Achievements of Truss Models for Reinforced Concrete Structures

Author(s)
P. G. Papadopoulos^{*},
H. Xenidis,
P. Lazaridis,
A. Diamantopoulos,
P. Lambrou,
Y. Arethas

ABSTRACT

Achievements are presented for truss models of RC structures developed in previous years: 1. Two constitutive models, biaxial and triaxial, are based on regular trusses, with bars obeying nonlinear uniaxial σ-ε laws of material under simulation; both models have been compared with test results and show a dependence of Poisson ratio on curvature of σ-ε law. 2. A truss finite element has been used in the nonlinear static and dynamic analysis of plane RC frames; it has been compared with test results and describes, in a simple way, the formation of plastic hinges. 3. Thanks to the very simple geometry of a truss, the equilibrium equations can be easily written and the stiffness matrix can be easily updated, both with respect to the deformed truss, within each step of a static incremental loading or within each time step of a dynamic analysis, so that to take into account geometric nonlinearities. So the confinement of a RC column is interpreted as a structural stability effect of concrete. And a significant role of the transverse reinforcement is revealed, that of preventing, by its close spacing and sufficient amount, the buckling of inner longitudinal concrete struts, which would lead to a global instability of the RC column. 4. The proposed truss model is statically indeterminate, so it exhibits some features, which are not met by the “strut-and-tie” model.

Achievements are presented for truss models of RC structures developed in previous years: 1. Two constitutive models, biaxial and triaxial, are based on regular trusses, with bars obeying nonlinear uniaxial σ-ε laws of material under simulation; both models have been compared with test results and show a dependence of Poisson ratio on curvature of σ-ε law. 2. A truss finite element has been used in the nonlinear static and dynamic analysis of plane RC frames; it has been compared with test results and describes, in a simple way, the formation of plastic hinges. 3. Thanks to the very simple geometry of a truss, the equilibrium equations can be easily written and the stiffness matrix can be easily updated, both with respect to the deformed truss, within each step of a static incremental loading or within each time step of a dynamic analysis, so that to take into account geometric nonlinearities. So the confinement of a RC column is interpreted as a structural stability effect of concrete. And a significant role of the transverse reinforcement is revealed, that of preventing, by its close spacing and sufficient amount, the buckling of inner longitudinal concrete struts, which would lead to a global instability of the RC column. 4. The proposed truss model is statically indeterminate, so it exhibits some features, which are not met by the “strut-and-tie” model.

Cite this paper

P. Papadopoulos, H. Xenidis, P. Lazaridis, A. Diamantopoulos, P. Lambrou and Y. Arethas, "Achievements of Truss Models for Reinforced Concrete Structures,"*Open Journal of Civil Engineering*, Vol. 2 No. 3, 2012, pp. 125-131. doi: 10.4236/ojce.2012.23018.

P. Papadopoulos, H. Xenidis, P. Lazaridis, A. Diamantopoulos, P. Lambrou and Y. Arethas, "Achievements of Truss Models for Reinforced Concrete Structures,"

References

[1] D. Ngo and A. C. Scordelis, “Finite Element Analysis of Reinforced Concrete Beams,” ACI Journal, Vol. 64, 1967, pp. 152-163.

[2] P. G. Bergan and I. Holand, “Nonlinear Finite Element Analysis of Concrete Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 17-18, 1979, pp. 443-467. doi:10.1016/0045-7825(79)90027-6

[3] A. C. Scordelis, Editor, ASCE Task Committee on Concrete and Masonry Structures, “State-of-the-Art Report on Finite Element Analysis of Reinforced Concrete,” ASCE Special Publication, 1982.

[4] J. H. Argyris, Organizer, International Conferences F.E.No.Mech. (Finite Elements in Nonlinear Mechanics). Institute for Statics and Dynamics, University of Stutgart, Germany, I.30 August-1 September 1978, II. 25-28 August 1981. III. 10-13 September 1984.

[5] W. F. Chen and E. C. Ting. “Constitutive Models for Concrete Structures,” Journal of Engineering Mechanics Division ASCE, Vol. 106, No. 1, 1980, pp. 1-19.

