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

Beam theory is overly conservative for continuous footings under concentrated column loads. Both nonlinear models show that soil stiffness governs load transfer and failure mechanisms, but:

• CSFM delivers a code-consistent, conservative, and practically usable prediction of capacity and failure modes.
• CDP predicts higher ultimate loads due to damage, dilation, and geometric nonlinearity, making it better suited for research, not routine design.

Bottom line:
CSFM captures the real mechanics of footing–soil interaction with the right level of conservatism; CDP confirms the physics but exceeds what is defensible for design.

This study rigorously examines the structural performance of a continuous footing supporting multiple columns under varying soil and foundation stiffness parameters. The primary aim is to elucidate the mutual interaction between columns and the underlying soil, and to evaluate how this interplay influences load distribution and the overall structural behavior of the footing. Both low-stiffness (LSS) and high-stiffness (HSS) soil conditions are systematically analyzed to determine their impact on displacement, stress distribution, and load transfer mechanisms, particularly in scenarios involving concentrated column loads.

The analysis utilizes the Compatible Stress Field Method (CSFM) in three dimensions. The outcomes derived from CSFM are meticulously validated against simulations conducted using the Concrete Damage Plasticity (CDP) model as well as traditional verification methodologies, ensuring a high degree of reliability and precision in the 3D predictions.

The results of this investigation offer an enhanced comprehension of footing-soil-structure interaction, identify limitations inherent in conventional design assumptions, and underscore the efficacy and robustness of the CSFM for designing and verifying continuous footings under localized loading and variable soil conditions. This research contributes to advancing foundation design methodologies and provides valuable insights for developing more resilient structural solutions across diverse geotechnical scenarios.

1) Introduction of the theme

The study investigates the structural response of continuous footings 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.

The analytical model follows the Euler–Bernoulli beam theory on a Winkler-type foundation, assuming an infinitely long beam subjected to a single concentrated load. This approach allows a direct comparison of deformation shapes and internal force gradients for different stiffness ratios between the foundation and the supporting soil.

Let us discuss the four possible combinations:

1. Low beam bending stiffness + Low soil stiffness 
2. High beam bending stiffness + Low soil stiffness (this verification article)
3. Low beam bending stiffness + High soil stiffness 
4. High beam bending stiffness + High soil stiffness (this verification article)

For the purpose of this verification, continuous footings with high bending stiffness were chosen for a study on numerical models.

Fig. 1 shows the four aforementioned combinations of footing systems.  

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) Analytical solution - infinite beam on elastic foundation

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

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

The most frequently employed solution adopted by structural engineers for the current model is a beam model integrated with code compliance checks in accordance with applicable standards. The setup of the testing model remains consistent across all levels of model complexity represents a column with a square cross-section measuring 500 x 500 mm and a length of 1000 mm (here represented by a thick bearing plate), a footing strip with a unit width of 1000 mm, and a length of 6000 mm. The height of the footing strip is a variable parameter. For the current verification, a height of 1000 mm is utilized. 

The distribution of forces is considered at an angle of 60°. This means that the force acting across a width of 500 mm is distributed across the beam axis over a width of 0.5 + (2 × 0.25) m = 1 m.

The bottom face of the footing strip is supported by compression-only springs with either low soil stiffness of 16000 kN/m³ or the high soil stiffness of 128000 kN/m³. Symmetry boundary conditions constrain the left and right ends of the footing strip. 

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

3) Dimensions and analytical 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 adheres to the minimal detailing requirements specified by EN 1992-1-1 [1]. A minimal reinforcement ratio is applied to both longitudinal bars and stirrups. The simulation is executed using a modulus of elasticity of 10 GPa, representing the secant modulus of the designated concrete material. Due to the hyperstatic nature of the structure, the modulus influences the redistribution of internal forces. 

The bending moment directly beneath the column reached the ultimate value of 950.8 kNm under an axial force in the column of -2520 kN. 

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

The second critical point is situated within the zone of maximum shear. To achieve a utilization ratio of less than 100%, only 1,350 kN of axial force can be applied. At this point, we can see the shear force of -553,3 kN and a corresponding bending moment of 369,6 kNm.

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

The most critical location on the structure is directly next to the column, and the failure mode involves the shear capacity and the tension in the longitudinal reinforcement bars. The concrete in compression is not critical for this case.  In the interaction code check, the theta angle for the concrete strut was adjusted to 40 degrees. The Eurocode permits the adjustment of the strut angle within the range of 21.5 to 45 degrees.

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

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 ultimate resistance in the column of -551,4 kN from the 2730 kN axial force is almost the same as that of low-stiffness soil.  

