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58-istam-sm-fp-30
Proceedings of 58th Congress of ISTAM (http://istam.iitkgp.ac.in)
Held at : BESU Shibpur; Howrah, W.B. (www.becs.ac.in)
IDENTIFICATION OF MATERIAL PARAMETERS OF PULTRUDED FRP
COMPOSITE PLATES USING FINITE ELEMENT MODEL UPDATING
Subhajit Mondal and Sushanta Chakraborty
Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur, India
Abstract: An automatic inverse material property determination algorithm has been
implemented using finite element program and experimental modal testing using a gradient
based inverse eigensensitivity method. The process depends upon the correlation between
these two approaches to extract the in-plane elastic parameters from globally measured
vibration responses of a pultruded FRP rectangular plate as specimen. The paper tries to
formalise the entire process through a real experimental case study so that it can be used as a
regular condition assessment and damage detection tool for pultruded FRP structures used in
infrastructure application.
Keywords: finite element model updating, composite laminate, experimental modal analysis,
material parameter.
Introduction
Apart from its usual weight sensitive aerospace applications, Fiber Reinforced Plastic (FRP)
composite structures are now being rapidly deployed in infrastructural type of applications
where cost and durability are more important. Most of this type of applications uses pultruded
FRP sections. Moreover, fabrication of FRP is totally different from conventional metal
structures, in the sense that the structural and material fabrications are a single process. Thus
the finally achieved material properties still varies widely from the initial guess made from
standard handbooks or from manufacturer’s average data. This makes considerable difference
in finding the dynamic performances of such structures or while finding if the structure has
developed any damage due to prolonged use. A large number of fabrication methods are in
use in fabricating FRP structures for various applications, such as- autoclave moulding, resin
transfer moulding, filament winding, pultrusion, out of which pultrusion process is mostly
common in infrastructural applications. The pultrusion creates continuous profile, like beams,
angles, tubes, plates etc by pulling raw constituent fibre and matrix through a shaping die and
hot cured. With high fibre content and consistent quality and also due to the fact that the
fibres are in tension while drawing, pultruded sections are much stronger and stiffer as
compared to ordinary fabrication and preferred in construction industry. As such FRP
pultruded sections has great application potentials where ordinary conventional materials like
metals have serious problems, such as corrosion near sea shore etc.
Investigators have proposed non-destructive techniques using finite element model updating
to resolve this issue by estimating the average material constants from experimental modal
tests data so that all subsequent analysis can be much relied upon, but the current literature
provides only limited experimental case studies. This is especially true for pultruded sections
deployed in infrastructural applications where such investigations perhaps are not existing. In
addition to this, for infrastructural applications and long time existence of structures, nondestructive periodical health monitoring and condition assessment exercise is mandatory for
pultruded FRP structures.
The current investigation employs finite element analysis technique using ABAQUS, a real
experimental modal testing and subsequent analysis using impact hammer type of excitation,
correlates and updates the finite element model taking in-plane homogenised equivalent
elastic parameters as the causes of discrepancies between these two models. Model updating
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software FEMTools implements the inverse eigensensitivity method. Although automatic, the
method is not straight forward and user intervention is needed in terms of application of
weights in Bayesian environment. At last, the computed material constants are verified by
actual quasi-static characterization tests in an UTM and the results are found to be
encouraging. The main aim of the current investigation is to establish a complete
experimental-numerical combined approach to estimate the material constants of pultruded
FRP composite plate type of structures non-destructively from dynamic responses. The
methodology demands very accurate measurement of natural frequencies and mode shape
data from the directly measured frequency response functions.
Literature Review
The investigations to determine average in-plane material constants from dynamic testing
date back to the mid-80s. The pioneer amongst them are Deobald and Gibson (1988),
Frederiksen (1997) Etc. Deobold and Gibson (1988) used modal analysis and Rayleigh-Ritz
technique to determine the material property orthotropic plate, they have identified that free –
free boundary condition is the best way to determine the elastic constant. More recently,
Hwang et al. (2000) investigated for both thin and thick carbon epoxy composite plates. Joel
et al. (2007) have used frequency and mode shape data to estimate properties of thick
laminated composite plate. The approaches of finite element model updating have been
summarised by the most referred paper of Mottershead and Friswell (1993). A very good
literature survey regarding material property determination of FRP can be obtained from
Rikards (2001) and Lauwagie (2005) and most recently from the paper of Ismail et al. (2013).
Even then, the current literature is very scanty about the infrastructural application of finite
element model updating. Experimental data for such inverse determination specific to
pultruded section of FRP is perhaps non-existent.
Mathematical Formulation and Numerical Implementation
An eight nodded shell element (S8R) is used for the finite modelling of the composite plate in
ABAQUS environment. A 12x12 mesh division was found to be adequate for proper
discretization and is used throughout the present investigation. The finite element program
requires initial values of all elastic parameters for modelling. These are selected tentatively
from the manufacturer’s manual or from established handbooks. Apart from the in-plane
elastic parameters, the finite element program also requires the transverse shear modulus
(G13, G23) which is kept as 5.73E9 N/m^2, and Poisson’s ratio which is taken as 0.15
throughout the investigation specified. The mass density is assumed as 2120 kg/m^3, which
were determined from actual physical measurement of similar samples.
The basic eigen value problem of the vibrating plate can be expressed as
Ku = ω2 Mu
(1)
Where K is the global stiffness matrix, M is the global mass matrix, U is the eigen-vector,
The linearized first order approximation of the relationship between changes in measured
modal properties (i.e. frequencies and mode shapes) and the changes in in-plane material
constants of FRP composites (to be estimated) can be related through a sensitivity matrix as (2)
{Δf } = [S ] {Δr}
Suitable changes are made to the initially guessed parameters {Δr} from the solution of the
above equations and the finite element model of the pultruded FRP plate is updated following
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{r}i +1 = {r}i + {Δr}i
(3)
The error between the experiment and finite element modelling is minimised in a weighted
least square sense through this IEM.
