Computation of Limit Loads for Bending Plates
Received: 8 January 2023 | Revised: 2 February2 2023 | Accepted: 17 February 2023 | Online: 2 April 2023
Corresponding author: Phu-Huan Vo Nguyen
Abstract
The purpose of this paper is to present a method for calculating the upper bound limit loads of plate bending using a conforming Hsieh-Clough-Tocher (HCT) element. These limit loads can be obtained from Koiter’s kinematic shakedown theorem for the case of one load vertex instead of using the kinematic limit theorem. When combining this theorem with the approximated displacement field, the limit analysis turns into an optimization problem and can be effectively solved by Second-Order Cone Programming (SOCP). Several benchmark plate problems such as square, rectangular, and L-shape plates are investigated to illustrate the effectiveness of the proposed solution. The results of the proposed method show good agreement with the results of previous studies. The maximum error is only 2.91% for the fully clamped rectangular plate problem.
Keywords:
limit analysis, second-order cone programming, plates, shakedown, plate bendingDownloads
References
W. Prager and P. G. Hodge Jr., Theory of Perfectly Plastic Solids, 1st ed. New York, NY, USA: John Wiley & Sons, 1951.
A. A. Abdulhussein and M. H. Al-Sherrawi, "Experimental and Numerical Analysis of the Punching Shear Resistance Strengthening of Concrete Flat Plates by Steel Collars," Engineering, Technology & Applied Science Research, vol. 11, no. 6, pp. 7853–7860, Dec. 2021. DOI: https://doi.org/10.48084/etasr.4497
A. N. Dalaf and S. D. Mohammed, "The Impact of Hybrid Fibers on Punching Shear Strength of Concrete Flat Plates Exposed to Fire," Engineering, Technology & Applied Science Research, vol. 11, no. 4, pp. 7452–7457, Aug. 2021. DOI: https://doi.org/10.48084/etasr.4314
P. C. Nguyen, "Local Instability Analysis for Axial Compressive Steel Columns," Engineering, Technology & Applied Science Research, vol. 12, no. 6, pp. 9527–9531, Dec. 2022. DOI: https://doi.org/10.48084/etasr.5238
P. H. V. Nguyen and P. C. Nguyen, "Effects of Shaft Grouting on the Bearing Behavior of Barrette Piles: A Case Study in Ho Chi Minh City," Engineering, Technology & Applied Science Research, vol. 11, no. 5, pp. 7653–7657, Oct. 2021. DOI: https://doi.org/10.48084/etasr.4389
P. C. Nguyen, D. D. Pham, T. T. Tran, and T. Nghia-Nguyen, "Modified Numerical Modeling of Axially Loaded Concrete-Filled Steel Circular-Tube Columns," Engineering, Technology & Applied Science Research, vol. 11, no. 3, pp. 7094–7099, Jun. 2021. DOI: https://doi.org/10.48084/etasr.4157
P. C. Nguyen, B. Le-Van, and S. D. T. V. Thanh, "Nonlinear Inelastic Analysis of 2D Steel Frames : An Improvement of the Plastic Hinge Method," Engineering, Technology & Applied Science Research, vol. 10, no. 4, pp. 5974–5978, Aug. 2020. DOI: https://doi.org/10.48084/etasr.3600
P. C. Nguyen, "Nonlinear Inelastic Earthquake Analysis of 2D Steel Frames," Engineering, Technology & Applied Science Research, vol. 10, no. 6, pp. 6393–6398, Dec. 2020. DOI: https://doi.org/10.48084/etasr.3855
F. Tin-Loi and N. S. Ngo, "Performance of the p-version finite element method for limit analysis," International Journal of Mechanical Sciences, vol. 45, no. 6, pp. 1149–1166, Jun. 2003. DOI: https://doi.org/10.1016/j.ijmecsci.2003.08.004
T. N. Trần, R. Kreißig, and M. Staat, "Probabilistic limit and shakedown analysis of thin plates and shells," Structural Safety, vol. 31, no. 1, pp. 1–18, Jan. 2009. DOI: https://doi.org/10.1016/j.strusafe.2007.10.003
D. K. Vu, A. M. Yan, and H. Nguyen-Dang, "A primal–dual algorithm for shakedown analysis of structures," Computer Methods in Applied Mechanics and Engineering, vol. 193, no. 42, pp. 4663–4674, Oct. 2004. DOI: https://doi.org/10.1016/j.cma.2004.03.011
