Ivo Senjanović Nikola Vladimir Dae Seung Cho ISSN 333-4 eissn 849-39 A SHEAR LOCKING-FREE BEAM FINITE ELEMENT BASED ON THE MODIFIED TIMOSHENKO BEAM THEORY Summary UDC 534-6 An outline of the Timoshenko eam theory is presented. Two differential equations of motion in terms of deflection and cross-section rotation are comprised in one equation and analytical expressions for displacements and sectional forces are given. Two different displacement fields are recognized, i.e. flexural and axial shear, and a modified eam theory with extension is worked out. Flexural and axial shear locking-free eam finite elements are developed. Reliaility of the finite elements is demonstrated with numerical examples for a simply supported, clamped and free eam y comparing the otained results with analytical solutions. Key words: Timoshenko eam theory, modified eam theory, flexural virations, axial shear virations, eam finite element, shear locking. Introduction Beam is used as a structural element in many engineering structures like frames and grillages. Also, the whole structure can e modelled as a eam to some extent, e.g. ship hulls, floating airports, etc. The Euler-Bernoulli theory is widely used for the simulation of slender eam ehaviour. The theory for thick eam was extended y Timoshenko [] in order to take the effect of shear into account. The shear effect is extremely strong in higher viration modes due to the reduced mode half wave length. The Timoshenko eam theory deals with two differential equations of motion in terms of deflection and cross-section rotation. Most papers use this theory, while a possiility to use only one equation in terms of deflection has een recognized recently [,3]. The Timoshenko eam theory has come into focus with the development of the finite element method and its application in practice. A large numer of finite elements have een worked out in the last decades [4-3]. They differ in the choice of interpolation functions for a mathematical description of deflection and rotation. The application of equal order polynomials leads to the so-called shear locking since the ending strain energy for a thin eam vanishes efore the shear strain energy [6,]. Various approaches have een developed in order to overcome this prolem, ut a unique solution has not een found yet []. The prolem is partially solved y some approaches such as the Reduced Integration Element TRANSACTIONS OF FAMENA XXXVII-4 (03)
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element (RIE), the Consistent Interpolation Element (CIE), and the Interdependent Interpolation Element (IIE), which are descried in [,3,4]. The aove short description of the state of the art has motivated further investigation into this challenging prolem. First, a physical aspect of the Timoshenko eam theory is analysed. It is found that actually two different displacement fields are hidden in the eam deflection and rotation, i.e. pure ending with transverse shear on one hand, and axial shear on the other. The latter is analogous to the ar stretching on an axial elastic support. The cross-section rotation and the axial shear slope are treated in the Timoshenko eam theory as one variale since they have the same stiffness in the analytical formulation. Understanding of the eam dynamic ehaviour makes the development of modified eam theory possile, which would e as exact as the Timoshenko eam theory. Based on the modified Timoshenko theory, a two-node eam finite element is developed y taking a static solution for interpolation functions. Also, a eam element is derived for axial shear virations. Both elements are shear locking-free. Illustrative examples are given and the otained results are compared with those otained in an analytical way.. Timoshenko eam theory. Basic equations The Timoshenko eam theory deals with the eam deflection and angle of rotation of cross-section, w and, respectively []. The sectional forces, i.e. ending moment and shear force, read w M D, Q S x x, () where D=EI is the flexural rigidity and S=kGA is the shear rigidity, A is the cross-section area and I is its moment of inertia, k is the shear coefficient, and E is Young's modulus and G E / is the shear modulus. The value of shear coefficient depends on the eam cross-section profile, [5] and [6]. Stiffness properties for a complex thin-walled