Numerical method for analysis and design of isolated square footing under concentric loading
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Abstract
Footing is an important part of a structure, because its repair is extremely difficult and cumbersome. Therefore, all structural parameters should be carefully considered when designing a footing. An isolated footing needs sufficient depth, when considering the fixed base assumption. Extensive research has previously been conducted to define and formalize the depth of a rigid footing, i.e., the depth required, such that footing behaves as a rigid. Alternatively, working stress method (WSM) and limit state method (LSM) provide a lower depth for low subgrades and a higher depth for high subgrades than that required by rigid condition if a unit width footing is considered for the design. This paper presents a simple approach for calculating the depth to meet the rigid condition under static loading. The proposed calculation method produced a better rigidity than the existing approaches and it correlated well to the finite-element method (FEM) for low subgrades. The reinforcement distribution is function of the bending moment (BM). Steel is uniformly embedded throughout the length or width of conventional footing methods, but this is inappropriate, because the bending moment is not uniform along the length or width of the footing. This paper proposes solutions to this by redefining the placement of steel in the central zone of the footing. The effective zone for reinforcement was based on the FEM results. This simple procedure was developed for calculating the maximum moment using the Diagonal Strip Method (DSM). DSM is a substitute for FEM, and it has been shown to correlate well. The BM at central zone as well as at the edges can be calculated to define the spacing of the reinforcements.
Keywords
Finite-element method Diagonal strip method Rigidity Static concentric load Rigid depthAbbreviations
- ACI
American Concrete Institute
- BM
Bending moment (kNm)
- DSM
Diagonal strip method
- E_{c}
Modulus of elasticity of the concrete (kN/m^{2})
- E_{s}
Modulus of elasticity of the soil (kN/m^{2})
- FEM
Finite-element method
- FS
Factor of safety
- K_{r}
Relative stiffness factor
- K_{s}
Subgrade reaction (kN/m^{2})
Introduction
Within the footing, stresses are developed due to different loading conditions, boundary conditions, and geometry. The forces acting on the footing are axial, shear, moment, and torsion. When the footing is loaded concentrically, the different modes of failure are bearing, flexure, and one-way and two-way shear. The effect of the stress level on the bearing capacity of the foundation, which is related to the foundation size, is very important (Jahanandish et al. 2012). ECP203 (2011), ACI318 (2008), and EC2 (2004) code provisions underestimate the structural failure loads of isolated column footings, while BS 8110.1 (1997) overestimates the failure loads for punching shear (Abdrabbo et al. 2016). Maximum bending moment (BM) is also an important factor for the calculation of footing depth. In the conventional method, the depth calculated for maximum BM is always less than that for one-way shear and punching shear, so it is neglected. Farouk and Farouk (2014a, b) recommended increasing the BM by 25% or increasing the reinforcement by 25% when calculating footing depth, due to an increasing edge stresses, which leads to an increase in the maximum moment.
Number of cases considered for study for validation
Case no. | Footing size in m | % Reinforcement in column | Axial load on column (kN) WSM | Axial load on column (kN) LSM | Column size (m) | K_{s} (kN/m^{3}) |
---|---|---|---|---|---|---|
1 | 2 × 2 | 0.8 | 650 | 725 | 0.3 × 0.3 | 20,000 |
2 | 2.2 × 2.2 | 1 | 700 | 775 | 0.3 × 0.3 | |
3 | 2.2 × 2.2 | 1.25 | 750 | 800 | 0.3 × 0.3 | |
4 | 2.3 × 2.3 | 1.5 | 800 | 850 | 0.3 × 0.3 | |
5 | 2.3 × 2.3 | 2 | 900 | 925 | 0.3 × 0.3 | |
6 | 2.6 × 2.6 | 0.8 | 1200 | 1300 | 0.4 × 0.4 | |
7 | 2.8 × 2.8 | 1 | 1300 | 1350 | 0.4 × 0.4 | |
8 | 2.8 × 2.8 | 1.25 | 1350 | 1425 | 0.4 × 0.4 | |
9 | 3 × 3 | 1.5 | 1400 | 1500 | 0.4 × 0.4 | |
10 | 3 × 3 | 2 | 1550 | 1650 | 0.4 × 0.4 | |
11 | 3.3 × 3.3 | 0.8 | 1900 | 2025 | 0.5 × 0.5 | |
12 | 3.5 × 3.5 | 1 | 2000 | 2125 | 0.5 × 0.5 | |
13 | 3.5 × 3.5 | 1.25 | 2100 | 2225 | 0.5 × 0.5 | |
14 | 3.5 × 3.5 | 1.5 | 2200 | 2350 | 0.5 × 0.5 | |
15 | 3.8 × 3.8 | 2 | 2450 | 2575 | 0.5 × 0.5 | |
