Structural Engineering and Mechanics

Techno-Press (테크노프레스)
권호 표지
DOI QR코드

DOI QR Code

Analysis of composite frame structures with mixed elements - state of the art

  • Ayoub, Ashraf (Department of Civil and Environmental Engineering, University of Houston)
  • Received : 2009.12.18
  • Accepted : 2011.12.13
  • Published : 2012.01.25
⟨ Previous Next ⟩

Abstract

The paper presents a review of the application of the newly proposed mixed finite element model for seismic simulation of different types of composite frame structures. To evaluate the performance of the element, a comparison with displacement-based and force-based models is conducted. The study revealed that the mixed model is superior to the others in terms of both speed of convergence and numerical stability, and is therefore considered the most practical approach for modeling of composite structures. In this model, the element is derived using independent force and displacement shape functions. The nonlinear response of the frame element is based on the section discretization into fibers with uniaxial material models. The interfacial behavior is modeled using an inelastic interface element. Numerical examples to clarify the advantages of the model are presented for the following structural applications: anchored reinforcing bar problems, composite steel-concrete girders with deformable shear connectors, beam on elastic foundation elements, R/C girders strengthened with FRP sheets, R/C beam-columns with bond-slip, and prestressed concrete girders. These studies confirmed that the model represents a major advancement over existing elements in simulating the inelastic behavior of composite structures.