girder are determined y the strip element method [7]. The eam is loaded with transverse inertia load per unit length, and the distriuted ending moment is expressed as w, x qx m m J t t, () where m A is the specific mass per unit length and J I is its moment of inertia. The equilirium of moments and forces M Q Q mx, qx x x leads to two differential equations w D S J 0 x x t (4) w w S m 0. x x t (5) (3) TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho From equation (5) one otains w m w x x S t and y sustituting (6) into (4) differentiated y x, one otains the eam differential equation of motion w J m w m J w x D S x t D t S t 4 4 w 0 4 Once (7) is solved, the angle of rotation is otained from (6) as where (6). (7) w m w d x f t x S, (8) t f t is the rigid ody motion. If w is extracted from (4) and sustituted into (5), the same type of differential equation as (7) is otained for, and as (8) for w.. General solution to natural virations In natural virations w Wsint and Ψ sint, and Eqs. (7) and (8) are reduced to the viration amplitudes 4 d W J md W m J W 0 4 dx D S dx D S (9) dw m Ψ Wx d C dx S. (0) x A solution to (9) can e assumed in the form W Ae that leads to a iquadratic equation where 4 a 0, () a Roots of () read where i and J m, m J D S D S. (),, i, i, (3) m J 4m m J S D D S D (4) m J 4m m J S D D S D. (5) TRANSACTIONS OF FAMENA XXXVII-4 (03) 3
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element Deflection function with its derivatives and the first integral can e presented in the matrix form W shx chx sin x cos x ch x sh x cos x sin x A W sin cos A W shx chx x x 3 3 3 3 ch x sh x cos x sin x A W 3 A 4 Wx d chx shx cos x sin x. (6) According to the solution to equations (9), (0) and (), eam displacements and forces read W Ash x Achx A sin x A cos x (7) 3 4 m m Ψ Achx Ashx S S m m A 3cosx A 4sinx S S m m M D Ash xach x A3sin xa4cos x S S m Q Ach xash xa3cos x A4sin x. (0) Relative values of constants A i, i,,3,4, are determined y satisfying four oundary conditions. Since there is no additional condition constant, C in (0) is ignored. Coefficient, Eq. (4), can e zero, in which case 0 S / J and S D m J 0 / /. Deflection function according to (7) takes the form W Ax A A sin x A cos x, () 3 0 4 0 where the first two terms descrie the rigid ody motion. If 0, then i, where (8) (9) m J m J 4m S D S D D () and the deflection function reads W A sin x A cos x A sin x A cos x. (3) 3 4 Expressions for displacements and forces, Eqs (7-0), have to e transformed accordingly. Hence, ch x cos x, sh x i sin x (an imaginary unit is included in constant A ),, instead of a single factor it is necessary to write, and finally all functions associated with A and A must have the same sign as those associated with A 3 and A 4. 4 TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho The aove analysis shows that the eam has a lower and higher frequency spectral response, and a transition one. Frequency spectra are shifted for the threshold frequency 0. This prolem is also investigated in [,8]. The asic differential equations (4) and (5) are x solved in [9] y assuming a solution in the form w Ae x and Be, and the same expressions for displacements (7) and (8) are otained. 3. Modified eam theory 3. Differential equations of motion Beam deflection w and the angle of rotation are split into their constitutive parts, Fig., w w w ws,,, (4) x Fig. Thick eam displacements: a total deflection and rotation w, ψ, pure ending deflection and rotation w, φ, c transverse shear deflection w s, d axial shear angle ϑ where w and w s are the eam deflections due to pure ending and transverse shear, respectively, and is the angle of cross-section rotation due to ending, while is the crosssection slope due to axial shear. Equilirium equations (4) and (5) can e presented in the form with separated variales w and w, and s 3 w w ws D J S D S J 3 x t x x x t (5) ws S m w wss x t x (6) Since only two equations are availale for three variales, one can assume that flexural and axial shear displacement fields are not coupled. In that case, y setting oth the left and the right hand side of (5) to zero, it follows that TRANSACTIONS OF FAMENA XXXVII-4 (03) 5
