A SHEAR LOCKING-FREE BEAM FINITE ELEMENT BASED ON THE MODIFIED TIMOSHENKO BEAM THEORY
|
|
- Beatrice Eaton
- 5 years ago
- Views:
Transcription
1 Ivo Senjanović Nikola Vladimir Dae Seung Cho ISSN eissn A SHEAR LOCKING-FREE BEAM FINITE ELEMENT BASED ON THE MODIFIED TIMOSHENKO BEAM THEORY Summary UDC 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)
2 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)
3 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
4 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 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, () 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)
5 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
6 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)
7 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 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
8 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)
9 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 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 (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 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
10 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 l l l l A C (58), (59) where C l l l l l (60) l l Bending deflection (5), y employing (59), yields W P A f, (6) 0 TRANSACTIONS OF FAMENA XXXVII-4 (03)
11 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 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 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)
12 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 l l l l l. (7) l Sym 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 l. (73) J l 3 80l l 30 l l Sym 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)
13 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* *.000* *Threshold f n n n TRANSACTIONS OF FAMENA XXXVII-4 (03) 3
14 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, *.000* *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, *.000* *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)
15 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* *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 ). 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) [4] R.E. Nickell, G.A. Secor, Convergence of consistently derived Timoshenko eam finite element, International Journal for Numerical Methods in Engineering 5 (97) [5] A. Tessler, S.B. Dong, On a hierarchy of conforming Timoshenko eam elements, Computers and Structures 4 (98) [6] G. Prathap, G.R. Bhashyam, Reduced integration and the shear flexile eam element, International Journal for Numerical Methods in Engineering 8 (98) TRANSACTIONS OF FAMENA XXXVII-4 (03) 5
16 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) [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) [9] J.J. Rakowski, A critical analysis of quadratic eam finite elements, International Journal for Numerical Methods in Engineering 3 (99) [0] Z. Friedman, J.B. Kosmatka, An improved two-node Timoshenko eam finite element, Computers and Structures 47 (993) [] 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) [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) [6] I. Senjanović, Y. Fan, The ending and shear coefficients ot thin-walled girders, Thin-Walled Structures 0 (990) [7] I. Senjanović, Y. Fan, A finite element formulation of ship cross-section stiffness parameters, Brodogradnja 4 (993), [8] B. Geist, J.R. McLaughlin, Doule eigenvalues for the uniform Timoshenko eam, Applied Math. Letters 0 (3) (997) [9] N.F.J. van Rensurg, A.J. van der Merve, Natural frequencies and modes of a Timoshenko eam, Wave motion 44 (006) [0] I. Senjanović, S. Tomašević, N. Vladimir, An advanced theory of thin-walled girders with application to ship virations, Marine Structures (3) (009) [] 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) [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)
Exact Shape Functions for Timoshenko Beam Element
IOSR Journal of Computer Engineering (IOSR-JCE) e-iss: 78-66,p-ISS: 78-877, Volume 9, Issue, Ver. IV (May - June 7), PP - www.iosrjournals.org Exact Shape Functions for Timoshenko Beam Element Sri Tudjono,
More informationASSESSMENT OF STRESS CONCENTRATIONS IN LARGE CONTAINER SHIPS USING BEAM HYDROELASTIC MODEL
ASSESSMENT OF STRESS CONCENTRATIONS IN LARGE CONTAINER SHIPS USING BEAM HYDROELASTIC MODEL Ivo Senjanović, Nikola Vladimir Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb,
More informationSIMILARITY METHODS IN ELASTO-PLASTIC BEAM BENDING
Similarity methods in elasto-plastic eam ending XIII International Conference on Computational Plasticity Fundamentals and Applications COMPLAS XIII E Oñate, DRJ Owen, D Peric and M Chiumenti (Eds) SIMILARIT
More informationHomework 6: Energy methods, Implementing FEA.
EN75: Advanced Mechanics of Solids Homework 6: Energy methods, Implementing FEA. School of Engineering Brown University. The figure shows a eam with clamped ends sujected to a point force at its center.
