Numerically Efficient Three-Dimensional Model for Non-Linear Finite Element Analysis of Reinforced Concrete Structures
Abstract
:1. Introduction
2. Proposed Solution Strategy
2.1. Constitutive Model of Concrete
2.2. Ultimate Surface of Concrete
2.3. Smeared Crack Model and Crushing
2.4. Generalised Constitutive Model of Steel
- Phase 1—elastic stage, in which concrete and steel behave in a linear manner and have equal strains;
- Phase 2—crack formation, which begins when stresses in concrete cover reach tensile strength;
- Phase 3—stabilised cracking; and
- Phase 4—yielding of steel, in which the TS effect vanishes due to the degradation of the bonding properties between reinforcing steel and concrete.
2.5. Implementation, Finite Elements, Solution Method, Convergence Criteria
- mesh size should be larger than approximately 1.5 the aggregate size (approximately 50 mm),
- mesh size should be smaller than typical crack spacing (approximately 150 mm).
3. Verification and Validation of the Proposed Strategy
3.1. CS1 Numerical Willam’s Test
- phase 1—uniaxial tension up to the crack formation, components of the strain increments vector in the Cartesian coordinate system have the following ratio: ;
- phase 2—biaxial tension and shear, components of the strain increments vector in the Cartesian coordinate system have the following ratio: .
- during the test, the larger principal stress does not exceed the tensile strength;
- at the end of phase 2 all stresses tend to zero.
3.2. CS2 Beam in Bending
3.3. CS3 Beam without Stirrups in Shear
3.4. CS4 Beam with Stirrups in Shear
4. Discussion
5. Conclusions and Future Work
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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No | Parameter | CS1 | CS2 | CS3 | CS4 |
---|---|---|---|---|---|
Concrete model | |||||
1 | (GPa) | 28 | 29 | 28 | 28 |
2 | 0.2 | 0.2 | 0.2 | 0.2 | |
3 | [1] | 0.2 | 0.2 | 0.2 | 0.2 |
4 | [1] | −0.08 | −0.05 | −0.05 | −0.005 |
5 | (MPa) | 24.0 | 27.0 | 22.5 | 24.1 |
6 | (MPa) | 2.0 | 3.2 | 2.2 | 2.5 |
Tension stiffening model | |||||
7 | (GPa) | - | 196 | 200 | 200 |
8 | (GPa) | - | 10 | 10 | 10 |
9 | (MPa) | - | 500 | 555 | 555 |
10 | [1] | - | 1.1 | 1.1 | 1.1 |
11 | [1] | - | 0.4 | 0.4 | 0.4 |
12 | [1] | - | 0.017 | 0.035 | 0.035 |
CS | [kN] | [kN] | [1] | [mm] | [mm] | [1] |
---|---|---|---|---|---|---|
2 | 41.0 | 40.9 | 0.998 | 10.0 | 12.0 | 1.200 |
3 | 330.0 | 329.0 | 0.988 | 6.6 | 8.9 | 1.348 |
4 | 467.3 | 471.2 | 1.008 | 13.8 | 16.6 | 1.203 |
mean | 0.998 | 1.250 | ||||
CoV | 0.008 | 0.057 |
Mesh | CS2 | CS3 | CS4 | |
---|---|---|---|---|
dimensions of model | ||||
variables | coarse | 690 | 1572 | 1980 |
medium | 1536 | 2160 | 2616 | |
fine | 2451 | 4443 | 5085 | |
very fine | 8781 | 14,730 | 16,818 | |
equations | coarse | 420 | 1320 | 1320 |
medium | 963 | 1848 | 1848 | |
fine | 1755 | 4005 | 4005 | |
very fine | 7440 | 13,950 | 13,950 | |
calculation effort | ||||
increments | coarse | 50 | 58 | 66 |
medium | 50 | 57 | 55 | |
fine | 70 | 82 | 87 | |
very fine | 145 | 216 | 262 | |
iterations | coarse | 129 | 676 | 691 |
medium | 231 | 709 | 450 | |
fine | 443 | 930 | 1046 | |
very fine | 1389 | 2496 | 3214 | |
calculation time [s] | coarse | 14 | 107 | 114 |
medium | 28 | 204 | 57 | |
fine | 70 | 361 | 276 | |
very fine | 770 | 1660 | 2130 |
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Dudziak, S. Numerically Efficient Three-Dimensional Model for Non-Linear Finite Element Analysis of Reinforced Concrete Structures. Materials 2021, 14, 1578. https://doi.org/10.3390/ma14071578
Dudziak S. Numerically Efficient Three-Dimensional Model for Non-Linear Finite Element Analysis of Reinforced Concrete Structures. Materials. 2021; 14(7):1578. https://doi.org/10.3390/ma14071578
Chicago/Turabian StyleDudziak, Sławomir. 2021. "Numerically Efficient Three-Dimensional Model for Non-Linear Finite Element Analysis of Reinforced Concrete Structures" Materials 14, no. 7: 1578. https://doi.org/10.3390/ma14071578