The effect of varying soil stiffness on a continuous "rigid" footing under concentrated load

The article provides follow-up insight and is linked to the current article submitted on our website. We highly recommend starting with the article that is available in the reference below:

Please refer to the article  The effect of varying soil stiffness on a continuous footing under concentrated load.    

1) Introduction of the theme

The study investigates the structural response of a continuous footing under concentrated loads resting on an elastic foundation. The analysis aims to verify the interaction between beam bending stiffness (foundation flexural rigidity) and subgrade stiffness (soil modulus), which together govern the deformation profile, bending moments, and shear force distribution along the footing.

Let us discuss the combinations:

1. High beam bending stiffness + Low soil stiffness 
2. High beam bending stiffness + High soil stiffness 

AI-generated illustrative photo to demonstrate the use-case.

01) Continues footing strip with multiple columns (use-case)

Material models

Material behaviour and properties have been adopted from EN 1992-1-1 [1]. The design properties of concrete grade C30/37 and the corresponding reinforcement B500B with hardening have been specified (Fig.2).

02) Material models

2) Linear beam model  with code-checks according to EN 1992-1-1

The most commonly used solution among engineers is a linear beam model with code-compliance checks in accordance with applicable standards. The setup of the testing model remains consistent across all levels of model complexity. It represents a column with a square cross-section measuring 500 x 500 mm and a length of 1000 mm (here represented by a 100 mm thick bearing plate), a footing strip with a unit width of 1000 mm, and a length of 6000 mm. For the current verification, a height of 1000 mm has been utilized. 

The bottom face of the footing strip is supported by compression-only springs with either low soil stiffness of 16000 kN/m³ or high soil stiffness of 128000 kN/m³. Symmetric boundary conditions constrain the left and right ends of the footing strip to emulate repetitive continuity of footing and columns under the same pattern. 

It is essential to note that all models are design models. For simulation and verification, the partial factors for materials have been applied.

3) Dimensions of the analyzed model

Linear beam model - Low-Stiffness-Soil (LSS)

Once the simulation is conducted on the beam model, the standard code checks can be employed. The designed reinforcement complies with the minimal detailing requirements specified in EN 1992-1-1 [1]. A minimal reinforcement ratio is applied to both longitudinal bars and stirrups. The simulation is executed with a modulus of elasticity of 10 GPa, corresponding to the secant modulus of the designated concrete material. Due to the structure's hyperstatic nature, the modulus influences the redistribution of internal forces. 

The critical point is situated within the zone of maximum shear. To achieve a utilization ratio below 100%, only -1350 kN of axial force can be applied. At this point, the shear force is 553 kN and a corresponding bending moment of 369 kNm.

4) Linear beam model - ultimate load for passing ULS checks - Max Vz / relevant My 

EN 1992-1-1 [1] permits the use of a range of theta angles from 21.5 to 45 degrees. When an angle of 21.5 degrees is specified, the maximum shear force is primarily recalculated for the longitudinal bars and causes failure across the depth of the footing on all horizontal bars in the tensile zone. When a 40-degree angle is used as the default, the failure mode changes. The failure shifts from being primarily on the horizontal bars to failure involving the stirrups.

5) Linear beam model - code-check for low-stiffness soil - angle 21.5°

6) Linear beam model - code-check for low-stiffness soil - angle 40°

Linear beam model - High-Stiffness-Soil (HSS)

The high-stiffness soil in this scenario, dense sand with a subgrade modulus of 128000 kN/m³, doesn't significantly alter the behavior of the structure. The internal forces, such as a 350 kNm bending moment and a 553 kN shear force, are caused by the submitted external force on the plate, which is -1405 kN, determining the ultimate bearing capacity of the footing.

7) Linear beam model - ultimate load for passing ULS checks - Max Vz / relevant My

In this case, the governing failure mode is the one of the stirrups. The longitudinal bars are exhibiting plasticity, but there is still some redundant capacity in the bars. On the other hand, the stirrups are fully utilized at a 40-degree angle, which clearly indicates the failure mode in shear under the current HSS setup.

8) Linear beam model - code-check for high-stiffness soils - angle 21,5°

9) Linear beam model - code-check for high-stiffness soils - angle 40°

3) Strut and Tie method

Assembling the Strut&Tie is quite complex. Some challenges connected with establishing Strut&Tie models:

• Thickness of the struts and ties 
• Clear unique load path 
• Linear models, redistribution dismissed
• Conservative solutions
• Ambiguous strut and tie analogies, no one clear solution
• Hyperstatic structures and stiffness sensitivity analysis of section stiffnesses

Topology optimization

Topology optimization is applied to reinforcement design of concrete discontinuity regions as an automated alternative to strut-and-tie models. The method minimizes strain energy (maximizes stiffness) using iterative linear FEM with density-based material redistribution under volume constraints. It identifies tension/compression paths for multiple load cases and supports efficient detailing, reducing engineering effort and reinforcement demand.

