dv + dw (48) u v w du = ds, dv = ds, δ w= ds (49) ds u v w u u v w λ = 1+ ε = = = ds (50) 739

8 Figure 3. Elongation of fibre ds. Let us consider the binomial theorem: 3 A A A 1+ A = for 8 16 and let us take into account onl the first two terms. hen we can write: and for ε u 1 u v w λ = u 1 u v w ε = λ 1 = A < 1 (51) If we want to be more accurate and take into account three terms of the binomial epansion, and if we neglect the third and higher powers of the derivatives of the displacement components, we get a more accurate epression for the stretch: (5) (53) and hence u 1 v w λ = (54) u 1 v w ε = + + For a 1D problem, therefore, this more accurate epression would be identical to the formula for from linear mechanics: Using the more accurate strain measure we obtain: (55) ε known u ε = (56) u 1 v ε = + = e + η (57) where e u =, 1 v η = 740

9 δ v v δη = = δ d GnewGnewd = δ d [ ] d = δ d d (58) l 0 l l G new instead of the standard G. l he linearized equation of the principle of virtual work (virtual displacement) modifies to: where is defined a new matri = [ ] δ EA et l δ d d + d d = l δ d f δ d (59) After transformation into global coordinate sstem and elimination of the vector of virtual displacements we get different geometric stiffness matri in the rotated and thus also in global coordinate sstem: K = new newd G Σ G = (60) l S SC S SC SC C SC C K = K = (61) l S SC S SC SC C SC C Resulting stiffness matri K derived from the principle of virtual work, using the more accurate strain measure, is the same as that derived from equilibrium conditions (18) and corresponds with Formula (1). It can be seen that the standard formula has produced a different geometric matri for the D truss element (7) than Formulae (18), (1) and (61) derived earlier and theoreticall unjustified geometric aial stiffness was also produced. his formula would lead to a poor convergence rate, inaccurac and even, in the case of etreme compression, to singularit. E.g. for = E, zero normal tangent stiffness would be obtained for the truss element, although there is no phsical reason for this. For < E the normal tangent stiffness would even be negative, which would be absurd. In the case of tension no stabilit problem would occur, but the low convergence problem is still present. E.g. when = E, the unbalanced nodal forces of 1/ of the load increment value would occur in the first iteration of the last increment. In the nd iteration it would be 1/4, and in the i-th iteration the unbalanced force of 1 i of the load increment value would still occur. hese problems are known, and therefore for the geometric stiffness of truss elements Formula (18) is widel used instead of Formula (7), which is derived from the general Formula (8) or (9). hen, in man computer programs different rates of convergence are obtained for a rod modeled b a truss element than in the case of a truss modeled b solid elements. o obtain the same geometric stiffness matri for the D truss element (18) as was derived above from the equilibrium, the influence of the member u must be omitted in the standard formula, i.e. the first row of the G matri must be filled in with zeros. 4. An Improved Formula for a Geometric Stiffness Matri Introducing a fibre of constant cross section area A in the direction of principal stress in a D or 3D continuum instead of a rod, and assuming onl nonzero strain in the direction of the fibre, and that all the other components of the strain tensor are zero, we can write a similar formula to (1): 741

