A walkthrough of the calculations to design an isolated footing (AS 3600 2018)

The foundation is an essential building system that transfers column and wall forces to the supporting soil. The engineer might opt for a shallow or deep foundation system based on the soil characteristics and building loads.

SkyCiv FoundationDesign Module includes analyzing and designing isolated footing conforming to the Australian Standards (AS 3600 2009 & 2018).

 

 

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Design of an Isolated Footing

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Dimension Requirements

To determine the dimensions of an isolated footing, service or unfactored loads, such as permanent action (G), imposed action (Q), wind action (W_u), earthquake action (E_u), and S_u will be applied using Load Combinations, as defined by AS 3600. Whichever Load Combination governs will be considered the design load, and is compared to the allowable soil pressure as shown in Equation 1.

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

q_a = allowable soil pressure
P_n = service level design loads
A = foundation area

The footing dimensions can be initially estimated by solving the foundation area (A) using Equation 1.

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

One-way Shear

The one-way shear limit state, also known as flexural shear, recognizes that the footing may fail in shear similar to a wide beam along a critical shear plane located at a distance “d” from the face of the column (Figure 1), 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 red area is indicated in Figure 1.

The One-way Shear Capacity or ϕVu_c is defined as the 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) 

ϕ = shear design factor
β₁= 1.1(1.6 – d_o/1000) ≥ 1.1 or 1.1(1.6(1-d_o/1000) ≥ 0.8
β₂ = 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
β₃ = 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 with Equation 3, calculates the one-way shear utility ratio (Equation 4) by taking Shear Demand over Shear Capacity.

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

Two-way Shear

The Two-way Shear limit state, also known as punching shear, extends its critical section to a distance “d/2” from the face of the column and around the perimeter of the column (Figure 2) based on AS3600 Clause 9.2.3(a).

Figure 2. Critical Shear Plane of Two-way shear

The Two-way Shear Demand or V_u occurs at the Critical Shear Plane, located at a distance of “d/2” where the (red) hatched area, as indicated in Figure 2.

The Two-way Shear Capacity or ϕV_uo is defined as the Ultimate shear strength and calculated using Equation 5 based on AS3600 Clause 9.2.3

\( \phi V_{uo} = \phi \times u \times f_{cv} \times d  \rightarrow \) Equation 5 (AS3600 Cl. 9.2.3(a)) 

f_cv = 0.17(1 + 2/β_h) √f’_c ≤ 0.34√f’_c
β_h  = ratio of the length of the column at the Z-axis over the X-axis

d = distance from extreme compression fiber to centroid of longitudinal tension reinforcement (mm)
u = length of the critical shear perimeter (mm)

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

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

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

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

Flexure

In an 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 section 0.7a_sup distance from the center of the column, where a_sup is half the width of the column.

 

Figure 3. Critical Flexure Section

The Flexural limit state occurs at the Critical Flexure Section, located 0.7a_sup from the center 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

Additional Verifications

Other verifications not mentioned by the code, including soil pressure checks, uplift, and other stability checks are also verified.

Soil Pressure

The determination of base pressure or the interaction between the soil and footing primarily relies on the footing dimensions and the resulting eccentricity of the applied loads. Depending on the positioning of this resulting eccentricity, the base pressure can induce full or partial compression on the footing. This assessment enables us to confirm whether the underlying soil can sustain the entirety of the loads transmitted from the footing.

For a detailed guide for manually calculating the soil pressure, please refer to this link: Pressure Distribution Under a Rectangular Concrete Footing

The utility ratio is evaluated by comparing the maximum soil pressure (serviceability state) with the allowable gross bearing soil capacity:

 

\( \text{Utility Ratio} = \frac{\text{Max. Soil Pressure}}{\text{Gross Allowable Soil Bearing Capacity}} \rightarrow \) Equation 14

Uplift

Checks the governing axial load acting on the footing. Sums all vertical loads including the user load and self-weights of the column, footing slab, soil, and buoyant force. If the column experiences an upward force, the self-weights specified must counterbalance the upward force; otherwise, the design risks failure due to instability.

Overturning

Overturning of the footing is checked by summing up all moments about a point in the footing including all the forces acting on it. All serviceability load combinations must be considered to check the governing overturning moment. Usually, a safety factor of 1.5-2 is employed to evaluate whether the footing passes the overturning check.

Sliding

To check for sliding, the sum of the horizontal resisting loads pointing to the right is divided by the sum of the loads pointing to the left.

• Resisting loads:
  • Horizontal force due to friction between the footing base and the substructure soil
  • Passive soil pressure (if included)
• Sliding loads:
  • The horizontal component of the active soil pressure
  • The horizontal component of the surcharge resultant pressure

Generally, a minimum factor of safety of 1.5 is used. If no horizontal force is acting on the footing, checking for sliding is not required.

SkyCiv Foundation Design Module

The Foundation Design Module is a powerful tool integrated with Finite Element Analysis (FEA), capable of conducting thorough soil pressure and wood armer analyses for detailed flexural checks. It performs all structural checks specified by ACI 318 and other verifications mentioned above and presents them in a comprehensive report.

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References

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