A tunnel of length L=150m, height H=7.2m, and width w=5.8m (with a flat roof) is to be constructed at a distance d=60m beneath the ground. The tunnel roof is be entirely supported by square steel columns, each with a cross sectional area of 960 cm^2. The mass of 1.0 cm^3 of the ground material is 2.8g. a) what is the total weight of the ground material the columns must support? b) How many columns are needed to keep the compressive stress on each column at one-half of its ultimate strength?

I am not sure where to begin. Please help.

(a) Calclate the weight of the soil above the roof. Its density is 2.8*10^3 kg/m^3. Multiply that by the roof area (5.8 m x 150 m), and the depth of the soil above (60 m). Then multiply by g to get the weight in Newtons.

(b) To do this part, you need to know the ultimate strength of the steel. You should have been given a nominal value for structural carbon steel. Set the working stress (half the ultimate stress) equal to the weight divided by the total cross section of N steel columns, and solve for N.

consider the pressure on a square cm due to the weight above.

Pressure= g*density*volume/area= density*height= 2.8g/cm^3 * 60m*100cm/m

then weight = pressure *(area roof in cm^2)
weight= pressure*150m*5.8m* (100cm/m)^2

check my thinking

To solve this problem, we need to calculate the weight of the ground material that the columns must support and then determine the number of columns needed.

a) To find the total weight of the ground material, we need to calculate the volume of the tunnel.

Volume of the tunnel = Length × Height × Width
= 150 m × 7.2 m × 5.8 m

Now, let's convert this volume to cubic centimeters (cm^3) since the mass of the ground material is given in grams per cubic centimeter.

1 m^3 = 100 cm × 100 cm × 100 cm = 1,000,000 cm^3

So, Volume of the tunnel = (150 m × 7.2 m × 5.8 m) × (1,000,000 cm^3 / 1 m^3)

Next, we can calculate the mass of the ground material using the given density.

Mass of the ground material = Volume of the tunnel × Density of the ground material

The density of the ground material is given as 2.8 g/cm^3.

So, Mass of the ground material = Volume of the tunnel × 2.8 g/cm^3

b) To find the number of columns needed, we need to determine the compressive stress on each column and compare it to one-half of its ultimate strength.

Compressive stress = Force / Area

The force acting on each column is the weight of the ground material that it must support.

So, Force = Mass of the ground material × Acceleration due to gravity

The acceleration due to gravity is approximately 9.8 m/s^2.

Ultimate strength is not given in the question, so we'll need that information to calculate the number of columns needed.

Please provide the ultimate strength of the square steel columns so we can proceed with the calculation.

To solve this problem, we need to break it down into smaller steps and calculate the necessary values. Let's start with part (a).

a) To find the total weight of the ground material the columns must support, we need to calculate the volume of the ground material and then determine its weight.

Step 1: Calculate the volume of the ground material.
The volume of the tunnel can be found by multiplying its length, height, and width:
Volume = Length × Height × Width = L × H × w

Step 2: Convert the volume of the tunnel into cubic centimeters.
As the unit of mass for the ground material is given in grams per cubic centimeter, we need to convert the volume to the appropriate unit:
Volume (in cm³) = Volume (in m³) × 100^3

Step 3: Calculate the weight of the ground material.
To find the weight of the ground material, we need to multiply the volume by the density of the ground material:
Weight (in grams) = Volume (in cm³) × Density (in g/cm³)

Step 4: Convert the weight to a more practical unit if needed.
If necessary, you can convert the weight to kilograms or any other desired unit.

b) To determine the number of columns needed to keep the compressive stress on each column at one-half of its ultimate strength, we need to calculate the compressive stress on each column and then divide it by half of the ultimate strength.

Step 1: Calculate the cross-sectional area of each column.
The cross-sectional area of each column is given to be 960 cm².

Step 2: Calculate the compressive stress on each column.
The compressive stress can be calculated by dividing the weight of the ground material (found in part a) by the total cross-sectional area of all columns.

Step 3: Divide the compressive stress by half of the ultimate strength.
Divide the compressive stress on each column by half of its ultimate strength to find the number of columns needed.

Finally, you can plug in the given values and perform the calculations to find the answers to both parts (a) and (b) of the problem.