Keywords

References

  1. Adekola, A.O. (1968), "Partial interaction between elastically connected elements of a composite beam", Int. J. Solids Struct., 4, 1125-1135. https://doi.org/10.1016/0020-7683(68)90027-9
  2. Aalami, B.O. (1990), "Load balancing-A comprehensive solution to post-tensioning", ACI Struct. J., 87(6), 662- 670.
  3. Aalami, B.O. (2000), "Structural modeling of post-tensioned members", J. Struct. Eng., ASCE, 126(2), 157-162. https://doi.org/10.1061/(ASCE)0733-9445(2000)126:2(157)
  4. Amadio, C. and Fragiacomo, M. (1993), "A finite element model for the study of creep and shrinkage effects in composite beams with deformable shear connections", Costruzioni Metalliche, 4, 213-228.
  5. Aprile, A., Spacone, E. and Limkatanyu, S. (2001), "Role of bond in RC beams strengthened with steel and FRP plates", J. Struct. Eng., ASCE, 127, 1445-1452. https://doi.org/10.1061/(ASCE)0733-9445(2001)127:12(1445)
  6. Arizumi, Y., Hamada, S. and Kajita, T. (1981), "Elastic-Plastic analysis of composite beams with incomplete interaction by finite element method", Comput. Struct., 14(5-6), 453-462. https://doi.org/10.1016/0045-7949(81)90065-1
  7. Attalla, M.R., Deierlein, G.G. and McGuire, W. (1994). ''Spread of plasticity−Quasi-plastic hinge approach.'' J. Struct. Eng., ASCE, 120(8), 2451-2473. https://doi.org/10.1061/(ASCE)0733-9445(1994)120:8(2451)
  8. Ayoub, A.S. and Filippou, F.C. (1999), "Mixed formulation of bond slip problems under cyclic loads", J. Struct. Eng., ASCE, 125(6), 661-671. https://doi.org/10.1061/(ASCE)0733-9445(1999)125:6(661)
  9. Ayoub, A.S. (1999), "Mixed formulation for seismic analysis of composite steel-concrete frame structures", Ph.D. Dissertation, University of California-Berkeley, USA.
  10. Ayoub, A.S. and Filippou, F.C. (2000), "Mixed formulation of non-linear steel-concrete composite beam element", J. Struct. Eng., ASCE, 126(3), 371-381. https://doi.org/10.1061/(ASCE)0733-9445(2000)126:3(371)
  11. Ayoub, A.S. (2001), "A two-field mixed variational principle for partially connected composite beams", Finite Elem. Analy. Des., 37, 929-959. https://doi.org/10.1016/S0168-874X(01)00076-2
  12. Ayoub, A.S. (2003), "Mixed formulation of non-linear beam on foundation elements", Comput. Struct., 81(7), 411-421. https://doi.org/10.1016/S0045-7949(03)00015-4
  13. Ayoub, A.S. (2005), "A force-based model for composite steel-concrete beams with partial interaction", J. Construct. Steel Res., 61(3), 387-414. https://doi.org/10.1016/j.jcsr.2004.08.004
  14. Ayoub, A.S. (2006), "Nonlinear analysis of reinforced concrete beam-columns with bond-slip", J. Eng. Mech., ASCE, 132(11), 1177-1186. https://doi.org/10.1061/(ASCE)0733-9399(2006)132:11(1177)
  15. Ayoub, A.S. and Filippou, F.C. (2010), "Nonlinear analysis of pre-tensioned prestressed concrete girders", J. Struct. Eng., ASCE, 136(4), 401-409. https://doi.org/10.1061/(ASCE)ST.1943-541X.0000132
  16. Ayoub, A.S. (2011), "Nonlinear finite element analysis of post-tensioned concrete bridge girders", J. Bridge Eng., ASCE, 16(3), 479-489. https://doi.org/10.1061/(ASCE)BE.1943-5592.0000157
  17. Babuska, I. (1973), "The finite element method with Lagrange multipliers", Numer. Math., 20, 179-192. https://doi.org/10.1007/BF01436561
  18. Baldwin, J.W., Henry, J.R. and Sweeny, C.M. (1965), "Study of composite bridge stringers, Phase II", Department of Civil Engineering, University of Missouri, Columbia, USA
  19. Bathe, K.J. (2001), "The Inf-Sup condition and its evaluation for mixed finite element methods", Comput. Struct., 79, 243-252. https://doi.org/10.1016/S0045-7949(00)00123-1
  20. Battini, J.M., Nguyen, Q.H. and Hjiaj, M. (2009), "Non-linear finite element analysis of composite beams with interlayer slips", Comput. Struct., 87(13-14), 904-912. https://doi.org/10.1016/j.compstruc.2009.04.002
  21. Bousias, S.N., Verzeletti, G., Fardis, M.N. and Guitierrez, E. (1995), "Load-Path effects in column biaxial bending and axial force", J. Eng. Mech., ASCE, 125(5), 596-605.
  22. Brezzi, F. (1974), "On the existence, uniqueness and approximation of saddle point problems arising from Lagrangian multipliers", RAIRO, 8-R2, 129-151.