10 d A d K = Acos Asin d α α l = + dl l = R. where represents the a principal stress. o evaluate the first part of the epression, a strain measure and pertinent constitutive relation must be chosen. his part represents material stiffness. he second part, which is the matter of our interest, represents geometric stiffness. he contributions of the two remaining principal stresses to the stiffness could be derived in a similar manner. Let us introduce the infinitesimal volume element of continuum d= ddd z, being the coordinate sstem in the principal stress aes given b the relation It was earlier shown that for a rod (see formula (1)) the first derivative of a displacement component with respect to the same direction does not generate geometric stiffness. For the D or 3D continuum a similar formula to (60) can be derived in a similar wa as in the case of a rod. ew measure of deformation ε defined in the principal aes can be, similarl as in the case of rod, divided into the linear and nonlinear parts as follows: where (6) ε = e + η (63) ê being the infinitesimal strain and e = R er (64) ˆ 1 uk uk ηij = i j ( 1 δij ) he linearized equation of the principle of virtual work (virtual displacement) for D or 3D continuum is similar t t t to (3) S + S = S + S = + S and reads: ( η) S = C : ε = C : e + C : e S : δε d= W ( S) δ ( e η ) + : + d= W S : δη 0 et δe : C : ed + : δηd = W : δed et et ielding its following form in terms of finite element matrices: where M et int (65) (66) δd K d + δd K d = δd f δd f (67) ( ) K + K d = f f (68) M et in = d K G Σ G (69) u he difference from the standard formula (8) lies in the fact that in the Gd epression the members, v w and are omitted. z A particular case where the standard formula was applied to a D truss element producing an unintentional change in the aial stiffness was presented earlier. his phenomenon can also be generall observed when the 74

11 standard formula is used. It is clear that the uniaial stress state will provide the same result regardless of the wa it is modeled, i.e. a truss member modeled as a 3D solid should provide the same result as one modeled b a truss member or b shell elements. o guarantee this and to improve the influence of the stress state on stiffness, u the members, v w and must be omitted in the standard formula. here is no reason wh a normal z stress component should influence the stiffness in the same direction. o ensure objectivit (independence from an arbitrar coordinate sstem) of the geometric stiffness matri, the omission of the above mentioned terms must be evaluated in the principal stress aes. hen, for the updated Lagrangian formulation, the following formulae for 3D solid elements hold: where g a 0 a, a, a, 0 0 = 0 0 a, az, az, 0 [,,,,, ] 1 a n (70) G = g g g g (71) Σ = I = z z I being the diagonal unit matri of the order ; is the stress tensor in the principal stress aes with,, z being the principal stresses. hen, the geometric stiffness matri in the aes of principal stresses is defined b the following formula: he component of the matri indicial notation: = (7) K G Σ Gd (73) K relating the element node a to the element node b can also be written in K = δ δ δ = abij a b kl k l ij ik jl ( 1 )( 1 ) d ( i, jkl,, 1,, 3) o obtain the geometric stiffness matri in the global coordinate sstem the following transformation must be performed: he transformation from the global coordinate sstem to the principal stress coordinate sstem can be defined as follows: (74) = R (75) hen, the relations between the first derivatives of the base functions and stresses are the following: z a, a, a, z a, = a, a, = R (76) a,z a,z a,z z z z z 743

12 0 0 τ τ z 0 0 = R τ τ z R (77) 0 0 z τz τ z z Illustration on a Quadrilateral Plane Stress Element For a quadrilateral plane stress element the following can be obtained: C S R = S C (78) = R (79) 0 1, 0, 0 3, 0 4, G = 1, 0, 0 3, 0 4, 0 (80) 0 τ C S τ C S Σ = 0 = R τ R = S C τ S C (81) K G Σ Gt da (8) = A where C = cos(α); S = sin(α); α is the angle between principal and global directions; t is the element thickness and A is the area of the element. A similar formula also holds for the total Lagrangian formulation for such an element, but the second Piola- Kirchhoff stress tensor is then used instead of the Cauch stress, integration is carried out on the undeformed bod 0 (in the original configuration), and the derivatives are performed b the material coordinates. 5. Eamples An application of the new formula for the geometric stiffness matri for large strain was demonstrated on the eample of a unit cube represented b different computational models (rod, shell, solid elements) with different orientations in space (see Figure 4) assuming isotropic hperelastic material with linear relation between the Figure 4. Different computational models (rod, shell, solid elements) with different orientations in space. 744

13 Figure 5. Magnitude of displacement due touniform normal load of the value E acting on two opposite sides. Figure 6. Magnitude of displacement due touniform normal load of the value-e acting on two opposite sides. 745

