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Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 841215, 7 pages http://dx.doi.org/10.1155/2013/841215 Research Article Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam 1 2 Abbas Moallemi-Oreh and Mohammad Karkon Department of Mechanical Engineering, Shahreza Branch, Islamic Azad University, Shahreza, Iran Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Correspondence should be addressed to Mohammad Karkon; karkon443@gmail.com Received 18 February 2013; Revised 1 April 2013; Accepted 1 April 2013 Academic Editor: K. M. Liew Copyright © 2013 A. Moallemi-Oreh and M. Karkon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beam. Cubic displacement polynomial and quadratic rotational fields are selected for this element. Moreover, it is assumed that shear strain of the element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energy and its stationing to shear strain. By exploiting these interpolation functions, beam elements’ stiffness matrix is also examined. Furthermore, geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation on stability and beam free vibration. At last, accuracy and efficiency of proposed element are evaluated through numerical tests. es Th e tests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis. 1. Introduction The first high-order element was proposed by Kapur with eight degrees of freedom [5]. Lees and o Th mas formulated a Twoversionsoftheorieshavebeendeveloped foranalysisof complex element by applying independent polynomial series beams.InEuler-Bernoullitheory,thedisplacementofbeamis for displacement and rotation fields [ 6, 7]. Also, this method considered without shear effects. This method gives appropri- has been used by Webster [8]. Rao and Gupta have examined ate and acceptable response in thin beam in which shear effect free vibration of rotating beams [9]. In some methods, like is insignificant. However, in this approach, by increasing the isoparametric formulation, displacement and rotation fields thickness of beam and shear eeff ct deformation, the error of are assumed dependently with the same order [10]. Based response is increasing [1]. Correspondingly, the eeff ct of shear on Euler-Bernoulli theory, Goncalves et al. have presented transformation is formulated in Timoshenko theory. er Th e- frequency equation and vibration modes for classical bound- fore, this method has a better result, especially in deep beams ary conditions such as clamped, free, pinned, and sliding in which shear effect is impressive. Although the rotational supports [11]. Leeand Schultzhaveconsidered free vibration inertia of thick beams was investigated by Rayleigh for the of Timoshenko beam through pseudospectral method [12]. first time, Timoshenko has developed this theory and formu- So far, very little research has been done on buckling of lated shear effect. Due to the complexity of the governing Timoshenko beam with respect to free vibration analysis. equations of the free vibration and stability of beams in Exploiting an approximate method based on n fi ite elements general, numerical methods such as finite element have been formulation, Wieckowski and Golubiewski examined beam developed profoundly. Up to now, many elements have been stability of Euler-Bernoulli and Timoshenko theories [13]. presented based on Timoshenko theory. es Th e elements are Also, Kosmatka has proposed a two-node element for stabil- classified into two groups which are simple and higher order ity and free vibration analysis of Timoshenko beam [14]. elements. Some researchers used simple two-node elements In this study, a new beam element is proposed for free with four degrees of freedom [2–4]. Thomas et al. have vibration and stability analysis of Timoshenko beams. In examined the elements proposed by other researchers [3]. order to compute shape function of the beam element, cubic 2 Advances in Acoustics and Vibration 𝑥 𝑠 In the present formula, the coecffi ients of the terms 𝑠 and 𝑠 are equivalent to zero. Therefore, in the succeeding lines, 𝛼 , 𝑖 0 𝜃 𝛽 are determined in terms of the unknown parameter 𝛾 : 𝜃 𝑗 1 0 𝑤 Γ= (𝑤 −𝑤 )−(𝜃 +𝜃 ) 𝑗 𝑗 𝑖 𝑖 𝑗 Figure 1: Timoshenko beam element. 𝛽 = (𝜃 −𝜃 ) 0 𝑖 𝑗 (3) 3 1 displacement polynomial and quadratic rotational fields are 𝛼 =− (𝛾 − Γ) 0 0 selected. In addition, shear strain of the element is assumed 2 2 constant. en, Th by exploiting the bending and shear strain energy of the beam and stationary with respect to constant 𝛽 = 𝛼 . 1 0 shear strain, interpolation function of this element has been At this stage, there is only one unknown constant 𝛾 ,which carefully calculated. In the following, by using these shape can be discovered through the condition of minimum strain functions, the stiffness matrix, geometric stiffness matrix, energy. It should be added that the structural strain energy is and mass matrix of the proposed element have been exactly the sum of bending and shear strain energy. Bending strain determined. Finally, several numerical tests are performed to energy is calculated in the following way: investigate the robustness of this element for free vibration and stability analysis of beams with dieff rent boundary 𝑙 1 2 2 (4) conditions. eTh ndings fi prove that the suggested element has 𝑈 = ∫ 𝜅 = ∫ 𝜅 𝑑𝑠. 2 4 0 −1 high level of accuracy and free of shear locking. In (4), 𝜅 represents curvature which is determined as follows: 2. Finite Element Formulation 𝜅=− ⋅ =𝜅 −6 𝑙 𝑙 In the ni fi te element method, displacement and rotation elds fi (5) of the element are associated with interpolation functions to 𝜅 = (𝜃 −𝜃 +3Γ𝑠 ). 