Buckling of a vertical column, Physics

Assignment Help:

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);


Related Discussions:- Buckling of a vertical column

Define the tea in thermos, Define the Tea in Thermos? One liter of tea...

Define the Tea in Thermos? One liter of tea at 90° C is poured keen on a vacuum-insulated container (thermos). The surface area of the thermos walls A=600 cm 2 . The volume am

Is the common end is important, Is the common end is important? Even th...

Is the common end is important? Even though manufacturers place extra attention on fibers and illuminators the common end is a very important piece for the correct operation of

What is the mass of the board - force, A board that is 1.5 m long is suppor...

A board that is 1.5 m long is supported in two places. If the force exerted by the first support is 25 N and the forced exerted by the second is 62 N. what is the mass of t

A high voltage generator, The electrostatic shielding may be found by prote...

The electrostatic shielding may be found by protecting and enclosing the sensitive instruments inside a hollow conductor due to inside hollow conductors, electric fields is zero.

Thermal equillibrium, a column of water contains fine metal particles of ra...

a column of water contains fine metal particles of radius 2*E(-8) m, which are in thermal equillibrium at 298 K. if there are 1000 particles per unit volume at a given height,how m

Projects, new ideas for physics projects ?

new ideas for physics projects ?

Determine the current taken by the capacitor, A motor has an output of 4.8k...

A motor has an output of 4.8kW, an efficiency of 80% and a power factor of 0.625 lagging when operated from a 240V, 50Hz supply. It is needed to improve the power factor to 0.9

What effect does this have on eclipses, Q. If the moons orbit is inclined s...

Q. If the moons orbit is inclined somewhat relative to the Earth's orbit what effect does this have on eclipses? Answer:- This makes eclipses rarer as well as if the moon's

Explain p-type substance, P-Type Substance: When a small quantity of bo...

P-Type Substance: When a small quantity of boron or Indium (group III A elements ) is added to silicon or Germanium, holes are formed in it and it becomes good conductor of pos

Write Your Message!

Captcha
Free Assignment Quote

Assured A++ Grade

Get guaranteed satisfaction & time on delivery in every assignment order you paid with us! We ensure premium quality solution document along with free turntin report!

All rights reserved! Copyrights ©2019-2020 ExpertsMind IT Educational Pvt Ltd