  23. Bursi, O.S. and Ballerini, M. (1996), "Behavior of a steel-concrete composite substructure with full and partial shear connection", Proceedings of 11th World Conference on Earthquake Engineering, Acapulco, Mexico.
  24. Bursi, O.S., Sun, F.F. and Postal, S. (2005), "Non-linear analysis of steel-concrete composite frames with full and partial shear connection subjected to seismic loads", J. Construct. Steel Res., 61(1), 67-92. https://doi.org/10.1016/j.jcsr.2004.06.002
  25. Cosenza, E. and Mazzolani, S. (1993), "Analisi in campo lineare di travi composte con conessioni deformabili: formule esatte e resoluzioni alla differenze", Proceedings of First Italian Workshop on Composite Structures, University of Trento, Italy.
  26. Cruz, P.J.S., Mari, A. and Roca, P. (1998), "Nonlinear time-dependent analysis of segmentally constructed structures", J. Struct. Eng., ASCE, 124(3), 278-287. https://doi.org/10.1061/(ASCE)0733-9445(1998)124:3(278)
  27. Dall'Asta, A. and Zona, A. (2002), "Nonlinear analysis of composite beams by a displacement approach." Comput. Struct., 80(27-30), 2217-2228. https://doi.org/10.1016/S0045-7949(02)00268-7
  28. Dall'Asta, A. and Zona, A. (2004a), "Three-field mixed formulation for the non-linear analysis of composite beams with deformable shear connection", Finite Elem. Anal. Des., 40(4), 425-448. https://doi.org/10.1016/S0168-874X(03)00071-4
  29. Dall'Asta, A. and Zona, A. (2004b), "Comparison and validation of displacement and mixed elements for the nonlinear analysis of continuous composite beams", Comput. Struct., 82(23-26), 2117-2130. https://doi.org/10.1016/j.compstruc.2004.04.009
  30. Dall'Asta, A. and Zona, A. (2004c), "Slip locking in finite elements for composite beams with deformable shear connection", Finite Elem. Anal. Des., 40(13-14), 1907-1930. https://doi.org/10.1016/j.finel.2004.01.007
  31. Daniel, B.J. and Crisinel, M. (1993), "Composite slab behavior and strength analysis. Part I: calculation procedure", J. Struct. Eng., ASCE, 119(1), 16-35. https://doi.org/10.1061/(ASCE)0733-9445(1993)119:1(16)
  32. Da Silva, A.R. and Sousa Jr., J.B.M. (2009), "A family of interface elements for the analysis of composite beams with interlayer slip", Finite Elem. Anal. Des., 45(5), 305-314. https://doi.org/10.1016/j.finel.2008.10.007
  33. De Veubeke, B.F. (1965), "Displacement and equilibrium models in finite element method", Chapter 9 of Stress Analysis, 145-197.
  34. El-Tawil, S., Sanz-Picon, C.F. and Deierlein, G.G. (1995), "Evaluation of ACI 318 and AISC (LRFD) strength provisions for composite beam-columns", J. Constr. Steel Res., 34, 103-123. https://doi.org/10.1016/0143-974X(94)00009-7
  35. El-Tawil, S. and Deierlein, G.G. (2001), "Nonlinear analyses of mixed steel-concrete moment frames. Part I: Beam-column element formulation", J. Struct. Eng., ASCE, 127(6), 647-655. https://doi.org/10.1061/(ASCE)0733-9445(2001)127:6(647)
  36. Eligehausen, R., Popov, E.P. and Bertero, V.V. (1983), "Local bond stress-slip relationships of deformed bars under generalized excitations", Report No. UCB/EERC 83/23, Earthquake Engineering Research Center, University of California, Berkeley, USA.
  37. Erkmen, R.E. and Bradford, M.A. (2011), "Treatment of slip locking for displacement-based finite element analysis of composite beam-columns", Int. J. Numer. Meth. Eng., 85(7), 805-826. https://doi.org/10.1002/nme.2990
  38. ETABS2000 (2002), CSI: Computers and Structures Inc., Berkeley, CA.
  39. Faella, C., Martinelli, E. and Nigro, E. (2010), "Steel-concrete composite beams in partial interaction: Closedform "exact" expression of the stiffness matrix and the vector of equivalent nodal forces", Eng. Struct., 32(9), 2744-2754. https://doi.org/10.1016/j.engstruct.2010.04.044
  40. Foraboschi, P. (2009), "Analytical solution of two-layer beam taking into account nonlinear interlayer slip", J. Eng. Mech., 135(10), 1129-1146. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000043
  41. Gara, F., Ranzi, G., and Leoni, G. (2006), "Displacement-based formulations for composite beams with longitudinal slip and vertical uplift", Int. J. Numer. Meth. Eng., 65(8), 1197-1220. https://doi.org/10.1002/nme.1484
