Numerical Analysis and Parametric Study on Multiple Degrees-of-Freedom Frames

George U. Alaneme, Alireza Bahrami, Uzoma Ibe Iro, Nakkeeran Ganasen, Obeten N. Otu, Richard C. Udeala, Blessing O. Ifebude, Emmanuel A. Onwusereaka


The design of multiple degrees-of-freedom frames is critical in civil engineering, as these structures are commonly used in various applications such as buildings, bridges, and industrial structures. In this study, a six-degrees-of-freedom beam-column element stiffness matrix was formulated by superposition of beam and truss elements stiffness matrices and was adapted to statically analyze indeterminate frame structures. The development of a numerical model for the frame structures was achieved using the finite element method in the current study. Also, the investigation of the effects of various parameters such as frame geometries, material properties, and loading conditions was conducted on the internal forces developed in the frame structures. Three different parametric study cases that presented the frame structures with varying geometries and loading conditions were analyzed utilizing this matrix approach for the sake of emphasis and to evaluate the flexibility and adequacy of this formula to analyze the indeterminate frames using the MATLAB software. The analysis method comprised the derivation of the system displacements employing the relationships between the stiffness matrix and fixed end forces as the force vector and taking the attained displacements, which would be transformed to the local coordinates to obtain the member forces. The computed results from the element stiffness matrix approach were further statistically compared with the results achieved from the finite element software (SAP2000) applying the analysis of variance (ANOVA). The statistical results showed a P-value > 0.05, which indicated a good correlation between the compared results and adequate performance for the derived beam-column element matrix formula method.


Doi: 10.28991/CEJ-2023-09-07-012

Full Text: PDF


Indeterminate Structural Analysis; Beam-Column Element; Analysis of Variance; Matrix Stiffness Method; MATLAB; SAP2000.


Xu, Q., Zhen, X., Zhang, Y., Han, M., & Zhang, W. (2022). Numerical simulation study of progressive collapse of reinforced concrete frames with masonry infill walls under blast loading. Modelling and Simulation in Engineering, 2022. doi:10.1155/2022/1781415.

Fu, F. (2013). Dynamic response and robustness of tall buildings under blast loading. Journal of Constructional steel research, 80, 299-307. doi:10.1016/j.jcsr.2012.10.001.

Rao, S. S. (2017). The finite element method in engineering. Butterworth-heinemann, Oxford, United Kingdom.

Kattner, M., & Crisinel, M. (2000). Finite element modelling of semi-rigid composite joints. Computers and Structures, 78(1), 341–353. doi:10.1016/S0045-7949(00)00064-X.

Anuntasena, W., Lenwari, A., & Thepchatri, T. (2019). Finite element modelling of concrete-encased steel columns subjected to eccentric loadings. Engineering Journal, 23(6), 299–310. doi:10.4186/ej.2019.23.6.299.

Wang, M., Shi, Y., Xu, J., Yang, W., & Li, Y. (2015). Experimental and numerical study of unstiffened steel plate shear wall structures. Journal of Constructional Steel Research, 112, 373-386. doi:10.1016/j.jcsr.2015.05.002.

Pepper, D. W., & Heinrich, J. C. (2005). The finite element method: basic concepts and applications. CRC Press, Boca Raton, United States.

Öchsner, A. (2016). Computational statics and dynamics: An introduction based on the finite element method. Springer, Singapore. doi:10.1007/978-981-10-0733-0.

Öchsner, A., & Merkel, M. (2018). One-dimensional finite elements: An introduction to the FE method. Springer, Cham, Switzerland. doi:10.1007/978-3-319-75145-0.

Zienkiewicz, O. C., & Taylor, R. L. (2000). The finite element method: solid mechanics (Volume 2). Butterworth-Heinemann, Oxford, United Kingdom.

Pelosi, G. (2007). The finite-element method, Part I: R. L. Courant “Historical Corner”. IEEE Antennas and Propagation Magazine, 49(2), 180–182. doi:10.1109/map.2007.376627.

Huebner, K. H., Dewhirst, D. L., Smith, D. E., & Byrom, T. G. (2001). The finite element method for engineers. John Wiley & Sons, Hoboken, United States.