More informationExact Free Vibration of Webs Moving Axially at High Speed
Eact Free Viration of Wes Moving Aially at High Speed S. HATAMI *, M. AZHARI, MM. SAADATPOUR, P. MEMARZADEH *Department of Engineering, Yasouj University, Yasouj Department of Civil Engineering, Isfahan
More informationDavid A. Pape Department of Engineering and Technology Central Michigan University Mt Pleasant, Michigan
Session: ENG 03-091 Deflection Solutions for Edge Stiffened Plates David A. Pape Department of Engineering and Technology Central Michigan University Mt Pleasant, Michigan david.pape@cmich.edu Angela J.
More informationChapter 5 Structural Elements: The truss & beam elements
Institute of Structural Engineering Page 1 Chapter 5 Structural Elements: The truss & beam elements Institute of Structural Engineering Page 2 Chapter Goals Learn how to formulate the Finite Element Equations
More informationResearch Article Analysis Bending Solutions of Clamped Rectangular Thick Plate
Hindawi Mathematical Prolems in Engineering Volume 27, Article ID 7539276, 6 pages https://doi.org/.55/27/7539276 Research Article Analysis Bending Solutions of lamped Rectangular Thick Plate Yang Zhong
More informationME 323 Examination #2
ME 33 Eamination # SOUTION Novemer 14, 17 ROEM NO. 1 3 points ma. The cantilever eam D of the ending stiffness is sujected to a concentrated moment M at C. The eam is also supported y a roller at. Using
More informationSection 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums Mass-Spring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationIn-plane free vibration analysis of combined ring-beam structural systems by wave propagation
In-plane free viration analysis of comined ring-eam structural systems y wave propagation Benjamin Chouvion, Colin Fox, Stewart Mcwilliam, Atanas Popov To cite this version: Benjamin Chouvion, Colin Fox,
More informationBuckling Behavior of Long Symmetrically Laminated Plates Subjected to Shear and Linearly Varying Axial Edge Loads
NASA Technical Paper 3659 Buckling Behavior of Long Symmetrically Laminated Plates Sujected to Shear and Linearly Varying Axial Edge Loads Michael P. Nemeth Langley Research Center Hampton, Virginia National
More informationACCURATE MODELLING OF STRAIN DISCONTINUITIES IN BEAMS USING AN XFEM APPROACH
VI International Conference on Adaptive Modeling and Simulation ADMOS 213 J. P. Moitinho de Almeida, P. Díez, C. Tiago and N. Parés (Eds) ACCURATE MODELLING OF STRAIN DISCONTINUITIES IN BEAMS USING AN
More informationA HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS
A HIGHER-ORDER BEAM THEORY FOR COMPOSITE BOX BEAMS A. Kroker, W. Becker TU Darmstadt, Department of Mechanical Engineering, Chair of Structural Mechanics Hochschulstr. 1, D-64289 Darmstadt, Germany kroker@mechanik.tu-darmstadt.de,
More informationUNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES
UNCONVENTIONAL FINITE ELEMENT MODELS FOR NONLINEAR ANALYSIS OF BEAMS AND PLATES A Thesis by WOORAM KIM Submitted to the Office of Graduate Studies of Texas A&M University in partial fulfillment of the
More informationIntroduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams.
Outline of Continuous Systems. Introduction to Continuous Systems. Continuous Systems. Strings, Torsional Rods and Beams. Vibrations of Flexible Strings. Torsional Vibration of Rods. Bernoulli-Euler Beams.
More informationVibration analysis of circular arch element using curvature
Shock and Vibration 15 (28) 481 492 481 IOS Press Vibration analysis of circular arch element using curvature H. Saffari a,. Tabatabaei a, and S.H. Mansouri b a Civil Engineering Department, University
More informationLecture 7: The Beam Element Equations.
