ESPI-MEASUREMENT OF STRAIN COMPONENTS ON A CFRPREINFORCED BENDING BEAM
E. Hack and A. Schumacher
EMPA – Swiss Federal Laboratories for Materials Testing and Research
Laboratory for Electronics/Metrology
Laboratory for structural analysis
CH-8600 Dübendorf
ABSTRACT
We describe a laboratory experiment to assess the stress transfer at the end of a CFRP plate, with a special focus on shear
and normal stresses. A small-scale steel I-beam reinforced with an adhesively bonded carbon fiber reinforced polymer (CFRP)
plate was subjected to four-point bending. Finite element analyses (FEA) of the bending deformations were carried out to
predict strain gradients near the end of the CFRP plate. In order to measure these strains, phase-stepping 3D-DSPI was
employed. The speckle pattern de-correlates rapidly due to the inevitable rigid body motion of the specimen. Evaluation of the
ESPI phase maps was possible only by shifting the final image by an appropriate number of pixels for each load step. Strain
values are evaluated using local polynomial fits to the measured in-plane displacements after correction of the rigid body
motion. They are compared to FE predictions. An uncertainty analysis is given.
Introduction
Reinforcement of structural elements using adhesively bonded carbon-fibre reinforced polymers (CFRP) is a well-advanced
strengthening method. In particular, concrete or steel beams are strengthened with CFRP plates. One of the failure modes is
the debonding of the plate from the beam, initiated at the plate end. Debonding is caused by high shear and normal interfacial
stresses derived from the transfer of plate tensile forces to the beam. In the past, work has been carried out to model
analytically the stress concentrations at the plate end [1, 2], and FE work have been used to verify these models [3].
Few quantitative comparisons have been made, however, between values derived numerically and those obtained
experimentally. For this reason, a model system was devised to assess the stress transfer between a CFRP plate and a
structural element both numerically and experimentally. A small-scale model steel beam was reinforced, tested and measured
with an experimental full-field method in the laboratory.
Non-contact optical methods are increasingly used as verification for FEA results in mechanical engineering [4-6]. Both moiré
interferometry and speckle methods have been used, e.g. to detect crack initiation from strain profiles in fibre-reinforced
composites [7], to assess strain in composite lap-joints [8] or to measure the strain concentration close to the weld toe of a
tubular steel joint [9]. For moiré interferometry a grid must be applied to the (flat) specimen surface. Digital speckle pattern
interferometry (DSPI) is well suited for strain analysis due to its high sensitivity, although it can suffer from speckle
decorrelation due to rigid body motion, especially for small areas of interest. Hence, an evaluation method that regains the
speckle correlation by image shifting was used to measure the displacement fields [10, 11]. Recently, we used this method to
measure the strain components on a CFRP reinforced concrete beam [12].
A Finite Element Analysis (FEA) of the strengthened beam was conducted using the commercial finite element code ABAQUS.
For quantitative analysis and comparison to experimental DSPI results the in-plane displacement data are interpolated along
selected lines and numerically differentiated. Different fitting strategies were used to average out the noise, among which is a
set of cubic splines within intervals along each line. The polynomial coefficients of the splines were calculated according to a
least squares criterion under the boundary condition of continuity of first and second derivatives of the resulting fit line at the
interval borders. The strain values were then calculated numerically from the fit lines. An uncertainty estimation will be given
based on the comparison of different fit strategies. The standard uncertainty in the strain values from the DSPI evaluation can
be estimated to be 20 x 10-6.
Experimental
For the reinforcement of the small-scale, 1.2 m long steel I-beam with 120 mm width we used CFRP strips of 50 mm width and
2.4 mm thickness. The plate was joined to the beam with adhesive from SIKA. The beam was loaded with manually controlled
hydraulic jacks in a four-point bending arrangement as seen in Figure 1. In order to allow interferometric measurements, the
experiment was performed on a vibration isolated optical table in a temperature stabilized laboratory.
Figure 1. Experimental set-up. Load-frame with hydraulic jacks and four-point bending beam placed on the optical table.
Green light is from the Nd:YVO laser.
Steelbeam
Adhesive
y
CFRP
x
2 mm
Figure 2. Field of view of 16x11 mm2 . The adhesive layer interfacing the CFRP plate and the steel beam is clearly seen.
