Figure 1
Analysis of a doubly reinforced rectangular beam [9]. (a) Cross-section, (b) Strain distribution, (c) Stress and forces, (d) Forces from Part 1, and (e) Forces from Part 2.
Figure 2
Alternate live loadings for minimum and maximum effects.
Figure 3
Bars in the compression area acting as stirrup hangers.
Figure 4
Minimum net tensile strain for tension-controlled sections.
Figure 5
Flowchart for the analysis of a doubly reinforced concrete section.
Figure 6
Schematic longitudinal and cross sections of the tested beams: (a) B1, (b) B2, and (c) B3 (all dimensions in mm).
Table 2
Compression steel in the tested beams.
| Beam ID | Details | |||
|---|---|---|---|---|
| B1 | 2Ø22 | 2Ø10 | 0.2 | Figure 6(a) |
| B2 | 2Ø22 | 3Ø10 | 0.3 | Figure 6(b) |
| B3 | 2Ø22 | 2Ø22 | 1.0 | Figure 6(c) |
Figure 7
Loading, shear force, and bending moment diagrams of the tested beams.
Figure 8
Concrete pouring of the beam specimens and cylinders.
Figure 9
Reinforcement cages of beam specimens.
Figure 10
Positions of strain gauges.
Figure 11
Test setup.
Figure 12
Compression-controlled failure of all the tested beams.
Figure 13
The relationship between load and maximum deflection for the tested beams.
Table 3
Comparison of experimental and proposed method results.
| Point of comparison | B1 beam | B2 beam | B3 beam | ||||||
|---|---|---|---|---|---|---|---|---|---|
| Experimental | Analytical | % | Experimental | Analytical | % | Experimental | Analytical | % | |
| Failure load (kN) | 118.4 | 105.9 | +11.8% | 122.0 | 111.1 | +9.8% | 166.0 | 150.2 | +10.5% |
| Nominal moment strength* (kN m) | 42.92 | 38.39 | +11.8% | 44.23 | 40.27 | +9.8% | 60.18 | 54.45 | +10.5% |
| Concrete compressive strain at failure | 0.001526 | 0.001507 | +1.3% | 0.001413 | 0.001507 | −6.2% | 0.001518 | 0.001507 | +0.7% |
| Strain in tension steel at failure | 0.001376 | 0.001365 | +0.8% | 0.001460 | 0.001446 | +1.0% | 0.001765 | 0.001729 | +2.1% |
| Strain in compressive steel at failure | 0.001344 | 0.001330 | +1.1% | 0.001211 | 0.001194 | +1.4% | 0.000979 | 0.001054 | −7.1% |
*Bending moment due to the weight of the beam is neglected.
Figure 14
The relationship between load and maximum concrete compressive strain for the tested beams.
Figure 15
The relation between load and strain in tensile steel for the tested beams.
Figure 16
The relation between load and strain in compressive steel for the tested beams.
Figure 17
Initial, tangent, and secant moduli of elasticity of concrete [9].
Figure 18
Typical concrete stress–strain curves in compression [9].
Figure 19
Analytical approximation to the compressive stress–strain curve for concrete [9].
Figure 20
Proposed analytical model for shallow beams. (a) Cross-section, (b) Strain distribution, and (c) Stress and forces.