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).

2317_BUCKLING OF A VERTICAL COLUMN.png

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)

1904_BUCKLING OF A VERTICAL COLUMN 2.png

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
High resistive materials: The materials which have their resistivity 10 -8 to 10 -6 ohm-m come under this category. These materials are used for making resistance elements fo

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 (

What are the types of optical fibres? Types of Optical fibres: Optical fibres are categorized within three main types based onto: I. Material II. The number of modes a

With Schematic diagram explain the working of a Michelson interferometer. Obtain the expression for radii of circular interference fringes. How shall you use it to measure waveleng

si el numero atomico del sodio es 11 y su numero de masa es 22,diga¿cuantos electrones,protones y neutrones respectivamente,tiene su atomo neutro?

The mass of 1 mole of electrons is: (1) 9.1x 10 -28 g (2) 1.008mg (3) 0.55mg (4) 9.1x 10 -27 g Ans:  (3)0.55mg

Advantages of fibre optics: Fibre-optic communications systems have a large bandwidth, e.g. 1 GHz.  The bandwidth is the maximum rate at which information can be transmitted.

VISUAL PHOTOMETRY: The branch of science which deals with measurement of brightness or luminous intensities of light emitted by different sources using certain standards and tech

The topic for the project is to design an Automotive Cruise Control System although you may choose a different title for your report. The aim is to research cruise control systems

Q. State the law of conservation of electric charges? The entirety charge in an isolated system always remains constant. For instance Uranium ( 92 U 238 ) can decay by emittin