Dynamic Behavior of Double Layer Cylindrical Space Truss Roofs

Transcription

1 Australian Journal of Basi and Applied Sienes, 5(8): 68-75, 011 ISSN Dynami Behavior of Double Layer Cylindrial Spae Truss Roofs 1, M. Jamshidi, T.A. Majid and 1 A. Darvishi 1 Faulty of Engineering, Islami Azad University, Chaloos Branh, Mazandaran, Iran. Shool of Civil Engineering, Engineering Campus, UniversitiSains Malaysia, Nibong Tebal, Penang, Malaysia. Abstrat: Reently, there are a lot of studies on dynami behaviors of double layer ylindrial spae trusses. In some ase, they used H/S (Rise to Span) or α (initial angle) as a main geometrial harateristi of struture and tried to depend the dynami behavior of these strutures to them (H/S, α). Moreover on those papers, they lassify their models base on these mentioned parameters. But in this paper, the effets of eah geometrial parameter suh as Rise, Span, length, H/S (Rise to Span),H*S*L(Rise multiplied by Span multiplied by length) on dynami behavior of vault trusses are studied. For this purpose, the main period time of struture as a dynami harateristi that inludes mass and stiffness matrix property is hosen, and the effetive of strutural geometry on the main period are studied. Key words: Barrel vaults, Double layer, Dynami study, Spae truss. INTRODUCTION Spae trusses are one of the lightest steel strutures with three-dimensional and omplex strutural behaviors made of thousands of steel tubular bars onneted together by nodes. The very high degree of indeterminay, their multiple redundanies and their appropriate three-dimensional geometrial forms provide additional margins of safety to prevent them from sudden ollapse in the ase of aidental loal failure of one or more elements, when the overall loading is below the servie load. They have beome widely popular as large span roof strutures, partiularly in areas suh as sport enters, exhibition halls and airport hangars. The main advantages of these strutures are that they are light in weight, have a high degree of indeterminay and great stiffness, simple prodution and fast assembly, are totally prefabriated, do not need site welding, are easily formed into various attrative geometrial surfaes, have the ability to over large areas with widely spaed olumn supports, have generally good response against earthquakes and are ost effetive. With their low weight and great stiffness, spae trusses are believed to attrat low fores during seismi ativities and an be onsidered to be amongst the least likely to suffer damage, when ompared to other large span roofs (Marsh, 000) Despite the earlier assumptions, reent studies show that where strong earthquakes are probable, these strutures are vulnerable to seismi failures, espeially when roofs are overed with snow. Kawaguhi (1997) reported damages due to earthquake. M.Rezaeei el al. (011) presented a new method to alulate the visous fititious damping for dynami relaxation on some strutures suh as spae truss. K. Koohestani and A. Kaveh (010) used a new method for vibration analysis of single and double layer shallow dome. J.G. Cai el al. (008) studied on the seismi performane of spae beam string struture. Coan and Plaut (Coan, 1983) determined the dynami response of lattie dome. Sadeghi (004) investigated the dynami behavior of double layer barrel vaults; and showed that they are vulnerable to earthquakes and have a brittle behavior. Zhang and Lan (000) have reviewed researh findings on dynami harateristis of spae trusses. Further studies on seismi behavior of spae strutures by Ishikawa et al (000). Kuneida et al. (011) studied the vibrational harateristis of some existing strutures and Ishikawa. The above studyshowed the importane and the need for arrying out dynami analysis in design of spae trusses. But it must regarded that dynami analysis of a spae truss is, in some manner,a umbersome proedure due to extreme omplexity of strutural onfiguration and numerous degree of freedoms, as well as a node to node distribution of mass whih would onsequently results in a ompliated dynami response. So some engineers refuse to perform the dynami analysis of spae truss in their designs. The main aim of this paper is to simplify the dynami study of this type of strutures. Corresponding Author: M. Jamshidi, Faulty of Engineering, Islami Azad University, Chaloos Branh, Mazandaran, Iran. 68

