Quasi-static Structural Analysis with LS-DYNA Merits and Limits
Authors:
Karl Schweizerhof*, Matthias Walz, Wilhelm Rust, Ulrich Franz
CAD-FEM GmbH, Stuttgart/Hannover, Germany
Markus Kirchner, University Karlsruhe,
* also Institute of Mechanics, University of Karlsruhe, Germany
Correspondence:
Karl Schweizerhof
LS-DYNA Group
CAD-FEM GmbH
Friedrich-List Str. 46
D-70771 Leinfelden-Echterdingen
Germany
Tel: +49-(0)711-9907450
Fax: +49-(0)0711-9907456
E-Mail: kaschweizerhof@cadfem.de
Abbreviations:
FE: finite element, CD: Central Difference scheme
Keywords:
Quasi-static analysis, nonlinear structural mechanics,
buckling analysis, roof crush analysis
1
ABSTRACT
In principle all events in structural mechanics can be considered as transient, however a static approach allows often much simpler analysis, in particular, if inertia or
friction effects do not play a major role and if the structural behavior remains mainly
linear. However, in the nonlinear regime the situation is rather different. Though
much success has been achieved in static analysis of mildly nonlinear structures,
computing buckling loads, ultimate loads with nonlinear material and even contact
problems with fairly constant contact partners, often convergence is very poor, even
if the algorithms are consistently developed e.g. concerning linearization. Convergence is often difficult to achieve, if postbuckling loads of complex structures and
contact problems with changing contact regimes involving friction have to be considered. Then treating the problems as transient - so-called "quasi-static" - resolves
many algorithmic and numerical obstacles, though other problems appear, in particular concerning the interpretation of the results for practical use. The highly sophisticated contact algorithms in addition to the large numbers of model features available in LS-DYNA make it a very valuable tool also for quasi-static analyses such as
of roof-crush, buckling and postbuckling behavior of shell structures. Even with the
rather small time steps needed for explicit time integration LS-DYNA proofs to deliver
reliable results, if some standard measures are taken into account and if the analyst
checks the results carefully for kinetic effects.
Within the analysis particular focus is on the sensitivity of the results concerning element type and hourglass control versus fully integrated elements, loading velocity
and mesh refinement. Also the computer time-saving effects of mass-scaling are
discussed. A final discourse shows the possibilities as well as the limits of the presented procedures.
INTRODUCTION - Why Quasi-Static Analysis and How ?
Though almost any problem in structural mechanics can be considered in principle
as a transient process, it is often much simpler to ignore all transient effects and stick
only with statics. The latter has proven to be the successful procedure for most
structural mechanics analyses in engineering. However, in particular when the tasks
are highly nonlinear then the simple static approach fails in areas of high interest
such as the computation of postbuckling branches [1] or in contact dominated problems including friction. Then the reliability of the computation to deliver an answer
with acceptable efficiency is rather limited; extremely small time steps resp. load
steps are needed. Convergence is hard to achieve, as either the matrices are bad
conditioned or the contact region changes within the iterations necessary for nonlinear solution methods. Then the standard Newton type methods loose their capability
of quadratic convergence with rather large load steps and other strategies may prove
more efficient.
Now the general procedure, how to choose an appropriate solution scheme is discussed. For a structural mechanics problem we get within the process of semidiscretization after the spatial discretization with Finite Elements (FE):
M  ü  Cu  R(u)  F(t)
2
Now either an explicit or an implicit time-integration scheme can be chosen. For a
general Newmark-type integration we get the following interpolation in time for a
solution in time step n, see [2]:
 n 1  R(un 1)  F(t n 1)  Fn 1
M  ün 1  C  u
 1

un 1  un  tu n  t 2 (   ) ün   ün 1
 2

u n 1  u n  t(1 -  )ün   ün 1
M is the mass matrix; C is the damping matrix; u, u , ü are the displacement,
velocity and acceleration vectors at the nodes; R is the vector of internal forces,
containing also parts from contact areas; F is the vector of external time dependent
forces;  t is the time step size, t n 1 is the current time after n+1 time steps. The
interpolation leads to an equation only dependent on the unknown displacements at
time t n 1 and the known quantities at t n :

1
M  un 1  R(un 1)  Fn 1  M  u
t  

1
un  t u n    1  1 ün
u 2
t  
 2

2
As the general case is nonlinear, usually a linearization is performed which finally
leads to the following iteration algorithm within the time-step n+1; damping is neglected for reasons of simplicity. First the solution of the linear equation system is
performed for  u, then the update of the displacements follows.
