Comparison of Lem, Nonlinear and FEM analyses
Comparison of LEM, Nonlinear, and FEM Analyses of a 10 m Deep Braced Excavation Using CPT-Based Soil Data
1. Introduction
Braced excavations require careful consideration of construction staging, soil–structure interaction, groundwater conditions, nearby surcharges, and the stiffness and capacity of the retaining walls and internal supports. Different analysis methods may predict different distributions of wall forces, movements, and support reactions because each method represents the soil and structural response differently.
This example presents the analysis of a 10 m deep excavation supported by two opposing retaining walls and three levels of temporary steel struts. The ground model is developed using an interpreted CPT profile and includes alternating drained and undrained soil layers. Groundwater and a 15 kPa surface surcharge beside the left wall are also included.
Three analysis methods available in DeepEX are considered :
Limit-equilibrium analysis (LEM);
Nonlinear soil-spring analysis (NL);
Two-dimensional finite-element analysis (FEM).
LEM is used to assess wall embedment and overall stability conditions. The NL and FEM analyses are compared in terms of wall movements, bending moments, shear forces, support reactions, and structural utilisation.
2. Analysis Description
2.1 Excavation Geometry
The excavation is 10 m deep and 10 m wide. The retaining walls extend to a depth of 16.5 m below the original ground surface, providing an embedment of 6.5 m beneath the final excavation level.
Three levels of cross-lot struts are installed at depths of 1, 4, and 7 m. The upper support is positioned 1 m below ground level, while the remaining levels are separated vertically by 3 m.
A uniformly distributed surcharge of 15 kPa is applied beside the left wall. This introduces an asymmetric loading condition that affects the pressures, internal forces, support reactions, and movements calculated on the two sides of the excavation.
Table 1. Principal excavation dimensions and loading conditions.
Parameter | Adopted value |
Excavation depth | 10 m |
Excavation width | 10 m |
Total wall length | 16.5 m |
Wall embedment below the excavation | 6.5 m |
Upper strut depth | 1 m |
Intermediate strut depth | 4 m |
Lower strut depth | 7 m |
Surface surcharge | 15 kPa |
Initial groundwater depth | 5 m |
Final internal groundwater depth | 10 m |
2.2 Ground Conditions
The soil stratigraphy is interpreted alongside the available CPT cone-resistance, qt, and sleeve-friction, fs, profiles. Five soil layers are included, comprising three predominantly frictional layers and two cohesive layers analysed under undrained conditions.
The upper S-8 layer extends from ground level to a depth of approximately 5.75 m. It is underlain by the CL-10 layer, which extends to a depth of 8 m, followed by the S-12 layer down to approximately 17 m. The retaining-wall toes are therefore located within the S-12 layer.
Below the wall toes, the profile includes the CL-13 layer between depths of approximately 17 and 19.8 m, followed by the deeper S-16 layer.
Table 2. Soil properties adopted in the analyses.
Soil layer | Approximate depth interval | γt (kN/m³) | c′ (kPa) | su (kPa) | φ′ (°) |
S-8 | 0–5.75 m | 18.3 | 0 | — | 39.8 |
CL-10, undrained | 5.75–8.0 m | 15.4 | — | 92.14 | 0 |
S-12 | 8.0–17.0 m | 19.7 | 1 | — | 39.0 |
CL-13, undrained | 17.0–19.8 m | 14.3 | — | 91.54 | 0 |
S-16 | Below 19.8 m | 19.5 | 0 | — | 37.4 |
Displaying the CPT results directly beside the adopted stratigraphy makes it easier to compare the selected layer boundaries and engineering parameters with the measured variation in cone resistance and sleeve friction.
2.3 Groundwater Conditions
The initial groundwater level is located 5 m below ground level on both sides of the excavation. As excavation progresses, the internal groundwater level is lowered until it reaches the final excavation depth of 10 m.
The difference between the internal and external groundwater levels is included in the analyses and contributes to the net pressures acting on the embedded portions of the retaining walls.
2.4 Retaining-Wall System
The two retaining walls extend to a depth of 16.5 m. Their section dimensions, stiffness, material properties, and structural resistance are defined through the wall-section properties assigned in DeepEX.
These properties are used in the NL and FEM analyses to calculate wall deformations and internal forces. They are also used in the structural verification of the calculated bending moments and shear forces.