16 | 4 × 4 | 0.8 | 2700 | 2925 | 0.6 × 0.6 | |
17 | 4.2 × 4.2 | 1 | 2850 | 3050 | 0.6 × 0.6 | |
18 | 4.2 × 4.2 | 1.25 | 3000 | 3200 | 0.6 × 0.6 | |
19 | 4.4 × 4.4 | 1.5 | 3200 | 3375 | 0.6 × 0.6 | |
20 | 4.5 × 4.5 | 2 | 3550 | 3700 | 0.6 × 0.6 |
For tension, if l_{c} > 0.75(C + 3d), then two-thirds of the reinforcement should be concentrated within a zone that extends on either side for a distance no more than 1.5d from the face of the column (Reynolds et al. 2007), where C is the column width, l_{c} is the distance from the center of a column to the edge of the pad, and d is depth. Seward (2014) recommended that the reinforcement for small pads can be uniformly distributed throughout the width of the foundation. However, if the width of the pad (L) exceeds 1.5(C + 3d), two-thirds of the reinforcement should be placed in a middle band at a width of (C + 3d). IS 456 (2000) recommended a central band equal to the width of the footing, placed along the length of the footing with the portion of the reinforcement determined in accordance with the equation below. This is limited to rectangular footing only:
Finite-element method (FEM) has become a powerful tool for the numerical simulation of engineering problems. FEM, therefore, is considered a benchmark, as its solutions are very accurate. The structural analysis program (SAP2000 2010) was used as an FEM tool for analysis purposes and the results of the proposed method are compared with FEM. In the present work, the diagonal strip method (DSM) was developed, which produces BM similar to finite-element method (FEM) results. This method provides reinforcement for a calculated bending moment without any increase in the percentage of steel.
Different approaches for rigid depth
The American Concrete Institute (ACI) Committee 336-2R (1988) suggested the use of a relative stiffness factor (K_{r}) developed by Meyerhof (1953) to determine whether the footing should be considered flexible or rigid; the proposed equation is:
Tabsh and Raouf Al-Shawa (2005) modified the stiffness factor K_{r} by considering all the dimensions of footing and column cross section. If K_{r} is less than or equal to 1, it can be considered flexible; otherwise, it will be considered rigid. The equation developed is:
Shehata (2016) conducted a study using Eq. (7) and recommended that, if the factor was less than or equal to 0.05, the footing is considered flexible, and otherwise, it is considered rigid.
The WSM and LSM for the analysis and design of footings were described in IS 456 (2000). For both the approaches, the depth is calculated for one-way shear, which is at distance d (effective depth of footing) from the face of column, for two-way or punching shear, which is at d/2 from the face of column and for maximum bending moment at column face. WSM is a traditional method used for design, which produces higher depths when compared to LSM. It was observed that both depths did not fulfill the rigidity requirement of the footing for all the types of soil or subgrade.
Depth must also be sufficient to accommodate the development length of a column bar in compression, i.e., L_{dc} as given in special publication (SP) 16 (1978) and special publication (SP) 34 (1987) of Indian standards. The development length of the bars must be sufficient to transfer compression to the supporting member, as per clause 12.3 of ACI 318 (2008). If the computed length is more than the distance available from the top of the footing to the tensile steel top, then the available solutions are to: (1) use a larger number of smaller bars in a column, (2) increase depth, and (3) construct a pedestal on top of the footing to provide extra length (McCormac and Brown 2015).
The depth of the base should be equal to or greater than 0.5a, where a is the distance from the face of column to the edge of the base (Reynolds et al. 2007). Depth calculated using this condition was close to depth calculated by the finite-element analysis for low subgrade or soft soil. However, this calculation was not valid for high subgrade or hard soil. The depth was considered rigid when it acted as solid block or contained very little reinforcement. The minimum reinforcement of 0.12% area must be specified if the calculated reinforcement is less than this minimum reinforcement (IS 456 2000). There are other conditions that give a rigid depth close to the FEM results. Some conditions are suitable for high subgrade but fail in lower subgrade, or vice versa.
