Introduction to Combined Pad Foundations

While isolated pad foundations support single columns, combined pad foundations are a type of shallow foundation designed to support two or more columns (or a column and a wall) on a single, larger footing. This solution is typically employed when:

• The columns are so closely spaced that their individual isolated footings would overlap.
• A column is located very near a property line, making it impossible to construct an isolated footing without encroaching beyond the boundary.
• There is a need to equalize settlement under adjacent columns, particularly when columns carry significantly different loads, or when the soil's bearing capacity is low.

The design of combined pad foundations, like isolated pads, aims to transfer column loads to the underlying soil without exceeding the soil's bearing capacity, while also ensuring structural adequacy, cost-effectiveness, and compliance with the safety and serviceability requirements specified in Eurocode 2 (EN 1992) and Eurocode 7 (EN 1997).

Principle of Combined Pad Foundation Design

The fundamental principle behind combined pad foundation design is to ensure that the resultant of all applied column loads (at serviceability limit state) passes as closely as possible through the centroid of the foundation's plan area. This helps to achieve a uniform or near-uniform bearing pressure distribution under the footing, thereby minimizing differential settlement, which is critical for the stability and integrity of the superstructure.

For two columns, the combined footing can typically be rectangular or trapezoidal in shape:

• **Rectangular combined footing**: Used when the column loads are approximately equal, or when moments are such that a rectangular shape allows the resultant load to be centralized.
• **Trapezoidal combined footing**: Often adopted when the loads from the two columns differ significantly, allowing the larger area to be under the heavily loaded column, thus ensuring the resultant load aligns with the footing's centroid.

As highlighted in resources like those by Prab Bhatt and Mosley & Bungey, the stiffness of the foundation must be sufficient to distribute the loads effectively and prevent excessive settlement or localized failure, adhering to Eurocode 7 (EN 1997-1) for geotechnical design.

Bearing Pressure Distribution

For combined pad foundations, careful consideration of bearing pressure is paramount to prevent overturning and excessive settlement. The objective is often to achieve a uniform bearing pressure to optimize the use of the soil's bearing capacity and mitigate differential settlement.

The general formula for bearing pressure, considering axial loads and bending moments, is applied, but with the combined effect of multiple columns:

where:

• \( \Sigma N \) is the sum of all axial loads from the columns.
• \( A \) is the total plan area of the combined footing.
• \( M_{xx} \) and \( M_{yy} \) are the moments about the x and y axes of the footing's centroid, respectively, resulting from the combined column loads and any applied moments.
• \( I_{xx} \) and \( I_{yy} \) are the second moments of area of the footing about its centroidal x and y axes.
• \( x \) and \( y \) are the distances from the centroid to the point where pressure is being calculated.

The design process emphasizes proportioning the footing such that the resultant of the applied loads passes through the centroid, ensuring a uniform pressure distribution. If a uniform distribution cannot be achieved (e.g., due to site constraints), the maximum and minimum bearing pressures must be checked against the allowable bearing capacity of the soil, ensuring that no tension develops at the soil-foundation interface. Eurocode 7 guidelines for effective foundation area under eccentric loading should be consulted.

Shear Design and Punching Shear Checks

Shear design for combined pad foundations involves checking for both one-way (beam) shear and two-way (punching) shear. The presence of multiple columns significantly complicates these checks compared to isolated footings.

One-Way Shear (Beam Shear)

One-way shear is checked at a distance \( d \) (effective depth) from the face of each column. The foundation is idealized as an inverted beam subjected to the upward soil pressure. The critical sections for shear are typically considered between the columns and at the cantilever ends. The design shear force \( V_{ED} \) must be less than the concrete's shear resistance \( V_{Rd,c} \) and the maximum shear resistance \( V_{Rd,max} \) as per Eurocode 2 (EN 1992-1-1) clause 6.2.2 and 6.2.3.

Two-Way Shear (Punching Shear)

Punching shear is a critical check around each column. The primary control perimeter for punching shear is typically located at a distance of \( 2d \) from the face of the column, though checks at the column face are also necessary. The design punching shear stress \( v_{ED} \) must be less than the concrete’s punching shear resistance \( v_{Rd,c} \) and the maximum allowable shear stress \( v_{Rd,max} \). For most combined footings, the punching shear control perimeter usually falls outside the footing perimeter, making punching shear checks difficult to verify

The resistance \( V_{RD,max} \) at the column face for combined footings is calculated for each column separately, considering the loaded perimeter. \[ V_{RD,max} = 0.2 \left(1 - \frac{f_{ck}}{250}\right) f_{ck} u_0 d \] where \( u_0 \) is the basic control perimeter at the column face.

For the perimeter at \( 2d \) from the column face, the critical shear stress \( v_{ED} \) is calculated by considering the total factored load acting outside the perimeter and dividing it by the perimeter length times the effective depth. Shear reinforcement may be required if \( v_{ED} \) exceeds \( v_{Rd,c} \), but not \( v_{Rd,max} \).

Flexural Design

Combined pad foundations are essentially continuous inverted slabs subjected to upward soil pressure and downward column loads. This results in significant bending moments that need to be resisted by reinforcement in both the longitudinal and transverse directions.

Longitudinal Bending

The footing is analyzed as a beam spanning between the columns, with cantilever portions extending beyond the outer columns. Both positive (sagging) and negative (hogging) moments will develop.

• Hogging Moments occur over the columns, requiring top reinforcement. The critical sections for hogging moments are typically at the column faces.
• Sagging moments occur in the spans between columns and in the cantilever portions, requiring bottom reinforcement. The critical sections for sagging moments are generally at the mid-span between columns and at the face of the outermost columns for cantilevers.

The design moment \( M_{ED} \) at critical sections is used to determine the required reinforcement area \( A_s \). \[ A_s = \frac{M_{ED}}{0.87 f_{yk} z} \] where \( f_{yk} \) is the characteristic yield strength of the reinforcement, and \( z \) is the lever arm, typically taken as \( 0.95d \) for initial calculations, as per Eurocode 2 (EN 1992-1-1).

Transverse Bending

Each column also acts as a concentrated load on a strip of the combined footing perpendicular to the longitudinal axis. These strips are treated as isolated pad footings cantilevered from the column faces. Reinforcement is designed for these transverse moments, typically at the bottom of the footing.

Detailed design procedures for combined footings, including the idealization as an inverted beam for longitudinal analysis and cantilever strips for transverse analysis, are extensively covered in texts like "Reinforced Concrete Design to Eurocode 2" by Mosley, Bungey, and Hulse, and "Reinforced Concrete Design" by Prab Bhatt. Minimum and maximum reinforcement limits, as well as detailing requirements (e.g., bar spacing, anchorage), must be strictly adhered to as per Eurocode 2, clause 9.8.