**An example walk-through of the calculations required to design an isolated footing (ACI 318-14)**

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 American Concrete Institute¹.

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**Design Example of an Isolated Footing**

**Dimension Requirements**

To determine the dimensions of a isolated footing, service or unfactored loads, such as dead (**D**), Live (**L**), Wind (**W**), Seismic (**E**), etc will be applied using Load Combinations, as defined by ACI 318-14. Whichever Load Combination governs will be considered the design load, and is compared to the allowable soil pressure as shown in Equation 1, as recommended in **Section 13.2.6 of ACI 318-14**.

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

where:

q_{a} = allowable soil pressure

P_{n} = unfactored design load

A = Foundation area

From Equation 1, **q _{a}** are interchange 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),

Figure 1. Critical Plane Shear 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) indicated in Figure 2, in accordance of

**ACI 318-14, Section 8.5.3.1.1**.

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

**ACI 318-14, Section 22.5.5.1**.

\(\phi\text{V}_{\text{c}} = \phi _{\text{shear}} \times 2 \sqrt{\text{f’}_{\text{c}}} \times \text{b}_{\text{w}} \times \text{d} \rightarrow \) Equation 2 (**ACI 318-14 Eq. 22.5.5.1** Imperial)

or

\(\phi\text{V}_{\text{c}} = \phi _{\text{shear}} \times 0.17 \sqrt{\text{f’}_{\text{c}}} \times \text{b}_{\text{w}} \times \text{d} \rightarrow \) Equation 2 (**ACI 318-14 Eq. 22.5.5.1** Metric)

where:

ϕ_{shear} = shear design factor

f’_{c} = specified concrete strength, psi or MPa

b_{w} = width of the footing, in or mm

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

Shear Demand and Shear Capacity must meet the following equation to meet the design requirements of ACI 318-14:

\(\text{V}_{\text{u}} \leq \phi\text{V}_{\text{c}} \rightarrow \) Equation 3 (**ACI Eq. 7.5.1.1(b)**)

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).

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, in accordance of

**ACI 318-14, Section 22.6.4**.

The **Shear Capacity **or** ϕV _{c}** is governed by the smallest value calculated using Equation 5, 6, and 7 per

**ACI 318-14, Section 22.6.5.2**

\(\phi\text{V}_{\text{c}} = \phi _{\text{shear}} \times 4 \times \lambda \times \sqrt{\text{f’}_{\text{c}}} \rightarrow \) Equation 5 (**ACI Eq. 22.6.5.2(a)** – Imperial)

\(\phi\text{V}_{\text{c}} = \left ( 2 + \frac{4}{\beta } \right ) \times \lambda \times \sqrt{f’_{c}} \rightarrow \) Equation 6 (**ACI Eq. 22.6.5.2(b)** – Imperial)

\(\phi\text{V}_{\text{c}} = \left ( 2 + \frac{\alpha _{s} \times d }{b{o}} \right ) \times \lambda \times \sqrt{f’_{c}} \rightarrow \) Equation 7 (**ACI Eq. 22.6.5.2(c)** – Imperial)

or

\(\phi\text{V}_{\text{c}} = \phi _{\text{shear}} \times 0.33 \times \lambda \times \sqrt{\text{f’}_{\text{c}}} \rightarrow \) Equation 5 (**ACI Eq. 22.6.5.2(a)** Metric)

\(\phi\text{V}_{\text{c}} = 0.17 \times \left ( 1 + \frac{2}{\beta } \right ) \times \lambda \times \sqrt{f’_{c}} \rightarrow \) Equation 6 (**ACI Eq. 22.6.5.2(b)** Metric)

\(\phi\text{V}_{\text{c}} = 0.0083 \times \left ( 2 + \frac{\alpha _{s} \times d }{b{o}} \right ) \times \lambda \times \sqrt{f’_{c}} \rightarrow \) Equation 7 (**ACI Eq. 22.6.5.2(c)** Metric)

**Note:** β is the ratio of long side to short side of the column, concentrated load, or reaction area and α_{s} is given by 22.6.5.3

where:

λ = modification factor to reflect the reduced mechanical properties of lightweight concrete relative to normalweight concrete of the same compressive strength

f’_{c} = specified compressive concrete strength, psi or MPa

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

Shear Demand and Shear Capacity must meet the following equation to meet the design requirements of ACI 318-14:

\(\text{V}_{\text{u}} \leq \phi\text{V}_{\text{c}} \rightarrow \) Equation 8 (**ACI Eq. 7.5.1.1(b)**)

SkyCiv Foundation, in compliance of Equation 8, calculates the two-way shear unity ratio (Equation 9) by taking Shear Demand over Shear Capacity.

\( \text{Unity Ratio} = \frac{\text{Shear Demand}}{\text{Shear Capacity}} \rightarrow \) Equation 9

**Flexure**

Figure 3. Critical Flexure Section

The **Flexural** limit state is occurs at** **the Critical Flexure Section, located at the face of the column on top of the footing (Refer to Figure 3).

The **Flexural Demand, **or **M _{u}** is located at the Critical Flexure Section (blue hatch area) indicated in Figure 3, and is calculated using Equation 10.

\( \text{M}_{u} = \text{q}_{u} \times \left ( \frac{l_{x}}{2} – \frac{c_{x}}{2} \right ) \times l_{z} \times \left ( \frac{\frac{l_{x}}{2} – \frac{c_{x}}{2} }{2} \right ) \rightarrow \) Equation 10

where:

q_{u} = factored soil pressure, ksf or kpa

l_{x} = footing dimension along the x-axis, in or mm

l_{z} = footing dimension along the z-axis, in or mm

c_{x} = column dimension along the x-axis, in or mm

The **Flexural Capacity, **or** ϕMn** is calculated using Equation 11.

\( \phi\text{M}_{n} = \phi_{\text{flexure}} \times A_{s} \times f_{y} \times \left( d – \frac{a}{2} \right) \rightarrow \) Equation 11

where:

ϕ = flexural design factor

l_{x} = footing dimension parallel to x-axis, in or mm

l_{z} = footing dimension parallel to z-axis, in or mm

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

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

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

fy = reinforcement strength, ksi or MPa

Moment Demand and Moment Capacity must meet the following equation to meet the design requirements of ACI 318-14:

\(\text{M}_{\text{u}} \leq \phi\text{M}_{\text{n}} \rightarrow \) Equation 12 (**ACI Eq. 7.5.1.1(b)**)

SkyCiv Foundation, in compliance of Equation 12, calculates the flexural unity ratio (Equation 13) by taking Flexural Demand over Flexural Capacity.

\( \text{Unity Ratio} = \frac{\text{Flexure Demand}}{\text{Flexure Capacity}} \rightarrow \) Equation 13

**Albert Pamonag**

Structural Engineer, Product Development

M.E. Civil Engineering

**References**

*Building Code Requirements for Structural Concrete (ACI 318-14) Commentary on Building Code Requirements for Structural Concrete (ACI 318R-14)*. American Concrete Institute, 2014.- McCormac, Jack C., and Russell H. Brown.
*Design of Reinforced Concrete ACI 318-11 Code Edition*. Wiley, 2014. - Taylor, Andrew, et al.
*The Reinforced Concrete Design Handbook: a Companion to ACI-318-14*. American Concrete Institute, 2015.