2018
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On the Geometrically Nonlinear Analysis of Composite Axisymmetric Shells
2
2
Composite axisymmetric shells have numerous applications; many researchers have taken advantage of the general shell element or the semianalytical formulation to analyze these structures. The present study is devoted to the nonlinear analysis of composite axisymmetric shells by using a 1D three nodded axisymmetric shell element. Both low and higherorder shear deformations are included in the formulation. The displacement field is considered to be nonlinear function of the nodal rotations. This assumption eliminates the restriction of small rotations between two successive increments. Both Total Lagrangian Formulation and Generalized Displacement Control Method are employed for analyzing the shells. Several numerical tests are performed to corroborate the accuracy and efficiency of the suggested approach.
1

402
419


Mohammad
RezaieePajand
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
Iran
rezaiee@um.ac.ir


E.
Arabi
Ferdowsi University of Mashhad, Iran
Ferdowsi University of Mashhad, Iran
Iran
arabii@um.ac.ir
Geometrical nonlinear analysis
Composite materials
Axisymmetric shells
Shear deformation
Large rotation
[[1] Librescu L., Refined geometrically nonlinear theories of anisotropic laminated shells, Quarterly of Applied Mathematics, 45(1), 1987, pp. 127.##[2] Dennis ST, Palazotte AN., Large displacement and rotational formulation for laminated shells including parabolic transverse shear, International Journal of NonLinear Mechanics, 25(1), 1990, pp. 6785.##[3] Alwar RS, Narasimhan MC., Axisymmetric nonlinear analysis of laminates orthotropic annular spherical shells, International Journal of NonLinear Mechanics, 27(4), 1992, pp. 611622.##[4] Birman V., Axisymmetric bending of generally laminated cylindrical shells, Journal of Applied Mechanics, 60(1), 1993, pp. 157162.##[5] Chandrashekhara K, Kumar BS., Static analysis of a thick laminated circular cylindrical shell subjected to axisymmetric load, Composite Structures, 23(1), 1993, pp. 19.##[6] Liu JH, Surana KS., Piecewise hierarchical pversion axisymmetric shell element for geometrically nonlinear behavior of laminated composites, Computers & Structures, 55(1), 1995, pp. 6784.##[7] Ziyaeifar M, Elwi AE., Degenerated plateshell elements with refined transverse shear strains, Computers & Structures, 60(6), 1996, pp. 428460.##[8] Argyris J, Tenek L, Olofsson L., TRIC: a simple but sophisticated 3node triangular element based on 6 rigidbody and 12 straining modes for fast computational simulations of arbitrary isotropic and laminated composite shells, Computer Methods in Applied Mechanics and Engineering, 145(12), 1997, pp. 1185.##[9] Argyris J, Tenek L, Papadrakakis M, Apostolopoulou C., Postbuckling performance of the TRIC natural mode triangular element for isotropic and laminated composite shells, Computer Methods in Applied Mechanics and Engineering, 166(34), 1998, pp. 211231.##[10] Pinto Correia IF, Barbosa JI, Mota Soares CM, Mota Soares CA., A finite element semianalytical model for laminated axisymmetric shells: statics, dynamics and buckling, Computers & Structures, 76(13), 2000, pp. 299317.##[11] Dumir PC, Joshi S, Dube GP., Geometrically nonlinear axisymmetric analysis of thick laminated annular plate using FSDT, Composites Part B: Engineering, 32(1), 2001, pp. 110.##[12] Pinto Correia IF, Mota Soares CM, Mota Soares CA, Herskovits J., Analysis of laminated conical shell structures using higher order models, Composite Structures, 62(34), 2003, pp. 383390.##[13] Santos H, Mota Soares CM, Mota Soares CA, Reddy J.N., A semianalytical finite element model for the analysis of laminated 3D axisymmetric shells: bending, free vibration and buckling, Composite Structures, 71(34), 2005, pp. 273281.##[14] Wu CP, Pu YF, Tsai YH., Asymptotic solutions of axisymmetric laminated conical shells, ThinWalled Structures,43(10), 2005, pp. 15891614.##[15] Smith TA., Analysis of axisymmetric shell structures under axisymmetric loading by the flexibility method, Journal of Sound and Vibration,318(3), 2008, pp. 428460.##[16] Reddy JN., Refined nonlinear theory of plates with transverse shear deformation, International Journal of Solids and Structures, 20(910), 1984, pp. 881896.##[17] Reddy JN., An evaluation of equivalentsinglelayer and layerwise theories of composite laminates, Copmosite Structures,25(14), 1993, pp. 2135.##[18] Mantari JL, Oktem AS, Guedes Soares C., Static and dynamic analysis of laminated composite and sandwich plates and shells by using a new higherorder shear deformation theory, Composite Structures, 94(1), 2011, pp. 3749.##[19] Han SC, Tabiei A, Park WT., Geometrically nonlinear analysis of laminated composite thin shells using a modified firstorder shear deformable elementbased Lagrangian shell element, Composite Structures, 82(3), 2008, pp. 465474.##[20] Reddy JN, Liu CE., A higherorder shear deformation theory of laminated elastic shells, International Journal of Engineering Science, 23(3), 1985, pp. 319330.##[21] Noor AK, ASCE M, Peters JM., Analysis of laminated anisotropic shells of revolution, Journal of Engineering Mechanics, 113(1), 1987, pp. 4965.