Rigid depths for K_{s} = 20,000 kN/m^{3}
Case no. | Length/width of footing (m) | Width of column (m) | Rigid depth (m) by Meyerhof/ACI 336-2R (when K_{r} = 0.5) | Rigid depth (m) by Tabsh and Al-shawa (when K_{r} = 1) | Rigid depth (m) by Hany Farouk/DIN 4018 (when K_{r} = 0.05) | Rigid depth (m) by WSM | Rigid depth (m) by LSM | Rigid depth (m) by Reynolds’s Handbook |
---|---|---|---|---|---|---|---|---|
1 | 2 | 0.3 | 0.330 | 0.184 | 0.194 | 0.400 | 0.360 | 0.425 |
2 | 2.2 | 0.3 | 0.363 | 0.214 | 0.220 | 0.400 | 0.380 | 0.475 |
3 | 2.2 | 0.3 | 0.363 | 0.214 | 0.219 | 0.450 | 0.400 | 0.475 |
4 | 2.3 | 0.3 | 0.380 | 0.229 | 0.233 | 0.500 | 0.450 | 0.500 |
5 | 2.3 | 0.3 | 0.380 | 0.229 | 0.233 | 0.500 | 0.450 | 0.500 |
6 | 2.6 | 0.4 | 0.429 | 0.260 | 0.274 | 0.500 | 0.470 | 0.550 |
7 | 2.8 | 0.4 | 0.463 | 0.292 | 0.303 | 0.550 | 0.500 | 0.600 |
8 | 2.8 | 0.4 | 0.463 | 0.292 | 0.303 | 0.550 | 0.500 | 0.600 |
9 | 3 | 0.4 | 0.495 | 0.325 | 0.332 | 0.600 | 0.520 | 0.650 |
10 | 3 | 0.4 | 0.495 | 0.325 | 0.332 | 0.600 | 0.550 | 0.650 |
11 | 3.3 | 0.5 | 0.545 | 0.359 | 0.377 | 0.650 | 0.580 | 0.700 |
12 | 3.5 | 0.5 | 0.578 | 0.393 | 0.407 | 0.660 | 0.600 | 0.750 |
13 | 3.5 | 0.5 | 0.578 | 0.393 | 0.408 | 0.700 | 0.620 | 0.750 |
14 | 3.5 | 0.5 | 0.578 | 0.393 | 0.408 | 0.750 | 0.640 | 0.750 |
15 | 3.8 | 0.5 | 0.628 | 0.446 | 0.455 | 0.750 | 0.670 | 0.825 |
16 | 4 | 0.6 | 0.660 | 0.465 | 0.488 | 0.750 | 0.680 | 0.850 |
17 | 4.2 | 0.6 | 0.693 | 0.501 | 0.521 | 0.800 | 0.700 | 0.900 |
18 | 4.2 | 0.6 | 0.693 | 0.501 | 0.521 | 0.800 | 0.720 | 0.900 |
19 | 4.4 | 0.6 | 0.727 | 0.539 | 0.554 | 0.850 | 0.750 | 0.950 |
20 | 4.5 | 0.6 | 0.743 | 0.558 | 0.572 | 0.900 | 0.800 | 0.975 |
Concept of modeling
Idealization of the model
In the present study, a square footing and column were considered. M25 grade concrete and Fe415 steel were used with a Poisson’s ratio of 0.15 for concrete. The size of the footing was evaluated using load and bearing pressure. Column cross-sectional dimensions included 0.3 m × 0.3 m, 0.4 m × 0.4 m, 0.5 m × 0.5 m, and 0.6 m × 0.6 m. The load carrying capacity of the column for percentage reinforcements of 0.8, 1, 1.25, 1.5, and 2% were considered when axially loaded. With these parameters, 20 cases were developed and analyzed to determine the rigid depth. These cases were considered for low-rise and medium-rise buildings, where isolated square footings are preferred. Values of the safe bearing capacity of soil included 200 kN/m^{2}, 500 kN/m^{2}, 1000 kN/m^{2}, and 2000 kN/m^{2}, which indicated variation of subgrades from low to high. The modulus of the subgrade was calculated as suggested by Bowles (1988):
New approach for rigid depth calculation
General equation for design of RCC as per Krishna and Jain (1959) is as follows:
The thickness of footing calculated with this method may be less to account the rigidity effect when compared with the FEM results. As the bending moment is maximum in central part, consideration of full width leads to an incorrect depth. To get the correct rigid depth, the central part of width should be considered with maximum moment. The number of cases has been analyzed with EFM to determine the effective central zone and is given by the following:
Diagonal strip method (DSM)
The diagonal strip method is simple and based on the unit width method. In the conventional method, the unit width is considered to be the horizontal direction or vertical direction. Footings are designed for loads from a column. When the connection between the column and the footing is rigid, the stresses develop at each point of the footing. The maximum distance from the face of the column is the diagonal distance, so using the horizontal distance or vertical distance from the column face is inappropriate. However, using the diagonal distance in the moment calculation gives the maximum moment at the column face. The diagonal distance to be considered is shown in Fig. 9.