 1



1
 t 2   M  KT (ui ) u  Fn 1  R ui   M u  t 2   ui 




ui 1  ui  u
With a proper selection of the parameters  and  various time integration schemes
can be chosen. The standard Newmark scheme, an implicit integration scheme, is
obtained with the following parameters  = 0.25 and  = 0.5. For  = 0 the equations
have to be slightly reformulated and the scheme becomes a recurrence scheme
even in the nonlinear case. Then the accelerations are the primary unknowns and
the displacements and velocities are computed with the general Newmark formulas:
 M   tC  ün 1  Fn 1  R(un 1)  C u n  t(1  )ün 


Implicit Methods
The interesting aspect inherent for implicit methods is that always a matrix factorization has to be performed for the solution of the equations within an iteration even
when the mass matrix is a diagonal matrix. An algorithm identical to standard statics
is obtained, if the time step is set to infinity, assuming then that the load is applied
very slowly. If the latter is true and the time step is chosen to be smaller, then we find
that the matrix used for the iteration in the standard Newmark algorithm is no longer
the stiffness matrix but is updated by the mass matrix and in the case of damping
also by the damping matrix. This has two consequences:
3


If a Newmark algorithm is used, then the condition of the iteration matrix is altered by adding a positive definite matrix e.g. M, often improving the convergence behavior.
The modification of the iteration matrix leads to a loss of information about the
status of the structure concerning stability, as the static stiffness matrix K is no
longer available and special measures have to be used to gain such an information [3]. In addition the algorithm leads to a forward type marching solution, thus
snap-back is not possible, as obtained with arc-length controlled schemes. Also
bifurcation points remain often undetermined and it is dependent on the time step
size, which branch is followed in a buckling prone system, as the iteration matrix
cannot be used directly to decide if there is a bifurcation point and which path
has to be followed.
Explicit Methods
If the parameters are chosen such that  = 0 and  = 0.5 then the so-called Central
Difference (CD) method is obtained. Though the method is usually written and implemented in a slightly different fashion, see [4], the general difference to the implicit
method is obvious. If the mass matrix M and the damping matrix C are diagonalized
as e.g. by mass lumping for the element matrices, then the scheme is called explicit
and the solution is very simple and efficient. The equations are uncoupled on the left
hand side and no factorization of a matrix is necessary. The penalty is the conditional
stability, as the time step has to be chosen appropriately to follow the CourantFriedrichs-Levy (CFL) condition. No iteration is needed and the algorithm is a simple
forward marching scheme even for the nonlinear case.
In order to achieve an optimal performance for large size models the spatial discretization in the so-called explicit codes such as LS-DYNA and others is performed using special, efficient element formulations with often reduced capabilities, in particular
so-called hourglass-controlled elements [5][6], based on fully reduced integration
with hourglass stabilization. The latter have to be used with some care and the users
have to control the deformations in the hourglass modes to detect possible instabilities. The latter is rather simple, as e.g. LS-DYNA reports the hourglass energies not
only in total but also for each material resp. part of the structure. However, the elements are extremely efficient and rather robust, in particular for meshes with large
element deformations. Also often the rate formulations used for the stress updates
are cited as possible sources of errors, which is true for some ("academic") cases,
but in most real world problems seem to be of less importance for the answers of
interest to the users at least in shell problems.
Implicit or Explicit Methods
In general it is common knowledge that implicit methods should be preferred for the
analysis of static and quasi-static problems. This is based on the higher efficiency of
implicit methods for problems of longer duration with rather low velocity loading.
Then the allowed time step size in implicit schemes is by far larger than the allowable
time step size in the explicit CD scheme outweighing the program efficiency of such
codes as LS-DYNA. Also the answers are smoothed out by the large time steps in
implicit schemes and high frequency damping may be achieved.