##[22] Sheinman I, Shaw D, Simitses GJ., Nonlinear analysis of axiallyloaded laminated cylindrical shells, Computers & Structures, 16(14), 1983, pp. 131137.##[23] Patel BP, Singh S, Nath Y., Postbuckling characteristics of angleply laminated truncated circular conical shells, Communications in Nonlinear Science Numerical Simulation, 13(7), 2008, pp. 14111430.##[24] Singh S, Patel BP, Nath Y., Postbuckling of laminated shells of revolution with meridional curvature under thermal and mechanical loads, International Journal of Structural Stability and Dynamics, 9(1), 2009, pp. 107126.##[25] Cagdas IU., Stability analysis of crossply laminated shells of revolution using a curved axisymmetric shell finite element, ThinWalled Structures, 49(6), 2011, pp. 732742.##[26] Wu CP, Chi YW., Threedimensional nonlinear analysis of laminated cylindrical shells under cylindrical bending, European Journal of Mechanics A/Solids. 24(5), 2005, pp. 837856.##[27] Bhaskar K, Varadan TK., A higherorder theory for bending analysis of laminated shells of revolution, Computers & Structures, 40(4), 1991, pp. 815819.##[28] Bhimaraddi A, Carr AJ, Moss PJ., A shear deformable finite element for the analysis of general shells of revolution, Computers & Structures, 31(3), 1989, pp. 299308.##[29] Chang TY, Sawamiphakdi K., Large deformation analysis of laminated shells by finite element method, Computers & Structures, 13, 1981, pp. 331340.##[30] RezaieePajand M, Arabi E., A curved triangular element for nonlinear analysis of laminated shells, Composite Structures, 153, 2016, pp. 538548.##[31] Xu CS., Buckling and postbuckling of symmetrically laminated moderatelythick spherical caps, International Journal of Solids and Structures, 28(9), 1991, pp. 11711184.##[32] Alankaya V, Oktem AS., Static analysis of laminated and sandwich composite doublycurved shallow shells, Steel and Composite Structures, 20(5), 2016, pp. 10431066.##[33] Sofiyev AH, Kuruoglu N., Buckling of nonhomogeneous orthotropic conical shells subjected to combined load, Steel and Composite Structures, 19(1), 2015, pp. 119.##[34] RezaieePajand M, Arabi E, Masoodi Amir R., A triangular shell element for geometrically nonlinear analysis, Acta Mechanica, 229(1), 2018, pp. 323342.##[35] Santos H, Mota Soarez CM, Mota Soarez CA, Reddy JN., A semianalytical finite element model for the analysis of cylindrical shells made of functionally graded materials, Composite Structures, 91(4), 2009, pp. 427432.##[36] Bich DH, Dung DV, Hoa LK., Nonlinear static and dynamic buckling analysis of functionally graded shallow spherical shells including temperature effects, Composite Structures, 94(9), 2012, pp. 29522960.##[37] Bich DH, Tung HV., Nonlinear axisymmetric response of functionally graded shallow spherical shells under uniform external pressure including temperature effects, International Journal of Nonlinear Mechanics,46(9), 2011, pp. 11951204##[38] Zozulya VV. Zhang CH., A high order theory for functionally graded axisymmetric cylindrical shells, International Journal of Mechanical Sciences, 60(1), 2012, pp. 1222.##[39] Viola E, Rossetti L, Fantuzzi N, Tornabene F., Static analysis of functionally graded conical shells and panels using the generalized unconstrained third order theory coupled with the stress recovery, Composite Structures, 112, 2014, pp. 4465.##[40] Arciniega RA, Reddy JN., Large deformation analysis of functionally graded shells, International Journal of Solids and Structures, 44(6), 2007, pp. 20362052.##[41] Kar VR, Panda SK., Nonlinear flexural vibration of shear deformable functionally graded spherical shell panel, Steel and Composite Structures, 18(3), 2015, pp. 693709.##[42] Wu CP, Liu YC., A state space meshless method for the 3D analysis of FGM axisymmetric circular plates, Steel and Composite Structures, 22(1), 2016, pp. 161182.##[43] Surana KS., Geometrically nonlinear formulation for the axisymmetric shell elements, International Journal for Numerical Methods in Engineering, 18(4), 1982, pp. 477502.##[44] Leon SE, Paulino GH, Pereira A, Menezes IFM, Lages EN., A unified library of nonlinear solution schemes, Applied Mechanics Reviews, 64(4), 2011, pp. 126.##]
Wave Motion and StopBands in Pipes with Helical Characteristics Using Wave Finite Element Analysis
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2
Pipes are widely used in many industrial and mechanical applications and devices. Although there are many different constructions according to the specific application and device, these can show helical pattern, such as spiral pipes, wirereinforced pipes/shells, springsuspension, and so on. Theoretical modelling of wave propagation provides a prediction about the dynamic behavior, and it is fundamental in the design process of these structures/devices and in structural health monitoring techniques. However, standard approaches have limitations in terms of difficulties in modelling and impossible computational cost at higher frequencies. In this study, the wave characteristics in waveguides with helical patterns are obtained using a Wave Finite Element (WFE) method. The method is described for a 1D and 2D waveguide with helical properties and it is illustrated by numerical examples. These include the optimization of stopbands for a fluidfilled pipe with concentrated masses and a cylindrical structure with helical orthotropy.
1