4.1 Beam Stiffness. A Beam: A long slender structural component generally subjected to transverse loading that produces significant bending effects as opposed to twisting or axial effects. MECH 40: Finite
More informationEE C245 ME C218 Introduction to MEMS Design
EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 16: Energy
More informationSolving Homogeneous Trees of Sturm-Liouville Equations using an Infinite Order Determinant Method
Paper Civil-Comp Press, Proceedings of the Eleventh International Conference on Computational Structures Technology,.H.V. Topping, Editor), Civil-Comp Press, Stirlingshire, Scotland Solving Homogeneous
More informationComposite Plates Under Concentrated Load on One Edge and Uniform Load on the Opposite Edge
Mechanics of Advanced Materials and Structures, 7:96, Copyright Taylor & Francis Group, LLC ISSN: 57-69 print / 57-65 online DOI:.8/5769955658 Composite Plates Under Concentrated Load on One Edge and Uniform
More informationStructural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian
Structural Dynamics Lecture Eleven: Dynamic Response of MDOF Systems: (Chapter 11) By: H. Ahmadian ahmadian@iust.ac.ir Dynamic Response of MDOF Systems: Mode-Superposition Method Mode-Superposition Method:
More informationPresented By: EAS 6939 Aerospace Structural Composites
A Beam Theory for Laminated Composites and Application to Torsion Problems Dr. BhavaniV. Sankar Presented By: Sameer Luthra EAS 6939 Aerospace Structural Composites 1 Introduction Composite beams have
More informationCIV-E1060 Engineering Computation and Simulation Examination, December 12, 2017 / Niiranen
CIV-E16 Engineering Computation and Simulation Examination, December 12, 217 / Niiranen This examination consists of 3 problems rated by the standard scale 1...6. Problem 1 Let us consider a long and tall
More informationLecture 15 Strain and stress in beams
Spring, 2019 ME 323 Mechanics of Materials Lecture 15 Strain and stress in beams Reading assignment: 6.1 6.2 News: Instructor: Prof. Marcial Gonzalez Last modified: 1/6/19 9:42:38 PM Beam theory (@ ME
More informationFREE VIBRATION OF AXIALLY LOADED FUNCTIONALLY GRADED SANDWICH BEAMS USING REFINED SHEAR DEFORMATION THEORY
FREE VIBRATION OF AXIALLY LOADED FUNCTIONALLY GRADED SANDWICH BEAMS USING REFINED SHEAR DEFORMATION THEORY Thuc P. Vo 1, Adelaja Israel Osofero 1, Marco Corradi 1, Fawad Inam 1 1 Faculty of Engineering
More informationAN IMPROVED EFFECTIVE WIDTH METHOD BASED ON THE THEORY OF PLASTICITY
Advanced Steel Construction Vol. 6, No., pp. 55-547 () 55 AN IMPROVED EFFECTIVE WIDTH METHOD BASED ON THE THEORY OF PLASTICITY Thomas Hansen *, Jesper Gath and M.P. Nielsen ALECTIA A/S, Teknikeryen 34,
More informationGeometry-dependent MITC method for a 2-node iso-beam element
Structural Engineering and Mechanics, Vol. 9, No. (8) 3-3 Geometry-dependent MITC method for a -node iso-beam element Phill-Seung Lee Samsung Heavy Industries, Seocho, Seoul 37-857, Korea Hyu-Chun Noh
More informationApplication of the assumed mode method to the vibration analysis of rectangular plate structures in contact with fluid
7 th International Conference on HYDROELASTICITY IN MARINE TECHNOLOGY Split, Croatia, September 16 th 19 th 015 Application of the assumed mode method to the vibration analysis of rectangular plate structures
More informationUsing the finite element method of structural analysis, determine displacements at nodes 1 and 2.
