Computer Project 3

Due: Wed. 28 April 1999

Adapted from Edwards and Penny - ODE

Consider a vertical column which consists of a circular steel rod of radius r and length L. The column is rigidly fixed at the base and is free at the top. As the length of the column is increased, a critical length is reached at which it buckles under it's own weight. Our goal is to find the critical length of the column that first leads to buckling. Referring to the figure below, let u(x) be the angular deflection so that u(x)=0 indicates no buckling. Buckling of the column will occur for the first value of L (as L increases from 0) that leads to a nonzero solution for u(x). The x-axis is oriented so that x=0 is the free end and x=L is the fixed bottom.

The theory of elasticity leads to the following equation (with boundary conditions) for the deflection:

u''(x) + a^2 x u(x) = 0
u'(0)=0
u(L)=0

where a^2 = gdA/(EI).

The constants are:

g= gravitational acceleration
d=volumetric density (d*g = 0.28 lb/in^3)
A=cross-sectional area (= Pi*r^2)
E=Young's modulus (= 2.8*10^7 lb/in^2)
I=cross-sectional moment of inertia = Pi*r^4/4
r= cross=sectional radius of column

The problem for the deflection is a Bounday Value Problem (BVP) since the auxiliary conditions are given at the two points 0 and L. BVP's are fundamentally different than IVP's and often more challenging as discussed in class.

Perform the following steps using Maple, to find the smallest length L that leads to buckling.

Handwritten Part: Classify all points of the equation.
Series Approximation Worksheet Part: Compute a series solution about x=0 using the boundary conditions u(0)=C and u'(0)=0. The extra condition at x=0 helps simplify the calculation without changing the proper value of L. The solution is of the form C*f( x) which must satisfy the boundary condition u(L)=C*f(L)=0. We must choose C=0 unless we can find an L such that f(L)=0.
If C=0 then there is no deflection. Thus the problem is to find the smallest value of L for which f(L)=0. We can set C=1 without affecting the location of the zero and solve f(x)=0.
- You can use the Maple commands
  Order := 8
  eqn := diff(u(x),x$2) + a^2*x*u(x)=0; (NEW)
  soln3 := dsolve({eqn,u(0)=1,D(u)(0)=0},u(x),type=series); (NEW)
  assign(soln3); (NEW)
  poly3 := convert(u(x),polynom); (NEW)
- For a column of circular radius of 0.5 in, compute the constant a and use fsolve to locate the roots and choose the first zero. Use the commands:
  a := "your value";
  # note you need to put an actual expression here in place of "your value". See the Exact Worksheet if any question. (NEW)
  soln3inches := fsolve(poly3=0,x); (NEW)
  # Convert the answer into feet.
  soln3feet := soln3inches/12; (NEW)
Exact Solution Worksheet Part: The solution constructed above is only an approximation since we used only the first three nonzero terms in the series for u(x). To find a more accuate solution, we need an exact solution to the ODE. Use the Vrel3 Worksheet OR Vrel5 Worksheet (The Vrel5 version is New and Revised to avoid Airy Function Error. For the Vrel3 version, save to disk (C on PC) and click file icon to open in Vrel4: Your mileage will vary on different computers) (NEW) to help with this construction. Answer the questions contained in the worksheet.

You are to turn in two (2) Maple Worksheets.

The first approximation is the one you construct using the series solution, plus the handwritten comments (or Maple comments) on the Equation Classification.
The second uses the exact solution.