However, in real practical simulations the situation is rather different. Often convergence, which is absolutely vital for implicit schemes is very hard to achieve, and very
small time steps are needed to obtain a solution at all. Often the condition number in
pure static analysis is too bad such that a transient algorithmic treatment is neces-
4
sary and also the active set of elements and nodes in the contact zones changes too
rapidly such that permanent adjustment of time steps is needed to carry on in the
simulation. Often, of course, a higher mesh resolution would have been required to
avoid high mesh distortion and bad conditioning; then convergence would have been
better. However, for current size models such high mesh resolutions appear to be
unacceptable for implicit algorithms due to limited computer resources and also due
to the limitations of current equation solvers.
In such situations such as roof crush, computation of postbuckling loads, situations
with high frequency response even under low velocity loading and many contact
problems explicit schemes have shown their best. The storage requirements are
small and the algorithms are fitting very well already now to parallel computers. Due
to the simple forward marching scheme the programs deliver answers to the problems in a decent time frame, if the model is set up even with only reasonable care.
Dynamic Relaxation
The idea to use the explicit CD scheme for the solution to nonlinear problems is
rather well known since long under the name "dynamic relaxation" (DR) [7][8]. This
means that the structure is loaded and the vibration is then damped to rest in the
displaced position of static equilibrium. Then the velocity and the corresponding kinetic energy is below a certain, very low tolerance. This works rather well, if the
damping parameters of the standard Rayleigh damping with diagonal mass matrix M
and diagonal damping matrix C = cM are chosen according to the frequencies excited. If the frequency band excited is rather small, then a solution is achieved with
few time steps. In general situations this is not the case and more sophisticated adjustments of the damping parameters c have to be used as suggested by Papadrakakis [9]; such algorithms are also implemented in LS-DYNA. If the mass density of
the elements is also adjusted - so-called "mass scaling" - then the width of the frequency band can be driven to a smaller value improving the effect of damping.
The experience with the application of the DR-algorithm to real size problems shows,
however, that a rather large number of steps and adjustments is needed to satisfy
the tolerance rendering it a tool with very limited capabilities.
Simple transient analysis with energy control
Thus it appears to be more efficient for large size simulation models to abandon the
goal of almost complete rest and to perform fully transient analyses with a control on
the kinetic energy of the system under load. As most real world problems under consideration even in the quasi-static case show some slightly transient behavior, some
kinetic energy appears to be bearable. This is the case in buckling and contact with
friction, where the buckling process as well as any sliding is connected with some
velocity. To which amount kinetic energy can be accepted is then depending on the
problem (and on the experience of the analyst with similar problems). Comparison to
modifications with different loading velocities are indispensable to judge the results.
Quasi-Static Nonlinear Analysis of Telescopic Cantilever
The structure to be considered is a telescopic cantilever beam - a crane arm - where
two structural tubes with rectangular cross section slide in each other and are loaded
by a single load at the upper end of the sliding tube. The interaction and load transfer
between the beams is via contact leading to a rather nonlinear behavior. It was important to get information about the failure loads and the failure behavior. The latter
takes place in the sliding beam in the vicinity of the transition. The model chosen for
5
presentation in this contribution shows only the characteristics of the real problem,
which cannot be presented due to proprietary reasons. Thus the FE model is restricted to the vicinity of this transition zone, see figure 1 and in order to get the correct loading the sliding part is enlarged by a rigid beam to full cantilever length. The
static analysis with implicit codes showed difficulties, when the failure and postfailure
behavior had to be captured, in particular for the rather complex buckling behavior of
the original structure. It was hard to distinguish between numerically and physically
based convergence problems.
Figure 1. Simplified FE model of critical part of telescopic cantilever tubes
Using LS-DYNA for a quasi-static analysis the force load at the cantilever was simulated by a prescribed displacement resp. prescribed velocity, which was increased
linearly from zero. In order to get a realistic behavior the velocity was chosen such
that the inertia forces remained negligible. This was controlled by checking the equilibrium - forces and moments - in various cross sections with the reaction force at the
displacement controlled cantilever end. Some two runs were needed to get reasonable hints about an efficient but not overly high velocity. To obtain an idea about the
failure loads, which is not possible directly with the perfect structure, the first eigenmode obtained from an eigenvalue analysis, which was performed in the implicit
static simulation, was taken to get an imperfect structure by adding it with a small
amplitude to the initial perfect geometry. Hourglass controlled Belytschko-Tsay elements were used for the spatial discretization.