420
428


Elisabetta
Manconi
Dipartimento di Ingegneria e Architettura, Universita degli Studi di Parma, Viale delle
Scienze 181/A, 43100 Parma, Italy
Dipartimento di Ingegneria e Architettura,
Iran
elisabetta.manconi@unipr.it


Sergey
Sorokin
Department of Materials and Production,
Aalborg University,
Fibigerstraede 16,
DK9220 Aalborg East,
Denmark
Department of Materials and Production,
Aalborg
Iran
svs@mp.auu.dk


Rinaldo
Garziera
Dipartimento di Ingegneria e Architettura, Universita degli Studi di Parma, Viale delle
Scienze 181/A, 43100 Parma, Italy
Dipartimento di Ingegneria e Architettura,
Iran
rinaldo.garziera@unipr.it


Alf
SoeKnudsen
Vestas Wind Systems A/S, Hedeager 42, DK8200 Aarhus North, Denmark
Vestas Wind Systems A/S, Hedeager 42, DK8200
Iran
alskn@vestas.com
Wave propagation
Dispersion curves
Helical structures
Stopbands
Pipes
[[1] Pearson, D., Dynamic behaviour of helical springs, The Shock and Vibration Digest, 20, 1988, pp. 39.##[2] Sorokin, S., Linear dynamics of elastic helical springs: Asymptotic analysis of wave propagation, Proceeding of the Royal Society A: Mathematical, Physical and Engineering Sciences, 465, 2009, pp.15131537.##[3] Maurin, E., Claeys, C., Van Belle, L., Desmet, W., Bloch theorem with revised boundary conditions applied to glide, screw and rotational symmetric structures, Computer Methods in Applied Mechanics and Engineering, 318, 2017, pp. 497513.##[4] Lee, J., Thompson, D.J., Dynamic stiffness formulation, free vibration and wave motion of helical springs, Journal of Sound and Vibration, 239, 2001, pp. 297320.##[5] Lee, J., Free vibration analysis of cylindrical helical springs by the pseudospectral method, Journal of Sound and Vibration, 302, 2007, pp. 185196.##[6] Sorokin, S., The Green’s matrix and the boundary integral equations for analysis of timeharmonic dynamics of elastic helical springs, Journal of the Acoustical Society of America, 129, 2011, pp. 13151323.##[7] Treyssède, F., Numerical investigation of elastic modes of propagation in helical waveguides, The Journal of the Acoustical Society of America, 121, 2007, pp. 3398–3408.##[8] Treyssède, F., Elastic waves in helical waveguides, Wave Motion, 45, 2008, pp. 457470.##[9] Liu, Y., Han, Q., Li, C., Huang, H., Numerical investigation of dispersion relations for helical waveguides using the Scaled Boundary Finite Element Method, Journal of Sound and Vibration, 333, 2014, pp. 19912002.##[10] Renno, J.M., Mace, B.R., Vibration modelling of helical springs with nonuniform ends, Journal of Sound and Vibration, 331, 2012, pp. 28092823.##[11] Paganini, L., Manconi, E., SøeKnudsen, A., Sorokin, S., Optimum design of a periodic pipe filter using waves and finite elements, EURODYN 2011, Eighth International Conference on Structural Dynamics, Leuven, Belgium, 46 July 2011.##[12] Manconi, E., Sorokin, S., Garziera, R., Wave propagation in pipes with helical patterns, COMPDYN 2017, 6th ECCOMAS thematic conference on computational methods in structural dynamics and earthquakes engineering, Rhodes Island, Greece, 1717 June 2017.##[13] Mace, B., Duhamel, D., Brennan, M., Hinke, L., Finite element prediction of wave motion in structural waveguides, Journal of the Acoustical Society of America, 117, 2005, pp. 28352843.##[14] Manconi, E., Mace, B.R., Wave characterization of cylindrical and curved panels using a finite element method, Journal of the Acoustical Society of America, 125, 2009, pp. 154163.##[15] Waki, Y., Mace, B.R., Brennan, M.J., Numerical issues concerning the wave and finite element method for free and forced vibrations of waveguides, Journal of Sound and Vibration, 327, 2009, pp. 92108.##[16] SøeKnudsen, A., Sorokin S., On accuracy of the wave finite element predictions of wavenumbers and power flow: A benchmark problem, Journal of Sound and Vibration, 330, 2001, pp. 26942700.##[17] Manconi, E., Mace, B.R., Estimation of the loss factor of viscoelastic laminated panels from finite element analysis, Journal of Sound and Vibration, 329, 2010, pp. 39283939.##[18] Manconi, E., Mace, B.R., Garziera, R., Wave finite element analysis of fluidfilled pipes, NOVEM 2009, Noise and Vibration: Emerging Methods, Oxford, UK, 0508 Apr 2009.##[19] Manconi, E., Mace, B.R., Garziera, R., The lossfactor of prestressed laminated curved panels and cylinders using a wave and finite element method, Journal of Sound and Vibration, 332, 2014, pp. 17041711.##[20] Renno, J.M., Mace, B.R., Calculating the forced response of twodimensional homogeneous media using the wave and finite element method, Journal of Sound and Vibration, 330, 2011, pp. 59135927.##[21] Mitrou, G., Ferguson, N., Renno, J., Wave transmission through twodimensional structures by the hybrid FE/WFE approach, Journal of Sound and Vibration, 389, 2017, pp. 484501.##[22] Zhong, W.X., Williams, F.W., On the direct solution of wave propagation for repetitive structures, Journal of Sound and Vibration, 181, 1995, pp. 485501.##[23] Shen, M.R., Cao, W.W., Acoustic bandgap formation in a periodic structure with multilayer unit cells, Journal of Physics DApplied Physics, 33, 2000, pp. 11501154.##[24] Domadiya, P.G., Manconi, E., Vanali, M., Andersen, L.V., Ricci, A., Numerical and experimental investigation of stopbands in finite and infinite periodic onedimensional structures, Journal of Vibration and Control, 22, 2014, pp. 920931.##[25] SøeKnudsen, A., Design of stopband filter by use of curved pipe segments and shape optimization, Journal Structural and Multidisciplinary Optimization archive, 44, 2011, pp. 863874.##[26] Auld, B.A., Acoustic fields and waves in solids, Krieger Publishing Company, Malabar, FL,1990.##[27] Price, W.L., Global optimization by controlled random search, Journal of Optimization Theory and Applications, 40, 1983, pp. 333347.##]
Mixed Strong Form Representation Particle Method for Solids and Structures
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2
In this paper, a generalized particle system (GPS) method, a general method to describe multiple strong form representation based particle methods is described. Gradient, divergence, and Laplacian operators used in various strong form based particle method such as moving particle semiimplicit (MPS) method, smooth particle hydrodynamics (SPH), and peridynamics, can be described by the GPS method with proper selection of parameters. In addition, the application of mixed formulation representation to the GPS method is described. Based on HuWashizu principle and HellingerReissner principle, the mixed form refers to the method solving multiple primary variables such as displacement, strain and stress, simultaneously in the FEM method; however for convenience in employing FEM with particle methods, a simple representation in construction only is shown. It is usually applied to finite element method (FEM) to overcome numerical errors including locking issues. While the locking issues do not arise in strong form based particle methods, the mixed form representation in construction only concept applied to GPS method can be the first step for fostering coupling of multidomain problems, coupling mixed form FEM and mixed form representation GPS method; however it is to be noted that the standard GPS particle method and the mixed for representation construction GPS particle method are equivalent. Two dimensional simple bar and beam problems are presented and the results from mixed form GPS method is comparable to the mixed form FEM results.
1

429
441


David
Tae
Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE. Minneapolis, MN, 55455, USA
Department of Mechanical Engineering, University
Iran
taexx007@umn.edu