Using a 3D-DSPI system (Steinbichler), measurements of the deformation at the end of the CFRP plate where stress
2
concentrations are expected were carried out. Figure 2 shows the field of view of about 16x11 mm . The object was
illuminated with a Nd:YVO-laser (VERDI from coherent) with λ = 532 nm sequentially from three directions. At each load state,
DSPI phase maps for the three illumination directions were taken with OPTOCAT Software (Breuckmann). A standard fourframe phase-stepping algorithm was applied. The phase maps of the displacement components for the three illumination
directions were transformed to Cartesian coordinates (u, v, w) using the sensitivity matrix which was calculated from the
geometry of the experimental set-up. This results in a sensitivity of 0.299 µm/fringe in-plane (u and v components) and
0.183 µm/fringe out-of-plane (w component).
One camera pixel corresponds to a 21-μm sided square on the object (teli camera with Matrix Vision frame grabber).
Measurements were performed within the linear-elastic behaviour range of the beam, with a load increment of 27 kN, up to a
deflection at mid-span of approximately 1.0 mm. After each major load step, a series of data was taken with additional small
load increments. This procedure guarantees that at least one pair of well-correlated speckle patterns is found for each of the
major load steps after an appropriate shift of the images by an integral number of pixels. The image pair with the best
correlation was chosen for subsequent analysis. The phase maps of each load step were added up to the total load of 27 kN.
Data evaluation
In order to assess the variations of the sensitivity matrix, it was calculated on a grid of 11x11 points across the field of view
(FOV). It was seen that the individual sensitivity matrix elements vary less than 1% from centre to edge of the FOV, and the
sensitivity could therefore be assumed constant. Although this variation seems negligible, the influence of the large rigid body
displacement on the displacement components through the off-diagonal elements of the sensitivity matrix had to be assessed.
This was done by simulation according to the following procedure.
First, the in-plane displacement of the object due to the loading is known from the image analysis (speckle pattern correlation).
It was found to be 5 pixels in x-direction, and 36 pixels in y-direction, corresponding to a displacement of 0.11 mm and
0.79 mm, respectively. This in-plane displacement introduces not only an offset in the fringe order (350 fringes in x, and 2640
fringes in y-direction), but it generates also a fringe field with a fringe density between 5 and 16 fringes/mm. These simulated
fringe fields had to be subtracted from the measured phase maps first. Next, a value of 1.5 mrad for the in-plane rigid body
rotation was determined from the phase map of the x- and y-displacements. Again, the phase map for this parameter was
simulated for all displacement components using the sensitivity matrix. The same was done for the out-of-plane rotation
components which were determined from the z-displacement field. However, the out-of-plane rotation does not affect the inplane phase-maps (x and y- components) in our configuration. The only parameter of the rigid body rotation that could not be
extracted from the images was the out-of plane translation. Simulation shows that out-of-plane translation does affect the xand y-components by introducing a virtual strain. This “strain” can be understood as being due to the change in object
magnification when the object moves to or from the camera. Hence, there is an unknown offset in the strain values exx and
eyy. It might be found from the Poisson ratio, or from comparison with FEA. After all, the remaining influence of the rigid body
motion adds at most 5 μm/m to the strain uncertainty. Hence, a diagonal sensitivity matrix is a good approximation for the
subsequent data evaluation.
The resulting phase maps after elimination of rigid body in plane translation and in-plane rotation fringes are displayed in Fig.3.
The stress transfer through the adhesive layer is clearly seen in the x-displacement field as a rapid change in fringe
orientation.
Figure 3. Fringe field for the in-plane displacements in x-direction(left) and in y-direction (right). One fringe corresponds to a
displacement increment of 0.3 μm.
It is well known that strain evaluation from a displacement measurement system such as DSPI is prone to high uncertainty
components. For DSPI, phase noise is the dominant uncertainty factor, and it is usual to reduce this noise by averaging or
interpolating the data. Other influence factors such as temperature instability, illumination point instability and sensitivity matrix
variations were modelled and their effect was estimated from simulations using sensitivity matrix analysis. These influences
were found to add a smooth, slowly varying systematic uncertainty component across the FOV, and their effect was negligible
when strain values are of interest. For strain evaluation local differentiation in displacement components were calculated,
causing most systematic effects of above influence factors to cancel out.