0 𝑖 𝑗 nodal degrees of freedom. Figure 1 shows proposed element with two nodes. eTh shape functions of the Timoshenko Substituting (5)into(4) leads to the following bending strain beam are calculated based on Figure 1. In order to compute energy: shape function of the beam in Figure 1,cubic displacement Γ𝛾 0 0 polynomial and quadratic rotational fields are selected. Addi- 𝑈 =𝑈 +6𝐼𝐸(− + ) 𝑏 0 𝑙 𝑙 tionally, it is assumed that shear strain has the constant value (6) of 𝛾 . Based on these assumptions, (1)can be writtenas follows: 𝑈 = ∫ 𝜅 𝑑𝑠. 0 0 −1 𝑤= (1−𝑠 )+ (1+𝑠 ) Besides, (7)shows theenergyofshear strain: 2 2 2 2 +𝛽 𝑙(1 − 𝑠 )+𝛽 (1 − 𝑠 ) 𝑈 = 𝛾 0 1 ∫ 2𝑓 𝜃 𝑗 𝑖 2 (1) 1 𝜃= (1−𝑠 )+ (1+𝑠 )+𝛼 (1 − 𝑠 ) 𝑙 (7) 2 2 = ∫ 𝛾 4𝑓 −1 𝛾=𝛾 = 𝛾 . 2𝑥 2𝑓 𝑠= −1. In this equation, 𝑓 is shear correction factor which depends In these equations, 𝛽 , 𝛽 , 𝛼 ,and 𝛾 are unknown parame- 1 0 0 0 on cross section shape of beam. This coefficient for rectan- ters. In order to determine their values, at first, the equation of gular section is 5/6. By adding the bending and shear strain shear strain for Timoshenko beam is established. By utilizing energy together, total strain energy is found as follows: the shear strain value equal to 𝛾 , the subsequent equations 6𝐸𝐼Γ 6𝐸𝐼 𝑙 2 2 will be available: 𝑈=𝑈 +𝑈 =𝑈 − 𝛾 + 𝛾 + 𝛾 . (8) 𝑏 𝑠 0 0 0 0 𝑙 𝑙 2𝑓 𝛾= −𝜃 = ⋅ −𝜃 Implementing 𝜕𝑈/𝜕𝛾 =0 will give the following results: 6𝐸𝐼Γ 2 𝑤 𝑖 2 𝛾 = =𝛿Γ (2) 0 𝛾 = (− + −2𝛽 𝑙𝑠 + 𝛽 𝑙−3𝛽 𝑙𝑠 ) 2 0 0 1 1 𝑙 /𝑓 + 12𝐸𝐼 𝑙 2 2 (9) 1−𝑠 1+𝑠 6𝜆 𝑓 2 𝑠 −𝜃 ( )−𝜃 ( )−𝛼 (1 − 𝑠 ). 𝛿= ,𝜆= . 𝑖 𝑗 0 2 2 𝑙 + 12𝜆 𝐺𝐴 𝐸𝐼 𝐺𝐴 𝑑𝑠 𝑑𝑥 𝑑𝑤 𝑑𝑤 𝐺𝐴 𝐺𝐴 𝑑𝑠 𝐺𝐴 𝑑𝑥 𝑙𝑠 𝐺𝐴 𝐸𝐼 𝑑𝑠 𝑠𝛾 𝑑𝜃 𝑑𝑥 𝐸𝐼 𝐸𝐼 Advances in Acoustics and Vibration 3 Substituting 𝛽 , 𝛽 , 𝛼 ,and 𝛾 into (1), the succeeding shape By calculating (13), the stiffness matrix of the proposed ele- 1 0 0 0 functions for Timoshenko beam can be as follows: ment is obtained in the succeeding forms: { } { } { } [𝐾 ]= 𝑤̃ 𝜃 𝑁 𝑁 𝑁 𝑁 0 𝑛𝑖 3 1 2 3 4 𝑙 + 12𝑙𝜆 { }=[ ] ̃ { } 𝜃 𝑁 𝑁 𝑁 𝑁 { } 5 6 7 8 𝑛 { } 12 6𝑙 −12 6𝑙 { } (15) 2 2 [ ] 6𝑙 4𝑙 + 12𝜆 −6𝑙 2𝑙 − 12𝜆 [ ] × . 3 [ ] −12 −6𝑙 12 −6𝑙 𝑁 = [2 + 𝑠 (1−2𝛿 )+𝑠 (−3 + 2𝛿 )] 2 2 6𝑙 2𝑙 − 12𝜆 −6𝑙 4𝑙 + 12𝜆 [ ] 2 3 𝑁 = [0.5 (1 − 𝑠 )+(𝑠 −𝑠) 0.5 − 𝛿 ] ( ) 4 3. Mass Matrix The kinetic energy, 𝑇 , of an elemental length, 𝑙 ,ofauniform 𝑁 = [2 − 𝑠 (1−2𝛿 )−𝑠 (−3 + 2𝛿 )] Timoshenko beam is given as follows [1]: (10) 2 3 𝑙/2 𝑙/2 𝑁 = [−0.5 (1 − 𝑠 )+(𝑠 −𝑠) (0.5 − 𝛿 )] 4 1 1 2 2 𝑇= ∫ 𝑦 ̇ + ∫ 𝜃 𝑑𝑥. (16) 2 2 −𝑙/2 −𝑙/2 𝑁 = [6 (1 − 𝑠 )(−1 + 2𝛿 )] 4𝑙 In this equation, 𝜌 is the mass density of the material of the beam and 𝐼 is the second moment of area of cross section. 𝑁 = [−1 + 𝑠 (−2 + 3𝑠 )+6(1 −𝑠 )𝛿] eTh refore, the mass matrix of the element has been two parts, one related to translations and the other related to rotations, 2 in the form of 𝑁 = [−6 (1 − 𝑠 )(−1 + 2𝛿 )] 4𝑙 [𝑀 ] =[𝑀 ]+[𝑀 ] 1 2 𝑁 = [−1 + 𝑠 2+3𝑠 +6(1 −𝑠 )𝛿]. ( ) 8 1 4 𝑇 = ∫ [𝑁 ] [𝑁 ]𝑑𝑠 𝑤 𝑤 (17) Innfi iteelementmethod,displacementsandstrainfieldofthe element can be related to the nodal degrees of freedom with 𝑙/2 interpolation functions. Hence, equations of finite element + ∫ [𝑁 ] [𝑁 ] 𝑑𝑠. 𝜃 𝜃 −𝑙/2 formulation can be written as follows: eTh translation mass matrix [𝑀 ] is achieved as follows: { }= [𝑁 ] {𝐷 } [𝑀 ]= (11) 210(12𝜆 + 𝑙 ) { } { } { } 𝜀 = = 𝐵 𝐷 , { } [ ] { } { } (18) 𝑚 𝑚 𝑚 𝑚 { } 11 12 13 14 − +𝜃 { } [ ] 𝑚 𝑚 −𝑚 𝑚 12 15 14 16 { } [ ] × . [ ] 𝑚 −𝑚 𝑚 −𝑚 13 14 11 12 where 𝑚 𝑚 −𝑚 𝑚 14 16 12 15 [ ] 0 In the following, the entries of this matrix are introduced: [ ] [𝐵 ] = [ ] [𝑁 ] (12) −1 6 2 2 4 [ ] 𝑚 = (1680𝜆 + 294𝑙 𝜆 + 13𝑙 ) and [𝑁] is the matrix of interpolation function. Also, {𝐷} 2 2 4 is nodal displacement and [𝐵] is strain matrix. Consequently, 𝑚 = (1260𝜆 + 231𝑙 𝜆 + 11𝑙 ) stiffness matrix of Timoshenko element can be obtained as follows: 9 2 2 4 𝑚 = (560𝜆 + 84𝑙 𝜆+3𝑙 ) (19) [𝐾 ]= ∫ 𝐵 [𝐷 ] 𝐵 𝑑𝑥, (13) [ ] [ ] 0 𝑚 2 2 4 𝑚 =− (2520𝜆 + 378𝑙 𝜆 + 13𝑙 ) 2𝑙 where, [𝐷 ]istheelasticitymatrixforTimoshenkobeamele- ment: 2 2 2 4 𝑚 = (126𝜆 + 21𝑙 𝜆 + 13𝑙 ) [ ] [𝐷 ]= . (14) 𝑚 2 2 4 𝑚 =− (168𝜆 + 28𝑙 𝜆 + 13𝑙 ). 𝑓 2 [ 𝑠 ] 2𝑙 𝐺𝐴 𝐸𝐼 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝜌𝐴 𝜌𝐼 𝜌𝐴 𝜌𝐼 𝑑𝑥 𝜌𝐴 𝑛𝑗 𝐸𝐼 4 Advances in Acoustics and Vibration In addition, the rotation mass matrix [𝑀 ] can be obtained 𝑑𝑠 as follows: [𝑀 ]= 𝑑𝑥 𝑑𝛿 30(12𝜆 + 𝑙 ) Figure 2: Straight and buckled form. (20) 𝑚 𝑚 −𝑚 𝑚 21 22 21 22 [ ] 𝑚 𝑚 −𝑚 𝑚 22 23 22 24 [ ] × . [ ] −𝑚 −𝑚 𝑚 −𝑚 21 22 21 12 By calculating this equation, the geometric stiffness matrix 𝑚 𝑚 −𝑚 𝑚 22 24 12 23 [ ] becomes as follows: The entries of this matrix are defined in the below form: 𝑃 [𝐾 ]= 60𝑙 𝑚 = 36𝑙 𝑘 𝑘 𝑘 𝑘 𝑔1 𝑔2 𝑔3 𝑔2 (25) [ ] 𝑘 𝑘 −𝑘 𝑘 𝑚 = −3 (60𝜆 − 𝑙 ) 𝑔2 𝑔4 𝑔2 𝑔5 [ ] × , [ ] 𝑘 −𝑘 𝑘 −𝑘 𝑔3 𝑔2 𝑔1 𝑔2 (21) 2 2 4 𝑘 𝑘 −𝑘 𝑘 [ 𝑔2 𝑔5 𝑔2 𝑔4 ] 𝑚 = (360𝜆 + 15𝑙 𝜆+3𝑙 ) where 2 2 4 𝑚 = (720𝜆 − 60𝑙 𝜆−𝑙 ). 