Figure 2. Progress of plastic buckling of telescopic cantilever (simplified model)
6
The failure behavior in the simulation was as expected; the structure behaves linearly
up to failure, then buckling and yielding of the material - elasto-plastic material 24 in
LS-DYNA - occurs almost concurrently, see figure 2. There is a clear buckling in the
side web. During failure the dynamic effects increase as in reality. Further studies
with the implicit code resulted in failure loads which were identical up to 1% to the
LS-DYNA results.
The analysis with LS-DYNA showed the known advantages of simple transient nonlinear analysis: Even the first attempt delivered good information about the real failure behavior. The analysis could be performed with little memory requirements in
about 10 CPU hours on an SGI Indigo 2. With minor adjustments the analysts could
report to the client and further studies could be performed with varying e.g. circular
cross sections.
Quasi-static Analysis of Cylinder Buckling
Axially loaded cylindrical shells, in particular, if they are empty, are prone to buckling
and imperfection sensitivity plays a great role. Nowadays, it is common knowledge,
how to compute the first bifurcation resp. snap-through load of such cylinders under
static load conditions. Unfortunately, these loads mostly are of little value for design
purposes, as geometric imperfections lower these loads considerably and the structures are sensitive to minor perturbations in loading and boundary conditions, thus
the lowest load level in the so-called post buckling is of interest. Since long it is a
major goal for many scientists to compute then all possible postbuckling branches in
order to obtain the so-called lowest postbuckling load, which could then be used for
design purposes. However, this was up to now only possible for structures with small
numbers of degrees of freedom. Real size problems become computationally very
difficult; the stiffness matrices are bad conditioned, often no longer positive definite
and convergence is almost impossible to achieve [1],[3]. As an alternative the transient behavior could be considered and surprisingly this approach delivered very
useful results [1][10]. For relatively large series of imperfect cylindrical silo shells with
and without content "transient" postbuckling loads could be computed which are very
close to design loads from current norms DIN 18800[11] and ECCS [12].
Figure 3: Axially loaded cylinder, material and geometry data of study.
7
Figure 4: Load deformation curve for quasi-static analysis with implicit program
FEAP-MeKa [13]
Figure 5: Load - deformation curve for loading velocity v = 150 mm/s; deformation at
various states
8
Implicit and explicit transient analyses have been performed. As in the experiments
the cylinders were loaded by the axial motion of a very stiff plate leading to a displacement controlled process, see figure 3.
The FE models were fixed in the radial and tangential directions at both boundaries
and allowed to move in axial direction, thus contact conditions had to be considered
in each step.
For low velocity loading e.g. 0.01 mm/s up to 1 mm/s, see e.g. figure 4 in detail, the
results were completely coincident with implicit static analysis up to the first buckling
load. Studies with varying loading velocity - up to 150 mm/s - showed as expected a
considerable increase in the first buckling load - which is the reaction force at the
loading plate - but the postbuckling loads remained almost unaltered, see figure 5.
The major difference to lower velocity loading is that the decrease of the load appears now over a wider deformation range. Concerning the quasi-static analysis
aspect the kinetic energies remained rather small, if the velocity was increased
smoothly at the beginning of the loading. After buckling higher frequencies are visible
in the reaction force at the loading plate, thus a smoothing of the response curve was
performed.
A major series of studies could be successfully performed using LS-DYNA with little
numerical effort for a wide range of silo situations with varying boundary and various
filling conditions. For a discussion see [1][10] and further work cited there.
Quasi-Static Roof Crush Analysis
General
Quasi-static roof crush analysis is a major application in automotive industry [14] and
a lot of experience was gained over the last years. The application of explicit programs is often disputed and sometimes also rejected by Bathe [14], who favors implicit algorithms. It will be shown that problems in performing a quasi-static analysis
are rather independent of the algorithm chosen.