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element w s D w J w S x S t. (7) By sustituting (7) into (6) differential equation for flexural virations is otained, which is expressed y a pure ending deflection 4 4 w J m w m J w S w 4. (8) x D S x t D t S t D x Disturing function on the right hand side in (8) can e ignored due to the assumed uncoupling. Once w is determined, the total eam deflection, according to (4), reads D w J w w w S x S t. (9) The right hand side of (5) represents a differential equation of axial shear virations S J x D D t 0. (30) 3. General solution to flexural natural virations Natural virations are harmonic, i.e. w Wsint and Θ sint, so that equations of motion (8) and (30) are related to the viration amplitudes 4 d W J md W m J W 0 4 dx D S dx D S (3) d Θ S J Θ 0. dx D S (3) The amplitude of total deflection, according to (9), reads W J d W D W S S dx. (33) Eq. (3) is known in literature as an approximate alternative of Timoshenko s differential equations, [3,0]. By comparing (3) with (9), it is ovious that the differential equation of flexural virations of the modified eam theory is of the same structure as that of the Timoshenko eam theory, ut they are related to different variales, i.e. to W and W deflection, respectively. Therefore, the general solution for W presented in Section. is valid for with all derivatives. In that case, flexural displacements and sectional forces read J D J D W B shxb chx S S S S J D J D B3 sinxb4 cosx S S S S (34) dw Φ Bchx BshxB3cos x B4sin x dx (35) W 6 TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho d W M D D B shxb chxb3 sin x B4 cos x (36) dx 3 dw dw J J Q D J D B 3 ch x B sh x dx dx D D J J B3 cos xb4 sin x. D D (37) Parameters and are specified in Section., Eqs (4) and (5), respectively. In this case, parameter can also e zero, which gives 0 S / J and S D m J 0 / /. By taking this fact into account, the ending deflection W is of the form (), while the total deflection according to (43), reads D W BxB 0 B3sin 0x B4cos 0x S, (38) where B and B are the new integration constants instead of B and B, which are infinite due to zero coefficients. Concerning the higher order frequency spectrum, the governing expressions for displacements and forces, Eqs. (34-37), have to e transformed in the same manner as explained in Section.. 3.3 General solution to axial shear natural virations Differential equation (3) for natural axial shear virations of eam reads Θ J S Θ d 0. (39) dx D D It is similar to the equation for rod stretching virations d u m 0 R u. (40) dx EA The difference is in the additional moment SΘ which is associated to the inertia moment JΘ and represents the reaction of an imaginary rotational elastic foundation with stiffness equal to the shear stiffness S, as shown in Fig.. Fig. Analogy etween an axial shear model and a stretching model TRANSACTIONS OF FAMENA XXXVII-4 (03) 7
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element Solution to (40) and a corresponding axial force du N EA read d x u C sin x C cos x (4) N EA C cos x C sin x, (4) where R m/ EA angle and moment one can write where. Based on the analogy etween (39) and (40), for the shear slope Θ C sinx C cosx (43) M D C cosx Csinx, (44) J S. (45) D D Between the natural frequencies of axial shear eam virations and stretching virations there is a relation, where 0 S / J elongs to the axial shear mode otained 0 R from (39), Θ A0 Ax, (which reminds us of a sheared set of playing cards). It is interesting that 0 is at the same time the threshold frequency of flexural virations, as explained in Section.. 4. Comparison etween the Timoshenko eam theory and the new theory 4. Dynamic response As elaorated in Section., the Timoshenko eam theory deals with two differential equations of motion with two asic variales, i.e. deflection and the angle of rotation. That system is reduced to a single equation in terms of deflection and all physical quantities depend on its solution. On the other hand, in the modified eam theory, Section 3., the total deflection is divided into the pure ending deflection and the shear deflection, while the total angle of rotation consists of pure ending rotation and axial shear angle. The governing equations are condensed into a single one for flexural virations with the ending deflection as the main variale and another variale for axial shear virations. Differential equations for flexural virations in oth theories are of the same