Question

A shear test is performed on a sample of sand, with \(c=0\) and \(\phi=40^{\circ}\), as determined in triaxial tests. The sand has been poured very carefully into the shear box, so that it can be expected that the horizontal stress in the sample is very low. The vertical normal stress is \(100 \mathrm{kPa}\). What is the maximum allowable shear stress on the sample?

Short answer

Answer: The maximum allowable shear stress on the sand sample is approximately 103.92 kPa.

Step-by-step solution

Convert the internal friction angle to radians

We're given the internal friction angle (\(\phi=40^{\circ}\)). To work with this angle in the Mohr-Coulomb failure criterion, we need to convert it to radians. To do this, we can use the formula: \(\phi_{rad} = \frac{\pi}{180} \cdot \phi_{deg}\) Now, we can substitute the given value of \(\phi_{deg}\): \(\phi_{rad} = \frac{\pi}{180} \cdot 40 = \frac{2\pi}{9}\)

Calculate the maximum allowable shear stress

Now, we can use the Mohr-Coulomb failure criterion to calculate the maximum allowable shear stress. The equation is: \(\tau = c + \sigma \cdot \tan{\phi}\) Substitute the given values and the radians value of \(\phi\): \(\tau = 0 + 100 \mathrm{kPa} \cdot \tan{\frac{2\pi}{9}} = 100 \mathrm{kPa} \cdot \tan{\frac{2\pi}{9}}\) Now, we can calculate the maximum allowable shear stress: \(\tau = 100 \mathrm{kPa} \cdot \tan{\frac{2\pi}{9}} \approx 103.92 \mathrm{kPa}\) Hence, the maximum allowable shear stress on the sample is approximately \(103.92 \mathrm{kPa}\).

Key concepts

Mohr-Coulomb Failure Criterion

Understanding the Mohr-Coulomb failure criterion is fundamental in soil mechanics, as it provides a simple yet effective way to describe the conditions under which soil and other cohesive-frictional materials will fail due to shear stress. In our exercise, the criterion takes center stage as we analyze the maximum allowable shear stress on a sand sample.

The Mohr-Coulomb failure criterion is expressed through the equation \( \tau = c + \sigma \cdot \tan{\phi} \), where \( \tau \) represents the shear stress, \( c \) the soil's cohesion, \( \sigma \) the normal stress, and \( \phi \) the internal friction angle. In the absence of cohesion (\(c=0\) for the sand), the shear strength is solely dependent on the internal friction angle and normal stress. When the shear stress exceeds the value calculated by this criterion, failure occurs along the plane receiving the stress.

By grasping this concept, students can predict the failure of earth materials under different loading conditions, a critical aspect for designing safe and effective engineering structures such as foundations, retaining walls, and slopes.

Internal Friction Angle

The internal friction angle, denoted \( \phi \), is a measure of the shear strength of soils and is derived from the physics of granular material behavior. This angle is pivotal to calculating the maximum shear stress that a soil can withstand before it begins to shear and deform.

Representing the resistance to sliding along internal planes, the internal friction angle depends on the properties of the material, including particle size, shape, and distribution. In our exercise, we were provided a significant internal friction angle of 40 degrees, indicative of a well-structured sand with substantial resistance to shearing forces.

The step-by-step solution converts the angle from degrees to radians before using it in calculations, ensuring accuracy within the Mohr-Coulomb framework. This underlines the importance of understanding how to properly incorporate the internal friction angle into predictive models of soil behavior.

Triaxial Shear Test

The triaxial shear test is a laboratory testing method used to assess the mechanical properties of soils and how they respond under stress. It plays a critical role in geotechnical engineering by providing key parameters such as shear strength, cohesion, and the internal friction angle that we previously discussed.

In this test, a cylindrical soil sample is enclosed in a latex membrane and subjected to controlled pressures, simulating different stress conditions. The process allows the determination of the soil's response to being compressed or sheared, information vital for safe and effective structural designs. In our exercise context, the sand's internal friction angle was derived from data obtained through triaxial tests, which speaks to the practical application and significance of these tests in engineering scenarios.

Studying the outcomes of triaxial shear tests prepares students for real-world applications where understanding soil behavior under stress is essential to foresee and mitigate potential groundwork issues.