C. V. Le, H. Nguyen-Xuan, and H. Nguyen-Dang, "Upper and lower bound limit analysis of plates using FEM and second-order cone programming," Computers & Structures, vol. 88, no. 1, pp. 65–73, Jan. 2010. DOI: https://doi.org/10.1016/j.compstruc.2009.08.011
S. Chen, Y. Liu, and Z. Cen, "Lower-bound limit analysis by using the EFG method and non-linear programming," International Journal for Numerical Methods in Engineering, vol. 74, no. 3, pp. 391–415, 2008. DOI: https://doi.org/10.1002/nme.2177
P. L. H. Ho and C. V. Le, "A stabilized iRBF mesh-free method for quasi-lower bound shakedown analysis of structures," Computers & Structures, vol. 228, Feb. 2020, Art. no. 106157. DOI: https://doi.org/10.1016/j.compstruc.2019.106157
P. L. H. Ho, C. V. Le, and T. Tran-Cong, "Limit state analysis of reinforced concrete slabs using an integrated radial basis function based mesh-free method," Applied Mathematical Modelling, vol. 53, pp. 1–11, Jan. 2018. DOI: https://doi.org/10.1016/j.apm.2017.08.006
C. V. Le, H. Askes, and M. Gilbert, "Adaptive element-free Galerkin method applied to the limit analysis of plates," Computer Methods in Applied Mechanics and Engineering, vol. 199, no. 37, pp. 2487–2496, Aug. 2010. DOI: https://doi.org/10.1016/j.cma.2010.04.004
C. V. Le, M. Gilbert, and H. Askes, "Limit analysis of plates using the EFG method and second-order cone programming," International Journal for Numerical Methods in Engineering, vol. 78, no. 13, pp. 1532–1552, 2009. DOI: https://doi.org/10.1002/nme.2535
H. V. Do and H. Nguyen-Xuan, "Limit and shakedown isogeometric analysis of structures based on Bézier extraction," European Journal of Mechanics - A/Solids, vol. 63, pp. 149–164, May 2017. DOI: https://doi.org/10.1016/j.euromechsol.2017.01.004
A. Capsoni and L. Corradi, "Limit Analysis of Plates-a Finite Element Formulation," Structural Engineering and Mechanics: An International Journal, vol. 8, no. 4, pp. 325–341, Oct. 1999. DOI: https://doi.org/10.12989/sem.1999.8.4.325
P. G. Hodge Jr. and T. Belytschko, "Numerical Methods for the Limit Analysis of Plates," Journal of Applied Mechanics, vol. 35, no. 4, pp. 796–802, Dec. 1968. DOI: https://doi.org/10.1115/1.3601308
T. N. Tran, "A dual algorithm for shakedown analysis of plate bending," International Journal for Numerical Methods in Engineering, vol. 86, no. 7, pp. 862–875, 2011. DOI: https://doi.org/10.1002/nme.3081
T. D. Tran, C. V. Le, D. C. Pham, and H. Nguyen-Xuan, "Shakedown reduced kinematic formulation, separated collapse modes, and numerical implementation," International Journal of Solids and Structures, vol. 51, no. 15, pp. 2893–2899, Aug. 2014. DOI: https://doi.org/10.1016/j.ijsolstr.2014.04.016
W. T. Koiter, "General theorems for elastic–plastic solids," in Progress in Solids Mechanics, I. N. Sneddon and R. Hill, Eds. Amsterdam , Netherlands: North-Holland Publishing Company, 1960.
C. V. Le, P. H. Nguyen, and T. Q. Chu, "A curvature smoothing Hsieh–Clough–Tocher element for yield design of reinforced concrete slabs," Computers & Structures, vol. 152, pp. 59–65, May 2015. DOI: https://doi.org/10.1016/j.compstruc.2015.02.009
K. D. Andersen, E. Christiansen, and M. L. Overton, "Computing Limit Loads by Minimizing a Sum of Norms," SIAM Journal on Scientific Computing, vol. 19, no. 3, pp. 1046–1062, Jan. 1998. DOI: https://doi.org/10.1137/S1064827594275303
J. Lubliner, Plasticity Theory. New York, NY, USA: Macmillan, 1990.
Downloads
How to Cite
License
Copyright (c) 2023 Trung-Dung Tran, Phu-Huan Vo Nguyen
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain the copyright and grant the journal the right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) after its publication in ETASR with an acknowledgement of its initial publication in this journal.