Now, bending moment at the face of column can be calculated by the following equation:
Bending moments calculated with the DSM were in agreement with FEM, proving that this method is an alternative for FEM. Figure 10 shows the comparison between DSM and FEM for maximum bending moment. Bending moment is maximum at the center and reduces towards the edge of the footing. The BM in the central zone (M_{avg} Fig. 7) is two-thirds the maximum BM, and the BM at the edge (M_{min} Fig. 7) is one-third the maximum BM. The maximum BM value can be calculated using DSM.
Proposed approach for distribution of reinforcement
The proposed method for distribution of reinforcement by Reynolds et al. (2007) and Seward (2014) was shown not to be valid for all the cases, and the reinforcement placement recommended by IS456 (2000) was not applicable to a square footing. On the other hand, when the reinforcement sizing is inconsistent with the importance of the structures, it can cause serious damage (Sadaoui and Bahar 2017). Providing uniform reinforcement throughout the width was not appropriate, because the BM varies across the width of the footing. At the central zone of the footing, the BM is a maximum, and FEM produces larger moments than the conventional method. The distribution of reinforcements must be as per the BM distribution. Figure 11 shows the details of the reinforcement for maximum BM, and the central zone can be calculated using the following:
The bending moment is more at the central zone, i.e., L/5 + b, and the remaining zone equals to L−(L/5 + b), is as shown in Fig. 11. The central zone covers 35–40% part of total length or width of footing. In addition, the excessive moment in this zone is found to be 40% more than average bending moment. Hence, 40% reinforcement of total reinforcement should be provided in this central zone.
Conclusions
The paper presented a simple approach to calculate the depth of and the placement of steel reinforcement for an isolated square footing under concentric loading. Observations were limited to square footings and columns with concentric loading. The working stress method (WSM) and the limit state method (LSM) failed to fulfill the rigidity requirements for a footing in the absence of sufficient width. The proposed procedure for calculating depth can be used in any case, because it gives a more conservative depth for lower subgrades and smaller depth for higher subgrades when compared to the existing methods. Depth calculated using the new approach agreed well with finite-element method (FEM), and also it satisfied the criterion for development length in compression (L_{dc}) for lower subgrades. For higher subgrades, depth should be calculated using the maximum of the LSM, the new approach, or minimum reinforcement approach. Reynolds’s handbook approach should not be used for higher subgrades, because it results in a lower depth than FEM. For the calculation of rigid depth, the total moment should be considered, which could be calculated by the conventional method or by Reynolds’s handbook equations.
Diagonal strip method (DSM) is not only a simple approach, but it is also an effective alternative to FEM for determining maximum bending moment (BM). The bending moments calculated with this method are nearly identical to the FEM results. Bending moments at the central zone and at the edge can be calculated using the bending moments of the DSM. Tensile reinforcement depends on the BM distribution. The proposed procedure of reinforcement distribution across the section of the footing is better than the conventional methods, which use uniform spacing of reinforcement, because this satisfies the BM distribution requirement. As per the proposed method, 40% of the total reinforcement must be provided at the central zone, because the BM distribution is highest here, and the remaining 60% should be placed in the remaining zones.
Notes
Compliance with ethical standards
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
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