𝑘 = 60 + 12𝛽 𝑔1 𝑘 =6𝑙𝛽 𝑔2 4. Geometric Stiffness Matrix 𝑘 = −60 − 12𝛽 𝑔3 The concept of the neutral state of equilibrium is used for buckling analysis of beam. In the mathematical formulation 2 2 (26) 𝑘 =5𝑙 +3𝑙𝛽 𝑔4 of elastic stability of beam, the neutral equilibrium is used assuming a bifurcation of the deformations. aTh t is, at the 2 2 𝑘 =3𝑙𝛽 −5𝑙 𝑔5 critical load, of the possible two paths of deformations (one associated with the stable equilibrium and the other pertinent to the unstable equilibrium condition, as shown in Figure 2), 𝛽= . (𝑙 + 12𝜆) the beam always takes the buckled form. In addition to the existence of this bifurcation of equilibrium paths, the elastic For stability analysis and determination of the magnitude of a stability analysis of plates assumes the validity of Hooke’s static compressive axial load that will produce beam buckling, law. By considering Figure 2 at the buckled form, the axial the following eigenvalue will be achieved: shortening of beam can be acquired as follows: det ([𝐾 ])= det ([𝐾 ]−𝑃[𝐾 ]) = 0⇒ 󳨐 𝑃 . (27) 0 𝑔 cr 2 2 = (𝑑𝑥 +𝑑𝑦 ) eTh lowest positive eigenvalue of this equation is the magni- tude of buckling load and the corresponding eigenvector is ≃𝑑𝑥 + ( ) (22) the deformed shape of the buckled beam. eTh exact solution of beam-column buckling load with shear deformation eeff ct 1 is obtained as succeeding form [15]: 󳨐⇒ 𝛿𝑑 = ( ) 𝑑𝑥. 𝑃 = cr As aresult, thestrainenergyofaxial load 𝑃 canbeexpressed eff (28) as follows: ×( ), 1+(𝜋 𝑓 ) / (𝐿 ) 𝑃 eff (23) Δ𝑊 = ∫ ( ) 𝑑𝑥. where 𝐿 istheeeff ctivebeamlengthinwhich (𝐿 =𝐿) and eff eff (𝐿 = 𝐿/2) areusedfor pinned-pinnedbeams andfixed- eff Basedon(23), geometric stiffness matrix of the element can fixed beams, respectively. be calculated as follows: 4.1. First Example (Vibration Analysis). The efficiency of pro- 𝑤 𝑤 posedelement is evaluatedbyanalyzing thefreevibration (24) [𝐾 ]= ∫ [ ] 𝑃[ ]𝑑𝑥. of beam with simply and clamped supports for various 𝑑𝑥 𝑑𝑥 𝑑𝑁 𝑑𝑁 𝑑𝑥 𝑑𝑦 𝐺𝐴 𝐸𝐼 𝐸𝐼 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑠 𝜌𝐼𝑙 𝑑𝑦 Advances in Acoustics and Vibration 5 Table 1: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.002 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1428 3.14158 3.14158 2 6.28319 6.2928 6.28310 6.28310 3 9.42478 9.4573 9.42449 9.42450 4 12.5664 12.6437 12.5657 12.5657 5 15.7080 15.8596 15.7066 15.7068 6 18.8496 19.1127 18.8473 18.8476 7 21.9911 22.4113 21.9875 21.9883 8 25.1327 25.7638 25.1273 25.1288 9 28.2743 29.1793 28.2666 28.2692 10 31.4159 32.6672 31.4053 31.4098 No.1 No.2 No.3 No.2 No.1 No.3 1 1 1 1 0.9 0.9 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0 0 0 0 0 0 0.5 −0.5 0 0.5 −0.2 0 0.2 −0.4−0.2 0 −0.5 0 0.5 −0.5 0 0.5 (a) (b) Figure 3: The first three modes of buckling for beam-column, (a) simply supported and (b) clamped supported. Table 2: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.01 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1425 3.14133 3.14133 2 6.28319 6.2908 6.28106 6.28106 3 9.42478 9.4503 9.41761 9.41764 4 12.5664 12.6271 12.5494 12.5496 5 15.7080 15.8267 15.6749 15.6754 6 18.8496 19.0552 18.7926 18.7937 7 21.9911 22.3186 21.9011 21.9034 8 25.1327 25.6231 24.9988 25.0034 9 28.2743 28.9749 28.0845 28.0927 10 31.4159 32.3806 31.1568 31.1705 6 Advances in Acoustics and Vibration Table 3: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.1 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1169 3.11568 3.11569 2 6.28319 6.0993 6.09066 6.09094 3 9.42478 8.8668 8.84052 8.84229 4 12.5664 11.3984 11.3431 11.3492 5 15.7080 13.7089 13.6132 13.6282 6 18.8496 15.8266 15.6790 15.7093 7 21.9911 17.7811 17.5705 17.6239 8 25.1327 19.5991 19.3142 19.3997 9 28.2743 21.3030 20.9325 21.0606 10 31.4159 22.9117 22.4441 22.6257 Table 4: Dimensionless frequency parameter 𝜆 for the clamped supported Timoshenko beam. ℎ/𝑙 = 0.002 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 4.73004 4.7345 4.72998 4.72998 2 7.85320 7.8736 7.85295 7.85296 3 10.9956 11.0504 10.9950 10.9950 4 14.1372 14.2526 14.1359 14.1360 5 17.2788 17.4888 17.2766 17.2768 6 20.4204 20.7670 20.4168 20.4174 7 23.5619 24.0955 23.5567 23.5578 8 26.7035 27.4833 26.6960 26.6980 9 29.8451 30.9398 29.8348 29.8382 10 32.9867 34.4748 32.9729 32.9786 Table 5: Dimensionless frequency parameter 𝜆 for the clamped Table 6: Dimensionless frequency parameter 𝜆 for the clamped 𝑖 𝑖 supported Timoshenko beam. supported Timoshenko beam. ℎ/𝑙 = 0.01 ℎ/𝑙 = 0.1 Mode Euler theory Mode Euler theory Lee and Lee and Ferreira Proposed Ferreira Proposed Schultz Schultz [10] element [10] element [12] [12] 1 4.73004 1 4.73004 4.7330 4.72840 4.72840 4.5835 4.57955 4.57962 27.85320 27.85320 7.8675 7.84690 7.84692 7.3468 7.33122 7.33193 3 10.9956 3 10.9956 11.0351 10.9800 10.9801 9.8924 9.85611 9.85918 4 14.1372 4 14.1372 14.2218 14.1062 14.1064 12.2118 12.1454 12.1540 517.2788 517.2788 17.4342 17.2246 17.2253 14.3386 14.2324 14.2513 6 20.4204 6 20.4204 20.6783 20.3338 20.3355 16.3046 16.1487 16.1841 723.5619 723.5619 23.9600 23.4325 23.4358 18.1375 17.9215 17.9807 8 26.7035 8 26.7035 27.2857 26.5192 26.5253 19.8593 19.5723 19.6641 929.8451 929.8451 30.6616 29.5926 29.6032 21.4875 21.1185 21.2523 10 32.9867 10 32.9867 34.0944 32.6514 32.6687 23.0358 22.5735 22.7598 lengths to thickness ratio. The Poisson’s ratio of this beam To compare other researchers’ results, frequency dimen- is ] = 0.3 and shear correction factor is taken 5/6. sionless parameter 𝜆 , defined in ( 29), has been shown in 𝑖 Advances in Acoustics and Vibration 7 Table 7: Critical load of the simply and clamped supported beam-column. Simply supported Clamped supported 𝑙/ℎ Analytical solution Ferreira [10] Proposed element Analytical solution Ferreira [10]Proposedelement 10 8013.8 8021.8 8013.86 29766 29877 29770 100 8.223 8.231 8.2225 32.864 32.999 32.864 1000 0.0082 0.0082 0.00822 0.0329 0.0330 0.0329 (Tables 1, 2, 3, 4, 5,and 6) for 10 rfi st frequencies of this beam References by exploiting 40 elements: [1] M. Petyt, Introduction of Finite Element Vibration Analysis, Cambridge University Press, 2nd edition, 2010. 