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
to measure the extension of an experimental wire due to different pulling forces using searle''s apparatues.hence, determine the young modules of the material of the wire.

explain with the aid of a diagram what is meant by the moment of a force about point

an object moves along the x axis according -to the equation x(t)=(3t^2-2t+3)m .determine (a) the average speed b/n t=2sec and t=3sec,(b) the instantaneous speed at t= 2s and t=3s (

deduction Newton law - KEPLUR law

photo electric cells, types ,how they work

what is expansion of solids

Artificial satellites 1. Satellites help to develop reliable communication links between different countries separated by large distances on the globe. The communications inclu

The conductivity of copper is less than that of silver by (A) 5 - 10 %. (B) 50 - 60 %. (C) 80 - 90 %. (D) 20 - 30 %. Ans: The conductivity of copper