14 logarithmic strain and Cauch stress tensors. Let E be the Young modulus and for simplicit let us assume zero Poisson ratio. he cube was eposed to uniform stress of the magnitude E or E normal on two opposite sides. A logarithmic strain value of 1 or 1 and the prolongation or shortening of the value of e 1 or 1 1 e (e being the base of the natural logarithm) should be obtained. Different computational models of the cube were tested, utilizing rod, shell and solid elements. Calculations were performed for three orientations in space (the basic configuration, a rotation of 30 degrees and a rotation of 45 degrees). Several orientations in space were also tested. In Figure 5 and Figure 6, which are graphical outputs from the RFEM program, it is shown that practicall eact results were obtained for all computational models and orientations Convergence of the Standard and ew Approach A Comparison Let the sequence { u n }. converge to u c. If such a positive number p 1 and constant λ > 0 λ < eists that: u u lim b lim n c n = = λ n n p u u n 1 then p is called the order of convergence of the sequence. he constant λ is called the asmptotic error. If p is large, the sequence { u n } converges rapidl to u c. If p = 1, the convergence is said to be linear. If p =, then convergence is quadratic, and if p = 3, it is cubic, etc. Most sequences converge linearl or quadraticall. Quadratic convergence is sufficient for computationall efficient numerical methods. 5.. he Standard Approach Figure 7 shows that the standard approach provides onl linear convergence for large strain. c (83) Figure 7. Investigation of linear ( p = 1, upper panel) and quadratic ( p =, lower panel) convergence, where un uc bn =. p u u n 1 c 5.3. he ew Approach Figure 8 shows that the new approach ields the quadratic convergence even for ver large strain. Figure 8. Investigation of linear ( p = 1, upper panel), quadratic ( p =, middle panel) and cubic ( p = 3 un uc convergence, where bn =. p u u n 1 c, lower panel) 746

15 he numerical solution of the presented eample has shown that to reach a sufficientl good result using the standard formula (ASYS etc.) 15 iterations were needed whereas using the improved approach presented in this paper (RFEM) onl 5 iterations were needed to obtain the same precision. 6. Conclusions he present formula for a geometric stiffness matri, which has been published in man books, is widel utilized, objective and simpl defined. However, stabilit and convergence problems occur when analzing large strains, or, what is more important in practice, in a case of ielding. If the ield criterion is satisfied, then the material stiffness decreases substantiall. he stress state remains high and in case of compression the tangent stiffness in the direction of the compression can become negative even with a small strain. his is caused b a theoreticall unsupported change in pressure stiffness in the direction of compression produced b the standard formula. his results in a correspondingl high nodal force unbalance, poor convergence and possibl also instabilit. he origin of the problem arises from the approimation of strain, in which onl the first two terms of the binomial series are applied. he presented algorithm is slightl more complicated, but remains objective and provides a solution with increased stabilit, a higher rate of convergence in the case of a large strain, or plastic ielding, and improved accurac over the present formula. In case of ver large strain, the number of iterations needed could be several times less using the new formula comparing to the standard formula. In man cases the new formula can even provide solutions in cases where the standard formula has failed. his new formula for a geometric matri has been implemented in the RFEM program and has been demonstrated to be much more stable and faster than the standard formula. Acknowledgements his outcome has been achieved with the financial support of the Czech Science Foundation (GACR) project S Aspects of the use of comple non linear material models. References [1] Bathe, K.-J. (1996) Finite Element Procedures. Prentice Hall, Upper Saddle River. [] Beltschko,., Liu, W.K., Moran, B. and Elkhodar, K. (000) onlinear Finite Elements for Continua and Structures. John Wile & Sons, Hoboken. [3] Bonet, J. and Wood, R.D. (008) onlinear Continuum Mechanics for Finite Element Analsis. Cambridge Universit Press, Cambridge. [4] Cook, R.D., Malkus, D.S., Plesha, M.E. and Witt, R.J. (00) Concepts and Applications of Finite Element Analsis. John Wile & Sons, Hoboken. [5] ěmec I., et al. (010) Finite Element Analsis of Structures: Principles and Prais. Shaker-Verlag, Aachen. [6] Redd, J.. (004) onlinear Finite Element Analsis. Oford Universit Press, Oford. [7] Zienkiewicz, O.C. and alor, R.L. (000) he Finite Element Method. Butterworth Heinemann, Oford. [8] de Souza eto, E.A., Periæ, D. and Owen, D.R.J. (008) Computational Methods for Plasticit: heor and Applications. John Wile & Sons, Hoboken. [9] Simo, J.C. & Hughes,.J.R. (008) ComputationalInelasticit. Springer, ew York, 39 p. [10] Wriggers, P. (008) onlinear Finite Element Methods. Springer-Verlag, ew York. [11] Lacarbonara, W. and Pacitti, A. (008) onlinear Modeling of Cables with Fleural Stiffness. Mathematical Problems in Engineering, 008, Article ID: [1] Xiao, H. and Chen, L.S. (00) Henck s Elasticit Model and Linear Stress-Strain Relations in Isotropic Finite Hperelasticit. Acta Mechanica, 157, [13] Xiao, H., Bruhns, O.. and Meers, A. (1998) Objective Corotational Rates and Unified Work-Conjugac Relation between Eulerian and Lagrangean Strain and Stress Measures. Archives of Mechanics, 50, [14] Meers, A. (1999) On the Consistenc of Some Eulerian Strain Rates. Zeitschrift fur Angewandte Mathematik und Mechanik, 79, [15] Simo, J.C. and Pister, K.S. (1984) Remarks on Rate Constitutive Equations for Finite Deformation Problem: Computational Implications. Computer Methods in Applied Mechanics and Engineering, 46,

