Vibration of Multi-span Thin Walled Beam Due to Torque and Bending Moment

  • Vinod Kumar VermaEmail author
Conference paper


In this paper, the flexural torsional vibration of a thin walled beam due to combined action of bending moment and torque has been analyzed to calculate the natural frequencies and corresponding eigenvectors under different combination of bending moment and torque. The aim is to account the effect of intermediate support on the dynamic behavior of beam compared to single span beam. Also, the interaction formula has been derived for axial force, bending moment and torque for single span and multiple span beams. In the case of lateral buckling of beams, the critical bending moment is inversely proportional to length, which is the condition of zero natural frequency. Therefore, the lowering of natural frequency due to increased length is expected. However, the effect of length on the ratio of torsional and flexural component of vibration is not so obvious in spite of the fact that a phenomenon of flexural torsional vibration is important in long span beam only. The calculation shows that as the length increases, given the bending moment, the torsional component increases as the length increases. This fact is important in case of long span bridges to show that in-plane flexural vibration will trigger the flexural torsional vibration. The problem of multi-span beam is also important in hydropower plants where due to large length of turbo generator shaft, to control the lateral vibration of shaft several intermediate guide bearings are provided to latterly support the shaft. This is a case of multi-span shaft with lateral constraint but torsionally unrestrained at intermediate support. The impact of intermediate support on the on the torsional component of flexural torsional vibration may warrant the consideration of flexural torsional vibration in the design of intermediate support.


Differential equation Thin walled beam Torque Vibration 


Am, Bm, Cm

Amplitude of mth harmonic of displacement


Cross sectional area


Young’s modulus


Modulus of rigidity


Polar moment of inertia


Moment of inertia about binormal


Moment of inertia about principal normal


Sectorial moment of inertia


St. Venent torsional constant of section


Length of the beam

m, m1

Rate of change of phase angle per unit length

Mn, Mb

Bending moment along unit vectors n and b


Twisting moment


Distributed torque

Qn, Qb

Shear force along unit vectors n and b


Axial force

Qy, Qx

Euler’s buckling load


Torsional buckling load

qt, qn, qb

Uniformally distributed load along unit vector t, n and b


Distance measured along the arc of the curve from the fiducial point

t, n, b

Unit vectors parallel to tangent, normal and binormal

ut, un, ub

Displacement along unit vectors t, n, b

x, y, z

Coordinate axes along b, n, t


Second order term of normal bending moment


Second order term of binormal bending moment


Second order term of twisting moment

θt, θn, θb

Rotation along unit vectors t, n, b

ρn, ρb

Change in curvature along unit vectors n, b


Mass per unit length


Vibration frequency in rad/s


  1. 1.
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  2. 2.
    Verma VK (1997) Geometrically nonlinear analysis of curved beams. In: 42nd congress of Indian Society of theoretical and applied mechanics (ISTAM), South Gujrat University, Surat, Gujarat (GJ), IndiaGoogle Scholar
  3. 3.
    Verma VK (2001) Vibration of beams under combined action of bending moment and torque. In: 46th congress of Indian Society of theoretical and applied mechanics (ISTAM), Regional Engineering College (now, National Institute of Technology), Hamirpur, Himachal Pradesh (HP), IndiaGoogle Scholar

Copyright information

© Springer India 2015

Authors and Affiliations

  1. 1.Regional OfficeNational Hydroelectric Power Corporation (NHPC)ChandigarhIndia

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