Foundation Engineering / Shape, depth and inclination
Inclined loads
Why a small horizontal component costs so much.
Lesson review: expert verification remains open
This lesson has received a full-context editorial review and a check of its technical explanations. Selected arithmetic has been checked separately. This is not independent engineering sign-off.
Corrections checked in this pass
- Corrected the zero-cohesion strip-footing limit and stated the factor family and horizontal-load direction.
Checks needed for project use
- A geotechnical reviewer must confirm drainage and stress basis, bearing-factor formulation, correlations and settlement influence factors. Partial factors and project-specific ground conditions are outside these teaching checks.
- Select compatible factor family, stress basis, design approach and National Annex; no full current-code verification claimed.
These are educational models and exercises. Check their assumptions before using the results. The full technical content and every diagram have not yet been independently verified.
Concrete and steel calculations retain their stated first-generation Eurocode teaching basis. Use the standards and amendments specified for your project; do not mix generations without an appropriate design basis.
Read the lesson register and outstanding review notesWhy 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°. This empirical factor comparison assumes the horizontal load acts along B. It is not a complete Eurocode verification.
The asymmetry that matters
- Check each correction term: omitting a shape factor below 1 can overestimate resistance.
- 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
- Omission of shape or inclination factors can overestimate resistance; assess each term and the limits of the adopted model.