The wall-definition images show the properties assigned to the retaining-wall sections and should therefore be presented as model-input information rather than analysis results.
2.5 Strut System
The excavation is supported by three levels of temporary steel pipe struts. The struts are spaced at 6 m in the out-of-plane direction and are assigned a free length of 20 m for structural verification.
A PM600×19 circular pipe section manufactured from A50 steel is adopted. No prestress is applied, and the struts are modelled as non-yielding members.
Table 3. Principal properties of the PM600×19 steel struts.
Property | Adopted value |
Outside diameter, D | 60 cm |
Wall thickness, tp | 1.9 cm |
Cross-sectional area, A | 346.77 cm² |
Yield strength, fy | 344.8 MPa |
Young’s modulus, E | 200,100 MPa |
Second moment of area, Ixx = Iyy | 146,488.5 cm⁴ |
Radius of gyration, rx = ry | 20.554 cm |
Horizontal spacing | 6 m |
Free length used for verification | 20 m |
Prestress | 0 kN |
Support classification | Temporary |
Nonlinear material behavior | Non-yielding |
3. Analysis Methodology
3.1 Limit-Equilibrium Analysis
The limit-equilibrium analysis is used to verify the adequacy of the wall embedment and evaluate the local and overall stability conditions associated with each retaining wall.
The reported LEM checks include :
Embedment stability;
Available passive resistance;
Rotational stability;
Required toe embedment;
Basal stability;
Hydraulic stability.
These results are considered separately from the wall movements, internal forces, and support reactions obtained from the NL and FEM analyses.
3.2 Nonlinear Soil-Spring Analysis
In the NL analysis, the retaining walls are represented by beam elements supported by nonlinear soil springs. The soil reactions depend on the calculated wall movements and the progressive mobilization of active and passive resistance.
The method considers :
Wall stiffness;
Nonlinear soil resistance;
Sequential excavation;
Activation of each strut level;
Groundwater changes;
Surface surcharge;
Redistribution of soil pressures as the walls deform.
This approach allows wall forces, support reactions, and deformation-related behavior to be evaluated throughout the excavation sequence.
3.3 Finite-Element Analysis
The FEM model represents the soil mass as a continuous two-dimensional domain discretized into finite elements. The retaining walls and struts interact directly with the surrounding soil as excavation and support installation proceed.
In addition to wall forces and support reactions, the FEM analysis provides the displacement field throughout the soil domain. This makes it possible to evaluate ground movements beneath the excavation and behind the retaining walls.
The same excavation geometry, ground conditions, structural sections, groundwater levels, surcharge, and construction stages are adopted in the NL and FEM models to provide a consistent basis for comparison.
3.4 Construction Staging
The excavation is constructed progressively. The retaining walls are first installed under the initial ground and groundwater conditions. Excavation then proceeds in stages, with each strut activated before significant excavation takes place below its level.
The general sequence comprises :
Establishment of the initial ground and groundwater conditions;
Installation of the retaining walls;
Excavation to the upper support level;
Installation of the upper struts;
Excavation to the intermediate support level;
Installation of the intermediate struts;
Excavation to the lower support level;
Installation of the lower struts;
Excavation to the final depth of 10 m.
3.5 Structural Verification
The retaining walls are checked for bending and shear resistance. Each strut is checked using its calculated axial force and corresponding structural capacity.
Structural utilization is reported through the structural ratio, STR. A ratio below 1.0 indicates that the calculated demand remains below the available resistance for the adopted section and design assumptions.
4. Results and Discussion
4.1 Limit-Equilibrium Stability Results
The LEM results confirm the required wall embedment and provide separate checks for passive resistance, rotation, basal stability, and hydraulic stability.
Table 4. Wall toe and stability results obtained from the LEM analysis.
Result | Left wall | Right wall |
Minimum reported wall factor of safety | 1.559 | 1.569 |
Embedment factor of safety | 1.559 | 1.569 |
Passive-resistance factor of safety | 2.919 | 100.767 |
Rotational factor of safety | 1.810 | 1.822 |
Required toe embedment for FS = 1.0 | 4.170 m | 4.144 m |
Corresponding toe elevation | −14.170 m | −14.144 m |
Basal-stability factor of safety | 2.514 | 2.694 |
Hydraulic factor of safety | 1.509 | 1.509 |
Toe verification | FS ≥ 1.0 | FS ≥ 1.0 |
The embedment verification governs the reported wall-stability results, giving factors of safety of 1.559 for the left wall and 1.569 for the right wall. The corresponding toe embedment required to obtain FS = 1.0 are approximately 4.17 and 4.14 m, respectively.
