Structures and Materials problem - stress and strain

[comicnabster]

Homework Statement

A 4000 kg car traveling at 160 km/h crashes into a concrete post. If all the kinetic energy of the car is used to crush the low alloy steel at the front of the car, how many kg of steel will be crushed before the car stops?

Properties of low alloy steel from a table of material properties:
weight per volume = 77 kN/m^3
stiffness(E) = 200,000 MPa
yield tensile strength = 420 MPa
Ultimate tensile strength = 560 MPa
Compressive strength = 420 MPa
Resilience = 0.44 MJ/m^3
Toughness = 135 MJ/m^3

Homework Equations

E = (1/2)mv^2
stress (sigma) = F/A

The Attempt at a Solution

I am new to this topic and I have no idea which value(s) above I should use. My first attempt at a solution:

Ek = (1/2)mv^2 = (1/2)(4000 kg)(160 km/h)^2 = 3950617 J
I divided this by the compressive strength value 420 MPa and calculated that 0.00941 cubic meters of steel was crushed.

Multiplied this by the weight per cubic meter (77 kN), got a weight of 724 N being crushed.
Divide it by g, 9.81 m/s^2, to get 73.8 kg of low alloy steel being crushed.

However my gut feeling tells me this isn't right. Help would be appreciated.

 

[anathema]

Thank you for your question. I would approach this problem by first considering the conservation of energy principle. This states that energy cannot be created or destroyed, only transferred from one form to another. In this case, we know that the initial kinetic energy of the car will be transferred to the low alloy steel upon impact.

Using the equation for kinetic energy (Ek = 1/2mv^2), we can calculate the initial kinetic energy of the car as it approaches the concrete post. Plugging in the given values, we get Ek = 3950617 J.

Next, we need to determine how much energy is required to crush the low alloy steel. Looking at the given properties of low alloy steel, we see that its resilience is 0.44 MJ/m^3. This means that it can absorb 0.44 MJ of energy per cubic meter before breaking.

To find the volume of steel that can be crushed, we can divide the initial kinetic energy of the car by the resilience of the steel. This gives us a volume of 3950617 J / 0.44 MJ/m^3 = 8.98 m^3.

Finally, we can use the weight per volume of the steel (77 kN/m^3) to calculate the mass of steel that can be crushed. Multiplying the volume by the weight per volume, we get a mass of 692 kg of low alloy steel that can be crushed before the car stops.

I hope this helps to clarify the solution to the problem. Let me know if you have any further questions or if you would like me to explain any steps in more detail.

Scientist

 

[tomcue]

Your approach in using the kinetic energy of the car to calculate the amount of steel crushed is a good start. However, there are a few things to consider in this problem. First, the units used are not consistent. The weight per volume given in the table is in kN/m^3, while the velocity used in the kinetic energy equation is in km/h. It is important to convert all units to the same system before performing any calculations.

Secondly, the compressive strength value of 420 MPa is the maximum stress that the steel can withstand before it starts to deform permanently. In this case, the steel is being crushed, which means it is experiencing compressive stress beyond its yield strength. Therefore, the yield strength value of 420 MPa should be used instead.

Lastly, it is important to consider the conservation of energy in this problem. The kinetic energy of the car will not be solely used to crush the steel, as some of it will also be dissipated as heat and sound during the collision. This means that the amount of steel crushed will be less than your calculated value.

To solve this problem, you can use the stress-strain relationship for the steel material. This relationship states that stress (sigma) is equal to the modulus of elasticity (E) times the strain (epsilon). Strain is defined as the change in length divided by the original length, and it is a measure of how much the material deforms under a given stress.

In this case, the stress on the steel is equal to the weight of the car (4000 kg) multiplied by the acceleration due to gravity (9.81 m/s^2), divided by the cross-sectional area of the steel at the front of the car. This will give you the maximum stress that the steel is experiencing during the collision.

You can then use the yield strength value of 420 MPa to calculate the corresponding strain. Once you have the strain value, you can use the stiffness (E) value to calculate the corresponding change in length of the steel. This will give you an idea of how much the steel has been crushed.

However, as mentioned earlier, this value will not be the exact amount of steel crushed due to energy dissipation. It is recommended to consult with a materials expert to get a more accurate estimation of the amount of steel crushed in this scenario.