Figure 1:
Scheme of a rectangular foundation.
Table 1:
Spring constants for a foundation with a rectangular base resting on elastic half-space [7], [8], [9].
| Motion | Spring constant | Reference | |
|---|---|---|---|
| Vertical |  | Barkan (1962) | |
| Horizontal |  | Barkan (1962) | |
| Rocking |  | Gorbunov-Possadov (1961) |
[i] where G – shear modulus of soil, v – Poisson’s ratio of soil, A– foundation area (A=2a2b), 2a– length of the foundation (along the axis of rotation for the case of rocking), 2b – width of the foundation (in the plane of rotation for the case of rocking), V, H, M – vertical, horizontal, and rocking forces, βz, βx, βϕ – nondimensional coefficients of spring constants for a rectangular foundation, and Kz, Kx, Kϕ – vertical, horizontal, and rocking spring constants.
Figure 2:
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].
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].
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).
Figure 6:
The analyzed 3D frame.
Figure 7:
Axonometric view of complex model A.
Figure 8:
Axonometric view of simple model B.
Figure 9:
Finite element mesh of complex model A.
Table 2:
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 |
Table 3:
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] |
Table 4:
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 |
Table 5:
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% |
Table 6:
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% |
Table 7:
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% |
Table 8:
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% |