The adopted embedment of 6.5 m therefore exceeds the minimum values calculated by LEM. Rotational factors of safety of 1.810 and 1.822 are obtained for the left and right walls, while the corresponding basal-stability factors are 2.514 and 2.694.
The hydraulic factor of safety is 1.509 for both walls. The adequacy of these values should ultimately be assessed against the acceptance criteria established for the project and the applicable design standard.
The passive-resistance factor is considerably higher for the right wall than for the left. This difference reflects the loading, geometry, and resistance conditions considered on each side and should not be interpreted as a direct measure of wall structural capacity.
4.2 Comparison of the NL and FEM Results
The two soil–structure interaction methods predict different distributions of wall forces, movements, and support reactions. Neither analysis produces the governing value for every response quantity.
Table 5. Comparison of the final-stage NL and FEM results.
Result | NL analysis | FEM analysis |
Maximum wall displacement | 0.95 cm | 1.72 cm |
Maximum settlement reported in the stage summary | 1.22 cm | 0 cm¹ |
Maximum absolute wall moment | 264.36 kN·m/m | 217.50 kN·m/m |
Maximum absolute wall shear | 147.81 kN/m | 163.73 kN/m |
Maximum wall structural ratio | 0.587 | 0.455 |
Governing support reaction | 189.34 kN/m | 226.92 kN/m |
Approximate governing support force at 6 m spacing | 1,136.0 kN | 1,361.5 kN |
Maximum support structural ratio | 0.472 | 0.547 |
Governing support level | Intermediate | Lower |
¹ The FEM stage summary reports zero settlement. However, the displacement contour shows a maximum total displacement magnitude of approximately 3.71 cm beneath the excavation. Total displacement represents the magnitude of the displacement vector and should not be interpreted directly as vertical ground-surface settlement.
4.3 Nonlinear Analysis Results
The NL analysis gives a maximum wall displacement of 0.95 cm, corresponding to 9.5 mm or approximately 0.095% of the excavation depth. The maximum reported ground-surface settlement is 1.22 cm.
The maximum absolute wall bending moment is 264.36 kN·m/m, while the maximum wall shear is 147.81 kN/m. The resulting maximum combined wall structural ratio is 0.587, indicating that the calculated wall demand remains below the available capacity.
The surcharge beside the left wall produces an asymmetric response, particularly in the calculated bending-moment distribution. The NL analysis accounts for this asymmetry through the different mobilization of the nonlinear soil springs on the two sides of the excavation.
The intermediate strut governs the support design, with a reaction of 189.34 kN/m and a structural ratio of 0.472. The upper and lower struts attract smaller reactions.
Table 6. Final-stage strut results from the NL analysis.
Strut level | Depth | Reaction | Force at 6 m spacing | STR |
Upper | 1 m | 1.01 kN/m | 6.06 kN | 0.098 |
Intermediate | 4 m | 189.34 kN/m | 1,136.04 kN | 0.472 |
Lower | 7 m | 43.03 kN/m | 258.18 kN | 0.179 |
The support loads are not distributed uniformly among the three levels. In the NL model, the intermediate strut provides the principal restraint, while the lower strut attracts a comparatively limited reaction.
4.4 Finite-Element Analysis Results
The FEM analysis gives a maximum wall displacement of 1.72 cm, or 17.2 mm, corresponding to approximately 0.172% of the excavation depth. The opposite wall reaches a similar maximum movement of approximately 1.60 cm in the other direction.
The maximum FEM wall displacement is approximately 81% greater than the NL value. This difference reflects the distinct ways in which the two methods represent soil stiffness, excavation-induced unloading, stress redistribution, and wall–soil interaction.
The FEM analysis produces a maximum absolute bending moment of 217.50 kN·m/m, approximately 18% lower than the NL value. Conversely, its maximum wall shear is 163.73 kN/m, approximately 11% higher than the NL result.
The maximum FEM wall structural ratio is 0.455, compared with 0.587 from the NL analysis. Although FEM predicts greater wall movement, the NL analysis governs the wall bending and combined structural verification.