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

Again, the extreme shear force has been displaced proximally to the column area and has reached a magnitude of -553 kN, with the corresponding moment being 350.0 kNm. The axial force in this scenarion is 1405 kN.

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

In the interaction code check, the theta angle for the concrete strut was again adjusted to 40 degrees. The critical mode of failure involves the shear and tension of the longitudinal reinforcement bars. 

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

4) Nonlinear solution - CSFM (plane stress)

Assumptions and model assembly

The theory employed in the nonlinear solution is called CSFM (Compatible Stress Field Method) and is traced in the theoretical background[2].

Assumptions and attributes of the model: 

• Plane stress model. 
• Compression-only line supports (low/high stiffness).
• Symmetry constraints are positioned on the left and right edges of the footing strip.
• A thick plate 100 mm on the top of the column to stem local stress concentration underneath the point force load.
• All the material properties for concrete C30/37 and reinforcement bars B500B are engaged as design values with partial factors according to EN 1992-1-1 [1]. 
• Mesh factor 1 - a minimum of four elements over the shortest edge.

10) 2D model + reinforcement bars layout

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

The maximum applied force capable of addressing the failure modes has reached -9469 kN. The vertical force has resulted in a contact stress of 0,172 MPa in the middle and 0.146 MPa at the ends. The displacement copies the shape of the stresses and reaches a maximum of 0,11 m.

The failure modes in concrete occurred in compression at the interface between the column edge and the face in contact with the footing. 

The failure modes in longitudinal reinforcement can be seen at the bottom of the footing.

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

12) Principal stress in compression, compression plastic strain, stress in reinforcements

The stirrups near the bearing plate exhibit a maximum stress of 443 MPa, indicating that this failure mode must be considered. 

13) Nonlinear deflections, stress in stirrups

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

The maximum load at which all governing failure mechanisms can still be resisted is –10645 kN. The corresponding vertical reaction induces a contact stress of 2.71 MPa at the footing–soil interface in the middle and 0,996 MPa at the end. The displacement copies the shape of the stresses and reaches a maximum of 0,02 m. 

The dominant failure mechanism is compressive crushing at the interface between the column edge and the loaded face of the footing. Simultaneously, tensile rupture of the bottom-layer longitudinal reinforcement and stirrups next to the bearing plate occurs.

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

15) Principal stress in compression, compression plastic strain, stress in reinforcements

The nonlinear deflections demonstrate little smaller displacements under higher loads compared to the LSS variants. Stress is predominantly concentrated beneath the column area. Also, the model exhibits evidence of local softening on the bottom face of the footing strip due to high tensile stress in reinforcement bars.

16) Nonlinear deflections, stress in stirrups, and localized compression softening

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

The theory used in the nonlinear solution is called 3D CSFM and is outlined in the theoretical background [3]. All the presumptions for the designed calculation procedure are explained in detail there.

Assumptions and attributes of the model: 

• 3D solution - volume elements.
• Mohr-Coulomb plasticity theory - zero angle of internal friction for concrete behaviour.
• Compression-only surface supports (low/high stiffness).
• Symmetry constraints are positioned on the left and right edges of the footing strip.
• A thick plate 100 mm on the top of the column to stem local stress concentration underneath the point force load.
• Bond model and tension stiffening are considered.
• Stress triaxiality and confinement effect.
• Compression softening is not a part of the implemented solution.
• Mesh factor 1 - recommended computational settings.

17) 3D model + reinforcement bars layout

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

The maximum axial force designated in the model reached -7138 kN due to failure modes involving tensile rupture of the longitudinal reinforcement at the bottom  under the bearing plate. Transverse compression forces are restrained by the stirrups, which in the column zone are utilized during yielding and contribute to additional failure mode of the horizontal stirrup legs caused by transverse tensile stress evolutions that cannot be captured in the plane stress solution. Over-compression and crushing of the concrete occur at the interface area between the bearing plate and footing. The confinement effect is localized in this region, based on the reinforcement effect and the stiffness of the footing strip. The failure mechanism involves concrete crushing, tensile rupture of the longitudinal reinforcement, and the horizontal legs of the stirrups in tension.

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

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

20) Compressive plastic strain and stress in reinforcements

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

22) Nonlinear deflections

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

The force absorbed by the footing strip reached -7838 kN, which is approximately the same as bearing capacity in LSS. The failure mode involves concrete crushing, tensile rupture of the longitudinal reinforcement, and the horizontal legs of the stirrups in tension.

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

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

25) Compressive plastic strain in concrete and stress in reinforcements

The maximum shear stress on the closed stirrups has reached a value of 460 MPa, which means that shear failure must also be taken into account. 