  42. Gara, F., Leoni, G., and Dezi, L. (2009), "A beam finite element including shear lag effect for the timedependent analysis of steel-concrete composite decks", Eng. Struct., 31(8), 1888-1902. https://doi.org/10.1016/j.engstruct.2009.03.017
  43. Hajjar, J.F., Schiller, P.H. and Moladan, A. (1997). "A distributed plasticity model for concrete-filled steel tube beam-columns with interlayer slip. Part I: slip formulation and monotonic analysis", Structural Engineering Report ST-97-1, Department of Civil Engineering, University of Minnesota, Minneapolis.
  44. Hirst, M.J.S. and Yeo, M.F. (1980), "The analysis of composite beams using standard finite element programs", Comput. Struct., 11(3), 233-237. https://doi.org/10.1016/0045-7949(80)90163-7
  45. Hjelmstad, K. and Taciroglu, E. (2002), "Mixed methods and flexibility approaches for nonlinear frame analysis", J. Construct. Steel Res., 58(5), 967-993. https://doi.org/10.1016/S0143-974X(01)00100-6
  46. Kang, Y.J. (1977), "Nonlinear geometric, material and time dependent analysis of reinforced and prestressed concrete frames", Report No. SESM 77-1, Division of Structural Engineering and Structural Mechanics, University of California, Berkeley.
  47. Kim K.D. and Engelhardt M.D. (2005), "Composite beam element for nonlinear seismic analysis of steel frames", J. Struct. Eng., ASCE, 131(5), 715-724. https://doi.org/10.1061/(ASCE)0733-9445(2005)131:5(715)
  48. Kodur, V.K.R. and Campbell, T.I. (1990), "Deformation controlled nonlinear analysis of prestressed concrete continuous beams", PCI J., Sept-Oct, 42-55.
  49. Liew, J.Y.R., Chen, H. and Shanmugam, N. (2005), "Inelastic analysis of steel frames with composite beams", J. Struct. Eng., ASCE, 127(2), 194-202.
  50. Limkatanyu, S. and Spacone, E. (2002), "Reinforced concrete frame element with bond interfaces. II: state determinations and numerical validation". J. Struct. Eng., ASCE, 128(3), 356-364. https://doi.org/10.1061/(ASCE)0733-9445(2002)128:3(356)
  51. Limkatanyu, S. and Spacone, E. (2006), "Frame element with lateral deformable supports: Formulations and numerical validation", Comput. Struct., 84(13-14), 942-954. https://doi.org/10.1016/j.compstruc.2005.12.002
  52. Lu, F. and Ayoub, A.S. (2008), "Effect of bond properties on the behavior of FRP-strengthened RC girders subjected to monotonic and cyclic loads", ACI Special Publication SP-258-08, 117-136.
  53. Lu, F. and Ayoub, A.S. (2011), "Evaluation of debonding failure of reinforced concrete girders strengthened with FRP laminates", J. Construct. Build. Mater., 25(4), 1963-1979. https://doi.org/10.1016/j.conbuildmat.2010.11.059
  54. Ma, K.P. (1973), "Inelastic analysis of composite beams with partial connections", M.Eng. Thesis, McMaster University, Hamilton, Ontario, Canada.
  55. Mari, A. (1984), "Nonlinear geometric, material and time dependent analysis of three dimensional reinforced and prestressed concrete frames", Report No. SESM 84-10, Division of Structural Engineering and Structural Mechanics, University of California, Berkeley.
  56. Mcgarraugh, J.B. and Baldwin, J.W. (1971), "Lightweight concrete-on-steel composite beams", Eng. J., AISC, 8(3), 90-98.
  57. Menegotto, M. and Pinto, P.E. (1973), "Method of analysis for cyclically loaded reinforced concrete plane frames including changes in geometry and nonelastic behavior of elements under combined normal force and bending", Proceedings of IABSE Symposium on Resistance and Ultimate Deformability of Structures Acted on by Well-Defined Repeated Loads, Final Report, Lisbon, Portugal.
  58. Mirza, S.A. and Skrabek, B.W. (1991), "Reliability of short composite beam-column strength interaction", J. Struct. Eng., ASCE, 117, 2320-2339. https://doi.org/10.1061/(ASCE)0733-9445(1991)117:8(2320)
  59. Mitchell, D., Cook, W.D. Khan, A.A. and Tham, T. (1993), "Influence of high strength concrete on transfer and development length of pretensioning strand", J. Pres. Concrete Ins., 14(4), 62-74
  60. Monti, G., Filippou, F. and Spacone, E. (1997), "Finite element for anchored bars under cyclic load reversals", J. Struct. Eng., ASCE, 123(5), 614-623. https://doi.org/10.1061/(ASCE)0733-9445(1997)123:5(614)
