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

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

Studia Geotechnica et Mechanica
Volume 46 (2024): Issue 4 (December 2024)
By: Wojciech Pakos and  Andrzej Helowicz  
Open Access
|Nov 2024

Figures & Tables

Figure 1:

Scheme of a rectangular foundation.
Scheme of a rectangular foundation.

Figure 2:

Coefficients βx, βz, and βϕ for a rectangular foundation base with regards to α=a/b.
Coefficients βx, βz, and βϕ for a rectangular foundation base with regards to α=a/b.

Figure 3:

Scheme of a foundation with a rectangular base according to Gorbunova-Possadov [12], [13].
Scheme of a foundation with a rectangular base according to Gorbunova-Possadov [12], [13].

Figure 4:

Graphs for determining the coefficients K1 (a) and K2 (b) used in equations (12) and (13). The graphs were prepared by the authors based on works [12] and [13].
Graphs for determining the coefficients K1 (a) and K2 (b) used in equations (12) and (13). The graphs were prepared by the authors based on works [12] and [13].

Figure 5:

Notations for the equation for the spring constant in the case of rocking motion: (a) for α=a/b=(1÷0.1); (b) for α=a/b=(1÷10).
Notations for the equation for the spring constant in the case of rocking motion: (a) for α=a/b=(1÷0.1); (b) for α=a/b=(1÷10).

Figure 6:

The analyzed 3D frame.
The analyzed 3D frame.

Figure 7:

Axonometric view of complex model A.
Axonometric view of complex model A.

Figure 8:

Axonometric view of simple model B.
Axonometric view of simple model B.

Figure 9:

Finite element mesh of complex model A.
Finite element mesh of complex model A.

Results for columns C1_

Column C1
Node 1Node 2
Model AModel BErrorModel AModel BError
Ux [mm]2.22.34.5%−0.4−0.525.0%
Uy [mm]3.53.75.7%−0.6−0.60.0%
Uz [mm]−5.0−5.714.0%−4.7−5.414.9%
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]161.0159.1−1.2%−7.9−5.3−32.9%
My [kN m]−124.3−123.0−1.0%8.45.6−33.3%

Results for columns C2_

Column C2
Node 3Node 4
Model AModel BErrorModel AModel BError
Ux [mm]2.12.24.8%0.20.20.0%
Uy [mm]3.23.33.1%−0.4−0.525.0%
Uz [mm]−6.5−7.312.3%−6.0−6.915.0%
N [kN]−532.2−532.20.0%−577.2−577.20.0%
Ty [kN]−23.6−23.1−2.1%−23.6−23.1−2.1%
Tx [kN]4.54.62.2%4.54.62.2%
Mx [kNm]167.0165.3−1.0%−21.5−19.2−10.7%
My [kNm]21.522.23.3%−14.7−14.5−1.4%

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

MotionSpring 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_

ModelA, B
SoilLoose sand [17]
Es [MN/m2]40 (Middle range value)
ν0.3
ϕ30 (Model A)
G [MN/m2]15.385
L × B [m]1.5 × 1.5
L × B [m]2.0 × 2.0
Frame structureConcrete C50/60
Ecm [MN/m2]37,000
v0.2

Results for beam B2_

Node 1Node 6Node 7
Model AModel BErrorModel AModel BErrorModel AModel BError
Ux [mm]2.22.34.5%2.22.34.5%2.22.34.5%
Uy [mm]3.53.75.7%3.53.75.7%3.53.62.9%
Uz [mm]−5.0−5.714.0%−17.0−17.84.7%−5.1−5.813.7%
N [kN]−30.8−30.2−1.9%−30.8−30.2−1.9%−30.8−30.2−1.9%
Ty [kN]152.4152.3−0.1%−4.0−4.12.5%−160.4−160.50.1%
Tx [kN]0.30.30.0%0.30.30.0%0.30.30.0%
Mx [kNm]−161.5−159.7−1.1%304.0305.10.4%−209.7−209.2−0.2%
My [kNm]1.62.025.0%0.00.00.0%−1.7−2.017.6%

Calculated spring constants used in model B_

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

Results for beam B1_

Node 1Node 5Node 3
Model AModel BErrorModel AModel BErrorModel AModel BError
Ux [mm]2.22.34.5%2.102.204.8%2.12.24.8%
Uy [mm]3.53.75.7%3.403.502.9%3.23.33.1%
Uz [mm]−5.0−5.714.0%−13.00−13.906.9%−6.5−7.312.3%
N [kN]−26.3−25.7−2.3%−26.30−25.70−2.3%−26.3−25.7−2.3%
Ty [kN]131.7131.6−0.1%2.502.40−4.0%−181.1−181.20.1%
Tx [kN]−0.4−0.40.0%−0.40−0.400.0%−0.4−0.40.0%
Mx [kNm]123.8122.4−1.1%−228.60−229.400.3%409.7409.3−0.1%
My [kNm]−1.8−2.011.1%−0.10−0.100.0%2.42.68.3%

Load applied to the structure_

Load typeValue
Self-weight of the concrete frame structure (SW)24 kN/m3
Uniform distributed load (UDL)20 kN/m
Concentrated force Px and Py10 kN
DOI: https://doi.org/10.2478/sgem-2024-0022 | Journal eISSN: 2083-831X | Journal ISSN: 0137-6365
Journal RSS Feed
Language: English
Page range: 302 - 314
Submitted on: May 3, 2024
|
Accepted on: Jul 25, 2024
|
Published on: Nov 13, 2024
Published by: Wroclaw University of Science and Technology
In partnership with: Paradigm Publishing Services
Publication frequency: 4 issues per year
Keywords:
design,
frame structure,
reinforced concrete,
foundation stiffness,
elastic subsoil
Related subjects:
Geosciences,
Geosciences, other,
Materials sciences,
Composites,
Porous materials,
Physics,
Mechanics and fluid dynamics

© 2024 Wojciech Pakos, Andrzej Helowicz, published by Wroclaw University of Science and Technology
This work is licensed under the Creative Commons Attribution 4.0 License.

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