**A. INTRODUCTION**

Deep excavation shoring systems must be designed to withstand a multitude of loads from soil, water, seismic loads, construction surcharges, adjacent structures, and more.

Wall beam analysis methods allow engineers to design the structural behavior of retaining systems during excavation, by modeling the system as a series of interconnected beams.

These methods allow us to assess the developed wall stresses and expected support reactions.

There are different types of beam analysis methods that are typically used when we analyze deep excavation systems with the classical limit equilibrium approach.

This article examines and contrasts three major wall analysis methods for a 30ft deep excavation with 2 levels of steel struts in DeepEX – shoring design software:

**- Blum’s method**

**- FHWA Simple span approach**

**- CALTRANS approach with 20% negative moments**

**B. MODEL IN DEEPEX**

First, we will generate a 30 feet excavation supported by AZ 19 sheet piles with 2 bracing levels (PP24x0.5 steel struts placed at 20ft spacing) with DeepEX.

The surface is at El. 0, the water table is at El. -10ft and the soil mass consists of frictional soils (sands and silts).

Lateral soil pressures are calculated with the FHWA apparent earth pressure method for all construction stages with supports, and the water pressures will be calculated with the simplified 1d flow method.

Figure 1 presents the last stage of the generated DeepEX model.

**Figure 1: Model of a braced deep excavation in DeepEX Software**

**C.1. – BLUM’S METHOD**

Blum's method has been developed by Professor Edward L. Blum, and it is essentially a continuous structural beam analysis.

This method provides an approximate solution for calculating the, moment and shear diagrams of a wall and the support reactions that can potentially even be solved by hand, after some time consuming calculations, using the moment redistribution method.

With Blum’s approach, the excavation wall is simulated and analyzed as a continuous beam with pinned supports.

The method assumes the existence of a virtual support below the excavation, at the point where the net pressure diagram becomes zero.

**Figure 2: Blum’s Method Concept**

Blum's method assumes linear behavior, neglecting the effects of shear deformation and rotational stiffness.

It is known to be quite a conservative method for the calculation of the expected support reactions, but it is not conservative in the calculation of the developed moments on the examined shoring system. Figure 3 presents the calculated moment diagram and support reactions on our examined deep excavation model, analyzed with the Blum’s approach.

**Figure 3: Net Pressures, Wall Moment Diagram & Support Reaction with Blum’s Method**

**C.2. FHWA SIMPLE SPAN APPROACH**

The FHWA (Federal Highway Administration) Simple Span method typically simplifies the analysis and design of excavation walls by considering them as simple beams spanning between supports.

The top wall part up to the second support level is analyzed as a continuous beam, and the rest of the support locations on the wall are considered to have formed plastic hinges (so by default it assumes that the developed moment on the support locations below the second support level is zero).

As with any limit-equilibrium method it neglects certain complex behaviors such as soil-structure interaction and lateral soil pressure distribution.

A virtual support is assumed at the excavation level, so the method does not consider any developed moments and shears below the excavation.

**Figure 4: FHWA Simple Span Method Concept**

As a method, it can be used for the preliminary assessment of deep excavation models, but it should be used with caution by designer engineers and contractors, since it tends to calculate smaller support reactions, especially for the bottom support row because of the placement of the virtual support exactly at the excavation level.

Figure 5 presents the calculated moment diagram and support reactions on our examined deep excavation model, analyzed with the FHWA simple span approach.

**Figure 5: Net Pressures, Wall Moment Diagram & Support Reactions with FHWA Simple Span Method**

**C.3. CALTRANS APPROACH WITH NEGATIVE MOMENTS**

The CALTRANS approach (introduced in the California Trenching and Shoring Manual), similarly to the FHWA approach analyzed above, assumes that the wall behaves as a continuous beam above the second support, and that all supports from the second and after formed a plastic hinge.

The method assumes the existence of a virtual support below the excavation, at the point of zero moment below the last support, balancing out both moments and shear forces.

For this method, instead of assuming zero moment at the support locations, we can define negative moments as a percentage of the simple span moment (typically 20%).

**Figure 6: CALTRANS Method Concept**

The simple span approach might be quite conservative for the calculation of positive moments, especially when we assume zero moment at the support locations.

The calculated support reactions are typically higher compared to the FHWA simple span approach, because of the placement of the virtual support below the excavation level.

Figure 7 presents the calculated moment diagram and support reactions on our examined deep excavation model, analyzed with the CALTRANS approach.

**Figure 7: Net Pressures, Wall Moment Diagram & Support Reactions with CALTRANS Method**

**D. CONCLUSION**

Table 1 summarizes and compares critical results for wall moments, wall shears, and support reactions for each beam analysis method.

The results clearly indicate significant variations when different methods are used. This underscores the importance for engineers to be knowledgeable about the advantages, disadvantages, and level of conservatism associated with each method.

It is crucial to select an appropriate method that can produce safe and reasonable results simultaneously.

**Table 1: Braced excavation - Critical results for different beam analysis methods**

Result/Method | Blum’s | FHWA Simple Span | CALTRANS (Neg. Moment) |

Max. Moment (K-ft/ft) | 22.8 | 20.05 | 40.1 |

Min. Moment (K-ft/ft) | -32.3 | -32.3 | -32.3 |

Max. Shear Force (kip/ft) | 10.1 | -14.1 | -14.1 |

Max. Top Support Reaction (klf) | 13.13 | 13.13 | 12.38 |

Max. Bottom Support Reaction (klf) | 15.53 | 9.58 | 13.57 |

All methods analyzed in this article can serve as useful tools for initial design considerations, conceptual understanding, and quick estimations of beam deflections, stresses and support reaction estimations while analyzing deep excavation systems with the classical limit equilibrium approach. However, for precise and accurate analysis, especially in complex structures or situations involving significant deflections, it is recommended to utilize more advanced analytical techniques (beam on elastoplastic foundations – finite element analysis approach).

DeepEX software implements all the above, allowing us to utilize different approaches and evaluate our models accordingly.

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