Reinforced Concrete Column Combined Footing Analysis and Design
Reinforced Concrete Column Combined Footing Analysis and Design
A combined footing was selected to support a 24 in. x 16 in. exterior column near a property line and a 24 in. x 24 in. Interior column. Each column carries the service dead and live loads shown in the following figure. The footing dimensions (25 ft 4 in. x 8 ft) were selected such that the centroid of the area in contact with soil coincides with the resultant of the column loads supported by the footing.
Check if the selected combined footing preliminary thickness of 36 in. is sufficient to resist twoway punching shear around the interior and exterior columns supported by the footing. Compare the calculated results with the values presented in the Reference and model results from spMats engineering software program from StructurePoint.
Figure 1 – Reinforced Concrete Combined Footing Geometry
Contents
1.1. Footing Cross Sectional Dimensions
2. TwoWay (Punching) Shear Capacity Check
3. OneWay Shear Capacity Check
4. Flexural Reinforcement Design
4.1. Negative Moment (Midspan)
4.2. Positive Moment (At Interior Column)
5. Combined Footing Analysis and Design – spMats Software
6. Design Results Comparison and Conclusions
Code
Building Code Requirements for Structural Concrete (ACI 31814) and Commentary (ACI 318R14)
Reference
Reinforced Concrete Mechanics and Design, 7^{th} Edition, 2016, James Wight, Pearson, Example 155
spMats Engineering Software Program Manual v8.12, StucturePoint LLC., 2016
Design Data
f_{c}’ = 3,000 psi normal weight concrete
f_{y} = 60,000 psi
Preliminary footing thickness = 36 in.
Dead load, D = 200 kips for exterior column and 300 kips for interior column
Live load, L = 150 kips for exterior column and 225 kips for interior column
Soil density, γ_{s} = 120 pcf
Concrete density, γ_{c} = 150 pcf for normal weight concrete
Allowable soil pressure, q_{a} = 5000 psf
Footing length = 25 ft 4 in.
Footing width = 8 ft
Preliminary footing depth = 36 in. with effective depth, d = 32.5 in.
The footing dimensions (25 ft 4 in. x 8 ft) were selected by the reference such that the centroid of the area in contact with soil coincides with the resultant of the column loads supported by the footing to achieve uniform soil pressures.
The factored net pressure that will be used in the design of the concrete and reinforcement is equal to:
The following Figure shows the shear and moment diagrams for the combined footing based on the factored columns loads and the factored net pressure.
Figure 2 – Shear Force and Bending Moment Diagrams
Twoway shear is critical on a rectangular section located at d/2 away from the face of the column as shown in the following Figures, Where:
b_{1} = Dimension of the critical section measured in the direction of the span for which moments are determined in ACI 318, Chapter 8.
b_{2} = Dimension of the critical section measured in the direction perpendicular to b_{1} in ACI 318, Chapter 8 (see Figure 5).
Figure 3 – Critical Shear Perimeters around Columns
The factored shear force (V_{u}) at the critical section is computed as the reaction at the centroid of the critical section minus the force due to soil pressure acting within the critical section (d/2 away from column face).
The factored unbalanced moment used for shear transfer, M_{unb}, is computed as the sum of the joint moments to the left and right. Moment of the vertical reaction with respect to the centroid of the critical section is also taken into account.
For the interior column, the location of the centroidal axis zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (8.4.2.3.2)
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the exterior column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
Since ϕv_{c} > v_{u} at the critical section, the slab has adequate twoway shear strength at this joint.
For the exterior column, the location of the centroidal axis zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (8.4.2.3.2)
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the exterior column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
Since ϕv_{c} < v_{u} at the critical section, the slab does not have adequate twoway shear strength at this joint.
Increase the footing thickness to 40 in. with effective depth, d = 36.5 in.
For the exterior column, the location of the centroidal axis zz is:
The polar moment J_{c} of the shear perimeter is:
ACI 31814 (8.4.2.3.2)
ACI 31814 (Eq. 8.4.4.2.2)
The length of the critical perimeter for the exterior column:
The twoway shear stress (v_{u}) can then be calculated as:
ACI 31814 (R.8.4.4.2.3)
ACI 31814 (Table 22.6.5.2)
Since ϕv_{c} > v_{u} at the critical section, the slab has adequate twoway shear strength at this joint.
Use a combined footing 25 ft 4 in. by 8 ft in plan, 3 ft 4 in. thick, with effective depth 36.5 in.
The critical section for oneway shear is located at distance d from the face of the column. The oneway shear capacity of the foundation can be calculated using the following equation:
ACI 31814 (22.5.5.1)
Where ϕ = 0.75 ACI 31814 (Table 21.2.1)
This example focus on the calculation of twoway shear capacity for combined foundation. For more details on the oneway shear check for foundation check “Reinforced Concrete Shear Wall Foundation (Strip Footing) Analysis and Design” example.
The critical section for moment is shown in the moment diagram in Figure 2. The design moment is:
Use d = 36.5 in.
To determine the area of steel, assumptions have to be made whether the section is tension or compression controlled, and regarding the distance between the resultant compression and tension forces along the footing section (jd). In this example, tensioncontrolled section will be assumed so the reduction factor ϕ is equal to 0.9, and jd will be taken equal to 0.95d. The assumptions will be verified once the area of steel in finalized.
Therefore, the assumption that section is tensioncontrolled is valid.
Depending of the method of analysis the minimum area of reinforcement shall be calculated using beam provisions or oneway slab provisions. In this case both beam and slab provisions will be illustrated.
For beam provisions:
ACI 31814 (9.6.1.2)
Use 17#8 top bars with A_{s} = 13.43 in.^{2} at midspan.
For slab provisions:
ACI 31814 (7.6.1.1)
Use 17#8 top bars with A_{s} = 13.43 in.^{2} at midspan.
In spMats, the slab provisions for minimum reinforcement can be used since the finite element analysis calculates the required area of steel in both the x (longitudinal) and y (transverse) direction independently.
For beam provisions:
Repeating the same process at Section 4.1, A_{s} = 3.3 in.^{2} << A_{s,min }= 11.7 in.^{2}. Thus, use 15#8 bottom bars with A_{s} = 11.9 in.^{2} at the interior column.
For slab provisions:
Repeating the same process at Section 4.1, A_{s} = 3.3 in.^{2} << A_{s,min }= 6.91 in.^{2}. Thus, use 9#8 bottom bars with A_{s} = 7.11 in.^{2} at the interior column.
Note code provisions permit the use of reinforcement of one third more than is required by analysis in some cases.
spMats uses the Finite Element Method for the structural modeling and analysis of reinforced concrete slab systems or mat foundations subject to static loading conditions.
The slab, mat, or footing is idealized as a mesh of rectangular elements interconnected at the corner nodes. The same mesh applies to the underlying soil with the soil stiffness concentrated at the nodes. Slabs of irregular geometry can be idealized to conform to geometry with rectangular boundaries. Even though slab and soil properties can vary between elements, they are assumed uniform within each element.
For illustration and comparison purposes, the following figures provide a sample of the input modules and results obtained from spMats models created for the reinforced concrete combined footing at a property line in this example. Two models were created for this example (the first model for the footing with 36 in. thickness that failed in punching shear check around the exterior column, and the second model for the same footing with a revised thickness of 40 in.).
Figure 4 –Defining Service Loads (spMats)
Figure 5 –Defining Columns Dimensions (spMats)
Figure 6 –Mesh Generation (spMats) Showing Node & Element Numbering
Figure 7 – Punching Shear Output 36 in. Strip Footing (spMats)
Figure 8 – Ultimate Moment Contour 40 in. Footing (spMats)
Figure 9 – Required Reinforcement Contour 40 in. Footing (spMats)
Figure 10 –Vertical Displacement Contour 40 in. Footing (spMats)
Table 1  Comparison of TwoWay (Punching) Shear Check Results for Footing with 36 in. Thickness 

