SMART MESHING OF IMPERFECT STRUCTURES FOR THE IMPROVED
PREDICTION OF BUCKLING AND POSTBUCKLING BEHAVIOUR
Carol A Featherston
Cardiff School of Engineering, Cardiff University
Queens Buildings, The Parade, Cardiff, CF24 3AA, UK.
ABSTRACT
The buckling loads and pre and postbuckling behaviour of thin shell structures are diminished, in some cases severely, by
imperfections such as geometric imperfections, load eccentricity and material properties. Of these, geometric
imperfections are thought to have the most dramatic effect. The paper presents a technique for the automatic generation
of finite element meshes representative of the real geometry of a structure from data obtained experimentally using digital
image correlation. The method is fast and accurate allowing it to be applied to ‘as manufactured’ components. The
meshes can then be used to analyse the behaviour of these structures on an individual basis to better predict their
behaviour. The method is applied to the case of a curved panel under combined compression and shear which is known to
be particularly sensitive to this type of effect and the results from the finite element analysis compared with those found
experimentally. Initial findings are promising and the technique is currently under further development.
Introduction
It is generally accepted, that the buckling load and postbuckling behaviour of many engineering components are
substantially impaired by the existence of small initial geometric imperfections such as deviations in shape, eccentricities
and local indentations. These imperfections can lead to large discrepancies between predicted and actual failure loads,
which can result in catastrophic failure unless large safety factors are applied, resulting in non-optimised structures. In
industries such as the aerospace industry, which rely on the use of minimum weight structures, this inability to accurately
model the effects of such imperfections acts to reduce the industry’s competitiveness.
Extensive work therefore has been based on modelling imperfections, thus allowing their effects to be assessed. This has
been summarised in reviews eg Simitses [1], and has been collated by several researchers to form imperfection data
banks (Arbocz [2]). Results have been obtained using a variety of techniques, all of which involve some degree of
approximation.
Initial work to develop a general theory of buckling and postbuckling which incorporated sensitivity to imperfections was
carried out by Koiter [3] and Arbocz [4] among others. This work was however, limited to a fairly small range of load and
boundary conditions. Fortunately since then, with the increasing sophistication of numerical buckling analysis software
combined with today’s high powered computers, it has become possible to model the buckling and postbuckling behaviour
of shells under complex load and boundary conditions whilst incorporating the effects of imperfections and other
nonlinearities such as elasto-plastic material behaviour. Nevertheless, difficulties still arise due to the need to model these
imperfections, and convert numerical buckling loads, based on any of several different types of buckling analysis available,
into a design load for a particular structure. Several approaches have been considered.
Two main types of approach exist. The first is based on a linear elastic bifurcation buckling analysis and applies reduction
factors to bifurcation loads to account for geometric imperfections (Samuelson and Eggwertz [5]). The alternative to this
approach is to carry out a fully non-linear analysis with geometric imperfections, plasticity and large deflections accurately
modelled. This requires the amplitude and form of the imperfection to be decided. The most accurate method is to base
any analysis on actual imperfections. Work has therefore been carried out to measure real imperfections using contact
techniques and then model these imperfections (Singer and Abromovich [7]).
In most cases detailed information on the amplitude and form of the actual imperfections in a structure is not available and
is uneconomical to obtain. In such cases researchers such as Speicher and Saal [6] have recommended that
imperfections whose form is based on the first bifurcation or eigenmode be used. Most commercially available finite
element codes recommend a similar approach to designers, setting the maximum amplitude of the imperfection equal to
that anticipated in the component itself. However since these models represent a worst case scenario, the use of this limit
in design calculations will always result in the component being over-engineered.
The technique presented in this paper overcomes these difficulties by using fast, accurate optical shape measurement
methods, to create finite element models representative of real structures, which can be analysed to provide accurate data
on its buckling and postbuckling behaviour.
Specimens
The work described considers the specific case of a curved panel built-in at both ends, simply supported along its two
longitudinal edges and loaded so as to introduce a combination of shear and in-plane bending which vary throughout the
structure (Figure 1). This example was selected as representative of a commonly found component of aerospace
structures (such as an aeroengine fan blade as shown in Figure 1), which is susceptible to failure by buckling, and which
has been previously found to have a buckling load which is sensitive to geometric imperfections (Featherston and
Ruiz[8,9]).
