CIVL2410 Chap.12 Elastic stress distribution and immediate settlement
Elastic stress distribution and immediate settlement
One dimensional analysis assumes no lateral strain, which is honest under a wide fill and false under a footing. Under a localised load the soil moves sideways as well as down, the applied stress spreads with depth instead of travelling straight through, and part of the settlement happens as the load is applied rather than over years.
Both effects are handled by treating the ground as an elastic half space, and this chapter is honest about what that costs.
What this chapter covers
- 01
Where one dimensional analysis stops, and what changes when it does
- 02
Two settlements added rather than chosen between: immediate and consolidation
- 03
Hooke's law written for a soil skeleton in effective stresses
- 04
Volumetric strain, mean stress, and the undrained condition that follows from them
- 05
Why the undrained Poisson's ratio is one half and the undrained modulus is larger
- 06
Boussinesq's point load solution, and the four cases obtained by integrating it
- 07
Line, strip and circular loads, and the geometry each needs
- 08
How the stress increase decays and spreads, and the depth of influence it implies
- 09
Superposition, and building any rectangle from corner solutions
- 10
Newmark's chart in five steps, for genuinely irregular areas
- 11
Turning stress increases into a settlement by summing over sublayers
- 12
The assumption list that has to accompany any elastic settlement number
Immediate settlement of a layered profile
- +1Top sublayer: 48 x 2.0 / 12000 = 8.0 mm.
- +1Middle sublayer: 30 x 2.0 / 15000 = 4.0 mm.
- +1Bottom sublayer: 18 x 2.0 / 20000 = 1.8 mm.
- +1Sum the three: 8.0 + 4.0 + 1.8 = 13.8 mm.
Key terms
- Elastic half space
- The idealisation used to obtain closed form stress distributions: a homogeneous, isotropic, linearly elastic material filling everything below a plane surface.
- Boussinesq solution
- The 1885 solution for a vertical point load on an elastic half space. Integrating it produces the line, strip, circular and rectangular cases used in practice.
- Stress bulb
- The pattern formed by contours of vertical stress increase beneath a loaded area. Its scale is set by the width of the load, which is why a wide raft reaches far deeper than a narrow footing at the same pressure.
- Undrained modulus
- The stiffness of a saturated soil loaded with no volume change, larger than the drained modulus because trapped water resists deformation. It is used with total stresses to give the immediate settlement.
- Newmark's chart
- A graphical integration of the point load solution that handles arbitrarily shaped loaded areas by counting blocks, each contributing an influence of 0.001.
Elastic stress distribution and immediate settlement FAQ
Why do the elastic stress solutions contain no soil properties?
Because for a homogeneous elastic half space the vertical stress distribution depends only on the geometry and the load. Not one of the four standard expressions involves a modulus, and only the off axis components involve Poisson's ratio. That is why the same influence chart serves a soft clay and a dense sand, and why getting a stiffness slightly wrong damages a settlement estimate far more than it damages a stress estimate.
Stresses from elastic theory can be used with reasonable confidence, and settlements only with stated reservations.
Is the immediate settlement instead of the consolidation settlement?
No, they are added. A foundation on clay settles twice: immediately as the load is applied, when the clay cannot change volume and so distorts, squeezing down under the footing and bulging slightly outside it; and then over months and years as water drains and the volume genuinely reduces.
The first is computed here with the undrained modulus and total stresses, the second from the oedometer parameters, and the total settlement is the sum of the two.
Exam move
Separate the two halves of an elastic settlement calculation and check each on its own: first the stress distribution, which is geometry and is trustworthy, then the strain, which needs a modulus and is not. Learn which method suits which shape, since choosing sensibly is itself examinable and reaching for Newmark's chart on a plain rectangle wastes time.
Use the attenuation of stress with depth to decide how deep to bother analysing, and say so in your answer rather than computing sublayers that contribute nothing. Above all, report the assumption list with the number, because a settlement figure quoted alone invites a confidence that the method does not support.
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