Pathak, M. V., & Bhaskar, G. B. (2016). Finite element analysis program of frames. International Journal for Technological Research in Engineering, 3(9), 2455–2459.

Alaneme, G. U., Ezeokpube, G. C., & Mbadike, E. M. (2020). Failure analysis of a partially collapsed building using analytical hierarchical process. Journal of Failure Analysis and Prevention. doi:10.1007/s11668-020-01040-3.

Rahemi, H., & Baksh, S. (2010). Finite difference impulsive response analysis of a frame structure-a Matlab computational project-based learning. Latin American and Caribbean Journal of Engineering Education, 4(2), 31-38.

Madenci, E., & Guven, I. (2015). The finite element method and applications in engineering using ANSYS®. Springer, New York, United States. doi:10.1007/978-1-4899-7550-8.

Cao, J. (2005). Application of a posteriori error estimation to finite element simulation of compressible Navier-Stokes flow. Computers and Fluids, 34(8), 991–1024. doi:10.1016/j.compfluid.2004.09.002.

Kattan, P. I. (2010). MATLAB guide to finite elements: an interactive approach. Springer Science & Business Media, Berlin, Germany. doi:10.1007/978-3-540-70698-4.

Nukala, P. K. V. V., & White, D. W. (2004). A mixed finite element for three-dimensional nonlinear analysis of steel frames. Computer Methods in Applied Mechanics and Engineering, 193(23–26), 2507–2545. doi:10.1016/j.cma.2004.01.029.

Li, Y., Lu, X., Guan, H., & Ren, P. (2016). Numerical investigation of progressive collapse resistance of reinforced concrete frames subject to column removals from different stories. Advances in Structural Engineering, 19(2), 314–326. doi:10.1177/1369433215624515.

Gao, Y. (2021). Seismic Performance Simulation of Magnetorheological Fluid Dampers with Single Degree-of-Freedom System. University of California, Irvine, United States.

Zhou, X., Chen, Y., Ke, K., Yam, M. C., & Li, H. (2022). Hybrid steel staggered truss frame (SSTF): A probabilistic spectral energy modification coefficient surface model for damage-control evaluation and performance insights. Journal of Building Engineering, 45, 103556. doi:10.1016/j.jobe.2021.103556.

Panahi, M., Zareei, S. A., & Izadi, A. (2021). Flexural strengthening of reinforced concrete beams through externally bonded FRP sheets and near surface mounted FRP bars. Case Studies in Construction Materials, 15, e00601. doi:10.1016/j.cscm.2021.e00601.

Fujii, K. (2014). Prediction of the largest peak nonlinear seismic response of asymmetric buildings under bi-directional excitation using pushover analyses. Bulletin of Earthquake Engineering, 12, 909-938. doi:10.1007/s10518-013-9557-x.

Liew, J. R. (2008). Survivability of steel frame structures subject to blast and fire. Journal of Constructional Steel Research, 64(7-8), 854-866. doi:10.1016/j.jcsr.2007.12.013.

Khalid, M. A., & Bansal, S. (2023). Framework for robust design optimization of tuned mass dampers by stochastic subset optimization. International Journal of Structural Stability and Dynamics, 2350155. doi:10.1142/S0219455423501559.

Ganjavi, B., & Hao, H. (2012). A parametric study on the evaluation of ductility demand distribution in multi-degree-of-freedom systems considering soil–structure interaction effects. Engineering Structures, 43, 88-104. doi:10.1016/j.engstruct.2012.05.006.

Ainsworth, M., & Oden, J. T. (2000). A Posteriori Error Estimation in Finite Element Analysis. John Wiley & Sons, Hoboken, United States. doi:10.1002/9781118032824.

Qiu, C., Zhang, A., Jiang, T., & Du, X. (2022). Seismic performance analysis of multi-story steel frames equipped with FeSMA BRBs. Soil Dynamics and Earthquake Engineering, 161, 107392. doi:10.1016/j.soildyn.2022.107392.

Yussof, M. M., Silalahi, J. H., Kamarudin, M. K., Chen, P.-S., & Parke, G. A. (2020). Numerical evaluation of dynamic responses of steel frame structures with different types of haunch connection under blast load. Applied Sciences, 10(5), 1815. doi:10.3390/app10051815.