  61. Monti, G. and Spacone, E. (2000), "Reinforced concrete fiber beam element with bond-slip", J. Struct. Eng., ASCE, 126(6), 654-661. https://doi.org/10.1061/(ASCE)0733-9445(2000)126:6(654)
  62. Mullapudi, T.R. and Ayoub, A.S. (2010a), "Nonlinear finite element modeling of beams on two-parameter foundations", Comput. Geotechnics, 37(3), 334-342. https://doi.org/10.1016/j.compgeo.2009.11.006
  63. Mullapudi, T.R. and Ayoub, A.S. (2010b), "Inelastic analysis of semi-infinite foundation elements", Mech. Res. Commun., 37(1), 72-77. https://doi.org/10.1016/j.mechrescom.2009.10.004
  64. Newmark, N.M., Siess, C.P. and Viest, I.M. (1951), "Tests and analysis of composite beams with incomplete interaction", Proceedings of the Society for Experimental Stress Analysis.
  65. Ngo, D. and Scordelis, A.C. (1967), "Finite element analysis of reinforced concrete beams", ACI Struct. J., 64(3), 152-163.
  66. Nguyen, Q.H., Hjiaj, M., Uy, B. and Guezouli, S. (2009), "Analysis of composite beams in the hogging moment regions using a mixed finite element formulation", J. Construct. Steel Res., 65(3), 737-748. https://doi.org/10.1016/j.jcsr.2008.07.026
  67. Nguyen, Q.H., Martinelli, E. and Hjiaj, M. (2011a), "Derivation of the exact stiffness matrix for a two-layer Timoshenko beam element with partial interaction", Eng. Struct., 33(2), 298-307. https://doi.org/10.1016/j.engstruct.2010.10.006
  68. Nguyen, Q.H., Hjiaj, M. and Guezouli, S. (2011b), "Exact finite element model for shear-deformable two-layer beams with discrete shear connection", Finite Elemen. Anal. Des., 47(7), 718-727. https://doi.org/10.1016/j.finel.2011.02.003
  69. Nilson, A.H. (1971), "Internal measurement of bond slip", ACI Struct. J., 69(7), 439-441.
  70. Pi, Y.L., Bradford, M.A. and Uy, B. (2006), "Second order nonlinear inelastic analysis of composite steelconcrete members. II: Applications", J. Struct. Eng, ASCE, 132(5), 762-771. https://doi.org/10.1061/(ASCE)0733-9445(2006)132:5(762)
  71. Picard, A., Massicotte, B. and Bastien, J. (1995), "Relative efficiency of external prestressing", J. Struct. Eng., ASCE, 121(12), 1832-1841. https://doi.org/10.1061/(ASCE)0733-9445(1995)121:12(1832)
  72. Providakis, C.P. (2008), "Pushover analysis of base-isolated steel-concrete composite structures under near-fault excitations", Soil Dyn. Earthq. Eng., 28(4), 293-304. https://doi.org/10.1016/j.soildyn.2007.06.012
  73. Ranzi, G., Gara, F. and Ansourian, P. (2006), "General method of analysis for composite beams with longitudinal and transverse partial interaction", Comput. Struct., 84(31-32), 2373-2384. https://doi.org/10.1016/j.compstruc.2006.07.002
  74. Ranzi, G. (2008), "Locking problems in the partial interaction analysis of multi-layered composite beams", Eng. Struct., 30(10), 2900-2911. https://doi.org/10.1016/j.engstruct.2008.04.006
  75. Razaqpur, A.G. and Nofal, M. (1989), "A finite element for modeling the nonlinear behavior of shear connectors in composite structures", Comput. Struct., 32(1), 169-174. https://doi.org/10.1016/0045-7949(89)90082-5
  76. Robinson, H. and Naraine, K.S. (1988), "Slip and uplift effects in composite beams", Proceedings of Engineering Foundation Conference on Composite Construction, ASCE.
  77. Sakr, M.A. and Sakla, S.S.S. (2008), "Long-term deflection of cracked composite beams with nonlinear partial shear interaction: I - Finite element modeling", J. Construct. Steel Res., 64(12), 1446-1455. https://doi.org/10.1016/j.jcsr.2008.01.003
  78. Salari, M.R., Spacone, E., Shing, B. and Frangopol, D. (1998), "Nonlinear analysis of composite beams with deformable shear connectors", J. Struct. Eng., ASCE, 124(10), 1148-1158. https://doi.org/10.1061/(ASCE)0733-9445(1998)124:10(1148)
  79. Schnabl, S., Planinc, I., Saje, M., Cas, B. and Turk, G. (2006), "An analytical model of layered continuous beams with partial interaction", Struct. Eng. Mech., 22(3), 263-278. https://doi.org/10.12989/sem.2006.22.3.263
  80. Schnabl, S., Saje, M., Turk, G. and Planinc, I. (2007), "Locking-free two-layer Timoshenko beam element with interlayer slip", Finite Elemen. Anal. Des., 43(9), 705-714. https://doi.org/10.1016/j.finel.2007.03.002
