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# Understanding Hoop Stresses in Circular Shaft Analysis

## 2D Model Analysis & 3D FEM Analysis

#### A. Introduction

Circular shafts play a fundamental role in various engineering applications, from simple mechanical systems to complex infrastructure projects. Understanding the behavior of circular shafts under different loading conditions is crucial for ensuring their structural integrity and performance. One key aspect of circular shaft analysis is the calculation of hoop stresses, which are essential for assessing the shaft's ability to withstand external forces and moments.

In this article, we will delve into the significance of calculating hoop forces in circular shaft analysis and discuss the procedures involved in determining these stresses. Additionally, we will touch upon on how you can use our DeepEX shoring design software handles such cases.

Figure 1: Circular Shaft – 3D FEM model in DeepEX

#### B. Calculating Hoop Stresses

Hoop stress, also known as circumferential stress, is a type of stress that acts tangentially to the circumference of a circular shaft. It arises primarily due to external loads such as torsion or bending. The calculation of hoop stresses is essential for evaluating the structural integrity of the shaft and determining its load-bearing capacity.

From simple pipe mechanics, if we were to cut a pipe subjected to uniform loading in half, the load applied over one diameter would be equal to:

P = w R

Where:              P = axial wall load

w = linear load normal to the wall

Because real walls don’t act as thin elements, 2D bending will also develop in addition to hoop forces. 2D bending becomes more profound for larger diameter excavations where the ratio of the shaft radius to the effective wall thickness becomes increasingly large.

C. Case Study - Circular Shaft Analysis with DeepEX

This section presents a case study illustrating how DeepEX - shoring design software can be used to simulate and analyze a circular shaft under various loading conditions. We will demonstrate the software's capabilities in modeling the shaft geometry, applying external loads, and calculating hoop stresses using the Non-Linear analysis with elastoplastic springs method.

#### C. 2D Model & Non-Linear Analysis in DeepEX

We will examine a 40 feet deep circular shaft supported by a reinforced concrete secant pile wall (2.5 feet diameter secant piles with rebars) and a cap beam. Circular shaft models can be generated easily with the DeepEX software Model Wizard. The following figures illustrate the wizard dialogs for the model generation, the generated model, and the Non-Linear analysis results respectively.

Because of installation tolerances, and the fact that the hoop ring is controlled by the intersection of the secant piles, DeepEX has two parameters for determining the effective wall thickness in the nonlinear analysis, a) the verticality tolerance as a percentage of the wall length, and b) a minimum out-of-place tolerance. DeepEX will project both values to the wall bottom and determine the equivalent wall thickness this way. These parameters are available in the advanced tab of the wall form.

Figure 2: Circular Shaft Dimensions - DeepEX model wizard

Figure 3: Soil Properties and Stratigraphy - DeepEX model wizard

Figure 4: Reinforced concrete secant pile wall - DeepEX model wizard

Figure 5: Circular shaft cap beam dimensions - DeepEX model wizard

Figure 6: Circular Shaft model – 2D section generated by the DeepEX model wizard

Figure 7: Circular Shaft model - Hoop force and wall moments in DeepEX – Non-Linear analysis method

When the 2D analysis is completed, we need to focus on the hydraulic heave safety factors, hoop stresses, and the hoop buckling safety factor. The hoop buckling safety factor is generally recommended by industry practice to be greater than 6. On the compressive hoop stresses, we are generally looking for an overall FS of 3 or more because many of these shafts are unreinforced. As a general piece of advice, one should look to specify larger Ko values as these result in increased hoop forces.

Figure 8: Maximum hoop force and buckling safety factor – DeepEX analysis summary table

#### D. 3D Model & 3D FEM Analysis

Our latest addition to DeepEX, our 3D Finite Element Analysis engine guarantees to provide you valuable insights in your excavation system behavior, considering full soil-structure interactions. The advantage of our 3D FEM analysis is that the 3D FEM models can be automatically generated in seconds with a click of a button, and the user can make real-time adjustments to the actual generated model.

When a circular shaft model is generated by the DeepEX model wizard, the software automatically generates the 2D cut section, the 3D model plan view, and the 3D Finite Element Analysis models. We can access the generated plan view of the 3D model and make real-time adjustments that pass automatically to the generated 3D FEM model. We can modify the model limits and set the desired FEM mesh density. DeepEX can generate the FEM mesh and perform the 3D FEM analysis in minutes with the click of a button. Depending on how detailed we wish our analysis to be, the program will generate several wall segments that approximate the circular excavation.

Figure 9: Circular Shaft – 3D model top view generated by the DeepEX model wizard

Figure 10: Circular Shaft – 3D Model – FEM Mesh in DeepEX

Figure 11: Circular shaft – 3D FEM analysis in DeepEX – Soil settlement shadings

Figure 12: Circular shaft – 3D FEM analysis in DeepEX – Wall moment shadings

In this case, because of the simplification of the circular shape with linear segments, the wall moments from the 3D FEM are greater vs. the 2D analysis. 3D FEM analysis can be particularly helpful when we need to consider unbalanced external loads, or unbalanced soil conditions.

#### E. Conclusion & General Recommendations

Hoop stresses are a critical aspect of circular shaft analysis, influencing the shaft's structural performance and load-bearing capacity. By accurately calculating these stresses, engineers can ensure the safety and reliability of circular shafts in diverse engineering applications. Specialized software tools like DeepEX provide advanced capabilities for analyzing circular shafts, making the task of hoop stress calculation more efficient and accurate.