A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam
<p>Strain compatibility between diagonally stressed concrete and the reinforcement in the cracked web of a reinforced concrete beam.</p> "> Figure 2
<p>Average stress profiles (<math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>a</mi> <mi>v</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>σ</mi> <mrow> <mi>s</mi> <mi>t</mi> <mo>,</mo> <mi>a</mi> <mi>v</mi> </mrow> </msub> </semantics></math>) for an embedded reinforcement constitutive model including several cracks.</p> "> Figure 3
<p>Computational efficiency indices.</p> ">
Abstract
:1. Introduction
2. Problem Statement
3. Development and Convergence of the Method
- (a)
- being ,
- (b)
- being ,
- when is the Banach space of real matrices, and is the set of linear operators defined in X.
4. Efficiency Indices
5. Numerical Performance
5.1. Example
Method | Iteration | e-Time | |||
---|---|---|---|---|---|
Newton | 8 | 2.0 | 0.98 | ||
O6 | 4 | 0.0 | 6.0 | 0.99 | |
FS6 | 4 | 6.0 | 1.02 | ||
PM1 | 4 | 6.0 | 0.99 | ||
4 | 5.0 | 1.00 | |||
4 | 6.0 | 1.02 |
5.2. Example
Method | Iterations | e-Time (Sec) | |||
---|---|---|---|---|---|
Newton | 11 | 2.0 | 9.99 | ||
O6 | 5 | 6.0 | 10.24 | ||
FS6 | 5 | 6.0 | 11.00 | ||
PM1 | 6 | 6.0 | 10.58 | ||
6 | 6.0 | 11.07 | |||
6 | 6.0 | 10.79 |
5.3. Example
Method | Iterations | e-Time (Sec) | |||
---|---|---|---|---|---|
Newton | 10 | 2.0 | 1.22 | ||
O6 | 5 | 6.0 | 1.24 | ||
FS6 | 4 | 6.0 | 1.20 | ||
PM1 | 5 | 6.0 | 1.27 | ||
5 | 5.0 | 1.39 | |||
5 | 6.0 | 1.39 |
Method | Iterations | e-Time (Sec) | ||
---|---|---|---|---|
Newton | 5 | 0.0342 | 18.3242 | |
O6 | 3 | 9.0862 | 21.6703 | |
FS6 | 3 | 218.18 | 22.1258 | |
PM1 | 4 | 3.8554 | 21.9594 | |
4 | 7.3926 | 28.0727 | ||
3 | 0.0423 | 22.4797 |
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Input | Range |
---|---|
[195,000, 205,000] | |
Method | CI |
---|---|
Newton | |
O6 | |
PM1 | |
FS6 |
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Padilla, J.J.; Chicharro, F.I.; Cordero, A.; Hernández-Díaz, A.M.; Torregrosa, J.R. A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam. Mathematics 2024, 12, 499. https://doi.org/10.3390/math12030499
Padilla JJ, Chicharro FI, Cordero A, Hernández-Díaz AM, Torregrosa JR. A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam. Mathematics. 2024; 12(3):499. https://doi.org/10.3390/math12030499
Chicago/Turabian StylePadilla, José J., Francisco I. Chicharro, Alicia Cordero, Alejandro M. Hernández-Díaz, and Juan R. Torregrosa. 2024. "A Class of Efficient Sixth-Order Iterative Methods for Solving the Nonlinear Shear Model of a Reinforced Concrete Beam" Mathematics 12, no. 3: 499. https://doi.org/10.3390/math12030499