  81. Scordelis, A.C., Ngo, D. and Franklin, H.A. (1974), "Finite element study of reinforced concrete beams with diagonal tension cracks", Proceedings of Symposium on Shear in Reinforced Concrete, ACI Publication SP-42.
  82. Sousa Jr., J.B.M. and Da Silva, A.R. (2007), "Nonlinear analysis of partially connected composite beams using interface elements", Finite Elemen. Anal. Des., 43(11-12), 954-964. https://doi.org/10.1016/j.finel.2007.06.010
  83. Spacone, E. and El-Tawil, S. (2004), "Nonlinear analysis of steel-concrete composite structures: State of the art", J. Struct. Eng., ASCE, 130(2), 159-168. https://doi.org/10.1061/(ASCE)0733-9445(2004)130:2(159)
  84. Sun, F. and Bursi, O. (2005), "Displacement-based and two-field mixed variational formulations for composite beams with shear lags", J. Eng. Mech., ASCE, 131(2), 199-210. https://doi.org/10.1061/(ASCE)0733-9399(2005)131:2(199)
  85. Tabatabai, H. and Dickson, T.J. (1993), "The History of the prestressing strand development length euation", J. Pres. Concrete Ins., 64-75.
  86. Thomsen, H., Spacone, E., Limkatanyu, S. and Camata, G. (2004) "Failure Mode Analyses of Reinforced Concrete Beams Strengthened in Flexure with Externally Bonded Fiber-Reinforced Polymers", J. Compos. Constr., 8, 123-131. https://doi.org/10.1061/(ASCE)1090-0268(2004)8:2(123)
  87. Tort, C. and Hajjar, J.F. (2010), "Mixed finite element for three-dimensional nonlinear dynamic analysis of rectangular concrete-filled steel tube beam-columns", J. Eng. Mech., 136(11), 1329-1339. https://doi.org/10.1061/(ASCE)EM.1943-7889.0000179
  88. Valipour, H.R. and Bradford, M.A. (2009), "A steel-concrete composite beam element with material nonlinearities and partial shear interaction", Finite Elemen. Anal. Des., 45(12), 966-972. https://doi.org/10.1016/j.finel.2009.09.011
  89. Viest, I.M., Colaco, J.P., Furlong, R.W., Griffis, L.G., Leon, R.T. and Wyllie, L.A. (1997), Composite Construction, Design for Buildings, McGraw Hill.
  90. Viwathanatepa, S., Popov, E.P. and Bertero, V.V. (1979), "Effects of generalized loadings on bond of reinforcing bars embedded in confined concrete blocks", Report No. UCB/EERC-79/22, Earthquake Engineering Research Center, University of California, Berkeley, USA.
  91. Wegmuller, A.W. and Amer, H.N. (1977), "Nonlinear response of composite steel-concrete bridges", Comput. Struct., 7(2), 161-169. https://doi.org/10.1016/0045-7949(77)90033-5
  92. Yassin, M.H.M. (1994), "Nonlinear analysis of prestressed concrete structures under monotonic and cyclic loads", Ph.D. Dissertation, University of California, Berkeley, USA.
  93. Zarnic, R., Gostic, S., Bosiljkov, V. and Bokan, V. (1999), "Improvement of bending load-bearing capacity by externally bonded plates", Proceedings of Creating with Concrete, Eds. Dhir, R.K. and Henderson, N.A., Thomas Telford, London.
  94. Zhou, F., Mosalam, K.M. and Nakashima, M. (2007), "Finite-element analysis of a composite frame under large lateral cyclic loading", J. Struct. Eng., 133(7), 1018-1026. https://doi.org/10.1061/(ASCE)0733-9445(2007)133:7(1018)
  95. Zienkiewicz, O.C. and Taylor, R.L. (1989), The Finite Element Method Volume 1. Basic Formulation and Linear Problems, Fourth Edition, McGraw Hill, London.
  96. Zona, A., Barbato, M. and Conte, J.P. (2008), "Nonlinear seismic response analysis of steel-concrete composite frames", J. Struct. Eng., ASCE, 134(6), 986-997. https://doi.org/10.1061/(ASCE)0733-9445(2008)134:6(986)
  97. Zona, A. and Ranzi, G. (2011), "Finite element models for nonlinear analysis of steel-concrete composite beams with partial interaction in combined bending and shear", Finite Elemen. Anal. Des., 47(2), 98-118. https://doi.org/10.1016/j.finel.2010.09.006

Cited by

  1. A computer method for nonlinear inelastic analysis of 3D composite steel–concrete frame structures vol.57, 2013, https://doi.org/10.1016/j.engstruct.2013.09.025
  2. Practical nonlinear inelastic analysis method of composite steel-concrete beams with partial composite action vol.134, 2017, https://doi.org/10.1016/j.engstruct.2016.12.017
  3. Review of Nonlinear Analysis and Modeling of Steel and Composite Structures vol.20, pp.4, 2012, https://doi.org/10.1142/s0219455420300037