The focus on the investigations within this contribution purely performed with LSDYNA vs. 940.2b is on the influence of important parameters in general modeling for
transient analysis - choice of element types, mesh size and time step size. The
choice of elements reflects the major goal in any analysis of large size problems in
dynamics: Efficiency. It is well known that the most efficient elements in the explicit
programs are underintegrated and suffer from hourglassing which can be controlled
to some extent. However, there are situations, when the hourglass control fails and
unusable results are obtained, e.g. as in [14]. Then, of course, the hourglass energies reported by LS-DYNA would point to the problem. Thus it has to be checked,
why the latter happens, and alternatives have to be found. Reasonably fine meshes
are a dominant factor to achieve reliable results, in particular, as contact is very sensitive to the surface modeling. The third aspect: time step size, has to be looked
upon in a very different fashion, when comparing implicit and explicit methods.
Whereas stability aspects and thus the highest frequency in the model - either a result of the mesh size or of the contact model and contact parameters - is determining
the time step size in the explicit scheme (the well known CFL-condition), in implicit
scheme it is often a matter of the physical quantities of interest in the application.
The latter leads often to much larger time steps than in explicit schemes, which has
9
also mostly a smoothing effect on results such as time history curves. To keep the
algorithm efficient, while obeying the requirements of fine meshing, so-called mass
scaling is often performed with explicit programs, which means that the density of the
elements leading to the high frequencies is increased in order to reduce the frequency. If the number of elements with high frequencies is small, then the added
mass and the resulting higher dynamic effects are very limited. A further aspect is to
increase the loading velocity considerably beyond the real velocity used in reality e.g.
experiments. This can be done as long as major dynamic effects artificially introduced are not present.
Roof Crush with Coarse Model
This model for the roof is part of the original TAURUS model from the NCAC database [15], which has been reduced for the roof analysis, see figure 6. The model was
NEVER designed to serve for this purpose, however, the rather coarse mesh allows
to make some effects concerning the use of different element technology more visible than with a fine mesh. For the complete structure self contact, type 26 in LSDYNA, was defined, as in particular the sides of the strut profiles collapse onto each
other in bending. Loading of the mesh, see figure 6 with 7708 shell elements, 8093
nodes, 174 constraints resp. rigid bodies, with boundaries completely fixed against
rotations and translations is performed via a frictionless rigid wall moving with a prescribed velocity.
Figure 6: Coarse (Reduced) model for roof crush based on NCAC model [ 15]
The latter is given by experiments with a value of about 10 mm/s. For a displacement
of 120 - 250 mm, as needed to judge the design, the loading duration would be in
between 12 and 25 sec - well above the duration usually obtained in crash problems.
The time step size limits of explicit algorithms would then rule out the use for such
type of analysis. However, the dynamics in the roof crush problem is very limited and
it can be viewed as a mainly geometry driven process. Thus some numerical "tricks"
(or engineering tools) can be applied to use explicit algorithms. The first is to
increase the velocity of the loading plate in the simulation and the second is to apply
"mass scaling", as described above. Both assumptions have to be carefully checked
by numerical analysis. Summarizing the following was obtained in the simulation:
Fixing the time step size to about 2 microsec, thus twice the size for the original
density, and increasing the loading velocity to 2000 mm/s showed no significant
difference in the results compared to a velocity of 200 mm/s without mass scaling.
10
This was also proven by the check of the quantities affected: The mass added to the
total structure was about .9 % of the total mass of the model and the peak value of
the kinetic energy at a velocity of 2000 mm/s was about 2 % of the corresponding
internal energy.
The focus in the first study was on the differences resulting from the element formulation. Thus two different element types, the fully underintegrated Belytschko-Tsay
element [5] (the "work horse" for crashworthiness analysis) and the fully integrated
element with assumed (natural) shear strain interpolation (ANS) were used and two
hourglass control methodologies, viscous and stiffness control were compared. In
short, viscous control is applied using the velocity differences at the nodes of an
element and stiffness control is based on the corresponding displacement differences. It is well known to experienced crash analysts that for low velocity loading
viscous control is often rather sensitive to perturbations and hourglassing may result.