Kumar K.
Tamma
Department of Mechanical Engineering, University of Minnesota, 111 Church St. SE. Minneapolis, MN, 55455, USA
Department of Mechanical Engineering, University
Iran
ktamma@umn.edu
Particle method
Mixed form
Structural mechanics/dynamics
[[1] S.N. Atluri, R.H. Gallagher, O.C. Zienkiewicz, Hybrid and Mixed Finite Element Methods. John Wiley Sons, 1983.##[2] O.C. Zienkiewicz, R.L. Taylor, The Finite Element Method for Solid and Structural Mechanics, Butterworthheinemann, 2005.##[3] K. Washizu, Variational Methods in Elasticity and Plasticity, Oxford; New York; Pergamon Press, 1982.##[4] E. Reissner, On a Variational Theorem in Elasticity, Studies in Applied Mathematics, 29(14), 1950, 9095.##[5] T. Belytschko, Y.Y. Lu, L. Gu, Element Free Galerkin Method, International Journal for Numerical Methods in Engineering, 37, 1994, 229–256.##[6] Y.Y. Lu, T. Belytschko, L. Gu, A New Implementation of the Element Free Galerkin Method, Computer Methods in Applied Mechanics and Engineering, 113, 1994, 397–414.##[7] S.N. Atluri, T. Zhu, A New Meshless Local PetrovGalerkin (MLPG) Approach in Computational Mechanics, Computational Mechanics, 22, 1998, 117–127.##[8] S.N. Atluri, T. Zhu, A New Meshless Local PetrovGalerkin (MLPG) Approach to Nonlinear Problems in Computer Modeling and Simulation, Computational Modeling and Simulation in Engineering, 3, 1998, 187–196.##[9] S. Koshizuka, Y. Oka, MovingParticle SemiImplicit Method for Fragmentation of Incompressible Fluid, Nuclear Science and Engineering, 123, 1996, 421–434.##[10] R.A. Gingold, J.J. Monaghan, Smoothed Particle Hydrodynamics: Theory and Application to Nonspherical Stars, Monthly Notices of the Royal Astronomical Society, 181, 1977, 375–389.##[11] M. Shimada, D. Tae, T. Xue, R. Deokar, K.K. Tamma, Second order Accurate Particlebased Formulations: Explicit MPSGS4II Family of Algorithms for Incompressible Fluids with Free Surfaces, International Journal of Numerical Methods for Heat & Fluid Flow, 26(3/4), 2016, 897–915.##[12] X. Zhou, K.K. Tamma, Design, Analysis, and Synthesis of Generalized Single Step Single Solve and optimal Algorithms for Structural Dynamics, International Journal for Numerical Methods in Engineering, 59, 2004, 597668.##[13] X. Zhou, K.K. Tamma, Algorithms by Design with Illustrations to Solid and Structural Mechanics/Dynamics, International Journal for Numerical Methods in Engineering, 66, 2006, 1738–1790.##[14] M. Shimada, K.K. Tamma, Explicit Time Integrators and Designs for First/Second Order Linear Transient Systems, Encyclopedia of Thermal Stresses, 3, 2013, 1524–1530.##[15] Y. Chikazawa, S. Koshizuka, Y. Oka, A Particle Method for Elastic and Viscoplastic Structures and Fluidstructure Interactions, Computational Mechanics, 27(2), 2001, 97–106.##]
TimeDiscontinuous Finite Element Analysis of TwoDimensional Elastodynamic Problems using Complex Fourier Shape Functions
2
2
This paper reformulates a timediscontinuous finite element method (TDFEM) based on a new class of shape functions, called complex Fourier hereafter, for solving twodimensional elastodynamic problems. These shape functions, which are derived from their corresponding radial basis functions, have some advantages such as the satisfaction of exponential and trigonometric function fields in complex space as well as the polynomial ones simultaneously, that make them a better choice than classic Lagrange shape functions, which only can satisfy polynomial function field. To investigate the validity and accuracy of the proposed method, three numerical examples are provided and the results obtained from the present method (complex Fourierbased TDFEM) and the classic Lagrangebased TDFEM are compared with the exact analytical solutions. According to them, using complex Fourier functions in TDFEM leads to more accurate and stable solutions rather than those obtained from the classic TDFEM.
1

442
456


Ebrahim
Izadpanah
Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran
Civil Engineering Department, Shahid Bahonar
Iran
ebrahim_cv@yahoo.com


Saeed
Shojaee
Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran
Civil Engineering Department, Shahid Bahonar
Iran
saeed.shojaee@uk.ac.ir


Saleh
HamzeheiJavaran
Civil Engineering Department, Shahid Bahonar University of Kerman, Kerman, Iran
Civil Engineering Department, Shahid Bahonar
Iran
s.hamzeheijavaran@uk.ac.ir
Timediscontinuous finite element method
Twodimensional elastodynamic analysis
Complex Fourier shape functions
Radial basis functions
[Agnantiaris, J.P., Polyzos, D., Beskos, D.E., Some studies on dual reciprocity BEM for elastodynamic analysis, Computational Mechanics, 17 (1996) 270–277.##Aksoy, H.G., Şenocak, E., Discontinuous Galerkin method based on peridynamic theory for linear elasticity, International Journal for Numerical Methods in Engineering, 88(7) (2011) 673692.##Argyris, J.H., Scharpf, D.W., Finite elements in time and space, Nuclear Engineering and Design, 10(4) (1969) 456464.##Bathe, K.J, Baig, M.M.I., On a composite implicit time integration procedure for nonlinear dynamics, Computers & Structures, 83(3132) (2005) 2513–2524.##Brebbia, C.A., Nardini, D., Dynamic analysis in solid mechanics by an alternative boundary elements procedure, International Journal of Soil Dynamics and Earthquake Engineering, 2 (1983) 228–233.##Bruch, J.C., Zyvoloski, G., Transient twodimensional heat conduction problems solved by the finite element method, International Journal for Numerical Methods in Engineering, 8(3) (1974) 481494.##Chung, J., Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalizeda method, Journal of Applied Mechanics, 60(2) (1993) 371–375.##Chien, C.C., Wu, T.Y., An improved predictor/multicorrector algorithm for a timediscontinuous Galerkin finite element method in structural dynamics, Computational Mechanics, 25(5) (2000) 430437.##Dominguez, J., Boundary element in dynamics. London: Computational Mechanics Publications, Southampton, Elsevier Applied Science, 1993.##Dyniewicz, B., Efficient numerical approach to unbounded systems subjected to a moving load, Computational Mechanics, 54 (2014) 321329.##Fried, I., Finiteelement analysis of timedependent phenomena, AIAA Journal, 7(6) (1969) 11701173.##Fung, T.C., Higherorder accurate leastsquares methods for firstorder initial value problems, International Journal for Numerical Methods in Engineering, 45(1) (1999) 7799.