The referenced article here: Topology optimization

S&T - LSS - 40 % of original volume

The topology optimization has successfully established the foundational template for the forthcoming S&T model. The maximum applied load of -2015 kN induces critical stress within the tensile zone (indicated in blue). According to the existing reinforcement configuration, the bottom layers of rebars within the blue zone have a capacity of 937 kN, while the stirrups assembly accommodates 221 kN (6 legs per row).

The boundary conditions for the Strut and Tie model have been respected and assigned according to the topology optimization model.

10) S&T for low-stiffness soils

An essential failure mechanism is observed within the horizontal reinforcements. This issue arises from foundation bending. Recalculations of the force transmitted from the tensile diagonal to the vertical stirrup layer revealed 72% utilization of the stirrups. The failure mode is occurring within the bottom horizontal layer of reinforcement bars.

11) S&T for low-stiffness soils - forces and tension

S&T - HSS - 40 % of original volume

The same model, with different support stiffness, yields a different optimized shape for 40% of the original volume. Instead of two ties, one continuous tie is created, merging more struts. The model has a higher bearing capacity. The maximum applied load of -2480 kN induced failure of the stirrups first, leaving some residual capacity for the bending reinforcement, which reached 96% of ultimate capacity. 

12) S&T for high-stiffness soils (40 %)

13) S&T for low-stiffness soils - forces and tension (40 %)

S&T - LSS - 60 % of original volume

The topology optimization with 60% effective volume yields a slightly different shape for the Strut-and-Tie model than the aforementioned model. The model has a higher bearing capacity than the 40% effective volume model. The maximum applied load of -2050 kN induced failure of the horizontal reinforcement due to bending. It has highlighted the same failure mode observed in 40 % of the effective volume.

14) S&T for low-stiffness soils - model and forces (60 %)

S&T - HSS - 60 % of original volume

The higher stiffness of the soil leads to lower deflections and a reduced level of bending. It is closely related to a higher bending capacity. The maximum applied load the model can withstand is -3000 kN. The failure mode is identified on the longitudinal bars. The change in the objective volume constraint and the different S&T model relate the failure mode to horizontal bars rather than shear stirrups, which are used for 40% of the volume. The effective volume and the complex geometry of the strut-and-tie system have led to failure modes occurring at different locations. This creates significant ambiguity in the selection and assembly of S&T.

15) S&T for high-stiffness soils - model and forces (60 %)

4) Nonlinear solution - CSFM (plane stress)

16) 2D model + reinforcement bars layout

2D CSFM - Low-Stiffness-Soil (LSS)

The stress redistribution in the concrete with dispersed continuous stress fields and activated longitudinal and shear reinforcement overcame a significant portion of the applied load -9469 kN.

• The model shows a fairly uniform contact stress range, from 1720 kPa in the middle to 1460 kPa at the end.
• The structure is displaced and deformed about 108 mm downwards.
• The failure modes occurred simultaneously during concrete crushing in the area where the load transmitter—bearing plate— is wedged into the concrete, with peaks appearing at the intersection of the plate edge and the top surface of the footing. The second failure mode aligns with the bending reinforcement longitudinal bars beneath the load application in the tensile area of the footing.

17) Maximal applied force, contact stress, and failure modes

The detailed results show an extreme compressive flow of forces outlining the "triangle" shape. Compression softening indicates areas with high principal tensile strains, where the concrete begins to soften and lose integrity under compression due to cracking and low reinforcement levels. The softening for current static setup indicates two critical areas:

• First area in the extreme bending zone and tensile zone, reinforced by the vertical contact pressure from soil.
• The second area exhibits significant shear force and increased use of stirrups in tension.

18) Principal stress in compression, compressive plastic strain, compression softening effect

The significant region of highly utilized longitudinal bars in the bending area at 465 MPa indicates that those bars failed in tension. The stirrups are also heavily utilized and are in a plastic state due to high shear and transverse tensile strains. 

19) Nonlinear deflections, stress in reinforcement

2D CSFM - High-Stiffness-Soil (HSS)

The model with higher soil stiffness transfers more load. The ultimate value of -10645 kN indicates the maximal utilization of longitudinal horizontal bars and also concrete crushing in the area of wedging the bearing plate into the mass of concrete.