16 [16] Curnier, A. and Rakotomanana, L. (1991) Generalized Strain and Stress Measures: Critical Surve and ew Results. Engineering ransactions, 39, [17] Chiskis, A. and Parnes, R. (000) Linear Stress-Strain Relations in onlinear Elasticit. Acta Mechanica, 146, [18] Farahani, K. and aghdabadi, R. (000) Conjugate Stresses of the Seth-Hill Strain ensors. International Journal of Solids and Structures, 37, [19] Darijani, H. and aghdabadi, R. (010) Constitutive Modeling of Solids at Finitede Formation Using a Second-Order Stress-Strain Relation. International Journal of Engineering Science, 48, [0] Hill, R. (1978) Aspects of Invariance in Solid Mechanics. Advances in Applied Mechanics, 18, [1] Farahani, K. and Bahai, H. (004) Hper-Elastic Constitutive Equations of Conjugate Stresses and Strain ensors for the Seth-Hill Strain Measures. International Journal of Engineering Science, 4, List of Variables A Cross section area of a beam C Material tangent moduli E Young modulus KM, K Material and geometric tangent stiffness matri, respectivel KM, K Material and geometric tangent stiffness matri in principal aes i Shape functions Matri of shape functions R Rotation tensor S Second Piola-Kirchhoff stress Wint, Wet Internal and eternal virtual work d Vector of nodal displacements ê Infinitesimal strain e Infinitesimal strain in principalaes f int Internal nodal forces f et Eternal nodal forces I Unit diagonal matri l Member length t ime u Displacement field uvw,, Displacements in the, and z directions respectivel z,, Spatial (Eulerian) coordinates,, z Coordinates in principal aces ε Modified strain tensor in principalaes η, ηij Quadratic terms of the modified strain tensor in principalaes Potential energ of geometrical stiffness Σ Σ = I Σ Σ = I, ij Cauch stress tensor Cauch stress tensor in principal aes, Domain of current (deformed), initial (undeformed) 0 748