2 2 [2] R. E. Nickel and G. A. Secor, “Convergence of consistently √ (29) 𝜆 =𝜔 𝑙 derived Timoshenko beam finite elements ,” International Jour- nal for Numerical Methods in Engineering,vol.5,no. 2, pp.243– 252, 1972. Tables 1, 2, 3,and 4, 5, 6 show dimensionless parameters [3] D.L.Thomas,J.M.Wilson, andR.R.Wilson, “Timoshenko of natural frequency of beam-free vibration for simply and beam finite elements,” Journal of Sound and Vibration,vol.31, clamped supports, respectively, considering three ratios of no.3,pp. 315–330, 1973. thickness to length. Results have been compared to other [4] D. J. Dawe, “A finite element for the vibration analysis of Timo- researchers’ studies. The mentioned tables demonstrate that shenko beams,” Journal of Sound and Vibration,vol.60, no.1, the accuracy of the proposed element is very high in analyz- pp. 11–20, 1978. ing free vibration of beam. [5] K. K. Kapur, “Vibrations of a Timoshenko beam using finite element approach,” Journalofthe Acoustical SocietyofAmerica, 4.2. Second Example (Buckling Analysis). The efficiency of vol. 40, pp. 1058–1063, 1966. theproposedelement is determinedinbucklinganalysis. For [6] A. W. Lees and D. L. Thomas, “Unified Timoshenko beam finite this propose, critical load of a simply and clamped supported element,” Journal of Sound and Vibration,vol.80, no.3,pp. 355– beam-column is computed. The modulus of elasticity, Pois- 366, 1982. son’s ratio, and length of this beam-column are 𝐸 = 10𝑒7 , ] = [7] A. W. Lees and D. L. Thomas, “Modal hierarchical Timoshenko 1/3,and 𝑙=1 , respectively. The buckling loads of simply beam finite elements,” Journal of Sound and Vibration,vol.99, and clamped supported beam-columns are listed in Table 7. no. 4, pp. 455–461, 1985. It demonstrates that the proposed element has high accuracy [8] J. J. Webster, “Free vibrations of shells of revolution using ring in buckling analysis. Figure 3 shows the first three modes of finite elements,” International Journal of Mechanical Sciences, buckling forsimply andclamped supportedbeam-column, vol. 9, no. 8, pp. 559–570, 1967. respectively. [9] S. S. Rao and R. S. Gupta, “Finite element vibration analysis of rotating timoshenko beams,” Journal of Sound and Vibration, vol. 242, no. 1, pp. 103–124, 2001. 5. Conclusion [10] A. J. M. Ferreira, MATLAB Codes for Finite Element Analysis, Thisstudy hasproposedanewbeamfinite elementformu- Springer, 2008. lation forthe stabilityand free vibrationanalysisofbeams [11] P. J. P. Goncalves, M. J. Brennan, and S. J. Elliott, “Numerical with shear effect deformation. For this purpose, displacement evaluation of high-order modes of vibration in uniform Euler- field of the element has been selected from the third degree, Bernoulli beams,” Journal of Sound and Vibration,vol.301,pp. rotation efi ld has been selected from the second degree, and 1035–1039, 2007. shear strain is assumed constant value. By employing the [12] J. Lee and W. W. Schultz, “Eigenvalue analysis of Timoshenko bending and shear strain energy of the element and stationary beams and axisymmetric Mindlin plates by the pseudospectral respect to unknown shear strain, this value is obtained. method,” JournalofSound andVibration,vol.269,no. 3–5, pp. In thefollowing,using theshear strain,the interpolation 609–621, 2004. functions for displacement, and rotation fields of element has [13] Z. Wieckowski and M. Golubiewski, “Improvement in accuracy been exactly calculated. en, Th these interpolation functions of the finite element method in analysis of stability of Euler- Bernoulli and Timoshenko beams,” Thin-Walled Structures ,vol. have been used and stiffness matrix, geometric stiffness 45, no. 10-11, pp. 950–954, 2007. matrix, and mass matrix of the proposed element have been [14] J. B. Kosmatka, “An improved two-node finite element for clearly obtained. Evaluating the efficiency and accuracy of the stability and natural frequencies of axial-loaded Timoshenko element for free vibration and stability analysis of beam with beams,” Computers and Structures, vol. 57, no. 1, pp. 141–149, simply and clamped supports, desirable results are obtained. eTh resultsshowhighaccuracyand ecffi iency of theproposed [15] B.Z.P.Bazant andL.Cedolin, Stability of Structures,Oxford element in calculating natural frequencies and critical load of University Press, New York, NY, USA, 1991. beam with different boundary conditions. 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Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam

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Copyright © 2013 Abbas Moallemi-Oreh and Mohammad Karkon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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Hindawi Publishing Corporation Advances in Acoustics and Vibration Volume 2013, Article ID 841215, 7 pages http://dx.doi.org/10.1155/2013/841215 Research Article Finite Element Formulation for Stability and Free Vibration Analysis of Timoshenko Beam 1 2 Abbas Moallemi-Oreh and Mohammad Karkon Department of Mechanical Engineering, Shahreza Branch, Islamic Azad University, Shahreza, Iran Department of Civil Engineering, Ferdowsi University of Mashhad, Mashhad, Iran Correspondence should be addressed to Mohammad Karkon; karkon443@gmail.com Received 18 February 2013; Revised 1 April 2013; Accepted 1 April 2013 Academic Editor: K. M. Liew Copyright © 2013 A. Moallemi-Oreh and M. Karkon. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. A two-node element is suggested for analyzing the stability and free vibration of Timoshenko beam. Cubic displacement polynomial and quadratic rotational fields are selected for this element. Moreover, it is assumed that shear strain of the element has the constant value. Interpolation functions for displacement field and beam rotation are exactly calculated by employing total beam energy and its stationing to shear strain. By exploiting these interpolation functions, beam elements’ stiffness matrix is also examined. Furthermore, geometric stiffness matrix and mass matrix of the proposed element are calculated by writing governing equation on stability and beam free vibration. At last, accuracy and efficiency of proposed element are evaluated through numerical tests. es Th e tests show high accuracy of the element in analyzing beam stability and finding its critical load and free vibration analysis. 1. Introduction The first high-order element was proposed by Kapur with eight degrees of freedom [5]. Lees and o Th mas formulated a Twoversionsoftheorieshavebeendeveloped foranalysisof complex element by applying independent polynomial series beams.InEuler-Bernoullitheory,thedisplacementofbeamis for displacement and rotation fields [ 6, 7]. Also, this method considered without shear effects. This method gives appropri- has been used by Webster [8]. Rao and Gupta have examined ate and acceptable response in thin beam in which shear effect free vibration of rotating beams [9]. In some methods, like is insignificant. However, in this approach, by increasing the isoparametric formulation, displacement and rotation fields thickness of beam and shear eeff ct deformation, the error of are assumed dependently with the same order [10]. Based response is increasing [1]. Correspondingly, the eeff ct of shear on Euler-Bernoulli theory, Goncalves et al. have presented transformation is formulated in Timoshenko theory. er Th e- frequency equation and vibration modes for classical bound- fore, this method has a better result, especially in deep beams ary conditions such as clamped, free, pinned, and sliding in which shear effect is impressive. Although the rotational supports [11]. Leeand Schultzhaveconsidered free vibration inertia of thick beams was investigated by Rayleigh for the of Timoshenko beam through pseudospectral method [12]. first time, Timoshenko has developed this theory and formu- So far, very little research has been done on buckling of lated shear effect. Due to the complexity of the governing Timoshenko beam with respect to free vibration analysis. equations of the free vibration and stability of beams in Exploiting an approximate method based on n fi ite elements general, numerical methods such as finite element have been formulation, Wieckowski and Golubiewski examined beam developed profoundly. Up to now, many elements have been stability of Euler-Bernoulli and Timoshenko theories [13]. presented based on Timoshenko theory. es Th e elements are Also, Kosmatka has proposed a two-node element for stabil- classified into two groups which are simple and higher order ity and free vibration analysis of Timoshenko beam [14]. elements. Some researchers used simple two-node elements In this study, a new beam element is proposed for free with four degrees of freedom [2–4]. Thomas et al. have vibration and stability analysis of Timoshenko beams. In examined the elements proposed by other researchers [3]. order to compute shape function of the beam element, cubic 2 Advances in Acoustics and Vibration 𝑥 𝑠 In the present formula, the coecffi ients of the terms 𝑠 and 𝑠 are equivalent to zero. Therefore, in the succeeding lines, 𝛼 , 𝑖 0 𝜃 𝛽 are determined in terms of the unknown parameter 𝛾 : 𝜃 𝑗 1 0 𝑤 Γ= (𝑤 −𝑤 )−(𝜃 +𝜃 ) 𝑗 𝑗 𝑖 𝑖 𝑗 Figure 1: Timoshenko beam element. 𝛽 = (𝜃 −𝜃 ) 0 𝑖 𝑗 (3) 3 1 displacement polynomial and quadratic rotational fields are 𝛼 =− (𝛾 − Γ) 0 0 selected. In addition, shear strain of the element is assumed 2 2 constant. en, Th by exploiting the bending and shear strain energy of the beam and stationary with respect to constant 𝛽 = 𝛼 . 1 0 shear strain, interpolation function of this element has been At this stage, there is only one unknown constant 𝛾 ,which carefully calculated. In the following, by using these shape can be discovered through the condition of minimum strain functions, the stiffness matrix, geometric stiffness matrix, energy. It should be added that the structural strain energy is and mass matrix of the proposed element have been exactly the sum of bending and shear strain energy. Bending strain determined. Finally, several numerical tests are performed to energy is calculated in the following way: investigate the robustness of this element for free vibration and stability analysis of beams with dieff rent boundary 𝑙 1 2 2 (4) conditions. eTh ndings fi prove that the suggested element has 𝑈 = ∫ 𝜅 = ∫ 𝜅 𝑑𝑠. 2 4 0 −1 high level of accuracy and free of shear locking. In (4), 𝜅 represents curvature which is determined as follows: 2. Finite Element Formulation 𝜅=− ⋅ =𝜅 −6 𝑙 𝑙 In the ni fi te element method, displacement and rotation elds fi (5) of the element are associated with interpolation functions to 𝜅 = (𝜃 −𝜃 +3Γ𝑠 ). 