# Development of a Cost-Based Design Model for Spread Footings in Cohesive Soils

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## Abstract

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## 1. Introduction

## 2. Literature Review

#### 2.1. Research Gap

#### 2.2. Aim and Objectives of the Study

- To present an optimized model comprising of cost-based foundation design charts which consider safety requirements, i.e., ultimate limit state (ULS) and serviceability limit state (SLS), along with the economy of the design project.
- To clarify the flaws in usual practices adopted to estimate bearing capacity by analyzing different case studies.
- To analyze cost estimation and to compare the economic effect while considering the variability of BC values.

## 3. Development of Conceptual Framework

_{f}). PLAXIS 2D is a finite element analysis tool and can be used to obtain the settlement of the foundation. In general, different numerical analysis tools were adopted for the scope of this study to obtain the foundation design charts which can be effectively used to obtain foundation design parameters that yield the lowest construction cost.

#### 3.1. Data Collection

#### 3.2. Bearing Capacity Calculations and Foundation Design

_{ult}(psf) is the ultimate bearing capacity of the soil, F (kips) is the applied vertical building load, B (ft) and L (ft) is the width and length of the foundation, and (SLS)

_{Δ}is the settlement that was estimated using PLAXIS 2D.

#### 3.2.1. Correlations between N-Value and S_{u}

_{u}) (kPa) was used to determine undrained shear strength. Equation (3). [27,28] is used to determine (s

_{u}) from SPT N values. The bearing capacity of the fine type of soil is a function of undrained shear strength as the friction angle (phi) is zero. The undrained condition exists in saturated clayey soil when subjected to loading in a short time interval. This is due to the reason that clayey soils have very low permeability and such type of soils do not permit water to dissipate under quick loading [29,30,31]. Thus, particle to particle interaction vanished due to an increase in pore pressure upon loading in undrained conditions. Hence, phi zero condition was developed.

#### 3.2.2. Bearing Capacity Equations

_{c}, N

_{q}, N

_{γ}can be found out using equations given below.

_{c}, S

_{q}, S

_{γ}are shape factors, d

_{c}, d

_{q}, d

_{γ}are depth factors, i

_{c}, i

_{q}, i

_{γ}are inclination factors, b

_{c}, b

_{q}, b

_{γ}are base inclination factors, and k is the coefficient of lateral earth pressure. Inclination factors are equal to 1 due to the concentric nature of the foundation. Other factors are given below.

#### 3.3. Calibration of Factor of Safety for Design

#### 3.4. CSI ETABS Modeling

#### 3.5. CSI SAFE Modeling

#### 3.6. PLAXIS 2D Modeling

#### 3.7. Cost Estimation Analysis

_{e}, R

_{f}, R

_{c}, R

_{c}, R

_{b}are unit rates for excavation, formwork, concrete, reinforcement, and compacted backfill, respectively, which can be calculated using Equations (20)–(24). The unit prices for shallow foundation construction are summarized in Table 3.

_{f}(m) are the width, length, and depth of the foundation, respectively, while L

_{o}and B

_{o}are over-excavation distances, respectively. Bo and Lo can be taken as 0.3 [39]. The quantity of formwork (Q

_{f}) can be calculated by Equation (21). For the concrete quantity width of footing (B), length (L), and thickness (T) are multiplied. From Equation (23), the quantity of concrete was determined by multiplying the unit weight of the bar (γ

_{Rebar}) with the width of footing (B), along with the total number of bars required for the footing. The quantity of the backfill after construction is determined from Equation (24) by subtracting the quantity of concrete (Q

_{c}) from the excavation quantity (Q

_{e}).

## 4. Results

#### 4.1. Results from CSI ETABS

#### 4.2. Results from CSI SAFE

_{s}) based on actual footing width, i.e., B

_{actual}. The reader is referred to Bowl’s book [31] for the details regarding the structural design of the footing.

#### 4.3. Results from PLAXIS-2D

#### 4.4. Bearing Capacities of Selected Sites

_{Design}, which is given by Equation (25) [28], where Ni represents the corrected SPT N value at each layer and I represent the number of layers at which SPT N values are available from footing base to influence zone of the footing. The bearing capacity was determined using three bearing capacity equations, i.e., Terzaghi, Meyerhof, and Vesic.

