A footing has to satisfy two separate criteria, and passing one tells you nothing about the other: the ground must not rupture beneath it, and it must not settle more than the structure can tolerate. This chapter is the first criterion.
Failure ModesTerzaghiMeyerhofWater Table
01
Two Criteria, Not One
A shallow foundation is one where the depth of embedment is roughly less than its width — it transfers load to soil near the surface. Designing one means satisfying two independent requirements, and the governing one is not always the one you would guess.
Criterion 1 — Strength
No bearing capacity failure
The soil must not rupture and let the footing punch into the ground. Governed by shear strength, and handled with a large factor of safety, typically 3.
Criterion 2 — Serviceability
Tolerable settlement
Covered in the settlement chapter. On clay this very often governs: the footing that is safe against collapse still settles 100 mm, which the structure cannot accept.
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Snowshoes. Stand on soft snow in boots and you punch through — a bearing capacity failure. Put on snowshoes and you don't, because spreading the same load over more area drops the pressure below what the snow can carry. But you still sink a little, evenly. That residual sinking is settlement, and no amount of extra area removes it entirely.
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Why it matters: engineers new to this often size a footing for bearing capacity, get a small answer, and stop. On soft clay the settlement check can demand a footing several times larger. Always run both.
02
Types of Shallow Foundation
Isolated / spread footing
One column, one pad
The default and cheapest option. A square or rectangular pad under a single column.
Strip / wall footing
Continuous under a wall
Long relative to its width, so it is analysed per metre run and uses the plane-strain (strip) bearing capacity factors.
Combined footing
Two or more columns on one pad
Used where columns are close together, or where a property line prevents centring a pad under an edge column.
Strap footing
Two pads tied by a beam
The strap beam transfers moment from an eccentrically loaded edge footing to an interior one, so neither pad has to resist it alone.
Mat / raft
One slab under everything
When individual footings would cover more than about half the plan area, merge them. A raft also bridges over soft spots and averages out differential settlement.
Choosing
Escalate only as needed
Start with isolated footings; move up the list only when loads, weak soil, or geometry force you. Each step costs more.
03
Three Modes of Bearing Failure
How the ground fails depends on how compressible it is — and only the first mode gives you a clear warning.
General shear
Dense sand, stiff clay
A continuous slip surface develops from the footing edge to the ground surface. The soil heaves visibly on both sides and the footing tilts. There is a clear, sudden peak load — catastrophic but at least identifiable.
Local shear
Medium dense
The slip surface is well developed under the footing but dies out before reaching the surface. Heave is slight. No sharp peak — the load-settlement curve just gets progressively steeper.
Punching shear
Loose sand, soft clay
No slip surface reaches the surface at all. The footing simply drives downward, compressing soil beneath it, with vertical shearing around the perimeter. Almost no surface warning — the danger is that failure is invisible until it is large.
Core Idea
Terzaghi's theory assumes general shear. For loose or soft soil, reduce the strength parameters before using it: \( c' = \tfrac{2}{3}c \) and \( \tan\phi' = \tfrac{2}{3}\tan\phi \). Applying the general-shear equation directly to loose sand overestimates capacity.
04
Terzaghi's Theory
Terzaghi (1943) idealised failure as a wedge of soil trapped beneath the footing, pushing outward against radial shear zones and passive wedges on either side — and produced the equation the entire subject is built on.
The Failure Mechanism
The footing cannot move down without pushing the wedge beneath it sideways, which in turn has to lift the soil either side. That is why burying a footing deeper raises its capacity so effectively — there is more soil to lift.
Three independent contributions: cohesion of the soil, surcharge q = γDf from the soil beside the footing, and the self-weight of the soil in the failure zone, which is the only term that grows with footing width B.
Strip footingqu = c'Nc + qNq + 0.5γBNγ
Square footingqu = 1.3c'Nc + qNq + 0.4γBNγ
Circular footingqu = 1.3c'Nc + qNq + 0.3γBNγ
Bearing factorsfunctions of φ' only
05
The General Bearing Capacity Equation
Terzaghi's form handles only vertical, centric loads on level ground with simple shapes. Meyerhof generalised it by attaching correction factors to each of the three terms.
Note how sensitive Nγ is to φ'. Going from φ' = 30° to 40° raises Nc by a factor of 2.5, Nq by 3.5, but Nγ by nearly 5. A few degrees of optimism in your friction angle becomes an enormous error in computed capacity — which is precisely why the SPT corrections in the previous chapter are worth the trouble.
06
The Water Table
Groundwater reduces bearing capacity, sometimes by nearly half, because both the surcharge term and the self-weight term depend on effective unit weight. There are three cases, determined purely by where the water table sits relative to the footing.
Case I — water above footing base
Water table at depth D₁ < Df. The surcharge becomes partly effective:
\[ q = D_1\gamma + (D_f-D_1)\gamma' \]
and use γ' in the last term.
Case II — within depth B below base
Surcharge is full q = γDf, but the γ in the third term is interpolated:
The water table lies below the entire failure zone, so it has no effect. Use moist unit weight throughout.
Core Idea
The mechanism is the same one from the slope stability chapter: submergence roughly halves effective unit weight (γ' ≈ 10 vs γ ≈ 19 kN/m³), and both the surcharge and self-weight terms scale with it. Always design for the highest water table the site will ever see, not the level on the day of drilling.
07
Net, Gross & Allowable
A persistent source of error. The soil at foundation level was already carrying the weight of the soil you excavated, so only the additional pressure is new.