0 𝑖 𝑗 nodal degrees of freedom. Figure 1 shows proposed element with two nodes. eTh shape functions of the Timoshenko Substituting (5)into(4) leads to the following bending strain beam are calculated based on Figure 1. In order to compute energy: shape function of the beam in Figure 1,cubic displacement Γ𝛾 0 0 polynomial and quadratic rotational fields are selected. Addi- 𝑈 =𝑈 +6𝐼𝐸(− + ) 𝑏 0 𝑙 𝑙 tionally, it is assumed that shear strain has the constant value (6) of 𝛾 . Based on these assumptions, (1)can be writtenas follows: 𝑈 = ∫ 𝜅 𝑑𝑠. 0 0 −1 𝑤= (1−𝑠 )+ (1+𝑠 ) Besides, (7)shows theenergyofshear strain: 2 2 2 2 +𝛽 𝑙(1 − 𝑠 )+𝛽 (1 − 𝑠 ) 𝑈 = 𝛾 0 1 ∫ 2𝑓 𝜃 𝑗 𝑖 2 (1) 1 𝜃= (1−𝑠 )+ (1+𝑠 )+𝛼 (1 − 𝑠 ) 𝑙 (7) 2 2 = ∫ 𝛾 4𝑓 −1 𝛾=𝛾 = 𝛾 . 2𝑥 2𝑓 𝑠= −1. In this equation, 𝑓 is shear correction factor which depends In these equations, 𝛽 , 𝛽 , 𝛼 ,and 𝛾 are unknown parame- 1 0 0 0 on cross section shape of beam. This coefficient for rectan- ters. In order to determine their values, at first, the equation of gular section is 5/6. By adding the bending and shear strain shear strain for Timoshenko beam is established. By utilizing energy together, total strain energy is found as follows: the shear strain value equal to 𝛾 , the subsequent equations 6𝐸𝐼Γ 6𝐸𝐼 𝑙 2 2 will be available: 𝑈=𝑈 +𝑈 =𝑈 − 𝛾 + 𝛾 + 𝛾 . (8) 𝑏 𝑠 0 0 0 0 𝑙 𝑙 2𝑓 𝛾= −𝜃 = ⋅ −𝜃 Implementing 𝜕𝑈/𝜕𝛾 =0 will give the following results: 6𝐸𝐼Γ 2 𝑤 𝑖 2 𝛾 = =𝛿Γ (2) 0 𝛾 = (− + −2𝛽 𝑙𝑠 + 𝛽 𝑙−3𝛽 𝑙𝑠 ) 2 0 0 1 1 𝑙 /𝑓 + 12𝐸𝐼 𝑙 2 2 (9) 1−𝑠 1+𝑠 6𝜆 𝑓 2 𝑠 −𝜃 ( )−𝜃 ( )−𝛼 (1 − 𝑠 ). 𝛿= ,𝜆= . 𝑖 𝑗 0 2 2 𝑙 + 12𝜆 𝐺𝐴 𝐸𝐼 𝐺𝐴 𝑑𝑠 𝑑𝑥 𝑑𝑤 𝑑𝑤 𝐺𝐴 𝐺𝐴 𝑑𝑠 𝐺𝐴 𝑑𝑥 𝑙𝑠 𝐺𝐴 𝐸𝐼 𝑑𝑠 𝑠𝛾 𝑑𝜃 𝑑𝑥 𝐸𝐼 𝐸𝐼 Advances in Acoustics and Vibration 3 Substituting 𝛽 , 𝛽 , 𝛼 ,and 𝛾 into (1), the succeeding shape By calculating (13), the stiffness matrix of the proposed ele- 1 0 0 0 functions for Timoshenko beam can be as follows: ment is obtained in the succeeding forms: { } { } { } [𝐾 ]= 𝑤̃ 𝜃 𝑁 𝑁 𝑁 𝑁 0 𝑛𝑖 3 1 2 3 4 𝑙 + 12𝑙𝜆 { }=[ ] ̃ { } 𝜃 𝑁 𝑁 𝑁 𝑁 { } 5 6 7 8 𝑛 { } 12 6𝑙 −12 6𝑙 { } (15) 2 2 [ ] 6𝑙 4𝑙 + 12𝜆 −6𝑙 2𝑙 − 12𝜆 [ ] × . 3 [ ] −12 −6𝑙 12 −6𝑙 𝑁 = [2 + 𝑠 (1−2𝛿 )+𝑠 (−3 + 2𝛿 )] 2 2 6𝑙 2𝑙 − 12𝜆 −6𝑙 4𝑙 + 12𝜆 [ ] 2 3 𝑁 = [0.5 (1 − 𝑠 )+(𝑠 −𝑠) 0.5 − 𝛿 ] ( ) 4 3. Mass Matrix The kinetic energy, 𝑇 , of an elemental length, 𝑙 ,ofauniform 𝑁 = [2 − 𝑠 (1−2𝛿 )−𝑠 (−3 + 2𝛿 )] Timoshenko beam is given as follows [1]: (10) 2 3 𝑙/2 𝑙/2 𝑁 = [−0.5 (1 − 𝑠 )+(𝑠 −𝑠) (0.5 − 𝛿 )] 4 1 1 2 2 𝑇= ∫ 𝑦 ̇ + ∫ 𝜃 𝑑𝑥. (16) 2 2 −𝑙/2 −𝑙/2 𝑁 = [6 (1 − 𝑠 )(−1 + 2𝛿 )] 4𝑙 In this equation, 𝜌 is the mass density of the material of the beam and 𝐼 is the second moment of area of cross section. 𝑁 = [−1 + 𝑠 (−2 + 3𝑠 )+6(1 −𝑠 )𝛿] eTh refore, the mass matrix of the element has been two parts, one related to translations and the other related to rotations, 2 in the form of 𝑁 = [−6 (1 − 𝑠 )(−1 + 2𝛿 )] 4𝑙 [𝑀 ] =[𝑀 ]+[𝑀 ] 1 2 𝑁 = [−1 + 𝑠 2+3𝑠 +6(1 −𝑠 )𝛿]. ( ) 8 1 4 𝑇 = ∫ [𝑁 ] [𝑁 ]𝑑𝑠 𝑤 𝑤 (17) Innfi iteelementmethod,displacementsandstrainfieldofthe element can be related to the nodal degrees of freedom with 𝑙/2 interpolation functions. Hence, equations of finite element + ∫ [𝑁 ] [𝑁 ] 𝑑𝑠. 𝜃 𝜃 −𝑙/2 formulation can be written as follows: eTh translation mass matrix [𝑀 ] is achieved as follows: { }= [𝑁 ] {𝐷 } [𝑀 ]= (11) 210(12𝜆 + 𝑙 ) { } { } { } 𝜀 = = 𝐵 𝐷 , { } [ ] { } { } (18) 𝑚 𝑚 𝑚 𝑚 { } 11 12 13 14 − +𝜃 { } [ ] 𝑚 𝑚 −𝑚 𝑚 12 15 14 16 { } [ ] × . [ ] 𝑚 −𝑚 𝑚 −𝑚 13 14 11 12 where 𝑚 𝑚 −𝑚 𝑚 14 16 12 15 [ ] 0 In the following, the entries of this matrix are introduced: [ ] [𝐵 ] = [ ] [𝑁 ] (12) −1 6 2 2 4 [ ] 𝑚 = (1680𝜆 + 294𝑙 𝜆 + 13𝑙 ) and [𝑁] is the matrix of interpolation function. Also, {𝐷} 2 2 4 is nodal displacement and [𝐵] is strain matrix. Consequently, 𝑚 = (1260𝜆 + 231𝑙 𝜆 + 11𝑙 ) stiffness matrix of Timoshenko element can be obtained as follows: 9 2 2 4 𝑚 = (560𝜆 + 84𝑙 𝜆+3𝑙 ) (19) [𝐾 ]= ∫ 𝐵 [𝐷 ] 𝐵 𝑑𝑥, (13) [ ] [ ] 0 𝑚 2 2 4 𝑚 =− (2520𝜆 + 378𝑙 𝜆 + 13𝑙 ) 2𝑙 where, [𝐷 ]istheelasticitymatrixforTimoshenkobeamele- ment: 2 2 2 4 𝑚 = (126𝜆 + 21𝑙 𝜆 + 13𝑙 ) [ ] [𝐷 ]= . (14) 𝑚 2 2 4 𝑚 =− (168𝜆 + 28𝑙 𝜆 + 13𝑙 ). 𝑓 2 [ 𝑠 ] 2𝑙 𝐺𝐴 𝐸𝐼 𝑑𝑥 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑦 𝜌𝐴 𝜌𝐼 𝜌𝐴 𝜌𝐼 𝑑𝑥 𝜌𝐴 𝑛𝑗 𝐸𝐼 4 Advances in Acoustics and Vibration In addition, the rotation mass matrix [𝑀 ] can be obtained 𝑑𝑠 as follows: [𝑀 ]= 𝑑𝑥 𝑑𝛿 30(12𝜆 + 𝑙 ) Figure 2: Straight and buckled form. (20) 𝑚 𝑚 −𝑚 𝑚 21 22 21 22 [ ] 𝑚 𝑚 −𝑚 𝑚 22 23 22 24 [ ] × . [ ] −𝑚 −𝑚 𝑚 −𝑚 21 22 21 12 By calculating this equation, the geometric stiffness matrix 𝑚 𝑚 −𝑚 𝑚 22 24 12 23 [ ] becomes as follows: The entries of this matrix are defined in the below form: 𝑃 [𝐾 ]= 60𝑙 𝑚 = 36𝑙 𝑘 𝑘 𝑘 𝑘 𝑔1 𝑔2 𝑔3 𝑔2 (25) [ ] 𝑘 𝑘 −𝑘 𝑘 𝑚 = −3 (60𝜆 − 𝑙 ) 𝑔2 𝑔4 𝑔2 𝑔5 [ ] × , [ ] 𝑘 −𝑘 𝑘 −𝑘 𝑔3 𝑔2 𝑔1 𝑔2 (21) 2 2 4 𝑘 𝑘 −𝑘 𝑘 [ 𝑔2 𝑔5 𝑔2 𝑔4 ] 𝑚 = (360𝜆 + 15𝑙 𝜆+3𝑙 ) where 2 2 4 𝑚 = (720𝜆 − 60𝑙 𝜆−𝑙 ). 