2 Aust. J. Basi & Appl. Si., 5(8): 68-75, double layer ylindrial spae truss (DLCST) are modeled and their dynami behavior are studied and the effet of strutural geometry on their behavior is investigated.. First Natural Period of Struture as a Dynami Charateristi: The free vibration dynami equilibrium equation for a spae frame with multi-degree of freedom (MDF) without visous damping is as follows... [ M][()] v t [ K][()] v t 0 Where, [M] and [K] are respetively the struture mass and stiffness matries. [v] and [v(t)]are the aeleration and displaement vetors of strutures. In order to alulate the natural j th mode shape of the free vibration with frequeny ω j it an be assumed that: (1) v () t sin( t) j j () And in a vetor form: [] v [ ]sin( t) (3) With replaing Eqn. 3 into Eqn. 1: {[ M][ ]( ) [ K][ ]}sin( t) 0 (4) Or [ K] [ M][ ] 0 (5) This is the equation alled "harateristi or frequeny equation" that used for a MDF system with 'N' as the number of freedom. Calulation of natural mode shapes of a Multi Degree Freedom struture is resulted to finding the eigenvalues of The Eq. 5. As soon as ω j, j=1 to N, are alulated, by using the Eqn. 6 the natural period of vibration an be obtained from natural irular frequeny of vibration: T (6) As it is shown, the duration period of struture in eah mode is depended on the mass and stiffness matrix. The dynami behavior of the struture is a funtion of these matrixes too (Eq. 1). So in this paper, the main period of struture is hosen as a dynami harateristi of spae frames. A omprehensive study on the effet of strutural onfiguration and geometry on main vibration mode is arried out..1. Effet of the Damping on the Main Period Time: The free vibration dynami equilibrium equation for single-degree of freedom (SDF) with visous damping is as follows. mu u ku 0 (7) k u u u 0 m m (8) Where, m and k are respetively the struture mass and stiffness and is a damping fator that represented the energy dissipation in a yle of amplitude or a period of fored harmoni vibration. If we onsider: 69

3 Aust. J. Basi & Appl. Si., 5(8): 68-75, 011 k n n m m Then the equation ( 8) will be: u u u n n 0 (9) Where ξ is alled, damping ratio and it's depend to mass and stiffness of the system. In most mention struture suh as buildings, bridges, dams, ξ is less than 0.1. By solving (Eq. 9) for the system with ξ < 1, displament funtion will be arrived as bellow: u(0) u (0) n ut () e t u(0)os n Dt sindt D (10) Where D n 1 that shows the natural frequeny for the system with damping fator is related to the natural frequeny for the system without damping. Moreover, damping detrated natural frequeny from ω n to the ω D and inreased period time from T n to the T D but for the system with ξ < 0%, it's effet on the ω and T is negleted (Chopra, 1995). As the damping ratio for the ommon struture are loated in this range, so ω D and T D are approximately equal to the ω n and T n... Calulating the Mass and Stiffness Matrix for Spae Trusses (Weaver, 1987): An element of a truss that is hinged in the joint j and k is shown in the Fig1. In this study, the onnetions are onsidered in the ideal manner so the rotations in the eah end of the element are negleted. The two main flexural surfaes are defined by the surfaes that are built from y and z with x (loal axis). In eah end, the translation in diretion x,y and z are shown with three numbered arrow. The stiffness matrix (6 6) for the prismati element in the loal diretion is represented as follow: Sym kjj kjk EA k kkj k kk L Fig. 1: An element of a truss. 70

4 Aust. J. Basi & Appl. Si., 5(8): 68-75, 011 Whereas, there isn't any stiffness due to onnetion at the joints in the perpendiular diretion to the truss axis, most of the terms in the k will be zero. As the same way, for the loal axis, the mass matrix will be shown as follow: 0 Sym Mjj Mjk AL 0 0 M Mkj M kk To form the rotation matrix, third point suh as P (in addition of J and K) is used for defining flexural surfaes. This point is loated in x -y surfae but do not lie on the x axis. If possible, this point must be onsidered as another joint of the struture that its oordinate are determined. The sentenes of the matrix will be driven by onsidering the properties of vetor multipliation. e z e e x x e e jp jp (11) e e e y z x (1) Where (e) is a vetor in its index diretion. For example e x is equal to: ex x y z (13) x x jk L y y jk L z z jk L xjk yjk zjk L (14) The same desription an be presented for the unite vetor by using the oordinate of j and p. If the rotational matrix is used for this three unite vetors, then: ex x y z R e y 1 3 e z The R operator with 6*6 is used to transfer the stiffness matrix into the strutural diretion: R 0 R 0 R By using the above operator, the stiffness matrix will be presented in the strutural diretion as follow: x x y y T EA x z y z z K R KR L x xy xz x x y y y z x y y x z y z z x z y z z 71

