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Dynamic response of a beam on elastic foundation of finite depth under a moving force

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In this paper, dynamic response of an infinitely long beam resting on a foundation of finite depth, under a moving force is studied. The effect of foundation inertia is included in the analysis by modelling the foundation as a series of closely spaced axially vibrating rods of finite depth, fixed at the bottom and connected to the beam at the top. Viscous damping in the beam and foundation is included in the analysis. Steady state response of the beam-foundation system is obtained. Detailed numerical results are presented to study the effect of various parameters such as foundation mass, velocity of the moving load, damping and axial force on the beam. It is shown that foundation inertia can considerably reduce the critical velocity and can also amplify the beam response.

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b :

width of the beam

C b :

coefficient of viscous damping for the beam

C f :

coefficient of viscous damping for the foundation

E :

Young's modulus

f :


H :

foundation depth

I :

moment of inertia

i :


K, k :

indexing variables

k f :

foundation modulus

m :

mass per unit length of the beam

N :

total number of frequency points in Eqs. (25) and (26)

n :

indexing variable

P :

moving force

Q :

axial force on the beam

q(x, t) :

foundation pressure per unit length of the beam


foundation pressure in the moving co-ordinate system

t :

time variable in sec.

U j (ξ):

generalized coordinate in Eq. (4)

U j *(f):

Fourier transform ofU j

u(y, t; x) :

axial displacement in the foundation at a particularx value

u(y, ξ):

foundation displacement in the moving coordinate system

\(\bar u(\bar y, \xi )\) :

\( = u(\bar y, \xi )/L\), nondimensionalized foundation deflection

v :

velocity in meters/sec.

v cr :

critical velocity corresponding to massless foundation

w(x, t) :

beam deflection


beam deflection in the moving coordinate system

\(\bar w(\xi )\) :

=w(ξ)/L nondimensionalized beam deflection

\(\bar w*(f)\) :

Fourier transform of\(\bar w(\xi )\)

x, y :

coordinate axis


velocity parameter

α cr :

critical velocity parameter


mass parameter


moving coordinate

η b :

beam damping parameter

η f :

foundation damping parameter

σ(y, t, x):

vertical stress in the foundation


Dirac delta function


foundation mass per unit depth per unit length of the beam


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Jaiswal, O.R., Iyengar, R.N. Dynamic response of a beam on elastic foundation of finite depth under a moving force. Acta Mechanica 96, 67–83 (1993).

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