After demodulation of the displacement phase maps, evaluation of the strain was carried out by local low order polynomial
fitting. Lines parallel to the lamella (x-direction) and perpendicular to it (y-direction) at different positions were extracted for
comparison with the FEA results. The uncertainty in the strain values from the DSPI evaluation is estimated from the standard
deviation over a range of constant strain values. It is found to be around 100 μm/m for values obtained from the more noisy ydisplacement field, and around 40 μm/m for values obtained from the x-displacment field..
FEA analysis
A FEA of the strengthened beam was conducted using the commercial finite element code ABAQUS [13]. The steel beam,
plate and adhesive were modelled with 8-node bi-quadratic plane strain quadrilateral elements with reduced integration. In
order to accurately capture the stress concentrations at the plate end, the mesh was refined along the material interfaces. Only
half of the beam was simulated due to symmetry. The overall geometry was modelled as rectangular, i.e. with flat boundary
surfaces and sharp 90° edges. The values for Young’s moduli of the materials were assumed to be E(steel) = 207 GPa,
E(adhesive) = 12.8 GPa (data sheet), E(CFRP) = 181 GPa (product specification), while Poisson ratios were set to 0.3 for all
materials. Results of the FEA analysis will be presented at the conference.
Results
Figure 4 shows the strain map for the longitudinal strain εxx in ppm. Strain levels are small in this component, but there is a
slight tendency for higher strain in the lamella. Figure 5 shows a plot a cross-sections through the interface for various xpositions, separated by a distance of 1 mm. A distinct transition from positive strain to compression across the adhesive layer
is seen. Strain measured during the experiment on top of the I-beam’s lower surface was 100 μm/m, in good agreement with
the strain levels measured by DSPI..
Fig.4. Longitudinal strain εxx in ppm.
Fig.5. Longitudinal strain εxx in ppm at various positions across the interface.
Figure 6 shows the strain map normal to the interface, εyy , in ppm. There is no distinct maximum in the strain field. Normal
strain has a value of about 200 μm/m.
Fig.6. Normal strain εyy in ppm.
Figure 7 shows the shear strain component εxy in ppm. There is clearly a maximum strain of around 1100 μm/m in the
adhesive layer, and a negative value of about -130 μm/m in the steel section. The counterpart, Fig.8 shows the shear strain εyx
in ppm. There is no prominent maximum near the adhesive layer. The field has an average value of +45 ppm with a standard
deviation of 105 ppm
Fig.7. Shear strain component εxy in ppm.
Fig.8. Shear strain component εyx in ppm.
In the following, we discuss the shear strain component εxy only, as the dominant component in the strain distribution, and a
debonding failure would be attributed to this component.
Figure 9 is a set of cross-sections parallel to the y-axis through the strain field εxy from Fig. 7. The lines are taken close to the
edge of the lamella and in distances increasing in steps of 1 mm. The strain value in the steel section remains constant at
about -130 μm/m with a standard deviation of 45 μm/m. For comparison, the cut at a distance of 7 mm from the edge of the
lamella is overlaid. It has an average value of -220 μm/m with a standard deviation of 40 μm/m.
Fig.8. Shear strain at different x-positions across the interface.
Conclusions
It was shown that phase-stepping DSPI can be used for quantitative experimental mechanics analyses, even if large rigid body
motions are present. This is subject to the condition that the loading is quasi-static, and that it is possible to retrieve the
speckle correlation by shifting the DSPI phase images an appropriate number of pixels. Adding up too small load steps
introduces a large variance into the final phase map due to the inherent speckle noise. For a subsequent strain analysis, noise
should be kept to a minimum in order to facilitate the interpolation and derivation of the displacement fields. However, because
it is not a priori known for which load level the best correlation will be retrieved when a shift of an integer number of pixels is
applied, it is still advisable to take measurements for a series of small load increments at each load state up to the final load.
The appropriate pixel shift can then be determined off-line.
The comparison to FEA calculations performed on the model system is conclusivet. A full-field comparison with experimental
results will be performed in the near future.
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