Innovative Equivalent Elastic Modulus Based Stress Calculation Methodology for Reinforced Concrete Columns
Abstract
:1. Introduction
2. Composite Elasticity Model
- The total cross-sectional area, AT, is composed of a concrete area, AC, and the steel bars’ area, AS;
- The steel bars’ area, AS, or the concrete area, AC, are expressible as follows;AS = AT − ACAC = AT – AS
- The total area can be calculatable provided that the cross-section shape of the column is given as circular, rectangular, or square with two-dimensional measures;
- As for the stress distribution, concrete and steel areas are subjected to the same stress amount;
- As for the strain, both materials will react equally, where concrete will be a dependent material to steel or vice versa. Thus, each material will have the same strain under the force, F, in a column (see Figure 1).
3. Application and Results
4. Discussion
- The applications indicate that the relative improvement percentages over the classical Hooke’s Law calculations vary between 6% and 27% (see Figure 6). The steel reinforcement consideration is the main improvement factor, together with areal and elastic modulus contributions of various cross-section shapes with different concrete qualities;
- As the concrete quality increases (from C16 to C50), the improvement percentage also increases, and other concrete quality improvement percentages are confined between these two concrete qualities;
- The minimum percentage improvement is with a square cross-sectional area coupled with C50, whereas the maximum is with a slightly rectangular cross-sectional area coupled with C16 concrete quality;
- Increasing the relative improvement percentage on behalf of the steel is possible by increasing the steel area percentage. Thus, there is an optimum reinforcement possibility for the column design. In such optimization work, the budget (economic) conditions also play a restrictive role;
- Steel reinforcement contribution calculations augment the strength of the column, and thus some part of the “safety factor” can be reduced according to the proposed methodological calculation. With the newly proposed methodology, the SF amount becomes closer to 1.
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Cross-Section No | Section Geometric Details | Rebar Details | Area Calculations | |||||||
---|---|---|---|---|---|---|---|---|---|---|
Shape | Width | Length | Diameter | Quantity | Diameter | AT | AS | AC | ρ | |
mm | mm | mm | Pieces | mm | mm2 | mm2 | mm2 | % | ||
1 | Circular | - | - | 600 | 20 | 22 | 282,743 | 7603 | 275,141 | 2.69% |
2 | Circular | - | - | 500 | 14 | 20 | 196,350 | 4398 | 191,951 | 2.24% |
3 | Circular | - | - | 550 | 18 | 22 | 237,583 | 6842 | 230,741 | 2.88% |
4 | Circular | - | - | 650 | 22 | 20 | 331,831 | 6912 | 324,919 | 2.08% |
5 | Circular | - | - | 450 | 12 | 18 | 159,043 | 3054 | 155,990 | 1.92% |