##Guo, P., Wu, W.H., Wu, Z.G., A time discontinuous Galerkin finite element method for generalized thermoelastic wave analysis, Acta Mechanica, 225(1) (2014) 299307.##Hamzehei Javaran, S., Khaji, N., Dynamic analysis of plane elasticity with new complex Fourier radial basis functions in the dual reciprocity boundary element method, Applied Mathematical Modelling, 38(14) (2014) 36413651.##Hamzeh Javaran, S., Khaji, N., Noorzad, A., First kind Bessel function (JBessel) as radial basis function for plane dynamic analysis using dual reciprocity boundary element method, Acta Mechanica, 218 (2011) 247258.##Hilber, H.M., Hughes, T.J., Taylor, R.L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthquake Engineering & Structural Dynamics, 5(3) (1977) 283292.##Hughes, T.J., Hulbert, G.M., Spacetime finite element methods for elastodynamics: formulations and error estimates, Computer Methods in Applied Mechanics and Engineering, 66(3) (1988) 339363.##Hulbert, G.M., Hughes, T.J., Spacetime finite element methods for secondorder hyperbolic equations, Computer Methods in Applied Mechanics and Engineering, 84(3) (1990) 327348.##Hulbert, G.M., Time finite element methods for structural dynamics, International Journal for Numerical Methods in Engineering, 33(2) (1992) 307331.##Khaji, N., Hamzehei Javaran, S., New complex Fourier shape functions for the analysis of twodimensional potential problems using boundary element method, Engineering Analysis with Boundary Elements, 37(2) (2013) 260272.##Lesaint, P., Raviart, P.A., On a finite element method for solving the neutron transport equation, in: C. de Boor ed., Mathematical Aspects of Finite Elements in Partial Differential Equations, Academic Press, NewYork, 89123, 1974.##Lewis, D.L., Lund, J., Bowers, K.L., The space–time SincGallerkin method for parabolic problems, International Journal for Numerical Methods in Engineering, 24(9) (1987) 16291644.##Li, X., Yao, D., Lewis, R.W., A discontinuous Galerkin finite element method for dynamic and wave propagation problems in nonlinear solids and saturated porous media, International Journal for Numerical Methods in Engineering, 57(12) (2003) 17751800.##Liu, Y., Li, H., He, S., Mixed time discontinuous spacetime finite element method for convection diffusion equations, Applied Mathematics and Mechanics, 29(12) (2003) 15791586.##Nguyen, H., Reynen, J., A spacetime leastsquare finite element scheme for advectiondiffusion equations, Computer Methods in Applied Mechanics and Engineering, 42(3) (1984) 331342.##Oden, J.T., A general theory of finite elements. II. Applications, International Journal for Numerical Methods in Engineering, 1(3) (1969) 247259.##Peters, D.A., Izadpanah, A.P., Hpversion finite elements for the spacetime domain, Computational Mechanics, 3(2) (1988) 7388.##Petersen, S., Farhat, C., Tezaur, R., A space–time discontinuous Galerkin method for the solution of the wave equation in the time domain, International Journal for Numerical Methods in Engineering, 78(3) (2009) 275295.##Reed, W.H., Hill, T.R., Triangular mesh methods for the neutron transport equation, Los Alamos Scientific Lab, N. Mex., USA, 1973.##Saedpanah, F., A posteriori error analysis for a continuous spacetime finite element method for a hyperbolic integrodifferential equation, BIT Numerical Mathematics, 53 (2013) 689716.##Shao, H.P., Cai, C.W., A three parameters algorithm for numerical integration of structural dynamic equations, Chinese Journal of Mechanical Engineering, 5(4) (1988) 76–81.##Tamma, K.K., Zhou, X., Sha, D., A theory of development and design of generalized integration operators for computational structural dynamics, International Journal for Numerical Methods in Engineering, 50 (2001) 1619–1664.##Tang, Q., Chen, C.M., Liu, L.H., Spacetime finite element method for schrödinger equation and its conservation, Applied Mathematics and Mechanics, 27 (2006) 335340.##Varoglu, E., Liam Finn, W.D., Spacetime finite elements incorporating characteristics for the burgers' equation, International Journal for Numerical Methods in Engineering, 16(1) (1980) 171184.##Wang, J.G., Liu, G.R., On the optimal shape parameters of radial basis functions used for 2D meshless methods, Computer Methods in Applied Mechanics and Engineering, 191(23) (2002) 26112630.##Wilson, E.L., Nickell, R.E., Application of the finite element method to heat conduction analysis, Nuclear Engineering and Design, 4(3) (1966) 276286.##Zhou, X., Tamma, K.K., Design, analysis, and synthesis of generalized single step single solve and optimal algorithms for structural dynamics, International Journal for Numerical Methods in Engineering, 59 (2004) 597668.##Zienkiewicz, O.C., Parekh, C.J., Transient field problems: Twodimensional and threedimensional analysis by isoparametric finite elements, International Journal for Numerical Methods in Engineering, 2(1) (1970) 6171.##Zienkiewicz, O.C., Taylor, R.L., The finite element method, London: McGrawHill, 1977.##Zienkiewicz, O.C., A new look at the Newmark, Houbolt and other time stepping formulas, A weighted residual approach, Earthquake Engineering & Structural Dynamics, 5(4) (1977) 413418. ##]
Experimental and Numerical Investigations on the Effect of Rectangular Openings’ Aspect Ratio on Outflow Discharge
2
2
Up to now, a few formulas have been suggested by scholars for the amount of discharge from openings, however, the effect of opening's geometry on the amount of discharge has not addressed thoroughly. In this study, to assess the effect of rectangular openings’ aspect ratio on the discharge amount, experimental and numerical investigations have been conducted on the discharge amount from rectangular openings at the bottom of tanks. In the experimental part of the study different water depths have been considered and the amounts of discharge have been measured for openings with identical area, but different aspect ratios. In the numerical part of the study the test results have been compared to those obtained from finitevolumebased numerical simulation. The experimental and numerical results are in good agreement, and both show that there is a trend of increase in the amount of discharge with increase of the opening’s aspect ratio. The amount of this increase is from 13% to 21% for hydraulic head varying between 0.3 to 0.6 meters. On this basis, the conventional orifice formula for calculation of the rectangular opening discharge needs modification.
1