structure, Eqs. (9) and (3), resulting in the same hyperolic and trigonometric functions in the solution for W and W. However, expressions for displacements and forces are different due to different coefficients associated to the integration constants. In order to otain the same expressions for displacements and forces in oth theories, the following relations etween the constants, ased on identical deflections, Eqs. (7) and (34), should exist J D Ai Bi, i, (46) S S J D Ai Bi, i 3,4. (47) S S Indeed, if relations (46) and (47) are sustituted into Eqs. (8), (9) and (0), and if expressions (4) and (5) for and are taken into account, expressions (8), (9), and 8 TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho (0) ecome identical to (35), (36), and (37). For illustration, let us check the identity of the first terms in expressions for shear forces, Eqs. (0) and (37) m J A D B. (48) D By taking (46) into account, (48) can e presented in the form J D J m S S D. (49) If (4) is sustituted for, relation (49) is satisfied. Based on the aove fact, flexural virations determined y the Timoshenko eam theory and its modification are identical. Therefore, axial shear virations, extracted from the Timoshenko eam theory, are not coupled with flexural virations, as assumed in Section 3.. 4. Static response One expects that expressions for static displacements can e otained straightforwardly y deducting dynamic expressions. In the case of the Timoshenko eam theory, the static term 3 3 of Eq. (9) leads to W A0 Ax Ax A3x, and Eq. (0) gives Ψ AAx 3Ax 3. That results in the zero shear force Q, Eq. (), which is also ovious from (0) if 0 is taken into account. Therefore, in order to overcome this prolem, it is necessary to return 3 3 ack to Eqs (4) and (5) with static terms. By sustituting (5) into (4), Dd Ψ /dx 0, i.e. 3 Ψ A A x3ax is otained. Based on the known Ψ, one otains from (4) 3 D dψ D W Ψ x A A Ax A x Ax A Ax S dx S 3 d 0 0 3 3 3. (50) On the other hand, the static part of Eq. (3) of the modified eam theory gives 3 W B B xb x B x, and from (33) it follows that 0 3 D d W 3 D W W B0 BxBx B3x B 3B3x, (5) S dx S which is an expression identical to (50). The angle of rotation is Φ = d W /dx B Bx 3Bx, which is the same as the aove Ψ in the Timoshenko eam theory. 3 5. Beam finite element ased on the modified eam theory The Timoshenko eam theory deals with two variales of flexural virations, W and Ψ, which are of the same importance. Therefore, in the development of the eam finite element, one takes into account the independent shape functions for W and Ψ of the same order. That leads to the shear locking prolem, which is remedied y an additional degree of freedom. That prolem can e avoided if the static solution for W and Ψ from Section 4., which includes relation (0), is used for the shape functions [3]. In that case, one otains the same eam finite element properties as for the modified eam theory. Derivation of the finite element y the latter theory is simpler and more transparent and that is why it is used here. TRANSACTIONS OF FAMENA XXXVII-4 (03) 9
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element A relatively simple two-node eam finite element can e derived in an ordinary way if the static solution is used for deflection interpolation functions, Section 4. where D / Sl 3 x x x W a0 a a a3 l l l x Ws a 3a 3 l 3 x x x x W a0 a a a3 a 3a3 l l l l dw x x Φ aa 3a3, (55) dx l l l and l is the element length. By satisfying alternatively the unit value for one of the nodal displacements and the zero value for the remaining displacements, one can write where CA (5) (53) (54), (56) C Inversion of (56) is W a0 Φ a, A, (57) W a Φ a 3 0 0 0 0 0 l. 6 3 0 l l l A C (58), (59) where C l l l l 6 4 3 6 6 l 0 0 0. (60) 3 3 3 6 l l Bending deflection (5), y employing (59), yields W P A f, (6) 0 TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho 3 where P x/ l x/ l x/ l and f P C (6) is the vector of ending shape functions. Referring to the finite element method [,], the ending stiffness matrix is defined as 0 0 f f D x dx dx l l x 3 l l l d d 4 T K D d x 4 C 0 03 dxc, (63) T where symolically C C K 0 0 D l T. After integration and multiplication one otains 6 3l 6 3l 66l 3l 6l 6 3l. (64) Sym. 66 l In a similar way, shear deflection (53) can e presented in the form s s s W P A f, (65) where P 0 0 3 x/ l and s s s f P C (66) is the vector of shear shape functions. The shear stiffness matrix reads [] l df df 6 0 0 dx dx l 0 l s s T K S dx S C 000dxC s. (67) 0 0 That leads to K s 36 S 4 l 4 l l l l. (68) l 4 l Sym. l Bending and shear stiffness matrices can e summed and the total stiffness matrix reads TRANSACTIONS OF FAMENA XXXVII-4 (03)
