Question

If the cohesion \(c\) is unequal to zero, it follows from eq. (33.3) that the minor principal stress \(\sigma_{3}\) can be zero, while the major principal stress \(\sigma_{1}\) is unequal to zero. This means that in a cohesive material an excavation can be made with vertical sides. What is the maximum depth of such an excavation, on the basis of this formula?

Short answer

To find the maximum depth of a vertical excavation in a material with non-zero cohesion, one must consider the Mohr-Coulomb failure criterion and the relationship between the principal stresses and the normal stress. The shear strength of the soil can be expressed as \(\tau = c + \sigma \tan(\phi)\), with the major principal stress acting as the normal stress (\(\sigma\)) on the vertical face of the excavation. The weight of the soil above the excavation balances the shear force acting on the excavation face. By solving the equation \(h_{max} = \frac{c + \sigma_1 \tan(\phi)}{\gamma}\) after determining the major principal stress and substituting known values, the maximum excavation depth can be calculated.

Step-by-step solution

Understand the given formula

Equation (33.3) is not provided in the prompt. However, in geotechnical engineering, the Mohr-Coulomb failure criterion is commonly used for such problems. The Mohr-Coulomb failure criterion relates the shear strength of a material to its normal stress, cohesion, and internal friction angle. The equation is given by: \( \tau = c + \sigma \tan(\phi)\) where \(\tau\) is the shear strength of the soil, \(c\) is the cohesion, \(\sigma\) is the normal stress, and \(\phi\) is the internal friction angle.

Relate the principal stresses to the normal stress

In this step, we need to find a relationship between the principal stresses and the normal stress. From the problem statement, we know that the minor principal stress \(\sigma_3 = 0\). Also, we have vertical excavation, so the major principal stress \(\sigma_1\) acts horizontally. Hence, the major principal stress, \(\sigma_1\), acts as the normal stress (\(\sigma\)) on the vertical face of the excavation.

Find the expression for the excavation depth

We can relate the excavation depth to the given condition by considering the vertical equilibrium of the soil above the excavation. The weight of the soil above the excavation balances the shear force acting on the excavation face. The shear strength of the soil can be expressed using the Mohr-Coulomb failure criterion. From Step 2, we have \(\sigma = \sigma_1\). Substituting in the Mohr-Coulomb failure criterion and considering the vertical equilibrium, we get: \(\tau = c + \sigma_1 \tan(\phi)\) Let's consider the excavation depth as \(h\). Then, the weight of the soil above the excavation (per unit length perpendicular to the plane of the cross-section we're considering) can be given as: \(W = \gamma h\) Where \(\gamma\) is the unit weight of the soil. Now, the shear strength of the soil, \(\tau\), is balanced by the weight of the soil above the excavation: \(c + \sigma_1 \tan(\phi) = \gamma h\)

Solve for the maximum excavation depth

To find the maximum depth, we need to solve the equation from Step 3 for \(h\). Rearranging the equation, we get the expression for the maximum excavation depth as: \(h_{max} = \frac{c + \sigma_1 \tan(\phi)}{\gamma}\) In this equation, we have the following variables: - Cohesion \(c\) is given - Internal friction angle \(\phi\) can be found from the soil properties - Unit weight of the soil \(\gamma\) can also be found from the soil properties - Major principal stress \(\sigma_1\) should be determined Note that for various excavation scenarios, different constraints might be imposed on the principal stresses (e.g., due to lateral support, groundwater table). Depending on these constraints, the resulting maximum excavation depth may also be different. After determining the major principal stress and substituting the known values in the above equation, the maximum excavation depth can be calculated.

Key concepts

Soil Mechanics and Excavation

Soil mechanics is a branch of engineering that delves into the behaviors of soil materials under varying conditions of stress and strain. Its relevance cannot be overstressed when planning excavations, as it provides insightful guidelines on how the ground will react to being dug up.

In the context of an excavation with vertical sides, we evaluate the stability of the soil structure as it directly impacts the safety and feasibility of the operation. Soil mechanics principles help determine critical factors such as the maximum excavation depth, which relies on understanding soil cohesion, weight, and the stresses it can withstand before failure occurs.

By calculating the maximum depth, engineers can ensure that the vertical walls of an excavation won't collapse inwards, which could be catastrophic. This is especially important in cohesive soils, where the presence of inter-particle forces allows for a certain depth of vertical excavation before additional support systems are necessary.

Mohr-Coulomb Failure Criterion

The Mohr-Coulomb failure criterion is a mathematical model that serves as a foundation for predicting the failure conditions of soils and rocks. It essentially correlates the maximum shear stress the soil can endure before failing with its normal stress, cohesion, and angle of internal friction.

By utilizing the equation \( \tau = c + \sigma \tan(\phi) \), we can depict the stress state at which a soil will likely shear, which is pivotal for understanding the stability of an excavation. With zero minor principal stress (\( \sigma_3 = 0 \) as presumed for a vertical excavation side), the major principal stress becomes a dominant factor in maintaining the equilibrium of the soil standing around the excavation.

Principal Stresses and Stability

Understanding principal stresses is crucial when dealing with the structural integrity of excavated soil. Principal stresses are the normal stresses acting on a plane where shear stresses are zero. In an excavation context, the minor principal stress (\( \sigma_3 \)) relates to the vertical plane and can be zero in cohesive soil, allowing for an unsupported vertical cut.

The major principal stress (\( \sigma_1 \)), however, acts horizontally in this scenario. It becomes a proxy for the normal stress on the vertical excavation face, which is used to calculate the intersection of shear strength and gravitational forces. This relationship aids in determining the maximum safe excavation depth before extra support or stabilization methods are necessary.

Cohesion in Soil and Its Role in Excavation

Cohesion in soil refers to the natural, chemical, and electrostatic forces that cause soil particles to stick together. This property is exceptionally relevant in excavations as it directly affects the soil's shear strength and, consequently, the depth to which an excavation can maintain vertical sides without external support.

In cohesive soils, the presence of cohesion (denoted as \( c \) in the Mohr-Coulomb criterion) allows for excavation with vertical walls to a certain point, meaning the sides can stand unsupported up to a depth where the weight of the soil above does not exceed its shear strength. This feature of cohesion facilitates the creation of deeper and steeper sides which would not be possible in non-cohesive soils such as sands and gravels.

The depth of such an excavation balances the cohesive strength with the gravitational forces acting downward, establishing a limit beyond which additional structural measures, such as shoring or benching, would be imperative for maintaining stability.