## A walk-through of the calculations required to design an isolated footing (AS 3600-09)

The foundation is an essential building system that transfers column and wall forces to the supporting soil. Depending on the soil properties and building loads, the engineer may choose to support the structure on a shallow or deep foundation system³.

SkyCiv Foundation includes the design of isolated footing conforming to the Australian Standards¹.

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## Design of an Isolated Footing

**Dimension Requirements**

To determine the dimensions of a isolated footing, service or unfactored loads, such as permanent action (**G**), imposed action** **(

**Q**), wind action

**(**

**), earthquake action**

**W**_{u}**(**

**)**

**E**_{u}**and**

**,****will be applied using Load Combinations, as defined by AS 3600-09. Whichever Load Combination governs will be considered the design load, and is compared to the allowable soil pressure as shown in Equation 1.**

**S**_{u}\(\text{q}_{\text{a}} = \frac{\text{P}_{\text{n}}}{\text{A}} \rightarrow \) Equation 1

where:

q_{a} = allowable soil pressure

P_{n} = service level design loads

A = foundation area

From Equation 1, **q _{a}** are interchanged with

**A**.

\(\text{A} = \frac{\text{P}_{\text{n}}}{\text{q}_{\text{a}}} \rightarrow \) Equation 1a

At this point, the dimensions of the footing can be back-calculated from the required area dimension, **A. **

**One-way Shear**

The** one-way shear **limit state, also known as **flexural shear, **is located at a distance “d” from the face of a column, at the Critical Shear Plane (Refer to Figure 1), and is based on **AS3600 Clause 8.2.7.1**

Figure 1. Critical Shear Plane of One-way shear

The **One-way ****Shear** **Demand ** or **V _{u}** is calculated assuming the footing is cantilevered away from the column where the area is (red) hatched, indicated in Figure 2.

The **One-way ****Shear** **Capacity **or** ϕVu _{c}** is defined as Ultimate shear strength and calculated using Equation 2 per

**AS3600-09 Cl 8.2.7.1**.

\( \phi \text{V}_{uc} = \phi \beta_{1} \times \beta_{2} \times \beta_{3} \times b_{v} \times d_{o} \times f_{cv} \times A_{st}^{\frac{2}{3}} \rightarrow \) Equation 2 (**AS3600 Eq. 8.2.7.1**)

where:

ϕ = shear design factor

β_{1}= 1.1(1.6 – d_{o}/1000) ≥ 1.1 or 1.1(1.6(1-d_{o}/1000) ≥ 0.8

β_{2} = 1, for members subject to pure bending; or

=1-(N^{*}/3.5A_{g}) ≥ 0 for member subject to axial tension; or

=1-(N^{*}/14A_{g}) for members subject to axial compression

β_{3} = 1, or may be taken as –

2d_{o}/a_{v} but not greater than 2

a_{v} = distance for the section at which shear is being considered to the face of the nearest support

f_{cv} = f’c^{1/3} ≤ 4 MPa

A_{st} = cross-sectional area of longitudinal reinforcement

Shear Demand and Shear Capacity must meet the following equation to meet the design requirements of AS 3600-09:

\(\text{V}_{\text{u}} \leq \phi\text{V}_{\text{uc}} \rightarrow \) Equation 3 (per **AS3600 Cl. 8.2.5**)

SkyCiv Foundation, in compliance of Equation 3, calculates the one-way shear unity ratio (Equation 4) by taking Shear Demand over Shear Capacity.

\( \text{Unity Ratio} = \frac{\text{Shear Demand}}{\text{Shear Capacity}} \rightarrow \) Equation 4

**Two-way Shear**

The **Two-way Shear** limit state, also known as **punching shear,** extends it critical section to a distance “d/2” from the face of the column and around the perimeter of the column. The Critical Shear Plane is located at that section of the footing (Refer to Figure 2) based **AS3600 Clause 9.2.3(a)**.

Figure 2. Critical Shear Plane of Two-way shear

The **Two-way S****hear Demand **or **V _{u}** occurs at the Critical Shear Plane, located a distance of “d/2” where the (red) hatched area, indicated in Figure 2.