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

27) Nonlinear deflections 

6) Summary and key takeaways 

This verification study presents a thorough comparative analysis of analytical beam solutions, linear code-based models, and sophisticated nonlinear simulations (CSFM in 2D and 3D). The findings consistently illustrate the critical interaction between beam stiffness and soil stiffness in determining the structural behavior of continuous footings subjected to concentrated loading. At all modeling levels, a distinct set of trends, engineering insights, and actionable design recommendations is identified, offering valuable guidance for practical applications.

Overview of results:

The findings indicate that the CSFM method is a solution in cases, where conventional approach is too conservative, but still aligned with code regulations. In addition, CSFM supports material nonlinearity.
While standard methods tend to yield overly conservative outcomes, this can be attributed to the use of an inappropriate approach for analyzing areas subjected to concentrated loads, which are likely discontinuity regions where beam solution assumptions do not apply and should be used rather Strut and Tie method.

With higher beam stiffness, we can observe smaller influence on the shape of the deformation and the load capacity. The soil stiffness has bigger influence mostly on the deformation - smaller stiffness means higher deformation. But due  to a rigid form of the footing, the shape of the deformation is almost constant.
On the other hand, we can see similar load capacity with different soil stiffness.

The CSFM method, particularly the 3D version, provides a viable alternative to the beam theory models integrated into the codes. Notably, its advantages surpass those of conventional solutions.

28)Results summary

29)Graph representation of results splitted for LSS and HSS

Key Technical Findings

Linear Beam Model (EN 1992-1-1 checks)

• The high stiffness of the subgrade does not increase the load-bearing capacity of the model. The subgrade modulus of 128000 kN/m³ in comparison with 16000 kN/m³, resulting in only 8 % increase in the magnitude of the applied force.
• Failure modes occur in the region directly next to the concrete column, where the shear capacity and the tension in the longitudinal reinforcement bars are critical.  

2D CSFM Solution

• The model accurately predicts identical failure modes as observed in the beam solution. Furthermore, the bearing capacity has been substantially enhanced for both LSS and HSS in comparison to the beam solution. This finding leads to the conclusion that beam theory is notably conservative when compared to a materially nonlinear solution using the 2D CSFM methodology.
• The concentrated load region is identified as the discontinuity region; therefore, beam theory is not valid for this solution in this case due to its overly conservative approach.

3D CSFM Solution

• Captures confinement, triaxial stress effects, and transverse reinforcement involvement—none of which are accessible in 2D.
• Failure modes are aligned with the two-dimensional plane stress solution. An additional failure mode arises due to the behavior in the transverse direction—stirrups are loaded up to the yield point, but this loading is limited to the horizontal bottom branches.
• Confirms that punching shear is not necessarily the governing mode even at high soil stiffness, provided adequate reinforcement is present.

Engineering Wisdom from the Study

• Soil stiffness is as critical as reinforcement detailing. Even heavily reinforced footings may fail prematurely due to soil-induced stress localization.
• Linear beam models are useful for predesign but insufficient for capturing true behaviour when compression softening, uplift, or confinement occur.
• Nonlinear models provide essential insight into failure mechanisms, especially when designing near capacity or verifying critical details.
• 3D effects matter. Transverse reinforcement and confinement significantly influence strength, ductility, and load redistribution.
• Punching shear is not automatically governing. Many footings reach failure due to combined bending and tension in longitudinal bars—even under high soil stiffness.
• Good modelling practice requires matching model complexity to the design question. Linear → 2D CSFM → 3D CSFM → CDP form a rational hierarchy of verification.

Positive Feedback and Recommendations for IDEA StatiCa Users

 2D CSFM Solution

• Excellent for capturing global deformation shape, contact stress redistribution, and reinforcement involvement.
• Provides clear and physically meaningful failure modes.
• Ideal for quick yet accurate verification of simple strip footing or wall–base scenarios.
• Highly efficient for comparing soil stiffness variants due to its low computational cost.

3D CSFM Solution

• Very strong at representing triaxial stress, confinement, transverse reinforcement action, and local crushing.
• Enables engineers to understand true spatial behaviour of complex details such as column–footing connections.
• Provides a realistic assessment of the contribution of stirrups and reinforcement legs in all directions.
• Particularly valuable where punching, shear, or confinement effects may interact.

 Final Recommendations for Practice

• Use linear beam models for early-stage sizing and code-check verification.
• Use 2D CSFM when uplift, nonlinear tension behaviour, or soil–structure interaction effects are critical.
• Use 3D CSFM for evaluating complex stress fields, confinement, or the influence of transverse reinforcement.
• Always evaluate soil stiffness in parallel with structural stiffness—this study confirms it is a decisive parameter.
• For safety-critical components, prefer nonlinear analysis to supplement code checks.

References

[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