Theoretical and numerical modeling of a shallow foundation stiffness based on the theory of elastic half-space

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Results for columns C1_

	Column C1
	Node 1			Node 2
	Model A	Model B	Error	Model A	Model B	Error
Ux [mm]	2.2	2.3	4.5%	−0.4	−0.5	25.0%
Uy [mm]	3.5	3.7	5.7%	−0.6	−0.6	0.0%
Uz [mm]	−5.0	−5.7	14.0%	−4.7	−5.4	14.9%
N [kN]	−299.2	−298.9	−0.1%	−344.2	−343.9	−0.1%
Ty [kN]	−21.1	−20.6	−2.4%	−21.1	−20.6	−2.4%
Tx [kN]	−16.6	−16.1	−3.0%	−16.6	−16.1	−3.0%
Mx [kN m]	161.0	159.1	−1.2%	−7.9	−5.3	−32.9%
My [kN m]	−124.3	−123.0	−1.0%	8.4	5.6	−33.3%

Results for columns C2_

	Column C2
	Node 3			Node 4
	Model A	Model B	Error	Model A	Model B	Error
Ux [mm]	2.1	2.2	4.8%	0.2	0.2	0.0%
Uy [mm]	3.2	3.3	3.1%	−0.4	−0.5	25.0%
Uz [mm]	−6.5	−7.3	12.3%	−6.0	−6.9	15.0%
N [kN]	−532.2	−532.2	0.0%	−577.2	−577.2	0.0%
Ty [kN]	−23.6	−23.1	−2.1%	−23.6	−23.1	−2.1%
Tx [kN]	4.5	4.6	2.2%	4.5	4.6	2.2%
Mx [kNm]	167.0	165.3	−1.0%	−21.5	−19.2	−10.7%
My [kNm]	21.5	22.2	3.3%	−14.7	−14.5	−1.4%

Spring constants for a foundation with a rectangular base resting on elastic half-space [7], [8], [9]_

Motion	Spring constant		Reference
Vertical	(1) Kz=G1−vβzA {K_z} = {G \over {1 - v}}{\beta _z}\sqrt A		Barkan (1962)
Horizontal	(2) Kx=2G1+vA βx {K_x} = 2G\left( {1 + v} \right)\sqrt A\, {\beta _x}		Barkan (1962)
Rocking	(3) Kϕ=G1−vβϕ 8ab2 {K_\phi } = {G \over {1 - v}}{\beta _\phi }\;8a{b^2}		Gorbunov-Possadov (1961)

Material properties used in the analysis_

Model	A, B
Soil	Loose sand [17]
Es [MN/m2]	40 (Middle range value)
ν	0.3
ϕ	30 (Model A)
G [MN/m2]	15.385
L × B [m]	1.5 × 1.5
L × B [m]	2.0 × 2.0
Frame structure	Concrete C50/60
Ecm [MN/m2]	37,000
v	0.2

Results for beam B2_

	Node 1			Node 6			Node 7
	Model A	Model B	Error	Model A	Model B	Error	Model A	Model B	Error
Ux [mm]	2.2	2.3	4.5%	2.2	2.3	4.5%	2.2	2.3	4.5%
Uy [mm]	3.5	3.7	5.7%	3.5	3.7	5.7%	3.5	3.6	2.9%
Uz [mm]	−5.0	−5.7	14.0%	−17.0	−17.8	4.7%	−5.1	−5.8	13.7%
N [kN]	−30.8	−30.2	−1.9%	−30.8	−30.2	−1.9%	−30.8	−30.2	−1.9%
Ty [kN]	152.4	152.3	−0.1%	−4.0	−4.1	2.5%	−160.4	−160.5	0.1%
Tx [kN]	0.3	0.3	0.0%	0.3	0.3	0.0%	0.3	0.3	0.0%
Mx [kNm]	−161.5	−159.7	−1.1%	304.0	305.1	0.4%	−209.7	−209.2	−0.2%
My [kNm]	1.6	2.0	25.0%	0.0	0.0	0.0%	−1.7	−2.0	17.6%

Calculated spring constants used in model B_

In the corner footing	In the inner footing	Units
βx =βy = 0.956
βz = 2.113
βϕX = βϕY = 0.49
kx = ky = 57,374	kx = ky = 76,498	[kN/m]
kz = 69,668	kz = 92,891
kϕX = kϕY = 36,381	kϕX = kϕY = 86,235	[kN/m/rad]

Results for beam B1_

	Node 1			Node 5			Node 3
	Model A	Model B	Error	Model A	Model B	Error	Model A	Model B	Error
Ux [mm]	2.2	2.3	4.5%	2.10	2.20	4.8%	2.1	2.2	4.8%
Uy [mm]	3.5	3.7	5.7%	3.40	3.50	2.9%	3.2	3.3	3.1%
Uz [mm]	−5.0	−5.7	14.0%	−13.00	−13.90	6.9%	−6.5	−7.3	12.3%
N [kN]	−26.3	−25.7	−2.3%	−26.30	−25.70	−2.3%	−26.3	−25.7	−2.3%
Ty [kN]	131.7	131.6	−0.1%	2.50	2.40	−4.0%	−181.1	−181.2	0.1%
Tx [kN]	−0.4	−0.4	0.0%	−0.40	−0.40	0.0%	−0.4	−0.4	0.0%
Mx [kNm]	123.8	122.4	−1.1%	−228.60	−229.40	0.3%	409.7	409.3	−0.1%
My [kNm]	−1.8	−2.0	11.1%	−0.10	−0.10	0.0%	2.4	2.6	8.3%

Load applied to the structure_

Load type	Value
Self-weight of the concrete frame structure (SW)	24 kN/m3
Uniform distributed load (UDL)	20 kN/m
Concentrated force Px and Py	10 kN