457
466


Abdolreza
Astaraki
Ph.D. Student, Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran 19537, Iran
Ph.D. Student, Structural Engineering Research
Iran
astaraki@iiees.ac.ir


Mahmood
Hosseini
Associate Professor, Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES), Tehran 19537, Iran
Associate Professor, Structural Engineering
Iran
hosseini@iiees.ac.ir


Aram
Soroushian
Assistant Professor, Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology (IIEES),Tehran 19537, Iran
Assistant Professor, Structural Engineering
Iran
a.soroushian@iiees.ac.ir


Mohammadreza
Jalili Ghazizadeh
Assistant Professor, Civil, Water and Environmental Engineering Department, Shahid Beheshti University, Tehran 1658953571, Iran
Assistant Professor, Civil, Water and Environmenta
Iran
m_jalili@sbu.ac.ir
Rectangular opening
Aspect ratio
Leakage discharge
Finite volume method
Orifice formula
[[1] Schussler, H., The water supply of San Francisco, California, before, during and after the earthquake of April 18th, 1906: and the subsequent conflagration. MB Brown Press, 1906.##[2] Manson, M., Harris De Haven, C., Ransom, T.W., Robinson, W.C., Reports on an auxilary water supply system for fire protection for San Francisco, California. Britton & Rey, 1908.##[3] Lawson, A.C., Reid, H.F., The California Earthquake of April 18, 1906: Report of the State Earthquake Investigation Commission. No. 87. Carnegie institution of Washington, 1908.##[4] Scawthorn, C., O’Rourke, T.D., Blackburn, F.T., The 1906 San Francisco earthquake and fire—Enduring lessons for fire protection and water supply, Earthquake Spectra, 22(S2) (2006) 135158.##[5] Steinbrugge, K.V., Schader, E.E., Bigglestone, H.C., Weers, C.A., San Fernando earthquake, February 9, Pacific Fire Rating Bureau, San Francisco, California, 1971.##[6] Jennings, P.C., Housner, G.W., Earthquake Damage to Water and Sewerage Facilities, San Fernando, California Earthquake of February 9, 1971, U.S. Department of Commerce, N.O.A.A., Washington, D.C., 2 (1973) 751973.##[7] Eguchi, R.T., Earthquake performance of water supply components during the 1971 San Fernando Earthquake, Prepared for the National Science Foundation. Redondo Beach, CA: JH Wiggins Company, 1982.##[8] Lund, L., Cooper, T., Water system." Northridge Earthquake: Lifeline Performance and Postearthquake Response, Technical Council on Lifeline Earthquake Engineering Monograph No 8, (1995) 96131.##[9] Hall, J.F., Northridge earthquake of January 17, 1994: reconnaissance report, Eeri, 1995.##[10] Eguchi, R.T., Chung, R.M., Performance of lifelines during the January 17, 1994 Northridge Earthquake. No. CONF9508226, American Society of Civil Engineers, New York, NY (United States), 1995.##[11] O'Rourke, T.D., Stewart, H.E., Jeon, S.S., Geotechnical aspects of lifeline engineering, Proceedings of the Institution of Civil EngineersGeotechnical Engineering, 149(1) (2001) 1326.##[12] Shi, P., O'Rourke, T.D., Seismic response modeling of water supply systems, Technical Report MCEER080016. May 5, 2006.##[13] Makar, J.M., Desnoyers, R., McDonald, S.E., Failure modes and mechanisms in gray cast iron pipe, International conference, Underground Infrastructure research: municipal, industrial and environmental applications; Kitchener, Canada, (2001) 303312.##[14] Neville, J., Hydraulic tables, coefficients, and formulae for finding the discharge of water from orifices, notches, weirs, pipes, and rivers, Lockwood & Co., London, 1875.##[15] Bovey, H.T., A treatise on hydraulics, New York, Wiley, 1909.##[16] Greyvenstein, B., Van Zyl, J.E., An experimental investigation into the pressureleakage relationship of some failed water pipes, Journal of Water Supply: Research and TechnologyAQUA, 56(2) (2007) 117124.##[17] White, F.M., Fluid Mechanics fourth edition, McGraw and Hill, International Edition, Singapore, 1994.##[18] Greyvenstein, B., An experimental investigation into the pressureleakage relationship of some failed water pipes in Johannesburg, Eng. Final Year Project Report, 2004.##[19] Cassa, A.M., Van Zyl, J.E., Laubscher, R.F., A numerical investigation into the behaviour of leak openings in uPVC pipes under pressure, CCWI2005 Water Management for the 21st Century (2005) 155160.##[20] Cassa, A.M., Van Zyl, J.E., Laubscher, R.F., A numerical investigation into the effect of pressure on holes and cracks in water supply pipes, Urban Water Journal, 7(2) (2010) 109120.##[21] Cassa, A.M., Van Zyl, J.E., Laubscher, R.F., A numerical investigation into the effect of pressure on holes and cracks in water supply pipes, Urban Water Journal, 7(2) (2010) 109120.##[30] Almohammadi, K.M., Ingham, D.B., Ma, L., Pourkashan, M., Computational fluid dynamics (CFD) mesh independency techniques for a straight blade vertical axis wind turbine, Energy, 58 (2013) 483493.##[22] Cassa, A.M., A numerical investigation into the behaviour of leak openings in pipes under pressure, PhD diss., University of Johannesburg, 2008.##[23] Brater, E.F., King, H.W., Handbook of hydraulics for the solution of hydraulic engineering problems, 1976.##[24] Michelotti, F.D., Sperimenti idraulici principalmente diretti a confermare la teorica e facilitare la pratica del misurare le acque correnti, Stamperia Reale, Turin, Italy, 1767.##[25] Groulx, D., Numerical study of nanoenhanced PCMs: are they worth it, In Proceedings of the 1st Thermal and Fluid Engineering Summer Conference, TFESC, New York City, USA, 2015.##[26] Shahangian, S.A., Experimental and numerical investigation on the leakage headdischarge relationship in the water distribution networks, M.Sc. Thesis, University of Tehran, 2015.##[27] Shahangian, S.A., Numerical study of leakage of submerged steel pipes in water based on laboratory results of nonsubmerged pipes, Iranian Journal of Water and Wastewater, 11(4) (2017) 100120 (In Persian).##[28] Temam, R., Navierstokes equations, Amsterdam: NorthHolland, 1984.##[29] Feistauer, M., Theory and Numerics for Problems of Fluid Dynamics, Charles University Prague, Faculty of Mathematics and Physics, 2006.##[30] Ondřej, S., Mohelníková, J., Plášek, J., Thermal CFD analysis of tubular light guides, Energies, 6(12) (2013) 63046321.##[31] Almohammadi, K.M., Ingham, D.B., Ma, L., Pourkashan, M., Computational fluid dynamics (CFD) mesh independency techniques for a straight blade vertical axis wind turbine, Energy, 58 (2013) 483493.##[32] Soroushian, A., Pseudo convergence and its implementation in engineering approximate computations, In Proceedings of 4th international conference from scientific computing to computational engineering (ICSCCE 2010), Athens, Greece. 2010.##[33] Soroushian, A., Equivalence between convergence and pseudo convergence when algorithmic parameters do not change geometrically, In Proceedings of 6th international conference from scientific computing to computational engineering (ICSCCE 2014), Athens, Greece. 2014.##[34] Liu, S., Valkó, P.P., Optimization of Spacing and Penetration Ratio for InfiniteConductivity Fractures in Unconventional Reservoirs: A SectionBased Approach, SPE Journal, 22(6) (2017) 116.##[35] Soroushian, A., Proper convergence, a concept new in science and important in engineering, In Proceedings of 4th international conference from scientific computing to computational engineering (ICSCCE 2010), Athens, Greece. 2010.##[36] Matthews, R.J., Liggett, J.A., Flow in a sharp corner, Journal of the Hydraulics Division, 93(6) (1961) 387410.##]
A General Rule for the Influence of Physical Damping on the Numerical Stability of Time Integration Analysis
2
2
The influence of physical damping on the numerical stability of time integration analysis is an open question since decades ago. In this paper, it is shown that, under specific very general conditions, physical damping can be disregarded when studying the numerical stability. It is also shown that, provided the specific conditions are met, analysis of structural systems involved in extremely high linearviscous damping is unconditionally stable. A secondary achievement is that, when the linearviscous damping increases, the numerical damping may increase or decrease.
1