6 | Circular | - | - | 350 | 8 | 14 | 96,211 | 1232 | 94,980 | 1.28% |
7 | Circular | - | - | 700 | 16 | 24 | 384,845 | 7238 | 37,7607 | 1.88% |
8 | Circular | - | - | 400 | 10 | 16 | 125,664 | 2011 | 123,653 | 1.60% |
9 | Rectangular | 300 | 300 | - | 8 | 16 | 90,000 | 1608 | 88,392 | 1.79% |
10 | Rectangular | 400 | 400 | - | 12 | 18 | 160,000 | 3054 | 156,946 | 1.91% |
11 | Rectangular | 500 | 400 | - | 14 | 22 | 200,000 | 5322 | 194,678 | 2.66% |
12 | Rectangular | 600 | 600 | - | 20 | 20 | 360,000 | 6283 | 35,3717 | 1.75% |
13 | Rectangular | 500 | 500 | - | 18 | 14 | 250,000 | 2771 | 247,229 | 1.11% |
14 | Rectangular | 500 | 600 | - | 20 | 16 | 300,000 | 4021 | 295,979 | 1.34% |
15 | Rectangular | 800 | 800 | - | 32 | 26 | 640,000 | 16,990 | 623,010 | 2.65% |
16 | Rectangular | 450 | 450 | - | 12 | 22 | 202,500 | 4562 | 197,938 | 2.25% |
17 | Rectangular | 450 | 400 | - | 12 | 24 | 180,000 | 5429 | 174,571 | 3.02% |
18 | Rectangular | 400 | 600 | - | 16 | 14 | 240,000 | 2463 | 237,537 | 1.03% |
19 | Rectangular | 500 | 300 | - | 14 | 16 | 150,000 | 2815 | 147,185 | 1.88% |
20 | Rectangular | 350 | 350 | - | 8 | 14 | 122,500 | 1232 | 121,268 | 1.01% |
Concrete Class | Compressive Strength | Tensile Strength | Modulus of Elasticity | Metal Alloy | Modulus of Elasticity | Shear Modulus | Poisson Ratio |
---|---|---|---|---|---|---|---|
MPa | MPa | MPa | GPa | GPa | - | ||
C16 | 16 | 1.4 | 27,000 | Aluminium | 69 | 25 | 0.33 |
C18 | 18 | 1.5 | 27,500 | Brass | 97 | 37 | 0.34 |
C20 | 20 | 1.6 | 28,000 | Copper | 110 | 46 | 0.34 |
C25 | 25 | 1.8 | 30,000 | Magnesium | 45 | 17 | 0.29 |
C30 | 30 | 1.9 | 32,000 | Nickel | 207 | 76 | 0.31 |
C35 | 35 | 2.1 | 33,000 | Cast iron | 120 | 46 | 0.30 |
C40 | 40 | 2.2 | 34,000 | Steel (rebars) | 207 | 83 | 0.30 |
C45 | 45 | 2.3 | 36,000 | Titanium | 107 | 45 | 0.34 |
C50 | 50 | 2.5 | 37,000 | Wolfram | 407 | 160 | 0.28 |
C16 | C18 | C20 | C25 | C30 | C35 | C40 | C45 | C50 | ||
---|---|---|---|---|---|---|---|---|---|---|
EQUIVALENT ELASTICITY MODULUS (×104 MPa) | ||||||||||
CROSS-SECTION AREA | 1 | 3.35 | 3.40 | 3.45 | 3.65 | 3.84 | 3.94 | 4.03 | 4.23 | 4.33 |
2 | 3.24 | 3.29 | 3.34 | 3.54 | 3.73 | 3.83 | 3.93 | 4.12 | 4.22 | |
3 | 3.40 | 3.45 | 3.50 | 3.69 | 3.89 | 3.98 | 4.08 | 4.27 | 4.37 | |
4 | 3.21 | 3.26 | 3.30 | 3.50 | 3.70 | 3.79 | 3.89 | 4.09 | 4.19 | |
5 | 3.17 | 3.22 | 3.26 | 3.46 | 3.66 | 3.76 | 3.85 | 4.05 | 4.15 | |
6 | 3.01 | 3.06 | 3.11 | 3.31 | 3.50 | 3.60 | 3.70 | 3.90 | 4.00 | |
7 | 3.16 | 3.21 | 3.26 | 3.45 | 3.65 | 3.75 | 3.84 | 4.04 | 4.14 | |
8 | 3.09 | 3.14 | 3.19 | 3.38 | 3.58 | 3.68 | 3.78 | 3.97 | 4.07 | |
9 | 3.13 | 3.18 | 3.23 | 3.43 | 3.63 | 3.72 | 3.82 | 4.02 | 4.12 | |
10 | 3.16 | 3.21 | 3.26 | 3.46 | 3.65 | 3.75 | 3.85 | 4.05 | 4.14 | |
11 | 3.35 | 3.40 | 3.44 | 3.64 | 3.83 | 3.93 | 4.03 | 4.22 | 4.32 | |
12 | 3.12 | 3.17 | 3.22 | 3.42 | 3.62 | 3.71 | 3.81 | 4.01 | 4.11 | |