5 Aust. J. Basi & Appl. Si., 5(8): 68-75, 011 Noted that there are a lot of parameter that influene in the stiffness matrix suh as how the struture is onneted to the earth or another struture as a support and how the element are onneted together, but when the stiffness matrix is written, the effet of these parameter are onsidered. So any variation on these parameters made a hange in the stiffness matrix and finally on the period time. Consequently, the stiffness matrix is affeted by the type of onnetion that will be mentioned in setion 3. As the same way, for the mass matrix: T M RMR 3. Method of Study: In this artile 50 double layer ylindrial spae truss roofs (DLCST) are modeled. The geometrial speifiations of these models are given in table 1 and Fig.. All end nodes of inner layers are hinged to the rigid supports. They have three rotational degrees of freedom, but their transitional degrees are restrained. All inner nodes used for the onnetion of strutural members have three transitional degrees of freedom. Fig. : General geometrial properties of models Mehanial Properties of Materials: It has been assumed that the same material is used for onstrution of all models. Mild steel material, with the Young's modulus of 06 GPa, Poisson's ratio of 0.3 and Yield stress of 43.5 MPa seleted for all members in of all models. The material behavior is proposed to be elasti perfetly. However in none of models the nonlinear behavior is allowed and only linear part of material behavior is ontributed in analysis. Table 1: Geometrial properties of models. Model name Span Rise Length Angl (α) Model name Span Rise Length Angl (α) (S) m (H) m (L) m degree (S) m (H) m (L) m degree M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M5 7

6 Aust. J. Basi & Appl. Si., 5(8): 68-75, Loading Condition: One of the most signifiant loads on spae struture is the snow load. In spae strutures, the ratio of snow to dead load is onsiderably greater than the one in ordinary building. In regular buildings the probability of oinidene of snow and earthquake loads does not play a signifiant role, beause the snow loads are usually a negligible fration of the total seismi weight. On the ontrary, in spae frames snow loads an easily reah or 3 times the self-weight of a spae strutures. Therefore, even a very small probability of experiening a strong seismi event when having heavy snow on the roof an lead to a severe onsequenes suh as ollapse of roof. Hene, it is essential to onsider the ombination of snow and earthquakes in design (Moghaddam, 000). In the analysis onduted in this paper the seismi weight is assumed to be inluded with the whole gravity weight of struture whih in a horizontal projetion is assumed to be 490 Pa plus 0 perent of live load whih is onsidered to be totally due to a snow load of 1370 kpa. Above ondition belongs to a weather zone with extreme old winters. The effet of milder situation in terms of less amount of snow is onsidered to have negligible effet on the dynami behavior of spae frame and is not inluded in analysis presented in this paper Method of Analysis: The mass and stiffness matrix are alulated for any DLCST by using of the equation that presented in setion.to evaluate the effet of eah geometrial harateristion the mass and stiffness matrix -the main parameter that defined the dynami behavior of eah struture- the natural period of vibration (T 0 ) is alulated by equations 5 and 6. And also and numerial analysis are fulfilled by finite element software- SAP000. Table : Main Period Time for Eah Model. Model name P.Time (T) se Model name P.Time (T) se Model name P.Time (T) se Model name P.Time (T) se M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M M Results: For any of DLCST, the natural period time are given in Tab.. Comparing the results from analytial analysis for eah DLCST with numerial analysis that performed by finite element software SAP000 shows the auray of the alulation-their differene is less than se. 5. Disussion: The orrelation oeffiient is alulated for eah seleted geometrial parameter to evaluate their effet on the dynami behavior and also period time of the struture. The orrelation is a mathemati oeffiient that determines the relation between two parameters. Two parameters are orrelated together when their value hange uniformly that means while one of the parameter is inrease or derease the other one is inrease or derease, too that their relation an be define by an equation. The orrelation oeffiient will be positive while these two parameters move in a same diretion, otherwise if they moved in ontrary diretion its value must be negative. In this paper the equation that presented by CARL PIRSON are used: r xxy y xx y y 1 (15) 73