• The contact stress reaches approximately 2710 kPa in the middle, with traction extending to the end of the model at 1000 kPa. Contact stress accumulation and a higher contact stress gradient are perceptible in those soils.
• The maximum displacement reaches 20 mm, which is approximately five times lower than in LSS.
• The failure modes are found simultaneously in concrete (local crushing) and in longitudinal tensile reinforcements, followed by secondary failure modes for the stirrups, as the stress level reaches a discrete portion of plasticity and axial stress. Soil pressure produces a degree of compression softening in the tensile area of the footing, reducing the concrete's bearing capacity in the stirrups. Soil pressure produces a degree of compression softening in the tensile area of the footing, reducing the concrete's bearing capacity in compression. However, the effects of compression softening are also evident in the high-shear zone, where extensive principal tensile strains dominate and compromise the integrity of concrete.

20) Maximal applied force, contact stress, and failure modes

21 Principal stress in compression, compressive plastic strain, compression softening effect

22) Nonlinear deflections, stress in reinforcement

5) Nonlinear solution - CSFM (Full 3D solution)

23) 3D model + reinforcement bars layout

3D CSFM - Low-Stiffness-Soil (LSS)

The volume model demonstrated how spatial compressive stress redistributes within the concrete and uncovered additional failure modes that were not accessible during 2D plane stress simulations. The maximum applied force was -7138 kN, about 25% less than the bearing capacity indicated by the 2D analysis. The main failure occurred near the bearing plate, where wedging led to local stress concentration and quickly triggered the 5% plastic strain threshold on concrete, which served as the stopping criterion.

24) Maximal applied force, failure modes, and transverse stress distribution

The declared critical area is indicated locally on the edge of the bearing plate. The model exhibited the triaxial stress state due to confinement. 

25) Minimal principal stress Sigma 3, confinement effect - ratio between triaxial vs uniaxial stress

The tensile stress in the longitudinal reinforcement reached its capacity threshold, indicating that the bearing capacity of the reinforcement was fully utilized. The plastic strain field exhibits a pronounced gradient, which can be attributed to the fact that even a small load increment causes the strain to exceed the 5% plastic strain threshold.

Although the current strain level is approximately 181e-4, the nonlinear material response leads to a rapid increase in plastic strain once the yielding region is reached.

26) Compressive plastic strain and stress in reinforcements

A limitation of the plane-stress formulation is its inability to capture stresses developing in the stirrups in the transverse direction. This behavior becomes visible only in the full 3D numerical model, where the spatial stress state can develop freely.

The 3D analysis revealed localized stresses in the transverse reinforcement beneath the bearing plate. These stresses originate from high localized transverse strains induced by the concentrated load transfer. Because the plane-stress model neglects the out-of-plane stress component, this mechanism was not captured, leading to an underestimation of the transverse reinforcement demand.

At first glance, the effect did not appear critical for the global behavior of the system; however, the 3D analysis demonstrates that localized stress concentrations may develop in this region and should be considered when assessing the detailed stress distribution in the reinforcement.

27) Detailed detection of critical stress on the longitudinal bars and stirrups 

28) Nonlinear deflections

3D CSFM - High-Stiffness-Soil (HSS)

The force resisted by the footing strip reached −7838 kN, which is approximately 9% higher than the capacity predicted by the LSS approach. The governing failure mode is characterized by concrete crushing, secondary tensile rupture of the longitudinal reinforcement, and yielding of the horizontal legs of the stirrups subjected to tension.

29) Maximal applied force, failure modes, and transverse stress distribution

30) Minimal principal stress Sigma 3, confinement effect - ratio between triaxial vs uniaxial stress

31) Compressive plastic strain in concrete and stress in reinforcements

The ultimate strength of longitudinal bars and stirrups on horizontal and vertical legs was reached. 

32) Detailed detection of critical stress on the longitudinal bars and stirrups 

33) Nonlinear deflections 

6) Summary and key takeaways 

This study investigates the structural behavior of a continuous reinforced concrete footing subjected to concentrated column load, considering the interaction between foundation bending stiffness and soil subgrade stiffness. The analysis compares several modeling approaches commonly used in structural engineering practice:

• Linear beam model with Eurocode EN 1992-1-1 code checks
• Strut-and-Tie (S&T) method supported by topology optimization
• Nonlinear finite-element modeling using CSFM in both 2D plane-stress and full 3D formulations

Two soil conditions were evaluated:

• Low-stiffness soil (LSS) — subgrade modulus 16000 kN/m³
• High-stiffness soil (HSS) — subgrade modulus 128000 kN/m³

The study highlights how soil stiffness significantly affects the deformation profile, stress redistribution, and governing failure mechanisms of the footing system.