Figure 1 Load Case
Tests were performed on a total of 5 specimens with radii of curvature 100mm. This radius was selected to give
specimens and which were relatively imperfection sensitive and which represented a typical component curvature. Each
specimen had an aspect ratio of 1:1 (again to increase imperfection sensitivity) and was manufactured to be 100mm wide
and 100mm long, thus providing reasonable dimensions for testing, and buckling and collapse loads within the capacity of
the available test equipment. Specimens were manufactured from material 0.55mm thick.
Due to the difficulties of reproducing simply supported boundary conditions using a test rig, an attempt was made to
introduce these conditions by modifying the shape of the test specimens. At the two straight edges of each panel,
o
therefore, an additional 10mm was folded over at 90 , thus allowing rotation about the longitudinal edge but reducing outof plane displacement. This was obviously not ideal, but could at least be represented exactly during the finite element
analysis.
Each specimen was manufactured from aircraft standard specification duraluminium BS1470 6082 – T6. This was
selected for its relatively high yield strength to Young’s Modulus ratio, to promote elastic behaviour during buckling and
early postbuckling. (However since elastic behaviour throughout the testing period could not be guaranteed the finite
element analysis included a full elastic-plastic material profile). The particular grade of duraluminium used, also made it
suitable for heat treatment which was essential for manufacture of the desired shape of specimen.
Specimens were manufactured by hot forming to ensure that the exact radii of curvature and folding angle were obtained
and to eliminate any residual stresses which might otherwise be incorporated during manufacture. Each specimen was
o
heated to a temperature of 250 prior to forming in a fly press using a matched pair of forming tools. Due to the heat
treatment the weight of the formers was sufficient to hold the specimens in the correct shape, and the press was therefore
used simply to align the two formers as the male tool was lowered into the female, on which the specimen was placed.
Cooling was achieved by heat transfer into the tools which were maintained at ambient temperature prior to forming.
Once manufactured, each specimen was impacted to create a series of different geometries with which to test the success
of the mesh generation and modeling technique.
Test Set-Up
The specimens were tested using the rig shown in Figures 2, 3 and 4. Each specimen was held firmly along its curved
edges using a pair of clamps. A series of four bolts which passed through the specimen were then used to hold the clamps
together. One set of clamps was bolted to the fixed end of the test rig imposing a fully clamped boundary condition. The
other set was attached to a loading plate which was then connected to the crosshead of the test machine. The
arrangement for this end can be seen in detail in Figure 3. In order to facilitate the application of the shear force, this end
of the plate was allowed to move vertically and to rotate in plane about its clamped end (i.e. about the x axis); however
lateral displacement was not permitted to prevent twisting of the plate (about the y axis). This was achieved by trapping
the loading plate between two uprights attached to the base plate. Ball bearings between the loading plate and the curved
edge of two vertical spacers allowed rotation about the y axis. The flatter external side of these spacers and the inside
surface of the uprights were hardened and ground, thus allowing them to slide against one another to facilitate movement
in the y direction and rotation about the z axis.
The base plate of the rig was bolted to the Howden universal testing machine and load was applied under displacement
control at a rate of 1mm/min through a loading arm attached to the crosshead. This loading arm was attached to a pin in
the loading plate via a spherical bearing. This resulted in a combination of shear and in-plane bending loads being
introduced into the specimen as shown in Figure 4. The test machine’s computer control software was used to program
the test, therefore ensuring consistency between individual experiments. The software also recorded the applied shear
load using a 10kN load cell and in plane displacement using a built in displacement transducer. Results were sampled at a
rate of 10 points per second.
direction
of loading
end plate
slides
bearings
end guides
folded
over edge
clamps
z
y
test panel
x
Figure 2 Test rig
Figure 3 Detail of the loading end
Applied
Load
Applied
Load
Figure 4 Load application
Digital Image Correlation
The shape of each of the specimens was captured prior to testing using the VIC3D Digital Image Correlation (DIC)
System. 3D image correlation uses two cameras to identify the position of each point on the specimen thereby producing a
coordinate cloud representative of its geometry. The cameras are first calibrated using a calibration plate containing a
regular grid of markers of known spacing as shown in Figure 5a. A number of images are taken of the plate in different
positions (rotations about the x and y axis). Since the position of the markers on the plate relative to each other is known,
this allows the position of the cameras relative to each other to be determined. A random speckle pattern is then applied to
the surface of the specimen using a spray can of black paint (applying this over an initial white basecoat helps to improve
contrast and prevent surface reflections). Sections of this random pattern are located within the image captured by each
camera using a similarity score or correlation function. From the position of each section of the pattern within the field of
view of each of the cameras, and the position of the cameras, its location in 3D space can be determined. This is repeated
over the surface of the specimen to give a full 3D profile.