Question 1 A pin-jointed plane frame, shown in Figure Q1, is fixed to rigid supports at nodes and 4 to prevent their nodal displacements. The frame is loaded at nodes 1 and by a horizontal and a vertical
More informationINTERNATIONAL JOURNAL OF APPLIED ENGINEERING RESEARCH, DINDIGUL Volume 2, No 2, 2011
Volume, No, 11 Copyright 1 All rights reserved Integrated Pulishing Association REVIEW ARTICLE ISSN 976 459 Analysis of free virations of VISCO elastic square plate of variale thickness with temperature
More informationGeneral elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More informationDiscontinuous Distributions in Mechanics of Materials
Discontinuous Distributions in Mechanics of Materials J.E. Akin, Rice University 1. Introduction The study of the mechanics of materials continues to change slowly. The student needs to learn about software
More informationNatural vibrations of thick circular plate based on the modified Mindlin theory
Arch. Mech., 66, 6, pp. 389 409, Warszawa 04 Natural vibrations of thick circular plate based on the modified Mindlin theory I. SENJANOVIĆ ), N. HADŽIĆ ), N. VLADIMIR ), D.-S. CHO ) ) Faculty of Mechanical
More informationCOMPARATIVE STUDY OF VARIOUS BEAMS UNDER DIFFERENT LOADING CONDITION USING FINITE ELEMENT METHOD
COMPARATIVE STUDY OF VARIOUS BEAMS UNDER DIFFERENT LOADING CONDITION USING FINITE ELEMENT METHOD A THESIS SUBMITTED IN PARTIAL FULLFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF Bachelor of Technology
More informationTransactions of the VŠB Technical University of Ostrava, Mechanical Series. article No Roland JANČO *
Transactions of the VŠB Technical University of Ostrava, Mechanical Series No. 1, 013, vol. LIX article No. 1930 Roland JANČO * NUMERICAL AND EXACT SOLUTION OF BUCKLING LOAD FOR BEAM ON ELASTIC FOUNDATION
More informationParametric study on the transverse and longitudinal moments of trough type folded plate roofs using ANSYS
American Journal of Engineering Research (AJER) e-issn : 2320-0847 p-issn : 2320-0936 Volume-4 pp-22-28 www.ajer.org Research Paper Open Access Parametric study on the transverse and longitudinal moments
More informationStress analysis of a stepped bar
Stress analysis of a stepped bar Problem Find the stresses induced in the axially loaded stepped bar shown in Figure. The bar has cross-sectional areas of A ) and A ) over the lengths l ) and l ), respectively.
More informationDesign and Analysis of Silicon Resonant Accelerometer
Research Journal of Applied Sciences, Engineering and Technology 5(3): 970-974, 2013 ISSN: 2040-7459; E-ISSN: 2040-7467 Maxwell Scientific Organization, 2013 Sumitted: June 22, 2012 Accepted: July 23,
More informationQuintic beam closed form matrices (revised 2/21, 2/23/12) General elastic beam with an elastic foundation
General elastic beam with an elastic foundation Figure 1 shows a beam-column on an elastic foundation. The beam is connected to a continuous series of foundation springs. The other end of the foundation
More information1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load
1859. Forced transverse vibration analysis of a Rayleigh double-beam system with a Pasternak middle layer subjected to compressive axial load Nader Mohammadi 1, Mehrdad Nasirshoaibi 2 Department of Mechanical
More informationMechanics of Inflatable Fabric Beams
Copyright c 2008 ICCES ICCES, vol.5, no.2, pp.93-98 Mechanics of Inflatable Fabric Beams C. Wielgosz 1,J.C.Thomas 1,A.LeVan 1 Summary In this paper we present a summary of the behaviour of inflatable fabric
More informationFLEXIBILITY METHOD FOR INDETERMINATE FRAMES
UNIT - I FLEXIBILITY METHOD FOR INDETERMINATE FRAMES 1. What is meant by indeterminate structures? Structures that do not satisfy the conditions of equilibrium are called indeterminate structure. These
More informationThe CR Formulation: BE Plane Beam
6 The CR Formulation: BE Plane Beam 6 Chapter 6: THE CR FORMUATION: BE PANE BEAM TABE OF CONTENTS Page 6. Introduction..................... 6 4 6.2 CR Beam Kinematics................. 6 4 6.2. Coordinate
More informationComb resonator design (2)
Lecture 6: Comb resonator design () -Intro Intro. to Mechanics of Materials School of Electrical l Engineering i and Computer Science, Seoul National University Nano/Micro Systems & Controls Laboratory
More informationFree vibration and stability of axially functionally graded tapered Euler-Bernoulli beams
Shock and Vibration 18 211 683 696 683 DOI 1.3233/SAV-21-589 IOS Press Free vibration and stability of axially functionally graded tapered Euler-Bernoulli beams Ahmad Shahba a,b, Reza Attarnejad a,b, and
More informationCIVL 8/7117 Chapter 12 - Structural Dynamics 1/75. To discuss the dynamics of a single-degree-of freedom springmass
CIV 8/77 Chapter - /75 Introduction To discuss the dynamics of a single-degree-of freedom springmass system. To derive the finite element equations for the time-dependent stress analysis of the one-dimensional
More informationAnalysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory
International Journal of Research in Advent Technology, Vol.4, No.5, May 16 E-ISSN: 1-967 Analysis of Thick Cantilever Beam Using New Hyperbolic Shear Deformation Theory Mr. Mithun. K. Sawant 1, Dr. Ajay.
