Module 5 · Lesson 5.2
Inclined loads
Why a small horizontal component costs so much.
Why this matters
Wind on a building, earth pressure on a wall, braking on a bridge bearing — horizontal load at foundation level is ordinary. It is also far more damaging than its size suggests, and treating it as a small perturbation on the vertical case is one of the more expensive mistakes available.
The failure mechanism of Module 3 assumes the load pushes straight down, so the wedge drives symmetrically into the ground and both passive zones resist equally. Tilt the load and that symmetry breaks. The mechanism is pushed sideways: on the downhill side the passive zone is smaller and easier to shift, and the whole failure surface rotates towards the direction of the horizontal component.
The inclination factors capture this, and they bite hard. At H/V = 0.2 — a horizontal component of only a fifth of the vertical — with φ′ = 30°:
- iq = 0.716
- iγ = 0.572
So a 20% horizontal load has removed about 28% of the surcharge term and 43% of the self-weight term.
Note also that iγ falls faster than iq. The self-weight term depends on the mechanism staying intact beneath the footing, and that is precisely what a horizontal push disrupts. So a foundation relying mostly on the self-weight term — a wide, shallow footing on sand — is the most vulnerable to inclination, which is exactly the case where a designer is most tempted to feel comfortable.
Practice
A pad carries a vertical load of 1200 kN and a horizontal load of 240 kN. The uncorrected surcharge term contributes 500 kPa. Using iq = 0.716, what does the surcharge term become, in kPa?
Check yourself
Which of these simplifications is NOT conservative?
Try it
Shape, depth and inclination
A 2.5 m footing in sand at φ′ = 30°. Watch which corrections help and which take capacity away.
The asymmetry that matters
- Shape and depth factors can be omitted — the result is conservative.
- Inclination factors cannot. Omitting them OVERSTATES capacity.
- sγ is below 1 for a square footing while sc and sq are above it. The effects genuinely oppose.
- sc
- 1.611
- sq
- 1.577
- sγ
- 0.600
- dq
- 1.173
- iq
- 0.854
- iγ
- 0.768
A horizontal component of just 10% of the vertical has cut iγ to 0.768. Leaving inclination out here would overstate capacity by 20%.
Summary
- Shape factors: sc and sq above 1, sγ below 1 — the effects genuinely oppose
- Depth factors credit the strength of soil beside the footing and grow with D/B
- Together they were worth 22% for an ordinary square pad at D/B = 0.6
- Inclination factors fall steeply — at H/V = 0.2, iγ was 0.572
- Shape and depth are safe to omit; inclination is not
This is educational material. It uses simplified examples to teach principles, and must not be relied on for real design or safety-critical decisions. Module overview and checkpoint