Figure 5 DIC set-up a) Calibration of cameras using calibration plate b) Shape measurement of specimen
Figure 6 Geometry test specimen 1
Smart Meshing
The information contained in the data clouds for each specimen, representing the x,y and z locations of each pixel along
with curvature in the x and y directions was imported into Matlab R2006b where it was processed to produce a mesh
suitable for finite element analysis (Figure 7). The aim of this procedure was to produce a responsive mesh with varying
density dependent on the local curvature of the specimen (high curvature, high density, low curvature, low density). This
was achieved in two stages. In the first stage each of the rows of data were considered. An initial estimate of the number
of elements in the y direction was made and this was used to divide the total number of pixels in the row into ‘cells’ of
equal size. The average of the reciprocal of the curvatures measured at each pixel of each cell was then calculated (to
give a low value for areas of high curvature and a high value for areas of high curvature). These averages were used to
reproportion the number of pixels in each cell between a stated maximum and minimum to avoid unsuitable or inaccurate
element geometries, to represent the curvature. (Although this process was initially performed using the maximum value of
curvature for each cell thus avoiding any ‘damping’ effects caused by taking the mean this was found to increase
sensitivity in areas of high curvature to such a point that the problem could not be converged). The process was repeated
with the new members of each cell until the cell sizes converged giving the position of the boundaries between each cell
and therefore each element in the row. This was repeated for each row. Once this had been completed for all the rows in
the data set the data from each row not representative of an element boundary was removed to form a reduced matrix in
the y direction. Having processed the data in each of the rows a second stage was entered in which individual columns of
data from the reduced matrix resulting from the first stage of the process were considered in exactly the same way. In this
way a set of data in the form of a matrix reduced from the original in both the y and z directions remained which
represented the corners of each of the elements ie the nodes of the finite element mesh.
Vic3D
Data file
.csv
Remove non-correlated data points
Convert to array format (a x b)
First row
Divide into n groups with
equal numbers of elements
d1,d2.. = a/n
Determine the average
curvature for each group
Resize the
groups in
inverse
proportion to
their maximum
curvature
between
maximum and
minimum limits
d1 = a ×
c xx
1 / c xx1
n
∑c
xx
1
Check for
convergence
Next row
Remove all columns except
the first in each group and the
last in the final group
Repeat for columns
Substitute x,y,z coordinates
for each of the
re
Additional data
ABAQUS
input file
.inp
Figure 7 Smartmeshing algorithm
Figure 8 Typical finite element mesh for standard analysis
Finally a series of elements were added around the periphery of the specimen aligned with those generated automatically
since the DIC method is not able to process information right to the edge of the component.
Model
The panels were modelled using 0.55mm thick S4R5 shell elements. The clamps and loading plate which were also
modelled to allow accurate representation of the boundary conditions on the loaded end of the plate, used C3D8R brick
elements as illustrated in Figure 8. For the panel itself a fully nonlinear experimentally determined elastic-plastic model
was used however for the clamps and loading plate a standard elastic model was considered sufficient.
The boundary conditions which were modelled directly on those found in the experiment can be described by reference to
Figure 9. Movement of edge 1 was restricted in all degrees of freedom, to represent a clamped end condition. Along the
o
longitudinal edges, additional elements representing folded over edges were added at 90 along both straight edges, to
represent simply supported boundary conditions (this has been shown to provide a reasonable approximation for initial
buckling after which the behaviour of these edges needs to be considered). On edge 2, boundary conditions prevented out
of plane displacement x and rotation about the y axis, but movement and rotation in all other directions was permitted thus
allowing shear and bending to be transmitted throughout the plate. Force was applied to the node on the loadplate
corresponding to the shear centre of the panel to avoid twisting.
Force
z
Edge 2
y
Edge 1
x
Figure 9 Boundary conditions
Analysis
Each model was analysed in ABAQUS/Standard using the fully nonlinear Riks analysis method which is particularly
suitable for solving unstable problems, where the load-displacement response is such that either the load or the
displacement may decrease as the solution evolves. This is achieved by automatically controlling step length according to
the curvature of the load versus displacement plot thus ensuring small steps at sudden changes of path direction allowing
the equilibrium in these areas to be closely followed.
For comparison, for each specimen a nonlinear analysis based on a ‘perfect’ structure with an imperfection in the form of
the first eigenvalue was performed. In each case this imperfection was scaled such that its maximum amplitude was equal
to that measured in the specimen.