More informationGREEN FUNCTION MATRICES FOR ELASTICALLY RESTRAINED HETEROGENEOUS CURVED BEAMS
The th International Conference on Mechanics of Solids, Acoustics and Virations & The 6 th International Conference on "Advanced Composite Materials Engineering" ICMSAV6& COMAT6 Brasov, ROMANIA, -5 Novemer
More informationCHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES
CHAPTER 14 BUCKLING ANALYSIS OF 1D AND 2D STRUCTURES 14.1 GENERAL REMARKS In structures where dominant loading is usually static, the most common cause of the collapse is a buckling failure. Buckling may
More informationASEISMIC DESIGN OF TALL STRUCTURES USING VARIABLE FREQUENCY PENDULUM OSCILLATOR
ASEISMIC DESIGN OF TALL STRUCTURES USING VARIABLE FREQUENCY PENDULUM OSCILLATOR M PRANESH And Ravi SINHA SUMMARY Tuned Mass Dampers (TMD) provide an effective technique for viration control of flexile
More informationERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA)
ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) General information Availale time: 2.5 hours (150 minutes).
More informationShape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression
15 th National Conference on Machines and Mechanisms NaCoMM011-157 Shape Optimization of Revolute Single Link Flexible Robotic Manipulator for Vibration Suppression Sachindra Mahto Abstract In this work,
More informationENGI Second Order Linear ODEs Page Second Order Linear Ordinary Differential Equations
ENGI 344 - Second Order Linear ODEs age -01. Second Order Linear Ordinary Differential Equations The general second order linear ordinary differential equation is of the form d y dy x Q x y Rx dx dx Of
More informationMODIFIED HYPERBOLIC SHEAR DEFORMATION THEORY FOR STATIC FLEXURE ANALYSIS OF THICK ISOTROPIC BEAM
MODIFIED HYPERBOLIC SHEAR DEFORMATION THEORY FOR STATIC FLEXURE ANALYSIS OF THICK ISOTROPIC BEAM S. Jasotharan * and I.R.A. Weerasekera University of Moratuwa, Moratuwa, Sri Lanka * E-Mail: jasos91@hotmail.com,
More informationNON LINEAR BUCKLING OF COLUMNS Dr. Mereen Hassan Fahmi Technical College of Erbil
Abstract: NON LINEAR BUCKLING OF COLUMNS Dr. Mereen Hassan Fahmi Technical College of Erbil The geometric non-linear total potential energy equation is developed and extended to study the behavior of buckling
More informationEigenvalues of Trusses and Beams Using the Accurate Element Method
Eigenvalues of russes and Beams Using the Accurate Element Method Maty Blumenfeld Department of Strength of Materials Universitatea Politehnica Bucharest, Romania Paul Cizmas Department of Aerospace Engineering
More informationFree vibration analysis of elastically connected multiple-beams with general boundary conditions using improved Fourier series method
Free vibration analysis of elastically connected multiple-beams with general boundary conditions using improved Fourier series method Jingtao DU*; Deshui XU; Yufei ZHANG; Tiejun YANG; Zhigang LIU College
More informationAalto University School of Engineering
Aalto University School of Engineering Kul-4.4 Ship Structural Design (P) ecture 6 - Response of Web-frames, Girders and Grillages Kul-4.4 Ship Structures Response ecture 5: Tertiary Response: Bending
More informationAN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES
Engineering MECHANICS, Vol. 15, 2008, No. 2, p. 115 132 115 AN EFFECTIVE SOLUTION OF THE COMPOSITE (FGM S) BEAM STRUCTURES Justín Murín, Vladimír Kutiš* The additive mixture rules have been extended for
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More information2014 International Conference on Computer Science and Electronic Technology (ICCSET 2014)