Support 
b_{1}, in. 
b_{2}, in. 
c_{AB}, in. 

Reference 
Hand 
Reference 
Hand 
Reference 
Hand 

Exterior 
32.25 
32.25 
32.25 
56.5 
56.5 
56.5 
8.6 
8.6 
8.6 
Interior 
56.5 
56.5 
56.5 
56.5 
56.5 
56.5 
28.25 
28.25 
28.25 


Support 
J_{c}, in.^{4} 
γ_{v} 
V_{u}, kips 

Reference 
Hand 
Reference 
Hand 
Reference 
Hand 

Exterior 
621,000 
620,710 
620,710 
0.335 
0.335 
0.335 
480 
480 
480 
Interior 
 
4,231,103 
4,231,103 
 
0.4 
0.4 
720 
720 
720 


Support 
M_{u,punching}, kipsft 
v_{u}, psi 
ϕv_{c, }psi 

Reference 
Hand 
Reference 
Hand 
Reference 
Hand 

Exterior 
579 
579 
943 
192 
192 
267 
164 
164.3 
164.3 
Interior 
0 
0 
0 
80.2 
80.2 
98 
164 
164.3 
164.3 
Table 2  Flexural Reinforcement Comparison  Longitudinal Direction 

M_{u}, kipsft 
A_{s,required}, in.^{2} 
A_{s,min}, in.^{2} 

Reference 
Hand 
Reference 
Hand 
Reference 
Hand^{*} 
spMats^{*} 

2100 
2100 
2132 
13.5 
13.4 
13.5 
11.7 
11.7 
11.7 
^{*} Using beam provisions to find A_{s,min} to be consistent with Reference approach. However, engineering judgment need to be taken to decide if the combined footing need to be treated as a oneway slab or beam. 
The results of all the hand calculations and the reference used illustrated above are in agreement with the automated exact results obtained from the spMats program except for v_{u} values.
In spMats, the factored unbalanced moment used for shear transfer, M_{unb}, is calculated as the sum of the moments at finite element nodes within the critical section (resisting zone). Moment of the vertical reaction with respect to the centroid of the critical section is also taken into account. The reference take into account only the moments of the vertical reaction and soil pressure with respect to the centroid of the critical section.