Investigation of Global and Local Free Vibration of Slender FRP Beams
Publication: Journal of Composites for Construction
Volume 27, Issue 3
Abstract
This paper addresses a theoretical investigation of the undamped free vibration of slender fiber-reinforced polymer (FRP) beams. While previous studies have focused on global vibration modes in the range of excitation frequencies reported in typical civil engineering problems, the present one discusses the influence of local modes, which can be useful for nondestructive indirect characterizations of materials and detection of damage in bridges and frames. Appropriate shape functions for flexural, flexural–torsional, torsional, and local vibration modes were chosen, and Lagrange’s equations of motion were derived after determination of the energy components. The formulation accounts for C and I sections, different end conditions, and enriched shapes for local modes, and transverse shear deformation is neglected. The results from the proposed equations for a C section were compared to those from generalized beam theory (GBT). Overall, an excellent agreement with GBT was achieved, except for global modes in beams of relatively short equivalent length, in which transverse shear deformation is relevant. Regarding local modes, the associated natural frequency tends to stabilize with increasing lengths and the transverse stiffness component plays a major role in the modal behavior. A study of the influences of flange width and cross-sectional shape (C and I) revealed that frequency drops significantly when flanges become wider. The influence of transverse shear was negligible for beams with a higher than 40 slenderness ratio. The formulation was successfully compared to a previous experiment performed by the authors.
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Data Availability Statement
Some data, models, or codes that support the findings of this study are available from the corresponding author upon reasonable request (data from GBT and calculation spreadsheets).
Acknowledgments
This study was partially financed by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior—Brasil (CAPES)—Finance Code 001 and by Brazilian funding agencies FAPERJ and Conselho Nacional de Desenvolvimento Científico e Tecnológico.
Notation
The following symbols were used in the paper:
- A
- cross-sectional area;
- A1, A2
- constants;
- B1, B2
- constants;
- b
- plate width;
- C1, C2
- constants;
- C
- constant;
- D11, D12, D22, D66
- plate stiffness parameters;
- EL
- longitudinal modulus of elasticity;
- EL,f
- plate longitudinal modulus in bending;
- ET
- transverse modulus of elasticity;
- ET,f
- plate longitudinal modulus in bending;
- F
- shape function for global vibration modes;
- f
- shape function;
- GLT
- in-plane shear modulus;
- g
- shape function;
- h
- thickness;
- Iw
- warping constant;
- IY
- moment of inertia about the minor axis;
- IZ
- moment of inertia about the major axis;
- J
- Saint-Venant torsional constant;
- [Ke]
- elastic stiffness matrix;
- k11, k12, k22
- elastic stiffness matrix components;
- L
- member length;
- Lef
- effective member length;
- [M]
- matrix of equivalent mass;
- m11, m12, m22
- equivalent mass matrix components;
- N
- number of constituent plates comprising the cross section;
- Nf
- number of flange plates comprising the cross section;
- n
- number of half-waves;
- q
- generalized coordinate;
- r0
- polar radius of gyration;
- rY
- radius of gyration about the minor axis;
- T
- kinetic energy;
- t
- time;
- U
- strain energy, displacement in the X-direction;
- u
- displacement in the x-direction;
- V
- displacement in the Y-direction;
- v
- displacement in the y-direction;
- W
- displacement in the Z-direction;
- w
- out-of-plane plate deflection;
- X, Y, Z
- global coordinates;
- x, y, z
- local coordinates;
- α, β
- auxiliary parameters for computation of natural frequencies;
- ρ
- density;
- νLT, νTL
- major and minor Poisson’s ratios;
- θ
- rotation about the shear center; and
- ω, ωL
- natural frequency and local natural frequency.
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Published In
Journal of Composites for Construction
Volume 27 • Issue 3 • June 2023
Copyright
© 2023 American Society of Civil Engineers.
History
Received: Jan 20, 2022
Accepted: Jan 25, 2023
Published online: Mar 28, 2023
Published in print: Jun 1, 2023
Discussion open until: Aug 28, 2023
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