04 International Conference on Computer Science and Electronic Technology (ICCSET 04) Lateral Load-carrying Capacity Research of Steel Plate Bearing in Space Frame Structure Menghong Wang,a, Xueting Yang,,
More informationFREE VIBRATIONS OF UNIFORM TIMOSHENKO BEAMS ON PASTERNAK FOUNDATION USING COUPLED DISPLACEMENT FIELD METHOD
A R C H I V E O F M E C H A N I C A L E N G I N E E R I N G VOL. LXIV 17 Number 3 DOI: 1.1515/meceng-17- Key words: free vibrations, Coupled Displacement Field method, uniform Timoshenko beam, Pasternak
More informationFlexural analysis of deep beam subjected to parabolic load using refined shear deformation theory
Applied and Computational Mechanics 6 (2012) 163 172 Flexural analysis of deep beam subjected to parabolic load using refined shear deformation theory Y. M. Ghugal a,,a.g.dahake b a Applied Mechanics Department,
More informationMODULE C: COMPRESSION MEMBERS
MODULE C: COMPRESSION MEMBERS This module of CIE 428 covers the following subjects Column theory Column design per AISC Effective length Torsional and flexural-torsional buckling Built-up members READING:
More informationStability of Simply Supported Square Plate with Concentric Cutout
International OPEN ACCESS Journal Of Modern Engineering Research (IJMER) Stability of Simply Supported Square Plate with Concentric Cutout Jayashankarbabu B. S. 1, Dr. Karisiddappa 1 (Civil Engineering
More informationAnalysis of Axially Loaded Non-prismatic Beams with General End Restraints Using Differential Quadrature Method
ISBN 978-93-84422-56-1 Proceedings of International Conference on Architecture, Structure and Civil Engineering (ICASCE'15 Antalya (Turkey Sept. 7-8, 2015 pp. 1-7 Analysis of Axially Loaded Non-prismatic
More informationThe Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation
Shiraz University of Technology From the SelectedWorks of Habibolla Latifizadeh 2011 The Homotopy Perturbation Method for free vibration analysis of beam on elastic foundation Habibolla Latifizadeh, Shiraz
More informationEffect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test
Effect of Specimen Dimensions on Flexural Modulus in a 3-Point Bending Test M. Praveen Kumar 1 and V. Balakrishna Murthy 2* 1 Mechanical Engineering Department, P.V.P. Siddhartha Institute of Technology,
More informationLaboratory 4 Topic: Buckling
Laboratory 4 Topic: Buckling Objectives: To record the load-deflection response of a clamped-clamped column. To identify, from the recorded response, the collapse load of the column. Introduction: Buckling
More informationDue Tuesday, September 21 st, 12:00 midnight
Due Tuesday, September 21 st, 12:00 midnight The first problem discusses a plane truss with inclined supports. You will need to modify the MatLab software from homework 1. The next 4 problems consider
More informationToward a novel approach for damage identification and health monitoring of bridge structures
Toward a novel approach for damage identification and health monitoring of bridge structures Paolo Martino Calvi 1, Paolo Venini 1 1 Department of Structural Mechanics, University of Pavia, Italy E-mail:
More informationBasic Energy Principles in Stiffness Analysis
Basic Energy Principles in Stiffness Analysis Stress-Strain Relations The application of any theory requires knowledge of the physical properties of the material(s) comprising the structure. We are limiting
More informationFinite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras. Module - 01 Lecture - 13
Finite Element Analysis Prof. Dr. B. N. Rao Department of Civil Engineering Indian Institute of Technology, Madras (Refer Slide Time: 00:25) Module - 01 Lecture - 13 In the last class, we have seen how