The linear beam approach predicted an ultimate capacity of approximately 1.35–1.40 MN, governed by shear and reinforcement utilization depending on the assumed strut inclination angle in Eurocode shear design.

The Strut-and-Tie models, derived with topology optimization, predicted higher capacities ranging approximately between 2.0 MN and 3.0 MN, with failure modes depending strongly on the selected topology and effective material volume. The results illustrate the ambiguity and sensitivity of S&T models in hyperstatic systems.

Strut-and-Tie models derived from linear analysis tend to be highly conservative and are consistent with the lower-bound theorem described in numerous design codes and literature sources. However, when the Strut-and-Tie model is developed based on nonlinear analysis results, the justification for constructing such a model becomes questionable from an engineering standpoint, since the nonlinear solution itself already provides a direct representation of the structural behavior.

In cases where the Strut-and-Tie system is derived from nonlinear stress fields, the predicted bearing capacity is approximately consistent with the capacity obtained from nonlinear simulations based on smeared-crack approaches. This indicates that the nonlinear model inherently captures the same load-transfer mechanisms that the Strut-and-Tie analogy attempts to idealize.

Nonlinear analysis using the Compatible Stress Field Method (CSFM) demonstrated significantly higher load-bearing capacity due to stress redistribution and activation of multiple reinforcement mechanisms:

• 2D CSFM results
  • LSS: approximately 9.5 MN
  • HSS: approximately 10.6 MN

However, full 3D modeling revealed additional mechanisms that are not captured by plane-stress analysis. The predicted ultimate load was reduced primarily because the adopted material model (Mohr–Coulomb plasticity with a zero internal friction angle) limits the confinement-related strength enhancement. In addition, the 3D formulation exposed failure mechanisms associated with the out-of-plane response of the reinforcement, particularly the activation and yielding of the transverse stirrup legs under shear.

• 3D CSFM results
  • LSS: 7.14 MN
  • HSS: 7.84 MN

The 3D analysis also revealed transverse stresses in the stirrups and confinement effects beneath the bearing plate, demonstrating the importance of spatial stress states in localized load-transfer regions.

The governing failure mechanisms included combinations of:

• Concrete crushing beneath the load introduction zone
• Tensile rupture of longitudinal reinforcement
• Plastic yielding of stirrups due to shear and transverse tensile stresses

Key Takeaways for Structural Engineers

1. Soil stiffness significantly influences structural response

Higher subgrade stiffness reduces deflections and bending demand but increases localized contact stresses and stress gradients near the load introduction region.

2. Linear beam models can significantly underestimate structural capacity

Code-based beam analysis predicted capacities 5–7 times lower than those from nonlinear simulations, due to its inability to capture stress redistribution in reinforced concrete and limited theory for current details.

3. Strut-and-Tie models depend strongly on engineering assumptions

Topology optimization can assist in generating S&T layouts, but the resulting capacity and failure mode are highly sensitive to geometry and assumed load paths, making interpretation ambiguous.

4. Nonlinear CSFM captures redistribution mechanisms

The nonlinear CSFM approach activates multiple reinforcement mechanisms simultaneously, providing a more realistic representation of reinforced concrete behavior under concentrated loads.

5. 2D plane-stress models may overestimate capacity

While useful for engineering design, 2D models neglect out-of-plane stresses, leading to an overestimation of capacity when localized three-dimensional stress states develop.

6. Full 3D modeling reveals additional critical mechanisms

The 3D model identified transverse stirrup stresses and confinement effects beneath the bearing plate that are not captured in plane-stress analysis.

7. Failure mechanisms are often combined

The governing failure mode in concentrated footing loads typically involves interaction of concrete crushing, longitudinal reinforcement rupture, and stirrup yielding, rather than a single isolated mechanism.

34)Results summary

[1] EN 1992-1-1:2004+A1:2014 – Eurocode 2: Design of concrete structures – Part 1-1: General rules and rules for buildings.
European Committee for Standardization (CEN), Brussels, 2014

[2] IDEA StatiCa, “Theoretical background for IDEA StatiCa Detail – Structural design of concrete discontinuities,” IDEA StatiCa Support Center. [Online]. Available: https://www.ideastatica.com/support-center/theoretical-background-for-idea-statica-detail 

[3] IDEA StatiCa, “IDEA StatiCa Detail – Structural design of concrete 3D discontinuities,” IDEA StatiCa Support Center. [Online]. Available: https://www.ideastatica.com/support-center/idea-statica-detail-structural-design-of-concrete-3d-discontinuities