7 Aust. J. Basi & Appl. Si., 5(8): 68-75, 011 Aording to table 3, dependene between the main period time as a main harateristi of the struture's dynami behavior and the Rise of the struture is more than other harateristi, although the effet of 'Span H *Rise* Length' and 'Span *Rise' is notieable. In table 3 it is also notieable that the influene of on the S period time is not signifiant but some of researhers tried to ategorize their model based on the (Moghaddam, H 000). It seems the dependene of the dynami behavior of struture to 'Rise to Span' is not reasonable S and it's better to study the variety of dynami behavior based on the Rise. Moreover, aording to table 3, it is not reasonable to relate the dynami property to α. Table 3: dependene between the main time period and geometrial parameter. Correlations H S L α H/S H/L S*H S H Pearson Correlation Proposed an Equation to Calulate Period Time for DLCST: If a diret line is drown between the sporadi point where the summation of the square deviation parallel to Y axis is minimum and then this line is alled regression of Y to X. If the equation of this line is then the value of a and b an be alulated by priniple of minimum square, as follow: y abxanbx (16) xy a xb x Where n is the number of the points and by solving equation 16 and 17, simultaneously, the value of a and b will be derived. The summation of the square regression is defined as follow whih is a riterion of sporadi for the predited Y value (derived from the regression line) from the average Y's value. if this value are near to 100% then the trust to the regression line will be inrease. For arriving the approximate equation to alulate the main period time of vault truss (DLCST), three statisti models are used. The auray of eah model is evaluated based on 'R square'. Table 4: "T" is a funtion of (S*H*L), (S*H) and H. Model a b Independent parameter S*H*L,S*H,H S*H, H H Dependent parameter T T T R Square Aording to table 4, we an see that the value of 'R square' in model '' is smaller than model 'a' and 'b', but it is equal in model 'a' and 'b', so for simplifying, the model 'b' is seleted to arrive an approximate equation, beause it has fewer variation than model 'a'. Based on alulated oeffiient from equations 16 and 17, the following equation is suggested: T = 0.061H S*H (17) The value of 'R square' in model '' has an appropriate auray too. So for more simplifying, the period time of these trusses are depended on the Rise of the struture only. So: T = H (18) (17) 74

8 Aust. J. Basi & Appl. Si., 5(8): 68-75, 011 Fig. 3: Main Period Time's Line. Conlusions: There are a lot of parameters that influene in the main period time of the struture. For example in the building strutures some parameter suh as length, wide,high,mass, onnetion and are affeted in the main period time of this struture but most of the building ode fous on the high as a prinipal parameter that almost an predit the dynami behavior of these strutures. In this paper, aording to the statistial studies, the main period time and onsequently the dynami behavior of vault truss (DLCST) is depended to the rise of the struture (H). So for dynami study it seems that it would be better to lassify these trusses H based on their rise and it's not reasonable to estimate their dynami behavior base on or α. S REFERENCES Cai, J., et al., 008. Study on the Seismi Performane of Spae Beam String Struture. International Assoiation for Earthquake Engineering. Coan, C.H. and R.H. Plaut, Dynami stability of a lattie dome. Earthquake Engineering & Strutural Dynamis, 11(): Chopra, A.K., Dynamis of strutures: theory and appliations to earthquake engineering Prentie Hall Upper Saddle River, New Jersey:. Kawaguhi, K., A report on large roof strutures damaged by the great Hanshin-Awaji earthquake. International Journal of Spae Strutures, 1(3): Koohestani, K. and A. Kaveh, 010. Effiient bukling and free vibration analysis of ylially repeated spae truss strutures. Finite Elements in Analysis and Design, 46(10): Ishikawa, K., et al., 000. Evaluation Method for Prediting Dynami Collapse of Double Layer Lattied Spae Truss Strutures due to Earthquake Motion. International Journal of Spae Strutures, 15(3): Marsh, C., 000. Some observations on designing double layer grids. International Journal of Spae Strutures, 15(3): Moghaddam, H.A., 000. Seismi Behaviour of Spae Strutures. International Journal of Spae Strutures. 15(): Rezaiee-pajand, M., et al., 001. A new method of fititious visous damping determination for the dynami relaxation method. Computers & Strutures. Sadeghi, A., 004. Horizontal Earthquake Loading and Linear/Nonlinear Seismi Behaviour of Double Layer Barrel Vaults. International Journal of Spae Strutures, 19(1): Weaver, W. and P.R. Johnston, Strutural dynamis by finite elements. Prentie-Hall. Zhang, Y. and T.T. Lan, 000. Researh on the Dynami Charateristis and Seismi Response of Spae Frames. International Journal of Spae Strutures, 15(3):