More informationCO-ROTATIONAL DYNAMIC FORMULATION FOR 2D BEAMS
COMPDYN 011 ECCOMAS Thematic Conference on Computational Methods in Structural Dynamics and Earthquake Engineering M. Papadrakakis, M. Fragiadakis, V. Plevris (eds.) Corfu, Greece, 5-8 May 011 CO-ROTATIONAL
More informationQuiz Samples for Chapter 7 Kinetic Energy and Work
Name: Department: Student ID #: Notice ˆ + ( 1) points per correct (incorrect) answer ˆ No penalty for an unanswered question ˆ Fill the lank ( ) with ( ) if the statement is correct (incorrect) ˆ : corrections
More informationNumerical procedure for the vibration analysis of arbitrarily constrained stiffened panels with openings
csnak, 04 Int. J. Nav. Archit. Ocean Eng. (04) 6:763~774 http://dx.doi.org/0.478/ijnaoe-03-00 pissn: 09-678, eissn: 09-6790 Numerical procedure for the vibration analysis of arbitrarily constrained stiffened
More informationSTRUCTURED SPATIAL DISCRETIZATION OF DYNAMICAL SYSTEMS
ECCOMAS Congress 2016 VII European Congress on Computational Methods in Applied Sciences and Engineering M. Papadrakakis, V. Papadopoulos, G. Stefanou, V. Plevris (eds. Crete Island, Greece, 5 10 June
More informationdw 2 3(w) x 4 x 4 =L x 3 x 1 =0 x 2 El #1 El #2 El #3 Potential energy of element 3: Total potential energy Potential energy of element 1:
MAE 44 & CIV 44 Introduction to Finite Elements Reading assignment: ecture notes, ogan. Summary: Pro. Suvranu De Finite element ormulation or D elasticity using the Rayleigh-Ritz Principle Stiness matri
More informationCHAPTER 5. Beam Theory
CHPTER 5. Beam Theory SangJoon Shin School of Mechanical and erospace Engineering Seoul National University ctive eroelasticity and Rotorcraft Lab. 5. The Euler-Bernoulli assumptions One of its dimensions
More informationChapter 2: Deflections of Structures
Chapter 2: Deflections of Structures Fig. 4.1. (Fig. 2.1.) ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 1 (2.1) (4.1) (2.2) Fig.4.2 Fig.2.2 ASTU, Dept. of C Eng., Prepared by: Melkamu E. Page 2
More informationFREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS
FREE VIBRATIONS OF FRAMED STRUCTURES WITH INCLINED MEMBERS A Thesis submitted in partial fulfillment of the requirements for the degree of Bachelor of Technology in Civil Engineering By JYOTI PRAKASH SAMAL
More informationME Final Exam. PROBLEM NO. 4 Part A (2 points max.) M (x) y. z (neutral axis) beam cross-sec+on. 20 kip ft. 0.2 ft. 10 ft. 0.1 ft.
ME 323 - Final Exam Name December 15, 2015 Instructor (circle) PROEM NO. 4 Part A (2 points max.) Krousgrill 11:30AM-12:20PM Ghosh 2:30-3:20PM Gonzalez 12:30-1:20PM Zhao 4:30-5:20PM M (x) y 20 kip ft 0.2
More informationDispersion relation for transverse waves in a linear chain of particles
Dispersion relation for transverse waves in a linear chain of particles V. I. Repchenkov* It is difficult to overestimate the importance that have for the development of science the simplest physical and
More informationTransactions on Modelling and Simulation vol 18, 1997 WIT Press, ISSN X
An integral equation formulation of the coupled vibrations of uniform Timoshenko beams Masa. Tanaka & A. N. Bercin Department of Mechanical Systems Engineering, Shinshu University 500 Wakasato, Nagano
More informationP = ρ{ g a } + µ 2 V II. FLUID STATICS
II. FLUID STATICS From a force analysis on a triangular fluid element at rest, the following three concepts are easily developed: For a continuous, hydrostatic, shear free fluid: 1. Pressure is constant
More informationDISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS
Nigerian Journal of Technology, Vol. 25, No. 2, September 2006 Osadebe and Mbajiogu 36 DISTORTION ANALYSIS OF TILL -WALLED BOX GIRDERS N. N. OSADEBE, M. Sc., Ph. D., MNSE Department of Civil Engineering