[6] Z. Mroz, V. A. Norris and O. C. Zienkiewicz, “Application of an Anisotropic Hardening Model in the Analysis of Elastic-Plastic Deformation of Soils,” Geotechnique, Vol. 29, 1979, pp. 1-34. doi:10.1680/geot.1979.29.1.1

[7] Z. P. Bazant and S. S. Kim, “Plastic-Fracturing Theory for Concrete,” Journal of Engineering Mechanics Division ASCE, Vol. 105, No. 3, 1979, pp. 407-428

[8] D. Darwin and D. A. Pecknold, “Analysis of Cyclic Loading of RC Structures,” Computers and Structures, Vol. 7, No. 1, 1977, pp. 137-147. doi:10.1016/0045-7949(77)90068-2

[9] K. J. Willam and E. P. Warnke, “Constitutive Model for the Triaxial Behavior of Concrete,” Proceedings of IABSE, Structural Engineering Report 19, Section III, 1975, pp. 1-30.

[10] N. J. Burt and J. W. Dougill, “Progressive Failure in a Model Heterogeneous Medium,” Journal of Engineering Mechanics Division ASCE, Vol. 103, 1977, pp. 365-376.

[11] P. G. Papadopoulos, “Biaxial Network Constitutive Model,” Journal of Engineering Mechanics ASCE, Vol. 110, No. 3, 1984, pp. 449-464. doi:10.1061/(ASCE)0733-9399(1984)110:3(449)

[12] P. G. Papadopoulos, “A Triaxial Network Constitutive Model,” Computers and Structures, Vol. 23, 1986, pp. 497-501. doi:10.1016/0045-7949(86)90093-3

[13] H. B. Kupfer, H. D. Hilsdorf and H. Rusch, “Behavior of Concrete under Biaxial Stresses,” ACI Journal, 1969, pp. 656-666.

[14] R. Palaniswamy and S. P. Shah, “Fracture and Stress- Strain Relationships of Concrete under Triaxial Compression,” Journal of Structural Division ASCE, Vol. 100, 1974, pp. 901-916.

[15] R. Scavuzzo, T. Stankowski, K. Gerstle and H.-Y. Ko, “Stress-Strain Curves for Concrete under Multiaxial Load Histories,” University of Colorado, Boulder, 1983.

[16] E. Absi, “Méthodes des Calcus Numerique en Elasticité,” Eyrolles, Paris, 1978.

[17] P. G. Papadopoulos, “Nonlinear Static Analysis of Reinforced Concrete Frames by Network Models,” Advances in Engineering Software, Vol. 110, No. 3, 1988, pp. 114- 122. doi:10.1016/0141-1195(88)90010-1

[18] P. G. Papadopoulos and C. G. Karayannis, “Seismic Analysis of R/C Frames by Network Models,” Computers and Structures, Vol. 28, No. 4, 1988, pp. 481-494. doi:10.1016/0045-7949(88)90022-3

[19] K. Stylianidis and G. Penelis, “Experimental Study of, bare and Infilled by Wall, One Story Frames under Cyclic shear Loading,” 7th Greek Conference on Concrete, Patra, Vol. 2, 1985, pp. 47-55.

[20] P. Hidalgo and R. W. Clough, “Earthquake Simulator Study of a Reinforced Concrete Frame,” EERC Report 74-13, University of California, Berkeley, 1974.

[21] S. C. Goel, B. Stojadinovicz and K. H. Lee, “Truss Analogy for Steel Moment Connections,” Engineering Journal, Second Quarter 1997, pp. 43-53.

[22] E. Schlangen and E. J. Garboczi, “Fracture Simulations of Concrete Using Lattice Models: Computational Aspects,” Engineering Fracture Mechanics, Vol. 57, No. 2-3, 1997, pp. 319-332. doi:10.1016/S0013-7944(97)00010-6

[23] F. Fraternali, M. Angelilo and A. Fortunato, “A Lumped Stress Method for Plane Elastic Problems and the Discrete Continuum Approximation,” International Journal of Solids and Structures, Vol. 39, 2002, pp. 6211-6240. doi:10.1016/S0020-7683(02)00472-9

[24] J. Schlaich, K. Sch?fer and M. Jennewein, “Towards a Consistent Design of Structural Concrete,” PCI Journal Special Report, Vol. 32, No. 3, 1987, pp. 75-150.