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element K D Mass matrix, according to definition [0] is where l 0 6 3l 6 3l 3l 3l 6l. (69) l 6 3l Sym. 3 l M m f f dx, (70) w m w w w f P C, (7) is the shape function of total deflection, and P x/ l x/ l x/ l 6 x/ l w integration, one otains M m ml 40. By sustituting (7) into (70) and after l 4 84 504 l 3378 50 l 56 358 060 384 504 l 46 50 l 56 358 060 46 50 54 5 0080 3378 50 l. (7) l Sym. 4 84 504 In a similar way, according to definition [0], one finds the mass moment of inertia matrix l df df M J dx J dx dx 0 l 36 3 80 36 3 80 l. (73) J 4 60 440 l 3 80l 60 70 l 30 l 36 3 80 l Sym. 4 60 440 l Beam axial shear virations are analogous to stretching virations, Section 3.3, and the vector of shape functions is f x/ l x/ l. The following stiffness and mass matrices are otained K M a a a D Sl l 6, (74) l J 6. (75) TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho 6. Numerical examples In order to evaluate the developed finite element, flexural virations of a simply supported, clamped and free eam are analysed and compared with analytical solutions [3] and D FEM results otained y NASTRAN [4]. The eam length is L 0 m and height H m. The D FEM model includes 50 eam elements and the D model 50x6=300 memrane elements. Fig. 3 The first four natural modes of a simply supported eam Tale presents the otained values of the frequency parameter / 0 for the simply supported eam, where 0 S / J is the threshold frequency otained from the last term in differential equation (3). It is well known that the simply supported eam exhiits a doule frequency spectrum for the same mode shapes shown in Fig. 3 [,8]. D FEM results follow very well the analytical solutions for the first spectrum up to the threshold frequency. D FEM results agree very well with the first spectrum of analytical solution. However, D and D FEM analyses cannot capture the second frequency spectrum. Tale Frequency parameter / 0 for a simply supported eam, h/ l 0., k 5/6 n Analytical FEM st spectrum, nd spectrum, D, D, f n 0.000* 0.055.064 0.05 0.055 0.89.7 0.83 0.9 3 0.36.445 0.35 0.366 4 0.549.693 0.535 0.558 5 0.74.959 0.74 0.755 6 0.935.37 0.96 0.953 6.335*.000* 7.8.54.064.5 8.3.86.08.346 9.5 3.3.3.538 0.70 3.44.30.77 *Threshold f n n n TRANSACTIONS OF FAMENA XXXVII-4 (03) 3
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element Values of frequency parameter for the clamped and the free eam are compared in Tale and 3, respectively. Very good agreement etween the D and D FEM results on the one hand and the analytical solutions on the other are shown. Tale Frequency parameter / 0 for a clamped eam, h/ l 0., k 5/6 Mode no. j Analytical, j FEM D, D, 0.06 0.0 0.07 0.4 0.36 0.47 3 0.404 0.394 0.4 4 0.577 0.564 0.590 5 0.758 0.74 0.775 6 0.94 0.93 0.960 *.000* 7.066.065.047 8.3.05.39 9.35.9. 0.34.95.33 *Threshold Tale 3 Frequency parameter / 0 for a free eam, h/ l 0., k 5/6 Mode no. j Analytical, j j FEM D, D, 0.7 0. 0.6 0.7 0.64 0.73 3 0.453 0.44 0.455 4 0.638 0.63 0.64 5 0.89 0.803 0.85 6 0.967 0.956 0.967 *.000* 7.070.075.07 8.097.087.094 9.7.60.63 0.79.65.69 *Threshold j Tale 4 shows the frequency parameter of axial shear virations. The finite element developed in Section 5, Eqs. (74) and (75), gives very reliale results. j j 4 TRANSACTIONS OF FAMENA XXXVII-4 (03)
A Shear Locking-Free Beam Finite Element I. Senjanović, N. Vladimir, D.S. Cho Tale 4 Frequency parameter / 0 for axial shear virations, h/ l 0., k 5/6 n Analytical, S n D FEM, 0.000*.050.050.88.88 3.387.388 4.65.68 5.888.894 6.67.77 7.455.47 8.75.776 9 3.05 3.088 0 3.356 3.407 *Threshold 7. Conclusion S n The Timoshenko eam theory deals with the total deflection and the cross-section rotation as two asic variales. The modified eam theory is an extension of the former from flexural to axial shear virations. The main variales are the pure ending deflection and the axial shear slope angle. The modified flexural eam theory is known in literature as