𝑘 = 60 + 12𝛽 𝑔1 𝑘 =6𝑙𝛽 𝑔2 4. Geometric Stiffness Matrix 𝑘 = −60 − 12𝛽 𝑔3 The concept of the neutral state of equilibrium is used for buckling analysis of beam. In the mathematical formulation 2 2 (26) 𝑘 =5𝑙 +3𝑙𝛽 𝑔4 of elastic stability of beam, the neutral equilibrium is used assuming a bifurcation of the deformations. aTh t is, at the 2 2 𝑘 =3𝑙𝛽 −5𝑙 𝑔5 critical load, of the possible two paths of deformations (one associated with the stable equilibrium and the other pertinent to the unstable equilibrium condition, as shown in Figure 2), 𝛽= . (𝑙 + 12𝜆) the beam always takes the buckled form. In addition to the existence of this bifurcation of equilibrium paths, the elastic For stability analysis and determination of the magnitude of a stability analysis of plates assumes the validity of Hooke’s static compressive axial load that will produce beam buckling, law. By considering Figure 2 at the buckled form, the axial the following eigenvalue will be achieved: shortening of beam can be acquired as follows: det ([𝐾 ])= det ([𝐾 ]−𝑃[𝐾 ]) = 0⇒ 󳨐 𝑃 . (27) 0 𝑔 cr 2 2 = (𝑑𝑥 +𝑑𝑦 ) eTh lowest positive eigenvalue of this equation is the magni- tude of buckling load and the corresponding eigenvector is ≃𝑑𝑥 + ( ) (22) the deformed shape of the buckled beam. eTh exact solution of beam-column buckling load with shear deformation eeff ct 1 is obtained as succeeding form [15]: 󳨐⇒ 𝛿𝑑 = ( ) 𝑑𝑥. 𝑃 = cr As aresult, thestrainenergyofaxial load 𝑃 canbeexpressed eff (28) as follows: ×( ), 1+(𝜋 𝑓 ) / (𝐿 ) 𝑃 eff (23) Δ𝑊 = ∫ ( ) 𝑑𝑥. where 𝐿 istheeeff ctivebeamlengthinwhich (𝐿 =𝐿) and eff eff (𝐿 = 𝐿/2) areusedfor pinned-pinnedbeams andfixed- eff Basedon(23), geometric stiffness matrix of the element can fixed beams, respectively. be calculated as follows: 4.1. First Example (Vibration Analysis). The efficiency of pro- 𝑤 𝑤 posedelement is evaluatedbyanalyzing thefreevibration (24) [𝐾 ]= ∫ [ ] 𝑃[ ]𝑑𝑥. of beam with simply and clamped supports for various 𝑑𝑥 𝑑𝑥 𝑑𝑁 𝑑𝑁 𝑑𝑥 𝑑𝑦 𝐺𝐴 𝐸𝐼 𝐸𝐼 𝑑𝑥 𝑑𝑦 𝑑𝑥 𝑑𝑥 𝑑𝑦 𝑑𝑠 𝜌𝐼𝑙 𝑑𝑦 Advances in Acoustics and Vibration 5 Table 1: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.002 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1428 3.14158 3.14158 2 6.28319 6.2928 6.28310 6.28310 3 9.42478 9.4573 9.42449 9.42450 4 12.5664 12.6437 12.5657 12.5657 5 15.7080 15.8596 15.7066 15.7068 6 18.8496 19.1127 18.8473 18.8476 7 21.9911 22.4113 21.9875 21.9883 8 25.1327 25.7638 25.1273 25.1288 9 28.2743 29.1793 28.2666 28.2692 10 31.4159 32.6672 31.4053 31.4098 No.1 No.2 No.3 No.2 No.1 No.3 1 1 1 1 0.9 0.9 0.9 0.9 0.9 0.9 0.8 0.8 0.8 0.8 0.8 0.8 0.7 0.7 0.7 0.7 0.7 0.7 0.6 0.6 0.6 0.6 0.6 0.6 0.5 0.5 0.5 0.5 0.5 0.5 0.4 0.4 0.4 0.4 0.4 0.4 0.3 0.3 0.3 0.3 0.3 0.3 0.2 0.2 0.2 0.2 0.2 0.2 0.1 0.1 0.1 0.1 0.1 0.1 0 0 0 0 0 0 0.5 −0.5 0 0.5 −0.2 0 0.2 −0.4−0.2 0 −0.5 0 0.5 −0.5 0 0.5 (a) (b) Figure 3: The first three modes of buckling for beam-column, (a) simply supported and (b) clamped supported. Table 2: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.01 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1425 3.14133 3.14133 2 6.28319 6.2908 6.28106 6.28106 3 9.42478 9.4503 9.41761 9.41764 4 12.5664 12.6271 12.5494 12.5496 5 15.7080 15.8267 15.6749 15.6754 6 18.8496 19.0552 18.7926 18.7937 7 21.9911 22.3186 21.9011 21.9034 8 25.1327 25.6231 24.9988 25.0034 9 28.2743 28.9749 28.0845 28.0927 10 31.4159 32.3806 31.1568 31.1705 6 Advances in Acoustics and Vibration Table 3: Dimensionless frequency parameter 𝜆 for the simply supported Timoshenko beam. ℎ/𝑙 = 0.1 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 3.14159 3.1169 3.11568 3.11569 2 6.28319 6.0993 6.09066 6.09094 3 9.42478 8.8668 8.84052 8.84229 4 12.5664 11.3984 11.3431 11.3492 5 15.7080 13.7089 13.6132 13.6282 6 18.8496 15.8266 15.6790 15.7093 7 21.9911 17.7811 17.5705 17.6239 8 25.1327 19.5991 19.3142 19.3997 9 28.2743 21.3030 20.9325 21.0606 10 31.4159 22.9117 22.4441 22.6257 Table 4: Dimensionless frequency parameter 𝜆 for the clamped supported Timoshenko beam. ℎ/𝑙 = 0.002 Mode Euler theory Ferreira [10]LeeandSchultz[12]Proposedelement 1 4.73004 4.7345 4.72998 4.72998 2 7.85320 7.8736 7.85295 7.85296 3 10.9956 11.0504 10.9950 10.9950 4 14.1372 14.2526 14.1359 14.1360 5 17.2788 17.4888 17.2766 17.2768 6 20.4204 20.7670 20.4168 20.4174 7 23.5619 24.0955 23.5567 23.5578 8 26.7035 27.4833 26.6960 26.6980 9 29.8451 30.9398 29.8348 29.8382 10 32.9867 34.4748 32.9729 32.9786 Table 5: Dimensionless frequency parameter 𝜆 for the clamped Table 6: Dimensionless frequency parameter 𝜆 for the clamped 𝑖 𝑖 supported Timoshenko beam. supported Timoshenko beam. ℎ/𝑙 = 0.01 ℎ/𝑙 = 0.1 Mode Euler theory Mode Euler theory Lee and Lee and Ferreira Proposed Ferreira Proposed Schultz Schultz [10] element [10] element [12] [12] 1 4.73004 1 4.73004 4.7330 4.72840 4.72840 4.5835 4.57955 4.57962 27.85320 27.85320 7.8675 7.84690 7.84692 7.3468 7.33122 7.33193 3 10.9956 3 10.9956 11.0351 10.9800 10.9801 9.8924 9.85611 9.85918 4 14.1372 4 14.1372 14.2218 14.1062 14.1064 12.2118 12.1454 12.1540 517.2788 517.2788 17.4342 17.2246 17.2253 14.3386 14.2324 14.2513 6 20.4204 6 20.4204 20.6783 20.3338 20.3355 16.3046 16.1487 16.1841 723.5619 723.5619 23.9600 23.4325 23.4358 18.1375 17.9215 17.9807 8 26.7035 8 26.7035 27.2857 26.5192 26.5253 19.8593 19.5723 19.6641 929.8451 929.8451 30.6616 29.5926 29.6032 21.4875 21.1185 21.2523 10 32.9867 10 32.9867 34.0944 