[25] T. T. C. Hsu, “Unified Theory of Reinforced Concrete,” CRC Press, 1993.

[26] F. J. Vecchio and M. P. Collins, “Compression Response of Cracked Reinforced Concrete,” Journal of Structural Engineering ASCE, Vol. 113, 1993, pp. 3590-3610. doi:10.1061/(ASCE)0733-9445(1993)119:12(3590)

[27] ASCE-ACI Committee 445 on Shear and Torsion, “Recent Approaches to Shear Design of Structural Concrete. State-of-the-Art Report,” Journal of Structural Engineering ASCE, Vol. 119, No. 12, 1998, pp. 1375-1417.

[28] P. G. Papadopoulos and H. C. Xenidis, “A Truss Model with Structural Instability for the Confinement of Concrete Columns,” Journal of EEE (European Earthquake Engineering), Part 2, 1999, pp. 57-79.

[29] P. G. Papadopoulos, H. Xenidis, C. Karayannis, A. Diamantopoulos and P. Lambrou, “Confinement of Concrete Column Interpreted as a Structural Stability Effect,” 6th GRACM (Greek Association of Computational Mechanics) Conference, Thessaloniki, 19-21 June 2008.

[30] P. G. Papadopoulos, H. Xenidis, D. Plasatis, P. Kiousis and C. Karayannis, “Concrete Stability Achieved by Confinement in a RC Column,” 12th International Conference on Civil, Structural and Environmental Engineering Computing, Coordinator B.H.V. Topping, Madeira, Portugal, 1-4 September 2009.

[31] K. Park, M. J. N. Priestley and W. D. Gill, “Ductility of Square Confined Concrete Columns,” Journal of Structural Division ASCE, Vol. 108, No. 4, 1982, pp. 929-950.

[32] S. Watson, F. A. Zahn and R. Park, “Confining Reinforcement for Concrete Columns,” Journal of Structural Engineering ASCE, Vol. 120, No. 6, 1984, pp. 1798- 1849.

[33] J. B. Mander, M. J. N. Priestley and R. Park, “Theoretical Stress-Strain Model for Confined Concrete,” Journal of Structural Engineering ASCE, Vol. 114, No. 8, 1988, pp. 1804-1826. doi:10.1061/(ASCE)0733-9445(1988)114:8(1804)

[34] Uniform Building Code 2, “Structural Engineering Design Provisions,” Chapter 19. Concrete, 19.2.1. Reinforced Concrete Structures Resisting Forces Induced by Earthquake Motions 19.2.14. Frame Members Subjected to Bending and Axial Load, 1994, pp. 237-239.

[35] New Zealand Standards 3101, “Code of Practice for the Design of Concrete Structures,” Chapter 17, Members Subjected to Flexure and Axial Loads, Additional Seismic Requirements, 1989.

[36] Eurocode 8, “Earthquake Resistant Design of Structures,” Part 1-3. General Rules and Rules for Buildings. 2, Specific Rules for Concrete Buildings. 2.8. Provisions for Columns, Brussels, 1993, pp. 35-46.

[37] P. G. Papadopoulos, “A Simple Algorithm for the Nonlinear Dynamic Analysis of Networks,” Computers and Structures, Vol. 18, No. 1, 1984, pp. 1-8. doi:10.1016/0045-7949(84)90074-9

[1] D. Ngo and A. C. Scordelis, “Finite Element Analysis of Reinforced Concrete Beams,” ACI Journal, Vol. 64, 1967, pp. 152-163.

[2] P. G. Bergan and I. Holand, “Nonlinear Finite Element Analysis of Concrete Structures,” Computer Methods in Applied Mechanics and Engineering, Vol. 17-18, 1979, pp. 443-467. doi:10.1016/0045-7825(79)90027-6

[3] A. C. Scordelis, Editor, ASCE Task Committee on Concrete and Masonry Structures, “State-of-the-Art Report on Finite Element Analysis of Reinforced Concrete,” ASCE Special Publication, 1982.