Net ultimate
\[ q_{u(net)} = q_u - q \]
Subtract the surcharge that was there before you built.
Allowable bearing capacity
\[ q_{all(net)} = \frac{q_u - q}{FS} \]
FS = 3 is standard for shallow foundations.
Allowable column load
\[ Q_{all} = q_{all(net)}\times B\times L \]
What the footing can actually carry.
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Apply the factor of safety to the net value, not the gross. Dividing qu by 3 and then forgetting to subtract q gives an answer that is wrong in the unsafe direction for deep footings, where the surcharge term is large.
08
Bearing Capacity Calculator
The full Meyerhof equation with shape and depth factors and all three water table cases. Drag the water table down and watch which case the calculator selects, and how capacity recovers.
09
Eccentric Loading
A column carrying moment, or sitting off-centre, produces a resultant that does not act at the middle of the footing. Meyerhof's effective area method handles this with one elegant idea.
Effective dimensions
\[ e = \frac{M}{Q}, \qquad B' = B - 2e \]
Pretend the footing is only as wide as the part symmetric about the load. Compute capacity using B' in place of B, then multiply by the effective area A' = B'L.
Why it works
The eccentric load is equivalent to a centric load on a narrower footing. Anything beyond that symmetric strip is not helping to resist the load — it is dead weight sitting on soil that is barely stressed. Same middle-third logic as the retaining wall chapter: keep e ≤ B/6 and the whole base stays in compression.
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Note the double penalty. Reducing B to B' shrinks the area and shrinks the 0.5γBNγ term, so a modest eccentricity costs disproportionately more capacity than you would expect. This is why designers work hard to centre footings under their loads.
10
Worked Examples
Example 1
Square Footing on Dry Sand
Problem: A 2 m × 2 m square footing is founded at Df = 1.5 m in sand with φ' = 30°, c' = 0, γ = 18 kN/m³. The water table is very deep. Find the allowable column load with FS = 3.
Answer: qall(net) = 389 kPa, Qall ≈ 1560 kN. Note the surcharge term supplies 80% of the capacity — embedment, not width, is doing most of the work here.
Example 2
The Same Footing with a High Water Table
Problem: Repeat Example 1 with the water table at the ground surface. γsat = 20 kN/m³.
1
Case I (water above the footing base, D₁ = 0): \( \gamma' = 20 - 9.81 = 10.19\text{kN/m}^3 \)
2
Effective surcharge: \( q = \gamma' D_f = 10.19(1.5) = 15.3\text{kPa} \) — down from 27.0
3
Self-weight term uses γ' too: \( 0.5(10.19)(2)(22.40)(0.600) = 137.0\text{kPa} \) — down from 241.9
Answer: Capacity falls from 1560 kN to 882 kN — a 43% loss from groundwater alone, with no change to the soil or the footing. This is why the design water table is one of the most consequential decisions in the report.
Example 3
Eccentric Load
Problem: The footing of Example 1 also carries a moment M = 300 kN·m alongside a vertical load Q = 1000 kN. Check whether it is still adequate.
1
Eccentricity: \( e = M/Q = 300/1000 = 0.30\text{m} \)
2
Middle-third check: \( B/6 = 2/6 = 0.333\text{m} > 0.30 \) — the whole base stays in compression ✓
3
Effective width: \( B' = B - 2e = 2 - 0.60 = 1.40\text{m} \), so \( A' = 1.40 \times 2 = 2.80\text{m}^2 \)
4
Recompute with B' = 1.40 m. Shape factors use B'/L = 0.70: \( F_{qs} = 1+0.70\tan30 = 1.404 \), \( F_{\gamma s} = 1-0.4(0.70) = 0.720 \). Depth factors keep the actual B: \( F_{qd} = 1.217 \). The self-weight term uses B':
\( q_u = 27(18.40)(1.404)(1.217) + 0.5(18)(1.40)(22.40)(0.720) = 848.6 + 203.2 = 1052\text{kPa} \)
5
Allowable: \( q_{all} = (1052-27)/3 = 342\text{kPa} \), so \( Q_{all} = 342 \times 2.80 = 957\text{kN} \) against an applied 1000 kN — inadequate.
Answer: Qall ≈ 957 kN < 1000 kN applied — the footing fails, despite passing the middle-third check. Enlarge the footing or reduce the eccentricity. Note it carried 1560 kN centrically: a 0.3 m eccentricity cost 39% of the capacity. (Shape factors take B', depth factors take the real B — a standard convention worth getting right.)
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Quick Reference & Quick Check
Terzaghi, stripc'Nc + qNq + 0.5γBNγ
Terzaghi, square1.3c'Nc + qNq + 0.4γBNγ
Nqe^(πtanφ')tan²(45+φ'/2)
Nc / Nγ(Nq−1)cotφ' / 2(Nq+1)tanφ'
Nc at φ' = 05.14
Surchargeq = γDf
Net ultimatequ(net) = qu − q
Allowable(qu − q)/FS, FS = 3
Local shear correctionc* = ⅔c, tanφ* = ⅔tanφ
Eccentric loadingB' = B − 2e, A' = B'L
1. A footing on soft clay passes the bearing capacity check with FS = 3.5. Is the design complete?
2. Why does raising the water table to the ground surface roughly halve the bearing capacity?
3. Which failure mode gives the least warning before it occurs?
4. Terzaghi's equation is applied directly to a footing on loose sand. What is the problem?