Three-Dimensional Force Flow Paths and Reinforcement Design in Concrete via Stress-Dependent Truss-Continuum Topology Optimization
Publication: Journal of Engineering Mechanics
Volume 141, Issue 1
Abstract
Strut-and-tie models (STMs) are widely used by RC designers. However, selection of a viable model is a challenging task, especially in complex three-dimensional (3D) design domains with irregular cutouts, which are common in building cores and shear walls. Therefore, topology optimization has been promoted as a means of automating the development of minimum strain energy STMs, which can lead to improved structural behavior. Current drawbacks of such methods are that solutions may be difficult to construct and may fail to properly account for tensile stresses resulting from force spreading. A two-dimensional hybrid truss-continuum topology optimization scheme was recently developed to overcome these challenges with the goal of reconfiguring traditional reinforcement layouts to automatically follow principal tensile stresses, reducing cracking at service loads and increasing strength and ductility at an ultimate limit state. That work is generalized and extended herein to 3D domains and mechanics models. Stiffness is formulated such that truss elements carry only tensile forces and thus represent straight steel rebar, while the continuum elements carry only compressive forces and thus represent concrete compression load paths. The latter is achieved using a stress-dependent orthotropic material model. The algorithm is demonstrated on several benchmark design examples.
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Acknowledgments
The authors thank Krister Svanberg for providing the MMA optimizer code.
References
Ali, M. A. (1997). “Automatic generation of truss models for the optimal design of reinforced concrete structures.” Ph.D. thesis, Cornell Univ., Ithaca, NY.
Ali, M. A., and White, R. N. (2000). “Formulation of optimal strut-and-tie models in design of reinforced concrete structures.” ACI SP 193-58, 979–998.
Ali, M. A., and White, R. N. (2001). “Automatic generation of truss model for optimal design of reinforced concrete structures.” ACI Struct. J., 98(4), 431–442.
Ambartsumyan, S. A., and Khachatryan, A. A. (1966). “The different-modules theory of elasticity.” Mech. Solids, 1(6), 64–67.
Amir, O., and Sigmund, O. (2013). “Reinforcement layout design for concrete structures based on continuum damage and truss topology optimization.” Struct. Multidiscip. Optim., 47(2), 157–174.
Baker, W., Beghini, A., Carrion, J., Mazeika, A., and Mazurek, A. (2008). “Numerical tools in structural optimization.” Proc., 6th Int. Conf. on Computation of Shell and Spatial Structures, Curran Associates, Red Hook, NY, 772–775.
Bendsøe, M. P. (1989). “Optimal shape design as a material distribution problem.” Struct. Optim., 1(4), 193–202.
Biondini, F., Bontempi, F., and Malerba, P. G. (1999). “Optimal strut-and-tie models in reinforced concrete structures.” Comput. Assist. Mech. Eng. Sci., 6(3), 279–293.
Biondini, F., Bontempi, F., and Malerba, P. G. (2001). “Stress path adapting strut-and-tie models in cracked and uncracked R.C. elements.” Struct. Eng. Mech., 12(6), 685–698.
Bogomolny, M., and Amir, O. (2012). “Conceptual design of reinforced concrete structures using topology optimization with elastoplastic material modeling.” Int. J. Numer. Methods Eng., 90(13), 1578–1597.
Bruggi, M. (2009). “Generating strut-and-tie patterns for reinforced concrete structures using topology optimization.” Comput. Struct., 87(23–24), 1483–1495.
Bruggi, M. (2010). “On the automatic generation of strut and tie patterns under multiple load cases with application to the aseismic design of concrete structures.” Adv. Struct. Eng., 13(6), 1167–1181.
Buhl, T., Pedersen, C. B. W., and Sigmund, O. (2000). “Stiffness design of geometrically nonlinear structures using topology optimization.” Struct. Multidiscip. Optim., 19(2), 93–104.
Darwin, D., and Pecknold, D. A. (1977). “Nonlinear biaxial stress-strain law for concrete.” J. Engrg. Mech. Div., 103(2), 229–241.
Gaynor, A. T., Guest, J. K., and Moen, C. D. (2013). “Reinforced concrete force visualization and design using bilinear truss-continuum topology optimization.” J. Struct. Eng., 607–618.
Guan, H. (2005). “Effect of sizes and positions of web openings on strut-and-tie models of deep beams.” Adv. Struct. Eng., 8(1), 69–84.
Guan, H., and Doh, J. H. (2007). “Development of strut-and-tie models in deep beams with web openings.” Adv. Struct. Eng., 10(6), 697–711.
Guest, J. K. (2009). “Imposing maximum length scale in topology optimization.” Struct. Multidiscip. Optim., 37(5), 463–473.
Guest, J. K., Asadpoure, A., and Ha, S. (2011). “Eliminating beta-continuation from Heaviside projection and density filter algorithms.” Struct. Multidiscip. Optim., 44(4), 443–453.
Guest, J. K., and Moen, C. D. (2010). “Reinforced concrete design with topology optimization.” Proc., Structures Congress Analysis and Computation Specialty Conf., ASCE, Reston, VA, 445–454.
Guest, J. K., Prevost, J. H., and Belytschko, T. (2004). “Achieving minimum length scale in topology optimization using nodal design variables and projection functions.” Int. J. Numer. Methods Eng., 61(2), 238–254.