Khaloo, A. R., & Khosravi, H. (2013). Modified fish-bone model: A simplified MDOF model for simulation of seismic responses of moment resisting frames. Soil Dynamics and Earthquake Engineering, 55, 195-210. doi:10.1016/j.soildyn.2013.09.013.

Muho, E. V., Qian, J., & Beskos, D. E. (2020). A direct displacement-based seismic design method using a MDOF equivalent system: application to R/C framed structures. Bulletin of Earthquake Engineering, 18, 4157-4188. doi:10.1007/s10518-020-00857-5.

Ou, C., Liu, J., Sun, L., Xiao, Z., Cheng, Y., Liu, M., Zhao, F., Zhen, M., & Wang, Y. (2021). Experimental and Numerical Investigation on the Dynamic Responses of the Remaining Structure under Impact Loading with Column Being Removed. KSCE Journal of Civil Engineering, 25(6), 2078–2088. doi:10.1007/s12205-021-1026-5.

Fragiacomo, M., Amadio, C., & Macorini, L. (2004). Seismic response of steel frames under repeated earthquake ground motions. Engineering Structures, 13(26), 2021-2035. doi:10.1016/j.engstruct.2004.08.005.

Lou, P., Dai, G. L., & Zeng, Q. Y. (2006). Finite-element analysis for a Timoshenko beam subjected to a moving mass. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science, 220(5), 669–678. doi:10.1243/09544062JMES119.

Diskin, B., & Thomas, J. (2012). Effects of mesh regularity on accuracy of finite-volume schemes. 50th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition. doi:10.2514/6.2012-609.

Hjelmstad, K. D., & Taciroglu, E. (2002). Mixed methods and flexibility approaches for nonlinear frame analysis. Journal of Constructional Steel Research, 58(5–8), 967–993. doi:10.1016/S0143-974X(01)00100-6.

Panahi, S., & Zahrai, S. M. (2021). Performance of typical plan concrete buildings under progressive collapse. Structures, 31, 1163-1172. doi:10.1016/j.istruc.2021.02.045.

Shan, S., & Pan, W. (2022). Collapse mechanisms of multi-story steel-framed modular structures under fire scenarios. Journal of Constructional Steel Research, 196, 107419. doi:10.1016/j.jcsr.2022.107419.

Ross, C. T. (1998). Advanced applied finite element methods. Woodhead, Sawston, United Kingdom. doi:10.1533/ 9780857099754.

Shan, S., & Pan, W. (2020). Structural design of high‐rise buildings using steel‐framed modules: A case study in Hong Kong. The Structural Design of Tall and Special Buildings, 29(15), e1788. doi:10.1002/tal.1788.

Van Loan, C. F., & Golub, G. (1996). Matrix computations. The Johns Hopkins University Press, Baltimore, United States.

ASME PTC 60/V&V 10-2006 (2006) Guide for Verification and Validation in Computational Solid Mechanics. The American Society of Mechanical Engineers Standards, New York, United States.

Spacone, E., Filippou, F. C., & Taucer, F. F. (1996). Fibre beam–column model for non‐linear analysis of R/C frames: Part I. Formulation. Earthquake Engineering and Structural Dynamics, 25(7), 711-725. doi:10.1002/(SICI)1096-9845(199607)25:7< 711::AID-EQE576>3.0.CO;2-9.

Ross, C. T., & Chilver, A. (1999). Strength of materials and structures (4th Ed.). Elsevier, Amsterdam, Netherlands.

Sharma, S., & Tiwary, A. K. (2021). Analysis of multi-story buildings with hybrid shear wall: steel bracing structural system. Innovative Infrastructure Solutions, 6, 1-12. doi:10.1007/s41062-021-00548-3.

Horr, A. M., & Schmidt, L. C. (1995). Closed-form solution for the Timoshenko beam theory using a computer-based mathematical package. Computers and Structures, 55(3), 405–412. doi:10.1016/0045-7949(95)98867-P.

Ferreira, A. J. (2009). MATLAB codes for finite element analysis. Springer, Cham, Switzerland. doi:10.1007/978-3-030-47952-7.

