12.2.1. Rectangular Approximation

Try these first

1. 1.

   What does the Dirac delta function approximate?

   Hint

   Think about how forces are applied in a very localized manner.

2. 2.

   Explain what happens to the height of the rectangular function as ϵ approaches 0.

   Hint

   Remember the area under the rectangle must always equal 1.

3. 3.

   What property does the rectangular approximation preserve?

   • Area equals 1
   • Height equals 0
   • Width increases

   Hint

   Think about how we defined the rectangle in our approximation.

4. 4.

   True or False: The rectangular approximation of the Dirac delta function can be used as an effective model for point loads in civil engineering.

   • True
   • False

   Hint

   Recall how Dirac delta functions are applied in engineering.

5. 5.

   A rectangular approximation for a load is given by δ(x) = 1/ϵ for |x|<ϵ. Calculate the total force exerted if ϵ = 0.1.

   Hint

   First, find the area of the rectangle using its height and width.

6. 6.

   Discuss the limitations of using the rectangular approximation to model a real-world point load. What might be the impact?

   Hint

   Consider how real-world forces differ from theoretical representations.