Modal Assurance Criteria (MAC) which is defined as the measure of similarity or
dissimilarity between two vectors is considered to check correlation between measured mode
and numerical mode shape. MAC value equal to 1 indicate good correlation between two
mode shapes.
The entire methodology is explained through a flow chart in Fig. 1 and is self explanatory.
The procedure stops when the error between the finite element model and the experiment
falls below a predetermined small quantity.
Initial guess of Parameter
Change in
Parameters
Experimental eigen values and
eigen vectors
Eigen solution of FE model
Correlation of Mac, Eigen data
Update FE Model
Converged?
Fig.1: Flow Chart of Model Updating Algorithm
No
Yes
Stop
Converged Value
Experimental Investigation
For this current work a rectangular pultruded Glass Fiber Reinforced Plastic (GFRP)
composite plate of size 300 mm x 400 mm having thickness of 10 mm has been fabricated.
Modal testing has been carried out by using Impact excitation from an Impact hammer (B&K
force transducer IEPE 8206-002) and the responses were picked up by accelerometer. Both
these digitised time signals were Fourier transformed in a B&K spectrum analyser 3560-GL4 and the Frequency Response Functions (FRFs) were estimated using the PULSE-LabShop
modal testing software. The FRFs were then curve fitted using the ME’Scope VES modal
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analysis software to extract the eigenvalues and mode shapes. Mode shapes are obtained upto
800 Hz frequency. Fig. 3(A) shows the Experimental setup for the modal testing.
To obtain the material property from quasi static tensile test, FRP coupons as per ASTM
standard (No.D3039/D3039M) has been performed. At least five nos. samples have been
used for each of the parameters. Shear Modulus has been determine using 450 samples (Jones,
1998).
Model updating to estimate material parameters
Fig. 3(B) shows the experimentally and numerical obtained mode shapes. MAC value on the
Fig. 2(B) shows good correlation between mode except 6th and 8th mode. The two modes are
not used in updating. Table 1 shows the comparison and errors of different modes. Fig.2 (A)
shows that transverse shear modulus and Poisson ration is very less sensitive.
Fig.2: (A) Sensitivity of Six Parameters (E1, E2, G12, G13, G23, υ12), (B) MAC value
Fig. 3: (A) Experimental. Set Up for Modal Test, (B) Numerical and Experimental Modes
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Table 1: Comparison of Eigen Values after updating
% in
Error
162.81
Updated
Eigen
values
163.02
2
255.84
256.15
0.12
3
417.42
416.63
0.18
4
480.98
480.98
0.00
5
574.8
574.53
0.04
6
658.95
NA
NA
7
709.67
709.34
0.04
8
743.51
NA
NA
Mode
No.
Experimental
Eigen values
1
0.1
Fig.4: Typical Convergence Curve for E1
Fig.5: (A) Typical Convergence Curves for E2, (B) Typical Convergence Curves for G12
Table 2: Updated Parameters
Trial: 1
Trial: 2
Trial:3
Initial Value
(GPa)
Initial Value
(GPa)
Initial Value
(GPa)
E1
5.00
5.00
2.00
35.50
33.05
E2
25.00
25.00
10.0
32.23
31.80
G12
45.00
45.00
20.00
7.11
5.73
Material
Property
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Updated
Parameter
(GPa)
Experimental
Value
(GPa)
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Fig. 4 and Fig.5 shows the converged elastic parameters from different initial values. Table 1
indicates that the eigenvalues of the updated model now matches exactly to the
experimentally observed modal properties, thus establishing that the updated model is a
proper representative model. The typical convergence curves which are monotonic from all
initial values and only a few iterations are required for convergence. In case of in-plane shear
modulus there is a little discrepancy between experimental result and updated results, though
the updated response is well matched with the experimental one. This discrepancy may be
due to the non availability of universal agreement on the best way to measure the shear
properties (Jones, 1998). More number of samples should be used for characterization test to
get proper average shear modulus. Moreover, updating of out of plane property and Poisson
ratio and the modes of higher frequency range should be included to accurately predict the
property of the composites
Conclusion:
The methodology can be replicated easily to correctly estimate the in-plane Young’s modulus
of pultruded FRP sections non-destructively uniquely. The accuracy of the current
methodology is verified by actual static testing. The present method can be implemented for
condition assessment of composites from time to time on a long term basis conveniently.
References
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Ismail, Z., Khov, H., and Li, W. L., Determination of material properties of orthotropic plates with
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(2013).
Joel Cugnoni, Thomas Gmur , Alain Schorderet, Inverse method based on modal analysis for
characterizing the constitutive properties of thick composite plates, Computers and Structures, 85,
1310–1320,(2007).
Lauwagie, Tom,Vibration-Based Methods for the Identification of the Elastic Properties of Layered
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Rikards, R., Chate, A. K., and Gailis, G., Identification of Elastic Properties of Laminates based on
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Shun-Fa Hwang,Chao-Shui Chang, Determination of elastic constants of materials by vibration testing
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Frederiksen, P. S., Experimental procedure and results for the identification of elastic constants of
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Mottershead, J. E., and Friswell, M. I., Model Updating in Structural Dynamics: A Survey, Journal of
Sound and Vibration, 167(2), 347–375, (1993).
Deobald LR, Gibson RF., Determination of elastic constant of orthotropic plate by modal
analysis/Rayleigh-Ritz technique, Journal of Sound and Vibration, 124:269-283, (1988).
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