467
481


Aram
Soroushian
Structural Engineering Research Center, International Institute of Earthquake Engineering and Seismology, S. Lavasani (Farmaiyeh, North Dibajee, West Arghavan, No. 21,Tehran 19537, Iran
Structural Engineering Research Center, Internatio
Iran
a.soroushian@iiees.ac.ir
Time integration
Numerical stability
Damping
Viscous
Numerical damping
Nonlinearity
[[1] IBC. International Building Code, International Code Council, USA, 2003.##[2] Hart, G.C., Jain, A., Performancebased wind evaluation and strengthening of existing tall concrete buildings in the Los Angeles region: dampers, nonlinear time history analysis and structural reliability, Struct. Des. Tall Spec. 23(16), 2014 12561274.##[3] Sassi, M.A., Nonlinear Dynamic Analysis of Wind Turbine Towers Subject to Design Wind and Seismic Loads, PhD Thesis, Colorado School of Mines, USA, 2016.##[4] Betsch, P., Steinmann, P., Inherently energy conserving time finite elements for classical mechanics, J. Comput. Phys. 160(1), 2000, 88116.##[5] Bruels, O., Golinval, J.C., The generalizedα method in mechatronic applications, Z. Angew. Math. Mech. 86(10), 2006, 748758.##[6] Eggl, S., Dvorak, R., An introduction to common numerical integration codes used in dynamical astronomy. In The Dynamics of Small Solar System Bodies and Exoplanets, Springer, USA, 2010, 431480.##[7] Faragó, I., Havasi, Á., Zlatev, Z. (Eds.) Advanced Numerical Methods for Complex Environmental Models: Needs and Availability, Bentham Science Publishers, ebook, 2013.##[8] Kontoe, S., Zdrakovic, L., Potts, D.M., An assessment of time integration schemes for dynamic geotechnical problems, Comput. Geotech. 35(2), 2008, 253–264.##[9] Kpodzo, K., Fourment, L., Lasne P., Montmotonnet, P., An accurate time integration scheme for arbitrary rotation motion: application to metal forming formulation, Int. J. Mater. Form. 9(1), 2016, 7184.##[10] Lemieux, J.F, Knoll, D.A., Losch, M., Girard, C., A secondorder accurate in time IMplicit–EXplicit (IMEX) integration scheme for sea ice dynamics, J. Comput. Phys. 263, 2014, 375392.##[11] Meijaard, J.P., Efficient numerical integration of the equations of motion of nonsmooth mechanical systems, Z. Angew. Math. Mech. 77(6), 1997, 419427.##[12] Paultre, P., Dynamics of Structures, John Wiley & Sons, USA, 2010.##[13] Soroushian, A., Integration step size and its adequate selection in analysis of structural systems against earthquakes. In The Computational Methods in Earthquake Engineering, Vol. 3, Springer, Norway, 2017, 285328.##[14] Tamma, K.K. and D'Costa, J.F., A new explicit variable time‐integration self‐starting methodology for computational structural dynamics, Int. J. Numer. Meth. Eng. 33(6), 1992, 11651180.##[15] Kadioglu, S.Y., and Knoll, D.A., A fully second order implicit/explicit time integration technique for hydrodynamics plus nonlinear heat conduction problems, J. Comput. Phys. 229(9), 2010, 32373249.##[16] Bonelli, A., Bursi O. S., Generalizedα methods for seismic structural testing, Earthq. Eng. Struct. D. 33(10) 10671102.##[17] Yousuf, M., Highorder time stepping scheme for pricing American option under Bates model, Int. J. Comput. Math. 96(1), 2019, 1832.##[18] Ghasemi, M., Sonner, S. Eberl, HY. J., Time adaptive numerical solution of a highly nonlinear degenerate crossdiffusion system arising in multispecies biofilm modeling, Eur. J. Appl. Math. 29(6), 2018, 10351061.##[19] Clough, R. W. Penzien, J., Dynamics of Structures, McGraw Hill, Singapore, 1993.##[20] Geradin, M., Rixen, D.J., Mechanical Vibrations: Theory and Applications to Structural Dynamics, John Wiley & Sons, USA, 2015.##[21] Gavin, H., Structural Dynamics, Duke University, USA, Class Notes CE 283, 2001.##[22] Hughes, T.J. R., Pister, K.S., Taylor R.L., Implicitexplicit finite elements in nonlinear transient analysis, Comput. Methods Appl. Mech. Eng. 17/18(1), 1979, 159–182.##[23] Wriggers, P., Computational Contact Mechanics, John Wiley & Sons, New York, 2002.##[24] Gear, C.W., Numerical Initial Value Problems in Ordinary Differential Equations, PrenticeHall, NJ, 1971.##[25] Soroushian, A., Wriggers, P., Farjoodi, J., Asymptotic upperbounds for the errors of Richardson Extrapolation with practical application in approximate computations, Int. J. Numer. Meth. Eng. 80(5), 2009, 565–595.##[26] Henrici, P., Discrete Variable Methods in Ordinary Differential Equations, PrenticeHall, NJ, 1962.##[27] Strikwerda, J.C., Finite Difference Schemes and Partial Differential Equations, Wadsworth & Books/Cole, Pacific Grove, CA, 1989.##[28] Richtmyer, R.D., Morton, K.W., Difference Methods for Initialvalue Problems, Interscience Publishers, USA, 1967.##[29] Bathe, K.J., Wilson E.L., Stability and accuracy analysis of direct integration methods, Earthq. Eng. Struct. D. 1(3), 1972, 283291.##[30] Lax, P.D., Richtmyer, R.D., Survey of the stability of linear finite difference equations, Commun. Pur. Appl. Math. 9(2), 1956, 267293.##[31] Wood, W.L., Practical Time Stepping Schemes, Oxford, New York, 1990.##[32] Greenberg, M.G., Advanced Engineering Mathematics, PrenticeHall, New Jersey, 1998.##[33] Belytschko, T., Hughes, T.J.R., (Eds.) Computational Methods for Transient Analysis, Elsevier, The Netherlands, 1983.##[34] Mengaldo, G., Wyszogrodzki, A., Diamantakis M., Lock, S. J. Giraldo F. X. and Wedi, N. P. Current and emerging timeintegration strategies in global numerical weather and climate prediction, Arch. Comput. Methods Eng. 122, 2018.##[35] NZS 1170, Structural Design Actions, Part 5: Earthquake ActionsNew Zealand, New Zealand, 2004.##[36] Hughes, T.J.R., The Finite Element Method: Linear Static and Dynamic Finite Element Analysis, PrenticeHall, NJ, 1987.##[37] Soroushian, A., Wriggers, P., Farjoodi, J., Time integration of nonlinear equations of motion  numerical instability or numerical inconsistency? In The Proc. 5th EUROMECH Nonlinear Oscillations Conference (ENOC 2005), European Mechanics Society, Eindhoven, The Netherlands, 2005.##[38] Han, B., Zdravkovic, L., Kontoe, S., Stability investigation of the Generalisedα time integration method for dynamic coupled consolidation analysis, Comput. Geotech. 64, 2015, 8395.##[39] Rashidi, S., Saadeghvaziri M.A., Seismic modeling of multispan simply supported bridges using Adina, Int. J. Comput. Struct. 64(5–6), 1997, 1025–1039.##[40] Xie, Y.M., Steven, G. P., Instability, chaos, and growth and decay of energy of timestepping schemes for nonlinear dynamic Equations, Commun. Numer. Methods Eng. 10(5), 1994, 393–401.