Given the following results of a direct shear test, plot the test data and least squares best fit line. What is the soil cohesion (y intercept) and what is the tangent of the friction angle (slope)?

Specimen                Normal Stress (psf)         Shear Stress (psf)

1                                   200                               450

2                                   400                              520

3                                   600                               590

4                                 1000                               740

The soil cohesion is 375.43 psf and the tangent of friction angle is 0.363.

Step-by-step explanation

To solve for this, we must determine the "best fit line" for the given normal and shear tress data.

The "least square method" will yield the most accurate result. In order to do it we must follow the steps below;

• Step 1: Calculate the mean of the x -values and the mean of the y -values:

• Step 2: The following formula gives the slope of the line of best fit:

• Step 3: Compute the y -intercept of the line by using the formula:

• Step 4: Use the slope m and the y-intercept b to form the equation of the line.

Here is the solution following the steps shown above:

• Here is the given data:

• Plot the points where Normal vs. Shear Stress:

• Calculate the means of the x -values and the y -values:

• Now calculate x_i− x?, y_i− y?, (x_i− x?)( y_i− y?) , and (x_i− x?)² for each i :
• Calculate the slope (tangent of friction angle):
• Use the formula to compute the y -intercept (soil cohesion, C):
• Use the slope and y -intercept to form the equation of the line of best fit:

The slope of the line is 0.363 and the y -intercept is 375.43 psf. Therefore, the equation is y = 0.363x + 375.43.

• Draw the line on the scatter plot:

Please see the attached file for the complete solution