Isolated Footing Design

Theory: Footing Design

How an isolated footing is designed: its plan by the soil, with eccentric loads and settlement, and its height and reinforcement by the concrete, as NBR 6122 and NBR 6118 ask.

A plan with equal overhangs

An isolated footing carries a column's load into the soil. Its plan is picked so the soil carries it safely and it settles little; with equal overhangs past the column's faces, the bending moments in both directions come out alike and the reinforcement economical.

B_x - a_x = B_y - a_y

a: the column's sides · B: the footing's · P: the weight of the footing and the soil over it

Eccentric loads

Moments move the resultant off the centre: e_x = M_x/N, e_y = M_y/N. While it stays in the kern, the whole base is in compression and the stress varies linearly; past it, part of the base lifts off, as the soil takes no tension.

\sigma = \frac{N}{B_x B_y}\left(1 \pm \frac{6 e_x}{B_x} \pm \frac{6 e_y}{B_y}\right)

The whole base in compression: the resultant in the kern's diamond

Past the kern, the stress is a plane over the area still in contact, found so that it balances the load and both moments. NBR 6122 asks for at least ⅔ of the base in compression.

For bearing capacity, Meyerhof replaces the base by an effective area centred on the resultant, B′ × L′ = (B_x − 2e_x)(B_y − 2e_y), loaded uniformly. With a given allowable stress, NBR 6122 lets the edge stress reach 1.3σ_adm.

Settlement

s = q_{\mathrm{net}}\, B\, \frac{1 - \nu^2}{E_s}\, I

q_net: the mean stress less the geostatic one at the base · B: the shorter side · I: a rigid footing's influence factor

L/B	1	1.5	2	3	5	10
I (rigid)	0.82	1.06	1.20	1.42	1.70	2.10

Rigid footings

NBR 6118 calls a footing rigid when its height is at least a third of its overhang. The load then goes down to the base in compressed struts, punching doesn't occur, and only the concrete's diagonal compression by the column needs checking.

h \ge \frac{B - a}{3}

u₀: the column's perimeter · d: the effective depth · F_Sd = 1.4N

Reinforcement by CEB-70

CEB-70 takes the moment at a reference section 0.15 of the column's side inside its face, from the soil's reaction on the overhang past it. The effective depth taken is at most 1.5 times that overhang.

A_s = \frac{1.4\, M_k}{0.85\, d\, f_{yd}} \ge 0.15\%\, A_c

A_c: the footing's section at the reference section · f_yd = f_yk/1.15

The bars are spread over the whole side, 10 to 20 cm apart, with the smallest diameter that keeps to that spacing.

References

ABNT. NBR 6122: Projeto e execução de fundações. Rio de Janeiro, 2019.
ABNT. NBR 6118: Projeto de estruturas de concreto. Rio de Janeiro, 2023.
Comité Européen du Béton. Recommandations particulières au calcul et à l'exécution des semelles de fondation. Bulletin d'Information n. 73, 1970.
Meyerhof, G. G. The bearing capacity of foundations under eccentric and inclined loads. Proc. 3rd ICSMFE, v. 1, p. 440–445, 1953.
Bowles, J. E. Foundation Analysis and Design. 5th ed. New York: McGraw-Hill, 1996.
Velloso, D. A.; Lopes, F. R. Fundações. New ed. São Paulo: Oficina de Textos, 2010.

Enter the column — its section, load and moments —, the footing's depth and the soil to design a rigid isolated footing: the plan with equal overhangs, checked for bearing, the area in compression and settlement, then its height, reinforcement and the concrete and steel it takes, seen in 3D with the stress under its base.

Column and loads

Side along x, a_xmSide along y, a_ymLoad, NkNMoment, M_xkN·mMoment, M_ykN·m

Characteristic loads, at the footing's top. M_x moves the resultant along x (e_x = M_x/N), M_y along y.

Footing

Plan

Depth, D_fm

The smallest plan with equal overhangs the soil carries. The height comes from rigidity and diagonal compression.

Soil

Allowable stress

Method

c′kPaφ′30°FSγ (above WT)kN/m³γ_sat (below WT)kN/m³Water table, D_wmγ_wkN/m³Modulus, E_sMPaPoisson's ratio, νAllowable settlementmm

q_u by the method picked, on the effective area; NBR 6122 asks for FS = 3 with theoretical methods. Leave the water table blank when it's deep.

Concrete and steel

f_ckMPaf_ykMPad′m

d′ runs from the base to the bars' centre. γ_f = γ_c = 1.4 and γ_s = 1.15.

A rigid footing (NBR 6118) with a rectangular base, faces sloping up to a 2.5 cm collar around the column and an edge h₀ ≥ h/3 and ≥ 20 cm. The plan is sized with equal overhangs, in 5 cm steps, from 0.60 m (NBR 6122).
The footing's weight (25 kN/m³) and the soil's over it add to the load on the base; buoyancy and side friction are left out.
The stress under the base is a plane, with the area in contact found by iteration when the resultant leaves the kern; the soil takes no tension. NBR 6122 asks for at least ⅔ of the base in compression.
By the bearing capacity equation, the total load acts on Meyerhof's effective area, B′ × L′ = (B_x − 2e_x)(B_y − 2e_y). With a given σ_adm, the mean stress must stay under it and the edge stress under 1.3σ_adm (NBR 6122).
Immediate settlement of a rigid footing on a homogeneous elastic half-space, with Bowles's influence factor; consolidation of clays isn't included.
Reinforcement by the CEB-70 method, at the section 0.15 of the column inside its face, from the soil's reaction to the column's load; CA-50 bars hooked up at both ends, to the edge's top less the cover. Diagonal compression at the column's face by NBR 6118 (τ_Rd2); the anchorage of the column's bars isn't checked.
The 3D is an illustration: the pit with 40 cm of working space, the backfill, the topsoil, the spacers and the column's starter bars and ties (φ12.5 and φ5) show the work as built but aren't designed here. The 2D drawing carries the dimensions, the reinforcement and the stresses.