#### 4.5. Design Charts

#### 4.5.1. Design Charts Based upon Terzaghi’s Equation

_{Design}value and ultimate bearing capacity determined from Terzaghi’s equation, with the increase in N

_{Design}value the ultimate bearing capacity also increased mainly due to the increase in stiffness of the soil. Likewise, in Figure 11b, the relation between ultimate bearing capacity and the actual width of the foundation was developed. It contains three trendlines for the allowable bearing capacity at the different factors of safety. The higher the FoS, the higher will be the B

_{Actual,}and the more conservative will be the approach for design. These trends decrease exponentially due to the enhancement of the soil properties, thereby increasing the ultimate bearing capacity. Figure 11c shows the relation between the actual width of footing and the required area of steel. It shows that the requirement for the area of the steel increases with the change in footing size. Bearing capacities of different FoS show a similar trend due to the minimum reinforcement criteria of the ACI-318 for the footing design. Figure 11d shows the relation between the actual width of footing and the amount of settlement calculated through the PLAXIS 2D software. This trend shows the increase in immediate settlement with the increase in the size of footing, while Figure 11e represents the estimated cost against the ultimate bearing capacity for different FoS values. It provides a quick total cost estimation for the footing including the material and labor costing. These charts can be used in steps to design an economical foundation in a sequence from part (a) to (e). For example, these design charts can be used in a way that once N

_{Design}is selected, ultimate bearing capacity is then determined using the procedure enlisted above. Then design chart (b) is used to find out the actual width of the footing. Then chart (c) is used to interpret the actual reinforcement details. Then, chart (d) is used to calculate the amount of settlement corresponding to the size of the footing and then compare it with the permissible limits. Finally, the economics of the designed foundation can be analyzed using the chart (e) based on different FoS values. Hence, optimized design is obtained quickly. It is worthwhile to mention that B

_{Actual}is the actual width of footing that is computed during the structural design of footing and it must not be confused with “B,” which is the assumed value of width used for the determination of bearing capacity during the geotechnical design of footing.

#### 4.5.2. Design Charts Based upon Meyerhof’s Equation

_{Design}value and ultimate bearing capacity determined from Meyerhof’s equation. Figure 12b represents the relation between ultimate bearing capacity and the actual width of the foundation. These trends also decrease exponentially with the increase in ultimate bearing capacity. Figure 12c shows the relation between the actual width of footing and the required area of steel. These trends increase with the surge in footing size. Lines also show a similar trend due to the minimum reinforcement criteria of the ACI-318 for the footing design. Figure 12d presents a correlation between the actual width of footing and the amount of settlement calculated through the PLAXIS 2D software. This shows immediate settlement increases with the increase in the size of the footing. Figure 12e presents a correlation of the cost of footing with the ultimate bearing capacity of the soil. Poor soil properties affect the bearing capacity, which causes an increment in the economics.

#### 4.5.3. Design Charts Based upon Vesic’s Equation

_{Design}value and ultimate bearing capacity determined from Vesic’s equation. Figure 13b represents the relation between ultimate bearing capacity and the actual width of the foundation. Figure 13c shows the relation between the actual width of footing and the required area of steel. Moreover, in this model, the minimum reinforcement criteria of the ACI-318 for the footing design governs causing approximately a similar trend for different bearing capacities. Figure 13d presents the correlation between immediate settlement and actual width of footing using PLAXIS 2D modeling. This trend shows the increase in immediate settlement with the increase in the size of the footing. Figure 13e presents a correlation of the cost of footing with the ultimate bearing capacity of the soil.

## 5. Discussion

#### 5.1. Optimization of Foundation Material

#### 5.2. Comparison between Traditional and Optimized Foundation Design

#### 5.3. Proposed Framework

_{Design}value along with the use of Vesic’s equation with a recommended lowest FoS. Then, obtained bearing capacity value will be utilized for the structural design of the footing. The SLS criteria should be thoroughly performed from PLAXIS-2D software, which relatively provides more accurate results.

- Step One: In the first step, N
_{Design}will be selected using Equation (25). Then, the ultimate bearing capacity will be determined using Figure 12a based on N_{Design}value. - Step Two: In the second step, q
_{u}computed in the first step will be utilized to estimate the actual width of footing corresponding to desired FoS value using Figure 12b. - Step Three: The value of B
_{Actual}computed in step two will be used to estimate the area of reinforcement from Figure 12c. - Step Four: The value of B
_{Actual}will be used to assess the settlement of the footing in quantitative terms using Figure 12d. - Step Five: The value of q
_{u}will be used to assess the construction cost of footing in quantitative terms using Figure 12e.