[4] J. H. Argyris, Organizer, International Conferences F.E.No.Mech. (Finite Elements in Nonlinear Mechanics). Institute for Statics and Dynamics, University of Stutgart, Germany, I.30 August-1 September 1978, II. 25-28 August 1981. III. 10-13 September 1984.

[5] W. F. Chen and E. C. Ting. “Constitutive Models for Concrete Structures,” Journal of Engineering Mechanics Division ASCE, Vol. 106, No. 1, 1980, pp. 1-19.

[6] Z. Mroz, V. A. Norris and O. C. Zienkiewicz, “Application of an Anisotropic Hardening Model in the Analysis of Elastic-Plastic Deformation of Soils,” Geotechnique, Vol. 29, 1979, pp. 1-34. doi:10.1680/geot.1979.29.1.1

[7] Z. P. Bazant and S. S. Kim, “Plastic-Fracturing Theory for Concrete,” Journal of Engineering Mechanics Division ASCE, Vol. 105, No. 3, 1979, pp. 407-428

[8] D. Darwin and D. A. Pecknold, “Analysis of Cyclic Loading of RC Structures,” Computers and Structures, Vol. 7, No. 1, 1977, pp. 137-147. doi:10.1016/0045-7949(77)90068-2

[9] K. J. Willam and E. P. Warnke, “Constitutive Model for the Triaxial Behavior of Concrete,” Proceedings of IABSE, Structural Engineering Report 19, Section III, 1975, pp. 1-30.

[10] N. J. Burt and J. W. Dougill, “Progressive Failure in a Model Heterogeneous Medium,” Journal of Engineering Mechanics Division ASCE, Vol. 103, 1977, pp. 365-376.

[11] P. G. Papadopoulos, “Biaxial Network Constitutive Model,” Journal of Engineering Mechanics ASCE, Vol. 110, No. 3, 1984, pp. 449-464. doi:10.1061/(ASCE)0733-9399(1984)110:3(449)

[12] P. G. Papadopoulos, “A Triaxial Network Constitutive Model,” Computers and Structures, Vol. 23, 1986, pp. 497-501. doi:10.1016/0045-7949(86)90093-3

[13] H. B. Kupfer, H. D. Hilsdorf and H. Rusch, “Behavior of Concrete under Biaxial Stresses,” ACI Journal, 1969, pp. 656-666.

[14] R. Palaniswamy and S. P. Shah, “Fracture and Stress- Strain Relationships of Concrete under Triaxial Compression,” Journal of Structural Division ASCE, Vol. 100, 1974, pp. 901-916.

[15] R. Scavuzzo, T. Stankowski, K. Gerstle and H.-Y. Ko, “Stress-Strain Curves for Concrete under Multiaxial Load Histories,” University of Colorado, Boulder, 1983.

[16] E. Absi, “Méthodes des Calcus Numerique en Elasticité,” Eyrolles, Paris, 1978.

[17] P. G. Papadopoulos, “Nonlinear Static Analysis of Reinforced Concrete Frames by Network Models,” Advances in Engineering Software, Vol. 110, No. 3, 1988, pp. 114- 122. doi:10.1016/0141-1195(88)90010-1

[18] P. G. Papadopoulos and C. G. Karayannis, “Seismic Analysis of R/C Frames by Network Models,” Computers and Structures, Vol. 28, No. 4, 1988, pp. 481-494. doi:10.1016/0045-7949(88)90022-3

[19] K. Stylianidis and G. Penelis, “Experimental Study of, bare and Infilled by Wall, One Story Frames under Cyclic shear Loading,” 7th Greek Conference on Concrete, Patra, Vol. 2, 1985, pp. 47-55.

[20] P. Hidalgo and R. W. Clough, “Earthquake Simulator Study of a Reinforced Concrete Frame,” EERC Report 74-13, University of California, Berkeley, 1974.

[21] S. C. Goel, B. Stojadinovicz and K. H. Lee, “Truss Analogy for Steel Moment Connections,” Engineering Journal, Second Quarter 1997, pp. 43-53.