Agor, C. D., Mbadike, E. M., & Alaneme, G. U. (2023). Evaluation of sisal fiber and aluminum waste concrete blend for sustainable construction using adaptive neuro-fuzzy inference system. Scientific Reports, 13(1), 2814. doi:10.1038/s41598-023-30008-0.

Rangel, R. L., & Martha, L. F. (2019). LESM—An object-oriented MATLAB program for structural analysis of linear element models. Computer Applications in Engineering Education, 27(3), 553–571. doi:10.1002/cae.22097.

Brenner, S. C., & Scott, L. R. (2008). The mathematical theory of finite element methods. Springer, New York, United States. doi:10.1007/978-0-387-75934-0.

Ghali, A., Neville, A. M., & Brown, T. G. (2009). Structural analysis: a unified classical and matrix approach (6th Ed.). CRC Press, London, United Kingdom. doi:10.1201/9781315273006.

Wunderlich, W., & Pilkey, W. D. (2004). Mechanics of structures. Variational and computational methods. Meccanica 39, 291–292. doi:10.1023/b:mecc.0000023038.64148.bc.

Bathe, K. J. (1996). Finite element procedures. Prentice Hall, Koboken, United States.

Trogrlic, B., & Mihanovic, A. (2008). The comparative body model in material and geometric nonlinear analysis of space R/C frames. Engineering Computations, 25(2), 155-171. doi:10.1108/02644400810855968.

Timoshenko, S.P. & Goodier J. N. (1970). Theory of Elasticity (3rd Ed.), McGraw-Hill, New York, United States.

Slivker, V. (2006). Mechanics of structural elements: theory and applications. Springer. doi:10.1007/978-3-540-44721-4.

Zhang, X. D., Trépanier, J. Y., & Camarero, R. (2000). A posteriori error estimation for finite-volume solutions of hyperbolic conservation laws. Computer Methods in Applied Mechanics and Engineering, 185(1), 1–19. doi:10.1016/S0045-7825(99)00099-7.

Najafgholipour, M. A., Dehghan, S. M., Dooshabi, A., & Niroomandi, A. (2017). Finite element analysis of reinforced concrete beam-column connections with governing joint shear failure mode. Latin American Journal of Solids and Structures, 14(7), 1200–1225. doi:10.1590/1679-78253682.

Duan, H., & Hueste, M. B. D. (2012). Seismic performance of a reinforced concrete frame building in China. Engineering Structures, 41, 77-89. doi:10.1016/j.engstruct.2012.03.030.

Shi, F., Wang, H., Zong, L., Ding, Y., & Su, J. (2020). Seismic behavior of high-rise modular steel constructions with various module layouts. Journal of Building Engineering, 31, 101396. doi:10.1016/j.jobe.2020.101396.

Lopes, P. C., Rangel, R. L., & Martha, L. F. (2021). An interactive user interface for a structural analysis software using computer graphics techniques in MATLAB. Computer Applications in Engineering Education, 29(6), 1505–1525. doi:10.1002/cae.22406.

Ibe Iro, U., Alaneme, G. U., Milad, A., Olaiya, B. C., Otu, O. N., Isu, E. U., & Amuzie, M. N. (2022). Optimization and simulation of saw dust ash concrete using extreme vertex design method. Advances in Materials Science and Engineering, 2022. doi:10.1155/2022/5082139.

Attah, I. C., Kufre Etim, R., Uwadiegwu Alaneme, G., Ufot Ekpo, D., & Usanga, I. N. (2022). Scheffe’s approach for single additive optimization in selected soils amelioration studies for cleaner environment and sustainable subgrade materials. Cleaner Materials, 5. doi:10.1016/j.clema.2022.100126.

Alaneme George, U., & Mbadike Elvis, M. (2019). Modelling of the mechanical properties of concrete with cement ratio partially replaced by aluminium waste and sawdust ash using artificial neural network. SN Applied Sciences, 1(11), 1514. doi:10.1007/s42452-019-1504-2.

Full Text: PDF

DOI: 10.28991/CEJ-2023-09-07-012


  • There are currently no refbacks.

Copyright (c) 2023 Alireza Bahrami

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.