32.6514 32.6687 23.0358 22.5735 22.7598 lengths to thickness ratio. The Poisson’s ratio of this beam To compare other researchers’ results, frequency dimen- is ] = 0.3 and shear correction factor is taken 5/6. sionless parameter 𝜆 , defined in ( 29), has been shown in 𝑖 Advances in Acoustics and Vibration 7 Table 7: Critical load of the simply and clamped supported beam-column. Simply supported Clamped supported 𝑙/ℎ Analytical solution Ferreira [10] Proposed element Analytical solution Ferreira [10]Proposedelement 10 8013.8 8021.8 8013.86 29766 29877 29770 100 8.223 8.231 8.2225 32.864 32.999 32.864 1000 0.0082 0.0082 0.00822 0.0329 0.0330 0.0329 (Tables 1, 2, 3, 4, 5,and 6) for 10 rfi st frequencies of this beam References by exploiting 40 elements: [1] M. Petyt, Introduction of Finite Element Vibration Analysis, Cambridge University Press, 2nd edition, 2010. 2 2 [2] R. E. Nickel and G. A. Secor, “Convergence of consistently √ (29) 𝜆 =𝜔 𝑙 derived Timoshenko beam finite elements ,” International Jour- nal for Numerical Methods in Engineering,vol.5,no. 2, pp.243– 252, 1972. Tables 1, 2, 3,and 4, 5, 6 show dimensionless parameters [3] D.L.Thomas,J.M.Wilson, andR.R.Wilson, “Timoshenko of natural frequency of beam-free vibration for simply and beam finite elements,” Journal of Sound and Vibration,vol.31, clamped supports, respectively, considering three ratios of no.3,pp. 315–330, 1973. thickness to length. Results have been compared to other [4] D. J. Dawe, “A finite element for the vibration analysis of Timo- researchers’ studies. The mentioned tables demonstrate that shenko beams,” Journal of Sound and Vibration,vol.60, no.1, the accuracy of the proposed element is very high in analyz- pp. 11–20, 1978. ing free vibration of beam. [5] K. K. Kapur, “Vibrations of a Timoshenko beam using finite element approach,” Journalofthe Acoustical SocietyofAmerica, 4.2. Second Example (Buckling Analysis). The efficiency of vol. 40, pp. 1058–1063, 1966. theproposedelement is determinedinbucklinganalysis. For [6] A. W. Lees and D. L. Thomas, “Unified Timoshenko beam finite this propose, critical load of a simply and clamped supported element,” Journal of Sound and Vibration,vol.80, no.3,pp. 355– beam-column is computed. The modulus of elasticity, Pois- 366, 1982. son’s ratio, and length of this beam-column are 𝐸 = 10𝑒7 , ] = [7] A. W. Lees and D. L. Thomas, “Modal hierarchical Timoshenko 1/3,and 𝑙=1 , respectively. The buckling loads of simply beam finite elements,” Journal of Sound and Vibration,vol.99, and clamped supported beam-columns are listed in Table 7. no. 4, pp. 455–461, 1985. It demonstrates that the proposed element has high accuracy [8] J. J. Webster, “Free vibrations of shells of revolution using ring in buckling analysis. Figure 3 shows the first three modes of finite elements,” International Journal of Mechanical Sciences, buckling forsimply andclamped supportedbeam-column, vol. 9, no. 8, pp. 559–570, 1967. respectively. [9] S. S. Rao and R. S. Gupta, “Finite element vibration analysis of rotating timoshenko beams,” Journal of Sound and Vibration, vol. 242, no. 1, pp. 103–124, 2001. 5. Conclusion [10] A. J. M. Ferreira, MATLAB Codes for Finite Element Analysis, Thisstudy hasproposedanewbeamfinite elementformu- Springer, 2008. lation forthe stabilityand free vibrationanalysisofbeams [11] P. J. P. Goncalves, M. J. Brennan, and S. J. Elliott, “Numerical with shear effect deformation. For this purpose, displacement evaluation of high-order modes of vibration in uniform Euler- field of the element has been selected from the third degree, Bernoulli beams,” Journal of Sound and Vibration,vol.301,pp. rotation efi ld has been selected from the second degree, and 1035–1039, 2007. shear strain is assumed constant value. By employing the [12] J. Lee and W. W. Schultz, “Eigenvalue analysis of Timoshenko bending and shear strain energy of the element and stationary beams and axisymmetric Mindlin plates by the pseudospectral respect to unknown shear strain, this value is obtained. method,” JournalofSound andVibration,vol.269,no. 3–5, pp. In thefollowing,using theshear strain,the interpolation 609–621, 2004. functions for displacement, and rotation fields of element has [13] Z. Wieckowski and M. Golubiewski, “Improvement in accuracy been exactly calculated. en, Th these interpolation functions of the finite element method in analysis of stability of Euler- Bernoulli and Timoshenko beams,” Thin-Walled Structures ,vol. have been used and stiffness matrix, geometric stiffness 45, no. 10-11, pp. 950–954, 2007. matrix, and mass matrix of the proposed element have been [14] J. B. Kosmatka, “An improved two-node finite element for clearly obtained. Evaluating the efficiency and accuracy of the stability and natural frequencies of axial-loaded Timoshenko element for free vibration and stability analysis of beam with beams,” Computers and Structures, vol. 57, no. 1, pp. 141–149, simply and clamped supports, desirable results are obtained. eTh resultsshowhighaccuracyand ecffi iency of theproposed [15] B.Z.P.Bazant andL.Cedolin, Stability of Structures,Oxford element in calculating natural frequencies and critical load of University Press, New York, NY, USA, 1991. beam with different boundary conditions. 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