##[41] Rose, I., Buffett, B., Heister, T., Stability and accuracy of free surface time integration in viscous flows, Phys. Earth Planet. Inter. 262, 2017, 90100.##[42] Lee, K., A short note on time integration stability of dynamic frictional contact problems of elastic bodies, P. I. Mech. Eng. KJ. Mul. 230(2), 2016, 113120.##[43] Cheng, M., Convergence and stability of stepbystep integration for model with negativestiffness, Earthq. Eng. Struct. D. 16(2), 1988, 227244.##[44] Elnashai, A.S., Sarno L.Di., Fundamentals of Earthquake Engineering, John Wiley & Sons, USA, 2008.##[45] Watkins, D. S., Fundamentals of Matrix Computation (2nd Ed.), John Wiley & Sons, USA, 2002.##[46] Houbolt, J. C., A recurrence matrix solution for the dynamic response of elastic aircraft, J. Aeronaut. Sci. 17(9), 1950, 540–550.##[47] Hart, G.C., Wong K., Structural Dynamics for Structural Engineers, John Wiley & Sons, USA, 1999.##[48] Soroushian, A., Development of an Algorithm and Computer Program to Evaluate the Numerical Stability and Consistency of New Time Integration Methods, International Institute of Earthquake Engineering and Seismology (IIEES), Iran, Report 7517, 2015. (in Persian)##[49] Piché, R., Nevalainen, P., Variable step Rosenbrock algorithm for transient response of damped structures, P. I. Mech. Eng. CJ. Mec. 213(2), 1998, 191198.##[50] Zhu, M. Zhu, J.Q., Studies on stability of stepbystep methods under complex damping conditions, J. Earthq. Eng. Eng. Vib. 21(4), 2001, 5962.##[51] Wang, J.T., Numerical stability of explicit finite element schemes for dynamic system with Rayleigh damping, Earthq. Eng. Eng. Vib. 22(6), 2002, 1824.##[52] Wu, B., Bao, H., Ou., J., Tian, S., Stability and accuracy analysis of the central difference method for realtime substructure testing, Earthquake Eng. Struct. Dyn. 34(7), 2005, 705718.##[53] Szabo, Z., Lukacs, A., Numerical stability analysis of a forced twodof oscillator with bilinear damping, J. Comput. Nonlin. Dyn. 2(3), 2007, 211217.##[54] RezaieePajand, M., KarimiRad, M., A new explicit time integration scheme for nonlinear dynamic analysis, Int. J. Struct. Stab. Dyn. 16(9), 2016, 1550054.##[55] Soroushian, A., A general rule for the effect of viscous damping on the numerical stability of time integration analyses. In The Proc. 12th World Congress on Computational Mechanics and 6th AsiaPacific Congress on Computational Mechanics (WCCM XII & APCOM VI), IACM, Seoul, South Korea, 2016.##[56] Soroushian, A., A general rule for the effect of arbitrary damping on the numerical stability of time integration analyses. In The Proc., 7th International Conference on Computational Methods (ICCM2016), University of California at Berkeley, Berkeley, USA, 2016.##[57] Ding, Z., Li, L., Hu Y., A modified precise integration method for transient dynamic analysis in structural systems with multiple damping models, Mech. Syst. Sig. Process. 98, 2018, 613633.##[58] Wood, W.L., On the effect of natural damping on the stability of a timestepping scheme, Commun. Appl. Numer. M. 32(2), 1987, 141144.##[59] Pourlatifi, S., A method to distinguish numerical and physical instability in analysis of structural systems, Msc Thesis, International Institute of Earthquake Engineering and Seismology, Iran, 2009.##[60] Zienkiewicz, O.C., Xie, Y.M., A simple error estimator and adaptive time stepping procedure for dynamic analysis, Earthq. Eng. Struct. D. 20(9) (1991) 871887.##[61] Apostol, T.M., Calculus, Vol. I, John Wiley & Sons, New York, 1967.##[62] Kardestuncer, H., Finite Element Handbook, McGraw Hill, USA, 1987.##[63] Newmark, N.M., A method of computation for structural dynamics, J. Eng. Mech. 85(3), 1959, 67–94.##[64] Clough, R.W., Numerical integration of equations of motion. In The Lectures on Finite Element Methods in Continuum Mechanics, Univ. of Alabama, Tuscaloosa, AL, 1973, 525533.##[65] Chung, J., Hulbert, G.M., A time integration algorithm for structural dynamics with improved numerical dissipation: The generalizeda method, J. Appl. Mech. 60(2), 1993, 371–375.##[66] Wilson, E.L. , A Computer Program for the Dynamic Stress Analysis of Underground Structures, University of California, Berkeley, Report 681,1968.##[67] Hilber, H.M., Hughes, T.J.R., Taylor R.L., Improved numerical dissipation for time integration algorithms in structural dynamics, Earthq. Eng. Struct. D. 5(3), 1977, 283–292.##[68] Hoff, C., Pahl, P.J., Development of an implicit method with numerical dissipation from a generalized singlestep algorithm for structural dynamics, Comput. Methods Appl. Mech. Eng. 67(3), 1988, 367385.##[69] Bathe, K.J., Conserving energy and momentum in nonlinear dynamics: a simple implicit time integration scheme, Int. J. Comput. Struct. 85(78), 2007, 437445.##[70] Bathe, K.J., Noh, G., Insight into an implicit time integration scheme for structural dynamics, Int. J. Comput. Struct. 9899(1), 2012, 16.##[71] Katsikadelis, J.T., A new direct time integration method for the equations of motion in structural dynamics, Z. Angew. Math. Mech. 94(9), 2014, 757774.##[72] RezaieePajand, M., KarimiRad, M., A family of secondorder fully explicit time integration schemes, Comp. Appl. Math. 37(3), 2017, 34313454.##[73] Lazan, B.J., Damping of Materials and Members in Structural Mechanics, Pergamon Press, UK, 1968.##[74] Allgower, E.L., Georg, K., Numerical Continuation Methods, An Introduction, Springer, New York, 1980.##[75] Kuhl D., Crisfield M.A., Energyconserving and decaying algorithms in nonlinear structural dynamics, Int. J. Numer. Meth. Eng. 45(5), 1999, 569–599.##[76] Soroushian, A., Wriggers, P., Farjoodi, J., Practical integration of semidiscretized nonlinear equations of motion: proper convergence for systems with piecewise linear behaviour, J. Eng. Mech., 139(2), 2013, 114145.##[77] Soroushian, A., Farjoodi, J., Bargi, K., Rajabi, M., Saaed, A., Arghavani, M., Sharifpour, M.M., Two versions of the Wilsonθ time integration method. In The Proc. 10th International Conference on Vibration Problems (ICOVP 2011), Prague, Czech Republic, 2011.##[78] RezaieePajand, M. KarimiRad, M., An accurate predictorcorrector time integration method for structural dynamics, Int. J. Steel Struct., 17(3), 2017, 10331047.##[79] Rio G., Soive A. Grolleau V., Comparative study of numerical explicit time integration algorithms, Adv. Eng. Software 36(4), 2005, 252265.##]