an approximate variant of the Timoshenko theory. By a linear transformation of expressions for displacements and sectional forces it is shown that the modified theory is exact as the Timoshenko theory. The developed sophisticated eam finite element ased on the modified and extended Timoshenko eam theory gives very good results, the same as the D FEM analysis, when compared to the analytical solutions. ACKNOWLEDGMENT This study was supported y the National Research Foundation of Korea (NRF) grant funded y the Korean Government (MEST) through GCRC-SOP (Grant No. 0-0030669). REFERENCES [] S.P. Timoshenko, On the transverse virations of ars of uniform cross section, Philosophical Magazine 43 (9) 5-3. [] X.F. Li, A unified approach for analysing static and dynamic ehaviours of functionally graded Timoshenko and Euler-Bernoulli eams, Journal of Sound and Viration 38 (008) 0-9. [3] L. Majkut, Free and forced virations of Timoshenko eams descried y single differential equation, Journal of Theoretical and Applied Mechanics, Warsaw, 47 () (009) 93-0. [4] R.E. Nickell, G.A. Secor, Convergence of consistently derived Timoshenko eam finite element, International Journal for Numerical Methods in Engineering 5 (97) 43-53. [5] A. Tessler, S.B. Dong, On a hierarchy of conforming Timoshenko eam elements, Computers and Structures 4 (98) 335-344. [6] G. Prathap, G.R. Bhashyam, Reduced integration and the shear flexile eam element, International Journal for Numerical Methods in Engineering 8 (98) 95-0. TRANSACTIONS OF FAMENA XXXVII-4 (03) 5
I. Senjanović, N. Vladimir, D.S. Cho A Shear Locking-Free Beam Finite Element [7] G. Prathap, C.R. Bau, Field-consistent strain interpolation for the quadratic shear flexile eam element, International Journal for Numerical Methods in Engineering 3 (986) 973-984. [8] P.R. Heyliger, J.N. Reddy, A higher-order eam finite element for ending and viration prolems, Journal of Sound and Viration 6 (988) 309-36. [9] J.J. Rakowski, A critical analysis of quadratic eam finite elements, International Journal for Numerical Methods in Engineering 3 (99) 949-966. [0] Z. Friedman, J.B. Kosmatka, An improved two-node Timoshenko eam finite element, Computers and Structures 47 (993) 473-48. [] J.N. Reddy, An Introduction to the Finite Element Method, nd edition, McGraw-Hill, New York, 993. [] J.N. Reddy, On locking free shear deformale eam elements, Computer Methods in Applied Mechanics and Engineering 49 (997) 3-3. [3] J.N. Reddy, On the dynamic ehaviour of the Timoshenko eam finite elements, Sãdhanã 4 (3) (999) 75-98. [4] G. Falsone, D. Settineri, An Euler-Bernoulli-like finite element method for Timoshenko eams, Mechanics Research Communications 38 (0) -6. [5] G.R. Cowper, The shear coefficient in Timoshenko's eam theory, Journal of Applied Mechanics 33 (966) 335-340. [6] I. Senjanović, Y. Fan, The ending and shear coefficients ot thin-walled girders, Thin-Walled Structures 0 (990) 3-57. [7] I. Senjanović, Y. Fan, A finite element formulation of ship cross-section stiffness parameters, Brodogradnja 4 (993), 7-36. [8] B. Geist, J.R. McLaughlin, Doule eigenvalues for the uniform Timoshenko eam, Applied Math. Letters 0 (3) (997) 9-34. [9] N.F.J. van Rensurg, A.J. van der Merve, Natural frequencies and modes of a Timoshenko eam, Wave motion 44 (006) 58-69. [0] I. Senjanović, S. Tomašević, N. Vladimir, An advanced theory of thin-walled girders with application to ship virations, Marine Structures (3) (009) 387-437. [] O.C. Zienkiewicz, The Finite Element Method in Engineering Science, McGraw-Hill, 97. [] I. Senjanović, The Finite Element Method in Ship Structure Analysis, University of Zagre, Zagre, 975. (Textook, in Croatian). [3] I. Senjanović, N. Vladimir, Physical insight into Timoshenko eam theory and its modification with extension, Structural Engineering and Mechanics 48 (4) (03) 59-545. [4] MSC. MD NASTRAN 00 Dynamic analysis user's guide. MSC Software, 00. Sumitted:.4.03 Accepted:..03 Ivo Senjanović ivo.senjanovic@fs.hr Nikola Vladimir nikola.vladimir@fs.hr Faculty of Mechanical Engineering and Naval Architecture University of Zagre Ivana Lučića 5, 0000 Zagre, Croatia Dae Seung Cho daecho@pusan.ac.kr Pusan National University, Dept. of Naval Architecture and Ocean Engineering 30 Jangjeon-dong, Guemjeong-gu, Busan, Korea 6 TRANSACTIONS OF FAMENA XXXVII-4 (03)