The **Two-way ****Shear Capacity **or** ϕV _{uo}** is defined as Ultimate shear strength and calculated using Equation 5 based

**AS3600 Clause 9.2.3**

\( \phi V_{uo} = \phi \times u \times d_{om} \left( f_{cv} + 0.3 \sigma_{cp} \right) \rightarrow \) Equation 5 (**AS3600 Cl. 9.2.3(1)**)

where:

f_{cv} = 0.17(1 + 2/β_{h}) √f’_{c} ≤ 0.34√f’_{c}

σ_{cp} = value of corner, edge and internal columns

d_{om} = mean value of do, averaged around the critical shear perimeter

β_{h} = ratio of length of column at Z-axis over X-axis

u = length of the critical shear perimeter

Shear Demand and Shear Capacity must meet the following equation to meet the design requirements of AS 3600-09:

\(\text{V}_{\text{u}} \leq \phi\text{V}_{\text{uo}} \rightarrow \) Equation 6 (per **AS3600 Cl. 8.2.5**)

SkyCiv Foundation, in compliance of Equation 6, calculates the two-way shear unity ratio (Equation 7) by taking Shear Demand over Shear Capacity.

\( \text{Unity Ratio} = \frac{\text{Shear Demand}}{\text{Shear Capacity}} \rightarrow \) Equation 7

**Flexure**

In a isolated footing, the upward soil pressure causes two-way bending with tensile stresses at the bottom surface. Bending moments are calculated in each direction at sections *0.7a _{sup}* distance from the centre of the column, where

*a*is half the width of the column.

_{sup}

Figure 3. Critical Flexure Section

The **Flexural** limit state occurs at the Critical Flexure Section, located *0.7a _{sup}* from the centre of the footing (Refer to Figure 3).

The **Flexural Demand **or **M _{u}** is located at the Critical Flexure Section indicated in Figure 3, and is calculated using Equation 8.

\( \text{M}^{*}= q_{u} \times D_{f} \times \left( \frac{ \frac{b_{f} – b_{c}}{2} }{2} \right)^{2} \rightarrow \) Equation 8

The **Flexural Capacity **or** ϕMn** is calculated using Equation 9.

\(M_{n} = A_{st} \times f_{sy} \times d \times \left(1- \frac{0.5}{\alpha_{s}} \times \frac{A_{st} \times f_{sy}}{b \times d \times f’_{c}} \right) \rightarrow \) Equation 9

where:

ϕ = flexural design factor

b = footing dimension parallel x-axis, in or mm

d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement, in or mm

A_{st} = reinforcement area, in^{2} or mm^{2}

a = depth of equivalent rectangular stress block, in or mm

fsy = reinforcement strength, ksi or MPa

Moment Demand and Moment Capacity must meet the following equation to meet the design requirements of AS 3600-09:

\(\text{M}_{\text{u}} \leq \phi\text{M}_{\text{n}} \rightarrow \) Equation 10 (per **AS3600 Cl. 8.2.5**)

SkyCiv Foundation, in compliance of Equation 10, calculates the flexural unity ratio (Equation 11) by taking Flexural Demand over Flexural Capacity.

\( \text{Unity Ratio} = \frac{\text{Flexure Demand}}{\text{Flexure Capacity}} \rightarrow \) Equation 11

**Reinforcement**

The amount of reinforcement required is determined by flexural strength requirements, with minimum reinforcement specified in Cl. 16.3.1.

\( \rho_{ \text{min} } = 0.19 \times \frac{D}{d}^{2} \times \frac{f’_{ct.f} }{ f_{sy} } \rightarrow \) Equation 12

The area of steel can be determined with the following equation:

\( \rho = \frac{ 2.7 \times M^{*} }{ d^{2} } \text{ or } \text{A}_{\text{st}} = \frac{ \text{M}^{*} }{ 370 \times \text{d} } \rightarrow \) Equation 13

As advised by AS 3600-09, a minimum concrete cover of *60 mm* for footing is recommended.

**Albert Pamonag**

Structural Engineer, Product Development

B.S. Civil Engineering

## References

- Council of Standards Australia. (2009) Australian Standard AS3600-2009.
- SJ Foster, AE Kilpatrick & RF Warner. (2011) Reinforced Concrete Basics 2nd Edition.
- Taylor, Andrew, et al.
*The Reinforced Concrete Design Handbook: a Companion to ACI-318-14*. American Concrete Institute, 2015. - YC Loo & SH Chowdhury. (2013) Reinforced & Prestressed Concrete.