More informationFinite element modelling of structural mechanics problems
1 Finite element modelling of structural mechanics problems Kjell Magne Mathisen Department of Structural Engineering Norwegian University of Science and Technology Lecture 10: Geilo Winter School - January,
More informationAbstract. keywords: finite elements, beams, Hermitian interpolation, shape functions
Generation of shape functions for straight beam elements Charles E. Augarde Originally published in Computers and Structures, 68 (1998) 555-560 Corrections highlighted. Abstract Straight beam finite elements
More informationBeam Model Validation Based on Finite Element Analysis
Beam Model Validation Based on Finite Element Analysis CARLA PROTOCSIL, PETRU FLORIN MINDA, GILBERT-RAINER GILLICH Department of Mechanical Engineering Eftimie Murgu University of Resita P-ta Traian Vuia
More informationSection 6: PRISMATIC BEAMS. Beam Theory
Beam Theory There are two types of beam theory aailable to craft beam element formulations from. They are Bernoulli-Euler beam theory Timoshenko beam theory One learns the details of Bernoulli-Euler beam
More informationCHAPTER -6- BENDING Part -1-
Ishik University / Sulaimani Civil Engineering Department Mechanics of Materials CE 211 CHAPTER -6- BENDING Part -1-1 CHAPTER -6- Bending Outlines of this chapter: 6.1. Chapter Objectives 6.2. Shear and
More informationFinite Element Nonlinear Analysis for Catenary Structure Considering Elastic Deformation
Copyright 21 Tech Science Press CMES, vol.63, no.1, pp.29-45, 21 Finite Element Nonlinear Analysis for Catenary Structure Considering Elastic Deformation B.W. Kim 1, H.G. Sung 1, S.Y. Hong 1 and H.J. Jung
More informationCOPYRIGHTED MATERIAL. Index
Index A Admissible function, 163 Amplification factor, 36 Amplitude, 1, 22 Amplitude-modulated carrier, 630 Amplitude ratio, 36 Antinodes, 612 Approximate analytical methods, 647 Assumed modes method,
More informationFree Vibration Analysis of Kirchoff Plates with Damaged Boundaries by the Chebyshev Collocation Method. Eric A. Butcher and Ma en Sari
Free Vibration Analysis of Kirchoff Plates with Damaged Boundaries by the Chebyshev Collocation Method Eric A. Butcher and Ma en Sari Department of Mechanical and Aerospace Engineering, New Mexico State
More informationDesign Variable Concepts 19 Mar 09 Lab 7 Lecture Notes
Design Variale Concepts 19 Mar 09 La 7 Lecture Notes Nomenclature W total weight (= W wing + W fuse + W pay ) reference area (wing area) wing aspect ratio c r root wing chord c t tip wing chord λ taper
More informationBEAM DEFLECTION THE ELASTIC CURVE
BEAM DEFLECTION Samantha Ramirez THE ELASTIC CURVE The deflection diagram of the longitudinal axis that passes through the centroid of each cross-sectional area of a beam. Supports that apply a moment
More informationAnalytical Strip Method for Thin Isotropic Cylindrical Shells
IOSR Journal of Mechanical and Civil Engineering (IOSR-JMCE) e-issn: 2278-1684,p-ISSN: 2320-334X, Volume 14, Issue 4 Ver. III (Jul. Aug. 2017), PP 24-38 www.iosrjournals.org Analytical Strip Method for
More informationJEPPIAAR ENGINEERING COLLEGE
JEPPIAAR ENGINEERING COLLEGE Jeppiaar Nagar, Rajiv Gandhi Salai 600 119 DEPARTMENT OFMECHANICAL ENGINEERING QUESTION BANK VI SEMESTER ME6603 FINITE ELEMENT ANALYSIS Regulation 013 SUBJECT YEAR /SEM: III
More informationTopic 5: Finite Element Method
Topic 5: Finite Element Method 1 Finite Element Method (1) Main problem of classical variational methods (Ritz method etc.) difficult (op impossible) definition of approximation function ϕ for non-trivial
More information