13 | 2.97 | 3.02 | 3.07 | 3.27 | 3.46 | 3.56 | 3.66 | 3.86 | 3.96 | |
14 | 3.03 | 3.08 | 3.12 | 3.32 | 3.52 | 3.62 | 3.72 | 3.91 | 4.01 | |
15 | 3.35 | 3.39 | 3.44 | 3.64 | 3.83 | 3.93 | 4.03 | 4.22 | 4.32 | |
16 | 3.25 | 3.30 | 3.35 | 3.54 | 3.74 | 3.83 | 3.93 | 4.13 | 4.22 | |
17 | 3.43 | 3.48 | 3.53 | 3.72 | 3.92 | 4.01 | 4.11 | 4.31 | 4.40 | |
18 | 2.95 | 3.00 | 3.05 | 3.25 | 3.44 | 3.54 | 3.64 | 3.84 | 3.94 | |
19 | 3.16 | 3.21 | 3.25 | 3.45 | 3.65 | 3.74 | 3.84 | 4.04 | 4.14 | |
20 | 2.94 | 2.99 | 3.04 | 3.24 | 3.44 | 3.54 | 3.64 | 3.84 | 3.93 |
C16 | C18 | C20 | C25 | C30 | C35 | C40 | C45 | C50 | ||
---|---|---|---|---|---|---|---|---|---|---|
RELATIVE IMPROVEMENT PERCENTAGE (RIP) | ||||||||||
CROSS-SECTION AREA | 1 | 24.20 | 23.71 | 23.24 | 21.51 | 20.00 | 19.31 | 18.67 | 17.48 | 16.93 |
2 | 20.16 | 19.75 | 19.36 | 17.92 | 16.66 | 16.09 | 15.55 | 14.56 | 14.10 | |
3 | 25.92 | 25.39 | 24.89 | 23.04 | 21.42 | 20.68 | 19.99 | 18.72 | 18.14 | |
4 | 18.75 | 18.37 | 18.00 | 16.66 | 15.49 | 14.96 | 14.46 | 13.54 | 13.12 | |
5 | 17.28 | 16.93 | 16.60 | 15.36 | 14.28 | 13.79 | 13.33 | 12.48 | 12.09 | |
6 | 11.53 | 11.29 | 11.07 | 10.25 | 9.52 | 9.20 | 8.89 | 8.32 | 8.06 | |
7 | 16.93 | 16.58 | 16.26 | 15.05 | 13.99 | 13.51 | 13.05 | 12.22 | 11.84 | |
8 | 14.40 | 14.11 | 13.83 | 12.80 | 11.90 | 11.49 | 11.11 | 10.40 | 10.08 | |
9 | 16.08 | 15.76 | 15.44 | 14.29 | 13.29 | 12.83 | 12.40 | 11.61 | 11.25 | |
10 | 17.18 | 16.83 | 16.50 | 15.27 | 14.20 | 13.71 | 13.25 | 12.41 | 12.02 | |
11 | 23.95 | 23.47 | 23.00 | 21.29 | 19.79 | 19.11 | 18.47 | 17.30 | 16.76 | |
12 | 15.71 | 15.39 | 15.08 | 13.96 | 12.98 | 12.53 | 12.11 | 11.34 | 10.99 | |
13 | 9.98 | 9.77 | 9.58 | 8.87 | 8.24 | 7.96 | 7.69 | 7.20 | 6.98 | |
14 | 12.06 | 11.82 | 11.58 | 10.72 | 9.97 | 9.63 | 9.30 | 8.71 | 8.44 | |
15 | 23.89 | 23.41 | 22.94 | 21.24 | 19.74 | 19.07 | 18.43 | 17.26 | 16.72 | |
16 | 20.28 | 19.87 | 19.47 | 18.02 | 16.76 | 16.18 | 15.64 | 14.64 | 14.19 | |
17 | 27.15 | 26.60 | 26.07 | 24.13 | 22.43 | 21.66 | 20.94 | 19.60 | 18.99 | |
18 | 9.24 | 9.05 | 8.87 | 8.21 | 7.63 | 7.37 | 7.12 | 6.67 | 6.46 | |
19 | 16.89 | 16.55 | 16.22 | 15.01 | 13.96 | 13.48 | 13.03 | 12.20 | 11.82 | |
20 | 9.05 | 8.87 | 8.69 | 8.05 | 7.48 | 7.22 | 6.98 | 6.54 | 6.33 |
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Şen, Z.; Mangir, A. Innovative Equivalent Elastic Modulus Based Stress Calculation Methodology for Reinforced Concrete Columns. Buildings 2023, 13, 1962. https://doi.org/10.3390/buildings13081962
Şen Z, Mangir A. Innovative Equivalent Elastic Modulus Based Stress Calculation Methodology for Reinforced Concrete Columns. Buildings. 2023; 13(8):1962. https://doi.org/10.3390/buildings13081962
Chicago/Turabian StyleŞen, Zekâi, and Atakan Mangir. 2023. "Innovative Equivalent Elastic Modulus Based Stress Calculation Methodology for Reinforced Concrete Columns" Buildings 13, no. 8: 1962. https://doi.org/10.3390/buildings13081962