The evidence that the model exhibits the same behavior becomes apparent when examining the phenomena under identical load levels. The maximum bearing capacity of the 3D CSFM will be compared with that of the CDP model.
Low-Stiffness-Soil (LSS)
The maximum bearing capacity of the 3D CSFM model has reached -980 kN of axial force acting on the column. The forces have been used as the benchmark level for comparison.
As observed, the minimum principal stress varies between output steps. This discrepancy arises from the nonlinear evolution of stress under compression, which depends on the constitutive behavior of the material. Due to triaxiality at the interface between the column and footing, the principal stress levels are higher than those in uniaxial compression.
In the 3D CSFM model, the deviatoric stress remains constant. The deviatoric stress is insensitive to the level of mean stress, same as for Tresca theory. Conversely, the CDP model employs a dilation angle of 30°, which generates volumetric expansion in compression and causes the deviatoric stress to evolve along the stress path, particularly under higher triaxiality. The peak compressive stress of −94.6 MPa in CDP corresponds to a local maximum associated with the sharp corner in the stress path, reflecting combined effects of triaxiality and dilatancy.
48)Minimal principal stress by level of load -980 kN
The difference between the stress at critical places of 3D CSFM compared to CDP.
- CDP approximately -70 MPa along the side of the column edge
- 3D CSFM - 60 MPa along the side
49)Detailed filtered stresses along the edge for CDP
The variation in stress observed in the reinforcements has been quantified at approximately 8% for rebars under tension and 28% for those under compression. The reduced stress in compression and the 28% discrepancy can be attributed to the concrete material model utilized for compression and dilation angle and the exclusion of the bond interaction between the rebars and concrete (perfect bond) within the CDP model. The 3D CSFM demonstrates a tendency towards conservative results, indicating elevated stress levels in both compression and tension.
50) Stress in reinforcements by the same level of load
Deformation level matches on 93 %.
51) Total deformation for the same level load
High-Stiffness-Soil (HSS)
The maximum bearing capacity of the 3D CSFM model has reached -2073 kN of loading force acting on the column. The forces have been used as the benchmark level for comparison.
The minimum principal stress for the CDP model reached −127 MPa at peak. This large compressive value is primarily the result of an increased level of deviatoric stress combined with strong dilatancy in compression (high dilation angle), which drives the stress path toward larger compressive principal stresses. Compared to the LSS case, the applied load was increased by approximately 211%, which explains the higher principal compressive stress in the CDP model.
By contrast, the 3D CSFM formulation used here assumes decoupling from hydrostatic pressure and does not employ an associated dilation law; as a result the effective frictional/dilatant response is different and the deviatoric stresses do not develop the same way as in the CDP . In that case, the minimum principal stress reached about −60 MPa (≈3× the uniaxial compressive strength), i.e. substantially lower compression than in the CDP. The stress differences between models will increase further if the mean (hydrostatic) stress becomes higher.
52)Minimal principal stress by level of load -2070 kN
The filtered stress distribution along the edge, with improved visualization and a properly scaled legend, indicates that the maximum compressive stress reaches approximately −70 MPa for the CDP model, compared with −60 MPa for the 3D CSFM model.
53)Detailed filtered stress along the edge for CDP
The variation in stress observed in the reinforcements has been quantified at approximately 8% for rebars under tension. The critical spot under tension has been identified in the exact location on the bottom longitudinal bars.
54) Stress in reinforcements by the same level of load
The evidence pertaining to the level of deformation corresponds to an 85% match.
55) Total deformation for the same level load