## 6. Conclusions

- This study presents an optimized model of foundation design comprising cost-based design charts which consider safety requirements, i.e., ULS, SLS, and economy at once. This developed optimized design approach provides a speedy method for foundation design for Silty Clay soils. The developed design charts can be used for an efficient economical design of shallow foundations considering safety requirements.
- This study concludes that the conventional design practices inculcate the use of a higher FoS to compensate for the uncertainties, inaccuracies, and unavailability of enough soil investigation data. Contrary to that, this study provides the design charts to estimate the design of shallow foundations with the use of an appropriate FoS, even with the non-availability of enough data. The design charts are also useful in designing a cost-effective shallow foundation considering safety requirements.
- Overconservative approaches may lead to an uneconomical situation. Therefore, it was observed that even the highest bearing capacity values, obtained from the utilized (Terzaghi, Meyerhof, Vesic) equations, can be recommended if they satisfy safety requirements. Thus, the highest bearing capacity computed can be recommended for the foundation design.
- The results show that savings could be as much as 37.5% in comparison with the cost obtained from conventional foundation design. The optimized construction cost may change depending upon the subsurface soil conditions, design requirements, and groundwater conditions.
- Statistical analysis shows there is a 0% probability of failure for low-rise buildings against the highest bearing capacity value and FoS equal to three or lesser can be used in general as the optimal value for design purposes to determine allowable bearing capacity. Since the average savings by changing FoS from 4.0 to 3.5 are 14.3% while changing FoS from 4.0 to 3.0 can be 27.3% can be achieved.
- It is observed that even the most conservative approaches for the foundation yield more than 2 Tsf bearing capacity for the silty clayey soil.
- Hence, future research may address other aspects of foundation design, such as seismic design parameters, variation of design loads, and validation of design charts through large-scale in-situ testing.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 11.**Design charts based on Terzaghi’s bearing capacity equation (

**a**) relation between the SPT N

_{Design}value and ultimate bearing capacity; (

**b**) relation between ultimate bearing capacity and the actual width of the foundation; (

**c**) relation between the actual width of footing and the required area of steel; (

**d**) relation between the actual width and the amount of settlement; and (

**e**) estimated cost against the different ultimate bearing capacity values.

**Figure 12.**Design charts based on Meyerhof’s bearing capacity equation (

**a**) relation between the SPT N

_{Design}value and ultimate bearing capacity; (

**b**) relation between ultimate bearing capacity and the actual width of the foundation; (

**c**) relation between the actual width of footing and the required area of steel; (

**d**) relation between the actual width and the amount of settlement; and (

**e**) estimated cost against the different ultimate bearing capacity values.

**Figure 13.**Design charts based on Vesic’s bearing capacity equation (

**a**) relation between the SPT N

_{Design}value and ultimate bearing capacity; (

**b**) relation between ultimate bearing capacity and the actual width of the foundation; (

**c**) relation between the actual width of footing and the required area of steel; (

**d**) relation between the actual width and the amount of settlement; and (

**e**) estimated cost against the different ultimate bearing capacity values.

**Figure 14.**Design model provides the optimization of different materials for the footing as compared to traditional design, and these include (

**a**) Concrete; (

**b**) Reinforcement; (

**c**) Formwork; and (

**d**) Excavation.

Material | Strength of Material | Usage |
---|---|---|

Concrete | 3000 Psi | For beams and slabs |

Concrete | 4000 Psi | For columns and footings |

Rebar | Grade 60 | Longitudinal (main) reinforcement |

Rebar | Grade 40 | Confinement bars (ties) |

Section | Size | Section | Size | Section | Size |
---|---|---|---|---|---|

Columns | 15″ × 15″ | Beams | 18″ × 15″ | Slab | 6″ Thick |

**Table 3.**Summary of unit prices for a shallow foundation [38].

Materials | Labor | Material | Unit |
---|---|---|---|

Concrete | PKR 4714 | PKR 5600 | Cubic Yard |

Reinforcement | PKR 12 | PKR 112 | Kilogram |

Formwork | PKR 18 | PKR 14 | Sq. ft/contact area |

Excavation | PKR 216 | PKR 0 | Cubic Yard |

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## Share and Cite

**MDPI and ACS Style**

Nawaz, M.M.; Khan, S.R.; Farooq, R.; Nawaz, M.N.; Khan, J.; Tariq, M.A.U.R.; Tufail, R.F.; Farooq, D.; Ng, A.W.M.
Development of a Cost-Based Design Model for Spread Footings in Cohesive Soils. *Sustainability* **2022**, *14*, 5699.
https://doi.org/10.3390/su14095699

**AMA Style**

Nawaz MM, Khan SR, Farooq R, Nawaz MN, Khan J, Tariq MAUR, Tufail RF, Farooq D, Ng AWM.
Development of a Cost-Based Design Model for Spread Footings in Cohesive Soils. *Sustainability*. 2022; 14(9):5699.
https://doi.org/10.3390/su14095699

**Chicago/Turabian Style**

Nawaz, Muhammad Muneeb, Shah Rukh Khan, Rashid Farooq, Muhammad Naqeeb Nawaz, Jamil Khan, Muhammad Atiq Ur Rehman Tariq, Rana Faisal Tufail, Danish Farooq, and Anne W. M. Ng.
2022. "Development of a Cost-Based Design Model for Spread Footings in Cohesive Soils" *Sustainability* 14, no. 9: 5699.
https://doi.org/10.3390/su14095699