[22] E. Schlangen and E. J. Garboczi, “Fracture Simulations of Concrete Using Lattice Models: Computational Aspects,” Engineering Fracture Mechanics, Vol. 57, No. 2-3, 1997, pp. 319-332. doi:10.1016/S0013-7944(97)00010-6

[23] F. Fraternali, M. Angelilo and A. Fortunato, “A Lumped Stress Method for Plane Elastic Problems and the Discrete Continuum Approximation,” International Journal of Solids and Structures, Vol. 39, 2002, pp. 6211-6240. doi:10.1016/S0020-7683(02)00472-9

[24] J. Schlaich, K. Sch?fer and M. Jennewein, “Towards a Consistent Design of Structural Concrete,” PCI Journal Special Report, Vol. 32, No. 3, 1987, pp. 75-150.

[25] T. T. C. Hsu, “Unified Theory of Reinforced Concrete,” CRC Press, 1993.

[26] F. J. Vecchio and M. P. Collins, “Compression Response of Cracked Reinforced Concrete,” Journal of Structural Engineering ASCE, Vol. 113, 1993, pp. 3590-3610. doi:10.1061/(ASCE)0733-9445(1993)119:12(3590)

[27] ASCE-ACI Committee 445 on Shear and Torsion, “Recent Approaches to Shear Design of Structural Concrete. State-of-the-Art Report,” Journal of Structural Engineering ASCE, Vol. 119, No. 12, 1998, pp. 1375-1417.

[28] P. G. Papadopoulos and H. C. Xenidis, “A Truss Model with Structural Instability for the Confinement of Concrete Columns,” Journal of EEE (European Earthquake Engineering), Part 2, 1999, pp. 57-79.

[29] P. G. Papadopoulos, H. Xenidis, C. Karayannis, A. Diamantopoulos and P. Lambrou, “Confinement of Concrete Column Interpreted as a Structural Stability Effect,” 6th GRACM (Greek Association of Computational Mechanics) Conference, Thessaloniki, 19-21 June 2008.

[30] P. G. Papadopoulos, H. Xenidis, D. Plasatis, P. Kiousis and C. Karayannis, “Concrete Stability Achieved by Confinement in a RC Column,” 12th International Conference on Civil, Structural and Environmental Engineering Computing, Coordinator B.H.V. Topping, Madeira, Portugal, 1-4 September 2009.

[31] K. Park, M. J. N. Priestley and W. D. Gill, “Ductility of Square Confined Concrete Columns,” Journal of Structural Division ASCE, Vol. 108, No. 4, 1982, pp. 929-950.

[32] S. Watson, F. A. Zahn and R. Park, “Confining Reinforcement for Concrete Columns,” Journal of Structural Engineering ASCE, Vol. 120, No. 6, 1984, pp. 1798- 1849.

[33] J. B. Mander, M. J. N. Priestley and R. Park, “Theoretical Stress-Strain Model for Confined Concrete,” Journal of Structural Engineering ASCE, Vol. 114, No. 8, 1988, pp. 1804-1826. doi:10.1061/(ASCE)0733-9445(1988)114:8(1804)

[34] Uniform Building Code 2, “Structural Engineering Design Provisions,” Chapter 19. Concrete, 19.2.1. Reinforced Concrete Structures Resisting Forces Induced by Earthquake Motions 19.2.14. Frame Members Subjected to Bending and Axial Load, 1994, pp. 237-239.

[35] New Zealand Standards 3101, “Code of Practice for the Design of Concrete Structures,” Chapter 17, Members Subjected to Flexure and Axial Loads, Additional Seismic Requirements, 1989.

[36] Eurocode 8, “Earthquake Resistant Design of Structures,” Part 1-3. General Rules and Rules for Buildings. 2, Specific Rules for Concrete Buildings. 2.8. Provisions for Columns, Brussels, 1993, pp. 35-46.

[37] P. G. Papadopoulos, “A Simple Algorithm for the Nonlinear Dynamic Analysis of Networks,” Computers and Structures, Vol. 18, No. 1, 1984, pp. 1-8. doi:10.1016/0045-7949(84)90074-9