The geometrically nonlinear equation of equilibrium (5) in Theory of Elasticity gives rise to geometrically nonlinear equation of motion, which demand that the velocity of propagation of elastic waves (both rotational and compression ) will be dependent upon initial stresses in an elastic medium. Due to gravitational forces, the nonuniform stresses are developed in earth interior, therefore velocity of propagation will be different at different depth depending upon the stresses due to gravitational forces. This may have some implication in geomechanics, as large cavities are created either naturally, or manmade as in underground mines. Similarly in machinery the components will be initially stressed due to load. The present problem is comparatively simple but a practical one and analogous to connecting rod of a steam locomotive and pile foundations. The frequency calculation of rod and arches under uniform compression has been shown elsewhere (1) and (4).

Differential equation and its solution: The geometrically nonlinear equation of equilibrium is as follows (5)

where Qt is compressive axial force in the rod and EA is axial rigidity. m is mass per unit length. Qt need not be constant throughout the span of rod. For uniform Qt the solution is unchanged except the velocity of propagation is now

Now suppose in a rod whose movement is constrained at both end i.e at s=0 and s=l u=0 and a load P is applied at s=l₁ then

DISCUSSION AND RESULTS:

A rod of length 5m and cross section ISMB400@61.6 kg/m has been considered with axial load at different locations and following results are obtained

Axial point load=0.57*105 Kg

Axial frequency=102.76 rad/sec (unchanged with shifting of point load)

Flexural frequency=1.818 radian/sec to .818 rad/sec with point load at 4.5 m

Torsional frequency= 4.209 rad/sec to 3.905 rad/sec

Following conclusions are drawn.

The impact of axial load is insignificant on the frequency of axial vibration due to very high axial rigidity.

The frequency calculated by energy methods have been compared. The flexural frequency is 1.478 rad/sec with load at 4.5 m which is very high compared to value of .818 obtained by present method. Even the critical load calculated by Euler’s method in case of axial load applied at midpoint is complex implied buckling is not possible.

As the axial load moves from one end to other end, the region of compressive force extend therefore frequency of flexural and torsional vibration reduces. When due to very high compressive load the propagation velocity becomes imaginary, it becomes a case of local structural instability.

In another case the axial load was uniformally distributed along the length. The Euler's critical load was 5.157*10⁴ Kg. A load twice of Euler's critical load was uniformly distributed along the length to get the flexural frequency of 1.294 rad/sec. Even though only the half rod was in compression and remaining half length was in tension. In this case the velocity of propagation was 0 only at one end. Thus it was a case of local instability. Beyond this load the velocity of propagation was imaginary. The torsional frequency also reduced from 3.905 to 3.882 rad/sec.

CONCLUSION:

The equations for axial, flexural and torsional vibration of straight rod under non-uniform axial stresses has been obtained. These expressions are exact solution of the differential equation of vibration. The case of local stability has been shown to exist. Such cases of rod partly in tension and partly in compression may not be unstable according to Euler’s theory due to complex critical load. Therefore such problems need to be analysed by propagation velocity criteria.

NOTATIONS:

A Cross sectional area

c velocity of wave propagation

E Young's modulus

G Modulus of rigidity

I0 Polar moment of inertia

In Moment of inertia about principal normal

Iω Sectorial moment of inertia

Kt St. Venent torsional constant of section

l Length of the beam

m Mass per unit length

Qt Axial force

s Distance measured along the axis of the rod from the fiducial point

t Thickness of thin walled beam

u Displacement vector

x,y,z Coordinate axes along b, n, t

ω Vibration frequency in rad/sec.

λ,μ Lame's elastic constants

REFERENCES:

1. Timoshenko, S., (1937), "Vibration Problems in Engineering". Second edition, D Van nostrand Company Inc, New York

2. Timoshenko, S. and Gere, J. M. (1961), "Theory of Elastic Stability". Second edition, McGraw Hill International Book Company, New York

3. Verma, V.K. (1997), "Geometrically Nonlinear Analysis of Curved Beams" Paper presented at 42nd congress of ISTAM held at South Gujrat University, SURAT

4. Verma, V.K. (2000), "Out of Plane Vibration of Circular Beam Under Uniform Compression and Uniform Bending" Paper presented at 45th congress of ISTAM held at Mepco Sheilnk College of Technology, Sivakasi, Tamil Nadu

5. Amenzade, Yu. A., (1979), "Theory of Elasticity" English translation, Mir Publishers, Moscow

Received on 22.12.2011 Accepted on 15.01.2012

Research J. Engineering and Tech. 3(2): April-June 2012 page108-111