The FEM displacement contour shows a maximum total displacement magnitude of approximately 3.71 cm beneath the excavation. Because this is the magnitude of the complete displacement vector, it should be evaluated separately from the vertical ground-surface settlement reported by the program.
4.5 Comparison of the Strut Reactions
The most significant difference between the NL and FEM support results occurs at the lower strut level.
Table 7. Comparison of final-stage strut reactions and structural ratios.
Strut level | NL reaction | NL STR | FEM reaction | FEM STR |
Upper | 1.01 kN/m | 0.098 | 3.78 kN/m | 0.100 |
Intermediate | 189.34 kN/m | 0.472 | 185.23 kN/m | 0.464 |
Lower | 43.03 kN/m | 0.179 | 226.92 kN/m | 0.547 |
The intermediate-strut reactions obtained from the two methods differ by only approximately 2%. However, the FEM model transfers substantially more load to the lower support.
Consequently, the intermediate strut governs the NL analysis, whereas the lower strut governs the FEM analysis. The maximum FEM reaction of 226.92 kN/m corresponds to a support force of approximately 1,361.5 kN at the adopted spacing of 6 m. Its structural ratio is 0.547, indicating that approximately 55% of the calculated support capacity is mobilized.
This difference demonstrates the importance of comparing the complete distribution of support forces. Although the two methods predict similar reactions at the intermediate support, they distribute the remaining demand differently among the upper and lower support levels.
4.6 Overall Interpretation
The three analysis methods provide different but complementary information.
LEM verifies the wall embedment and evaluates passive resistance, rotation, basal stability, and hydraulic stability. It does not provide the same soil–structure interaction response as the NL and FEM analyses and should therefore be presented separately.
Among the soil–structure interaction methods, NL gives the more critical results for :
Maximum wall bending moment;
Combined wall structural utilization;
Reported ground-surface settlement.
FEM gives the more critical results for :
Maximum wall displacement;
Maximum wall shear;
Governing support reaction;
Maximum strut utilization.
The NL analysis produces a maximum wall ratio of 0.587, compared with 0.455 from FEM. Conversely, FEM gives a maximum support ratio of 0.547, compared with 0.472 from NL. All reported wall and strut utilization ratios remain below 1.0.
The comparison demonstrates that relying on a single result or analysis method may overlook an important aspect of the excavation response. For this model, LEM governs the wall-embedment verification, NL governs the wall structural check, and FEM governs wall movement and lower-strut demand.
5. Conclusions
This example compares LEM, nonlinear soil-spring, and two-dimensional FEM analyses of a 10 m deep excavation supported by three levels of temporary steel struts.
The excavation is formed between two 16.5 m long retaining walls, providing an embedment of 6.5 m below the final excavation level. The ground model is interpreted alongside CPT cone-resistance and sleeve-friction profiles and includes both drained frictional layers and cohesive layers analyzed under undrained conditions.
The LEM analysis gives embedment factors of safety of 1.559 and 1.569 for the left and right walls, respectively. The required toe embedment for FS = 1.0 are approximately 4.17 and 4.14 m, which are less than the adopted embedment of 6.5 m. The rotational, basal-stability, and hydraulic checks also produce factors of safety greater than 1.0.
The NL analysis gives the higher wall bending demand, with a maximum moment of 264.36 kN·m/m and a maximum combined wall ratio of 0.587. FEM gives a lower maximum moment of 217.50 kN·m/m and a wall ratio of 0.455.
FEM predicts the greater wall movement, with a maximum value of 1.72 cm compared with 0.95 cm from NL. It also produces the higher maximum wall shear: 163.73 kN/m compared with 147.81 kN/m from NL.
The two methods produce markedly different support-load distributions. The intermediate strut governs the NL analysis, with a reaction of 189.34 kN/m and a structural ratio of 0.472. In the FEM analysis, the lower strut governs, with a reaction of 226.92 kN/m and a structural ratio of 0.547.
All reported wall and strut structural ratios remain below 1.0 for the adopted sections. Nevertheless, the comparison confirms that the different methods govern different aspects of the design. LEM controls the embedment verification, NL controls the wall structural demand, and FEM controls the predicted wall displacement and lower-strut demand.
Using the three methods together provides a broader understanding of the excavation response and allows the stability, structural capacity, support loading, and deformation performance of the system to be assessed within a consistent DeepEX model.
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