Compression-only ground spring

Results.

Example	Reference	Records	Largest deviation
Rigid footing, eccentricity outside the kern and partial contact	Published source	6	0.871%
Raft, column load outside the kern and partial contact	Published source	6	2.27%

Problem. A compression-only soil spring drops out in tension. Once the eccentricity of a footing leaves the kern, contact becomes partial. For a strip footing and a square raft the contact length and the peak soil pressure are compared with the closed-form solution.

Reference document. The classical closed-form solution for partial contact under a rigid footing is used.

The contact length and the maximum ground pressure are computed from these relations. They hold when the eccentricity exceeds one sixth of the length.

a = 3\left(\dfrac{L}{2} - e\right),\qquad q_{\mathrm{max}} = \dfrac{2P}{b\,a}

Numerical model. The strip footing is modelled with twenty frame elements and the raft with twenty by twenty shell elements, with a compression-only spring at every node. The contact length is read where the settlement crosses zero, and the soil pressure as the subgrade modulus times the settlement at a node.

Rigid footing, eccentricity outside the kern and partial contact

Published source. The classical solution for partial contact

a = 3(L/2 - e),\qquad q_{max} = 2P/(ba)

Quantity	Condition	Reference	Computed	Deviation
Contact length $a$	L = 6 · e/L = 0.2 · springs = 21 · iterations = 4	5.4	5.39619	0.0706%
Maximum ground pressure	L = 6 · e/L = 0.2 · springs = 21 · iterations = 4	222.222	220.287	0.871%
Contact length $a$	L = 6 · e/L = 0.25 · springs = 21 · iterations = 4	4.5	4.51037	0.231%
Maximum ground pressure	L = 6 · e/L = 0.25 · springs = 21 · iterations = 4	266.667	264.798	0.701%
Contact length $a$	L = 6 · e/L = 0.3 · springs = 21 · iterations = 5	3.6	3.61905	0.529%
Maximum ground pressure	L = 6 · e/L = 0.3 · springs = 21 · iterations = 5	333.333	330.815	0.755%

Raft, column load outside the kern and partial contact

Published source. The classical solution for partial contact

a = 3(L/2 - e),\qquad q_{max} = 2P/(ba)

Quantity	Condition	Reference	Computed	Deviation
Contact length $a$	B = 6 · t = 0.8 · e/B = 0.2 · springs = 441 · iterations = 5	5.4	5.42007	0.372%
Maximum ground pressure	B = 6 · t = 0.8 · e/B = 0.2 · springs = 441 · iterations = 5	185.185	187.29	1.14%
Contact length $a$	B = 6 · t = 0.8 · e/B = 0.25 · springs = 441 · iterations = 5	4.5	4.54853	1.08%
Maximum ground pressure	B = 6 · t = 0.8 · e/B = 0.25 · springs = 441 · iterations = 5	222.222	226.776	2.05%
Contact length $a$	B = 6 · t = 0.8 · e/B = 0.3 · springs = 441 · iterations = 6	3.6	3.65887	1.64%
Maximum ground pressure	B = 6 · t = 0.8 · e/B = 0.3 · springs = 441 · iterations = 6	277.778	284.096	2.27%

Note. The raft is a realistic concrete plate and is not fully rigid. As the closed form assumes a rigid footing, the remaining difference of one to two per cent comes from the flexibility of the raft.

The contact pressure distribution. Once the eccentricity leaves the kern, the contact is partial.

Rigid footing, eccentricity outside the kern and partial contact#

Raft, column load outside the kern and partial contact#

Rigid footing, eccentricity outside the kern and partial contact

Raft, column load outside the kern and partial contact