Results
The buckling loads determined for each of the five specimens tested are presented in Table 1. A degree of scatter can be
seen, as anticipated due to the variation in the amplitude and form of the imperfection introduced in addition to possible
variations in boundary conditions (again due to the geometries of the specimens) and load eccentricities.
Test no.
1
2
3
4
5
Mean
SD
Buckling Load (N)
873
846
730
725
779
791
60
Table 2 Experimentally determined buckling loads
The experimentally determined plot of load versus in-plane displacement (at the point of load application) for specimen 1,
is presented in Figure 10. It is compared with the results of the two FEA analyses for this specimen ie those based on a
mesh derived from the process described earlier (shown in Figure 11) and a perfect model with an imperfection in the form
of the first eigenmode and having an amplitude representative of the maximum amplitude found in the specimen itself.
1
0.9
0.8
Load (kN)
0.7
0.6
0.5
0.4
0.3
0.2
Riks analysis - Smartmesh
Riks analysis - Standard mesh with eigenmode imperfection
Experimental results
0.1
0
0
0.5
1
1.5
2
2.5
3
3.5
4
Displacement (mm)
Figure 10 FEA versus experimentally determined load versus in-plane displacement for test specimen 1
Figure 11 Smartmesh test specimen 1
Discussion
The experimentally detemined buckling loads presented in Table 1 indicate, when compared with those predicted by either
the analytically determined buckling loads for ‘equivalent’ perfect structures (3076N (Featherston and Ruiz [8])) or those
found from linear eigenvalue analysis (1715N [8]) that the existence of geometric imperfections within the panels tested
substantially decreases their structural performance to a degree which varies with the magnitude, form and position of
these imperfections (indicated by the degree of scatter).
The commonly suggested solution to this problem, in which the effects of these imperfections are incorporated by
introducing an imperfection in the form of the first eigenmode, with amplitude equivalent to the maximum amplitude
anticipated into a ‘perfect’ model of the structure and performing a fully nonlinear finite element analysis can be seen by
the results presented in Figure 10. Here the imperfection amplitude has been determined from the VIC 3D digital image
correlation data and incorporated into the finite element mesh. Comparison of the results obtained with those found
experimentally indicate that in this case, this method is conservative, underestimating both the buckling load (calculated as
784N which compares with the experimentally determined value of 873N) and the prebuckling stiffness. The method also
overestimates the postbuckling performance, although it is believed that this is in part due to inadequate modeling of the
effects of plasticity which needs to be addressed for future models.
The results obtained using the method presented in this paper to derive a mesh representative of the exact geometry of
the specimen with a mesh density which varies according to local curvature in order to obtain an efficient solution in terms
of the trade-off between accuracy and processing time are also presented in Figure 10. These can be seen to provide a
more accurate estimate of the buckling load (857N), although in this case the prebuckling stiffness is overestimated. It is
felt that this is due to inaccurate modeling of the boundary conditions, particularly along the top edge of the specimen
which is key to the overall behaviour of the specimen. Further work needs to be carried out to incorporate shape data on
the folded over edges, already acquired prior to testing (again using the VIC3D digital image correlation system), into the
model. These edges are currently modeled as having perfect geometry which will clearly result in an overestimate of the
stiffness of the overall structure. In addition as before the incorporation of material plasticity needs to be improved.
The technique has been shown to be relatively straightforward to use once an algorithm has been written for the geometry
of interest. However further work needs to be done in a number of areas as indicated and greater automation is needed.
Conclusions
The existence of geometric imperfections in addition to other factors such as boundary conditions and load eccentricities
can significantly impair the buckling behaviour of thin shells. This is not generally reflected in the results obtained by
applying many of the existing design rules obtained from analytical solutions to the governing differential equations, or
linear eigenvalue analyses of the structures involved. One possible alternative is to perform a fully nonlinear analysis for
example using finite element techniques, on a perfect model to which an imperfection in the form of the first eigenmode
with an amplitude representative of the maximum imperfection anticipated has been introduced. However this is
computationally expensive, and, as it provides a lower bound, often conservative leading to non optimized structures. The
method described in this paper has been shown to provide a suitable alternative which, whilst accurately representing the
behaviour of an ‘as manufactured’ specimen is relatively simple to apply. Further work now needs to be carried out in
order to test the method further and to introduce improved modeling of boundary conditions and material plasticity.
Acknowledgments
The author acknowledge the contributions made by Mr M Rumfitt, Mr M Eaton and Mr S Mead in carrying out this work
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