Modeling Deflections in Beams

When a load is placed on a beam, the formerly-straight horizontal (centroidal) axis of the beam is deformed into a curve. For each point on the centroidal axis of the deformed beam, we let x denote the horizontal distance of that point from the left-hand end of the beam, and we define w(x) to be the vertical displacement of the beam. The graph of the function w is called the deflection curve for the beam.

The goal of mathematical modeling is to represent natural processes by mathematical equations, to analyze the mathematical equations, and then to use the mathematical model to better understand and predict the natural process. In this module, we are interested in predicting and understanding the deflection of loaded beams by mathematically modeling their deflection curves.

Before beginning our mathematical analysis, let us review the ingredients that will go into it. As we have already seen, the shape of the deflection curve will depend on several factors. The four factors that enter into our mathematical model are:

The material properties of the beam as measured by the modulus of elasticity.
The beam's cross section as measured by its centroidal moment of inertia.
The load on the beam, described as a function of the position along the beam.
The way the beam is supported, which is captured by the boundary conditions of the differential equation in our model.

Experiments show that the deflection curve depends inversely on the modulus of elasticity, E , and also depends inversely on the centroidal moment of inertia of the beam's cross section, I . The way the deflection depends on the applied load, q(x) , and on the manner in which the beam is supported is more complicated, as we shall see.

Under the assumption that the deflections of the beam are small, an understanding of certain physical principles and geometric concepts allows one to derive a fourth-order differential equation that the deflection function w(x) must satisfy:

               q(x)
    w''''(x) = ----                (1)
                EI

We will call this equation the static beam equation . In words, the beam equation tells us that the deflection function is a function whose fourth derivative at every point, x , is equal to the load at that point divided by a constant quantity that depends only on the material properties and shape of the beam.

The importance of this equation is that in principle, we can determine the quantities on the right-hand side of the equation. Thus, the static beam equation enables us to mathematically solve for the deflection function!

Group Discussion

How do you expect each of the four main factors to affect the (as yet unknown) deflection function? Does your intuition seem to qualitatively agree with what the static beam equation tells you about the deflection function? What are some other factors that our model does not take into account?

Our goal now is to explore the mathematical relationship between the various parts of the static beam equation. We begin by examining the relationship between the beam's support information and its deflection function.

Question 5

Through careful measurements of a loaded beam of length L , you determine that the deflection of the beam is approximately given by

         x^4 - 4L x^3 + 6L^2 x^2
w(x) = ---------------------------
               24 E I

for 0 <= x <= L . (You may wish to verify that this function is a solution to the static beam equation.)

What is the deflection of the beam at x=0 ?
What is the slope of the deflected beam at x=0 ?
The second derivative of deflection tells us how much torsion (also called the bending moment ) the beam feels. Find the bending moment at x=L .
The third derivative of deflection tells us how much shearing force the beam feels. Find the shearing force at x=L .
Graph the deflection function for 0 < x < L . (Hint: we are only concerned about the shape of the function, so you may set L = E = I = 1 for this problem.) Remember that a positive deflection means that the beam sags down, so turn your graph upside down (or plot -w ) to see what the deflection of this beam looks like. Can you identify the geometric characteristics of the graph that correspond to w(0), w'(0) , and w''(L) ?

In the previous question, you gathered four pieces of information about the deflection and its derivatives. Two of the pieces give data about conditions of the beam at the left-hand endpoint (or boundary); the other two pieces give conditions of the beam at its right-hand boundary. Collectively, these data form a set of boundary conditions that tell us how the beam is supported. In this case, the conditions actually tell us that the beam is cantilevered!

The important point to note here is that the deflection function of a beam contains information about the beam's support, and we can extract that information from it in the form of boundary conditions. Conversely, starting with the boundary conditions and the other quantities on the right-hand side of the static beam equation, we can construct the deflection function that encodes them.

Question 6

Assume for this question that we are considering a cantilevered beam with a uniform load. Then the right-hand side of the beam equation is a constant, q/(EI) . As you found out above, the cantilevered beam corresponds to specifying the following boundary conditions:

     w(0)=0,  w'(0)=0,  w''(L)=0, w'''(L)=0.

Integrate both sides of the static beam equation with respect to x , and don't forget the constant of integration! (Hint: the integral of the fourth derivative is the third derivative!)
Apply one of the boundary conditions to solve for the constant of integration, and cross off that boundary condition to show that it has been "used". This enables you to write down an equation for the third derivative w''' that does not have any arbitrary constants.
Integrate both sides of the above equation for w''' with respect to x , and, again, don't forget the constant of integration!
Again, apply one of the boundary conditions (then cross it off the list!) to solve for the constant of integration. You now have an equation that w'' must satisfy.
Two more integrations followed by the application of the remaining boundary condition should give you an equation for the deflection function. Did you get back the function from the previous group activity?

Congratulations, you have solved a differential equation.

A particularly interesting thing to note is that solving the static beam equation only required four boundary conditions in addition to E , I , and q(x) . Mathematically, this is because the static beam equation is a fourth order differential equation. To solve the equation, we can use any four boundary conditions. This mathematical insight can sometimes be used to simplify a particular problem, by choosing a "good" set of four boundary conditions.

Next: Boundary Conditions
Return to: Moments of Inertia
Up: Outline

The Geometry Center Calculus Development Team