Mathematical modeling of suspension bridge

Mathematical modeling of suspension bridge Presented by Yumnam Girish Singh12/CE/419Under the guidance of Dr Diptesh Das

1Introduction

Mathematical modeling can be concerned for the computation of deflection in the deck, moment created due to static load.In the recent days mathematical modeling is concerned with the dynamic behavior of the bridgeThe collapse of Tacoma narrow bridge stimulate the mathematical modeling of suspension bridge

The collapse of Tacoma bridgeBridge was collapsed just after 4 month of opening.The collapse was stimulated by small force causing large deformation due to non-linear property of the cableBoth the vertical and torsional oscillation occurred in the failureThe bridge was oscillating about the 45 min before failing.

Fig: oscillation at tacoma bridge and collapse after the oscillation

At first the scientist suggest the failure was due to the resonance.But actually it took five decades to solve the mystery of collapse of the bridge.Mathematician McKenna first published a paper describing the failure of Tacoma bridge. ObjectiveMathematical modeling of suspension bridge under large amplitude oscillation.Mathematical modelingLinear modeling is insufficient to explain the large oscillatory behaviorFor the dynamic behavior one dimensional modeling is considered.

Mathematical modeling under large-amplitude periodic oscillations in suspension bridges

A. Single beam equation

B. Spring beam system

Single beam equationConstruction holding the cable stays is solid and immovable.

The non linear equation is given by

Where m=mass per unit length of the bridge E= youngs modulusi= moment of inertia of the cross sectionb= damping co efficientw= weight per unit length of the bridge = external periodic forces such as wind etcL= length of the centre span of the bridge.

The first term in the equation represents an inertial force, the second term is an elastic force and the last term on the left-hand side describes a viscous damping. On The right-hand side, we have the influence of the cable stays, the gravitation force and the external force Due to the wind (time-periodic). The cable stays can be taken as one-sided springs, Obeying Hooke's law, with a restoring force proportional to the displacement if they are stretched, and with no restoring force if they are compressed.

Spring beam systemIn this case the construction holding the cable stays as an immovable object

In this case two non linear equation was obtain

With boundary condition

Future workI will be studying the dynamic behavior of the suspension bridge under the seismic load.REFERENCESGabriela Holubova Tajcova : Mathematical modeling of suspension bridges , Mathematics and Computers in Simulation 50 (1999) 183-197A. C. Lazer; P. J. McKenna: Large-Amplitude Periodic Oscillations in Suspension Bridges: Some New Connections with Nonlinear Analysis , SIAM Review, Vol. 32, No. 4. (Dec., 1990), pp. 537-578.P.J. McKenna : Large Torsional Oscillations in suspension bridge Revisted:Fixing and old approximation, Dr. Richard Ohene Kwofie: Mathematical Model of a Suspension bridge case study: adomi bridge , master of science thesis

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