Buckling of a vertical column, Physics

Assume that the column is a homogeneous circular cylinder with small cross sections loaded by its own weight. The free end of the column will bend away when its height exceeds a certain critical value. Let x=0 be at the top free end and x=L at the base (assume the column is embedded in concrete).


diagram 1:Edwards Penny p.255          

Let θ(x) be the angle between the tangent to the column at the arbitrary point x and the vertical, i.e. the angular deflection of the column.

Question 1.  What other variables and constants do we need to consider? (See the appendix for a derivation of our basic equation (1).  E=Young's modulus, I= moment of inertia of column, ρ is the linear density)


Question 2. Evaluate γ2  E=Young's modulus =2.8x107 lb/in2 for a steel column. I=moment of inertia of a circular disk= .25π r4. ρ=δA, so g ρ=gδA, A is the cross sectional area. Take gδ=.28 lb/in3. Put this into your answer to question 4 to find the "buckling point" or critical height of a flagpole of diameter 1 in.

Question 3. Transform the differential equation (1) to a Bessel differential equation by first using the substitution z=γ2/3x  to obtain the differential equation θ''+z θ=0 and then make the substitution  θ(z)=u(t)z1/2 where t=2/3z3/2. Be careful in using the chain rule. You should get the Bessel differential equation : t2u''+tu'+(t2- 1/9)u=0. Show that the solution to our problem in terms of the Bessel function would be the zeros of J-1/3 (2/3 γx3/2)

Use Maple (or any other CAS) software to find roots of approximate equation for critical height in Question 2 and to find the zero of the Bessel's function in Question 6. Explain the discrepancy between critical height found in Qiestion2 and Question3.

In Maple use the following commands:

> plot(BesselJ(-1/3,x),x=0..16);

> r1:=RootOf(BesselJ(-1/3,x)=0,x,2);

> h:=solve((2/3)*gam*x^(3/2)=r1,x);

Posted Date: 2/25/2013 7:24:35 AM | Location : United States

Related Discussions:- Buckling of a vertical column, Assignment Help, Ask Question on Buckling of a vertical column, Get Answer, Expert's Help, Buckling of a vertical column Discussions

Write discussion on Buckling of a vertical column
Your posts are moderated
Related Questions
Q The refractive index of a medium is √3. Compute the angle of refraction if the un-polarised light is incident on it at the polarizing angle of the medium. μ = tan i p = √3

what is Gravitational field at a distance r from the infinitely long rod with liner mass density d?  Please drive it Ans) use Gauss'' law of electrostatics, but replace (1/4πε0)

Q. Show the condition for dispersion without deviation? Q. A thin convex length of focal length 0.1m and a thin concave lens of focal length 0.1m are placed coaxially on contact

SMALL ALKALINE CELLS: Hermetically sealed Ni-Cad cells are produced in the same size and shape as their primary counterparts. They are small, portable and maintenance free, but

Reflection and Refraction Describing reflection and refraction, involving total internal reflection. You might drag the control at the top to modify the angle of incidence and

.how can we prove that sn=u+a(1/2)

(i) It is the length of single wave. (ii) Wavelength is similar to the distance covered by the wave during the time in which any one particle of the medium done one vibration ab

The earth’s magnetic field: The earth acts as a magnet and the lines of force produced by it follow the pattern shown in the diagram below. If the Earth were completely symm

Question Why are polarized sunglasses are effective? Answer: If it weren't for the fact that surfaces that produce glare have a tendency to absorb vertically polariz

Alloying: When an impurity atom of a different size compared to the host atom is placed into the crystal lattice, the impurity atom distorts the resin around it, either by pushing