Kim, H. A., and Baker, G. (2001). “Topology optimisation of reinforced concrete structures.” Proc., 1st Asian-Pacific Congress on Computational Mechanics, Elsevier Science, Oxford, U.K., Vol. 2, 1251–1256.
Kim, H. A., and Baker, G. (2002). “Topology optimization for reinforced concrete design.” Proc., 5th World Congress on Computational Mechanics, Elsevier Science, Oxford, U.K., 1–9.
Kuchma, D., Yindeesuk, S., Nagle, T., Hart, J., and Lee, H. H. (2008). “Experimental validation of strut-and-tie method for complex regions.” ACI Struct. J., 105(5), 578–589.
Kumar, P. (1978). “Optimal force transmission in reinforced concrete deep beams.” Comput. Struct., 8(2), 223–229.
Kwak, H. G., and Noh, S. H. (2006). “Determination of strut-and-tie models using evolutionary structural optimization.” Eng. Struct., 28(10), 1440–1449.
Lee, H.-J., Choi, W., and Lee, J.-J. (2007). “Development of the stress path search model using triangulated irregular network and refined evolutionary structural optimization.” J. Korean Soc. Agric. Eng., 49(6), 37–46.
Leu, L.-J., Huang, C.-W., Chen, C.-S., and Liao, Y.-P. (2006). “Strut-and-tie design methodology for three-dimensional reinforced concrete structures.” J. Struct. Eng., 929–938.
Liang, Q. Q. (2005). Performance-based optimization of structures: Theory and applications, Routledge, London.
Liang, Q. Q. (2006). “Performance-based optimization of strut-and-tie models in reinforced concrete beam-column connections.” Proc., 10th East Asia-Pacific Conf. on Structural Engineering and Construction: Materials, Experimentation, Maintenance and Rehabilitation (EASEC-10), AIT, Bangkok, Thailand, 1–6.
Liu, S., and Qiao, H. T. (2011). “Topology optimization of continuum structures with different tensile and compressive properties in bridge layout design.” Struct. Multidiscip. Optim., 43(3), 369–380.
Liu, X. B., and Meng, Q. C. (2002). “On the convergence of finite element method with different extension-compression elastic modulus.” J. Beijing Univ. Aeron. Astron., 28(2), 231–234.
Luo, Y., and Kang, Z. (2013). “Layout design of reinforced concrete structures using two-material topology optimization with Drucker–Prager yield constraints.” Struct. Multidiscip. Optim., 47(1), 95–110.
Marti, P. (1980). “On plastic analysis of reinforced concrete.” Rep. No. 104, Institute of Structural Engineers, Zurich, Switzerland.
Marti, P. (1985). “Basic tools of reinforced concrete beam design.” ACI Struct. J., 82(1), 46–56.
Moen, C. D., and Guest, J. K. (2010). “Reinforced concrete analysis and design with truss topology optimization.” Proc., 3rd Int. FIB Congress and Exhibition, Curran Associates, Red Hook, NY, 1–11.
Mörsch, E. (1909). Concrete-steel construction, Trans. E. P. Goodrich, McGraw Hill, New York.
Nagarajan, P., and Pillai, T. M. M. (2008a). “Analysis and design of simply supported deep beams using strut and tie method.” Adv. Struct. Eng., 11(5), 491–499.
Nagarajan, P., and Pillai, T. M. M. (2008b). “Development of strut and tie models for simply supported deep beams using topology optimization.” Songklanakarin J. Sci. Technol., 30(5), 641–647.
Ritter, W. (1899). “Die bauweise hennebique.” Schweiz. Bauzeitung, 33(7), 59–61.
Sarkisian, M., Long, E., Shook, D., and Doo, C. S. (2012). “Optimization tools for the design of structures.” Proc., Structures Congress 2012: 20th Analysis and Computation Specialty Conf., J. Carrato and J. G. Burns, eds., ASCE, Reston, VA, 219–230.
Schlaich, J., and Schafer, K. (1991). “Design and detailing of structural concrete using strut-and-tie models.” Struct. Eng., 69(6), 113–125.
Schlaich, J., Schaefer, K., and Jennewein, M. (1987). “Toward a consistent design of structural concrete.” PCI J., 32(3), 74–150.
Stromberg, L. L., Beghini, A., Baker, W. F., and Paulino, G. H. (2011). “Application of layout and topology optimization using pattern gradation for the conceptual design of buildings.” Struct. Multidiscip. Optim., 43(2), 165–180.
Svanberg, K. (1987). “The method of moving asymptotes—A new method for structural optimization.” Int. J. Numer. Methods Eng., 24(2), 359–373.
Zhu, M., Yang, Y., Gaynor, A. T., and Guest, J. K. (2014). “Considering constructability in structural topology optimization.” Proc., Structures Congress 2014, ASCE, Reston, VA, 2754–2764.
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Published In
Journal of Engineering Mechanics
Volume 141 • Issue 1 • January 2015
Copyright
© 2014 American Society of Civil Engineers.
History
Received: Nov 12, 2013
Accepted: May 5, 2014
Published online: Jun 11, 2014
Published in print: Jan 1, 2015
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