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Effect of structure–ground interaction on shrinkage stresses in foundation reinforced concrete elements

Jacek Grosel

Jacek Grosel

Wroclaw University of Science and Technology, Faculty of Civil EngineeringWrocław, Poland
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Grosel, Jacek
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Wojciech Pakos

Wojciech Pakos

Wroclaw University of Science and Technology, Faculty of Civil EngineeringWrocław, Poland
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Pakos, Wojciech
  
04 giu 2025
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Categoria dell'articolo: Original Study
Pubblicato online: 04 giu 2025
Pagine: 121 - 133
Ricevuto: 11 ott 2024
Accettato: 13 apr 2025
DOI: https://doi.org/10.2478/sgem-2025-0013
Parole chiave
© 2025 Jacek Grosel et al., published by Sciendo
This work is licensed under the Creative Commons Attribution 4.0 International License.

Figure 1:

Cross section of reinforced concrete beam with reinforcing bar arrangement, dimensions in millimetres.
Cross section of reinforced concrete beam with reinforcing bar arrangement, dimensions in millimetres.

Figure 2:

The section under analysis, deformations of cross section, forces due to strain.
The section under analysis, deformations of cross section, forces due to strain.

Figure 3:

Stress S33 = Szz (kPa) in the concrete beam.
Stress S33 = Szz (kPa) in the concrete beam.

Figure 4:

Stresses S33 = Szz (kPa) in the cross section at the mid-span of a concrete beam.
Stresses S33 = Szz (kPa) in the cross section at the mid-span of a concrete beam.

Figure 5:

Stress S11 = Szz (kPa) in the reinforcement.
Stress S11 = Szz (kPa) in the reinforcement.

Figure 6:

Structure model diagrams adopted for numerical analyses.
Structure model diagrams adopted for numerical analyses.

Figure 7:

Stress S11 = σx (kPa) in the concrete slab founded on soil (Winkler model: kz = 50,000 kN/m3, kx = ky = 4000 kN/m3).
Stress S11 = σx (kPa) in the concrete slab founded on soil (Winkler model: kz = 50,000 kN/m3, kx = ky = 4000 kN/m3).

Figure 8:

Stress S22 = σy (kPa) in the concrete slab founded on soil (Winkler model: kz = 50,000 kN/m3, kx = ky = 4000 kN/m3).
Stress S22 = σy (kPa) in the concrete slab founded on soil (Winkler model: kz = 50,000 kN/m3, kx = ky = 4000 kN/m3).

Figure 9:

Stress S11 = σx (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.1, kz = 50,000 kN/m3.
Stress S11 = σx (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.1, kz = 50,000 kN/m3.

Figure 10:

Stress S22 = σy (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.1, kz = 50,000 kN/m3.
Stress S22 = σy (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.1, kz = 50,000 kN/m3.

Figure 11:

Stress S11 = σx (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.5, kz = 10,000 kN/m3.
Stress S11 = σx (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.5, kz = 10,000 kN/m3.

Figure 12:

Stress S22 = σy (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.5, kz = 10,000 kN/m3.
Stress S22 = σy (kPa) in the concrete slab resting on the lean concrete substructure: friction coefficient μ = 0.5, kz = 10,000 kN/m3.

Figure 13:

Maximum stress σx = S11 and σy = S22 depending on stiffness kz and friction coefficient μ.
Maximum stress σx = S11 and σy = S22 depending on stiffness kz and friction coefficient μ.

Figure 14:

Change of maximum stresses Δσx and Δσy in relation to stresses for kz =10,000 kN/m3.
Change of maximum stresses Δσx and Δσy in relation to stresses for kz =10,000 kN/m3.

Figure 15:

Change of maximum stresses Δσx and Δσy in relation to stresses for μ = 0.1.
Change of maximum stresses Δσx and Δσy in relation to stresses for μ = 0.1.

Figure 16:

Maximum stress σx depending on the friction coefficient μ and stiffness kz.
Maximum stress σx depending on the friction coefficient μ and stiffness kz.

Figure 17:

Maximum stress σy depending on the friction coefficient μ and stiffness kz.
Maximum stress σy depending on the friction coefficient μ and stiffness kz.

Maximum stresses S11 = σx and S22 = σy depending on stiffness kz and friction coefficient μ_

kz σx σy
(kN/m3) (MPa) (MPa)
μ (-) 0.1 0.5 0.1 0.5
10,000 4.761 7.107 3.113 6.028
50,000 4.977 7.204 3.452 6.543
100,000 5.023 7.221 3.522 6.616

Change of maximum stresses Δσx and Δσy in relation to stresses for kz =10,000 kN/m3_

kz Δσx Δσy
(kN/m3) (%) (%)
μ (-) 0.1 0.5 0.1 0.5
10,000 0.0% 0.0% 0.0% 0.0%
50,000 4.5% 1.4% 10.9% 8.5%
100,000 5.5% 1.6% 13.1% 9.8%

Results from analytical and numerical FEM solution_

Stresses (MPa)
Analytical solution σA Numerical solution (FEM) σF Relative error Δσ (%) Remarks
Concrete, the lower edge of the section 1.95 1.896 2.8% Tension
Concrete, the upper edge of the section −0.80 −0.748 6.5% Compression
Reinforcing steel bars −38.30 −38.38 0.2% Compression

Stresses S11 = σx and S22 = σy depending on stiffness kz and friction coefficient μ_

Numerical μ kz S11 = σx (MPa) S22 = σy (MPa)
model (-) (kN/m3) Max. Min. Max. Min.
Model ‘A’ 0.08 50,000 0.4198 -0.0199 0.1949 0.0228
Model ‘A’ 0.1 50,000 0.5191 -0.0245 0.2427 0.0282
Model ‘B’ 0.1 10,000 4.7610 0.2423 3.1130 0.2116
Model ‘B’ 0.5 10,000 7.1070 0.3046 6.0280 0.3023
Model ‘B’ 0.1 50,000 4.9770 0.3585 3.4520 0.3234
Model ‘B’ 0.5 50,000 7.2040 0.5170 6.5430 0.5246
Model ‘B’ 0.1 100,000 5.0230 0.4017 3.5220 0.3636
Model ‘B’ 0.5 100,000 7.2210 0.6327 6.6160 0.6270

Range of modulus of subgrade reaction kz based on [13]_

Soil kz (kN/m3)
Loose sand 4800–16,000
Medium dense sand 9600–80,000
Dense sand 64,000–128,000
Clayey medium dense sand 32,000–80,000
Silty medium dense sand 24,000–48,000
Clayey soil:
qa < 200 kPa 12,000–24,000
200 < qa < 800 kPa 24,000–48,000
qa > 800 kPa >48,000

Change of maximum stresses Δσx and Δσy in relation to stresses for μ = 0_1_

kz Δσx Δσy
(kN/m3) (%) (%)
μ (-) 0.1 0.5 0.1 0.5
10,000 0.0% 49.3% 0.0% 93.6%
50,000 0.0% 44.7% 0.0% 89.5%
100,000 0.0% 43.8% 0.0% 87.8%
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