This was hardly obtained in this example and the energy plots showed no considerable increase in the hourglass energy, see figure 7 left. However, a jump in the internal energy and the sliding interface energy (negative) at a displacement of 180 revealed that problems in the contact appeared; the coarse model lead to fairly large
intrusions of parts of the structure into the loading plate and to major problems in the
self contact zones in the struts, which resulted in large contact forces. This effect is
not present in the viscous controlled analysis, see figure 7 right, but even stronger in
the analysis with fully integrated ANS-elements. However, in all cases a minor mesh
modification or alternatively a reduction of the time step size resolved the problem of
hourglassing, leading to the smooth energy curves. A closer look at the rigid-wall
(stonewall) energies, which are rather large (the negative values), also reveals that
the penetrations into the rigid loading plate are too large to be accepted; the latter is
due to the coarse mesh not allowing for a smooth contact.
Figure 7: Roof crush analysis of coarse model. Energy time history plots for
Belytschko-Tsay element with stiffness and viscous hourglass control.
With problems in the contact zone - stiffness control (left)
and without problems - viscous control (right).
The force-displacement results for the analysis with the fully integrated elements
(type 16) show a considerably larger crush force at the end of the analysis, see figure
8 . In their basic form the curves show some waviness due to the dynamic and other
more local effects and are plotted in filtered form. Though the curves are very close
to each other for the small deformation regime - as known from comparison on small
size problems and from the cylinder buckling problems discussed above- the stiffer
response of the type 16 elements seems to sum up over time. However, the impor-
11
tant effects as the first buckling of the struts and the second buckling of the roof are
captured at the almost identical deformation states as in the analyses using the elements with hourglass control. This very much pronounced difference is reduced considerably with more refined meshes as e.g. then elements are not yet warped so
much in the initial geometry. However, this is a major point to be considered in further studies.
Figure 8: Roof crush analysis of coarse model. Crush force vs. displacement of
loading plate for different element types and different hourglass control.
Remark: In larger simulation models with fairly coarse meshes the variation of the
elements seems to lead to contradictory results. In the example above, as in other
examples too, the stiffer response of the fully integrated elements, which are also
known to behave slightly too stiff in-plane, is enlarged by the effect that the contact
surface has - as it is based on the same mesh - a fairly coarse resolution. With finer
or rather converged meshes - hardly ever used in large scale simulations up to date this effect is diminished. Or described vice versa; the artificial softening effect of
hourglass elements - more pronounced in coarse meshes - is somehow a good
match for the corresponding effect due to the coarse resolution of the contact zones.
The deformation states for the two different element types are given in figure 9 for
the final deformation state and show some difference, in particular visible in the roof
close to the rear strut which is fairly coarsely meshed.
Figure 9: Roof crush analysis of coarse model. Deformation plots at final state for
hourglass controlled Belytschko-Tsay (left) / fully integrated element with ANS (right).
12
Roof Crush with Consistently Refined Model
As second large example the roof crush analysis of a different roof however with
consistently refined mesh closer to those used nowadays in automotive industry is
shown, figure 10. The mesh consists of 27712 shell elements and 26115 nodes. All
model points at the lower boundary are fixed. For the contact within the roof internal
contacts of type 13 were applied. The loading by the steel plate with 10 mm/s is
simulated with different velocities - 200 mm/sec, 2000 mm/s and 10000 mm/s - to
demonstrate the effect of overly enlarged dynamic effects. No friction was assumed
between the roof and the loading plate. The analysis was performed up to about 120
mm resp. 170 mm. As in the coarse model the effect of the choice of the element
type and the hourglass control type was studied using the same variations. Mass
scaling was also applied allowing the analysis time to be reduced considerably. Only
1.1 kg compared to a total weight of 118 kg of this model part were added to adjust
the time step size to about twice the size of the original model.
Figure 10: Roof crush analysis of fine model. Mesh, loading plate and loading direction (arrow).
The following results discussion is concerned with the energy curves, figures 11 and
12. The internal energy is slightly increased when the velocity is increased from 200
to 2000 mm/s. No difference is visible between the two hourglass control types. The
increase of the internal energy is considerably larger for the high loading velocity of
10000 mm/s. The curves for the kinetic energies show also a considerable increase
in kinetic energy for the latter case well above 25% of the internal energy, whereas
the maximum value remained below 2% of the internal energy for 2000 mm/s and is
almost not visible for 200 mm/s. However, in all cases the hourglass energies remained well below 1 % of the corresponding internal energy value indicating that the
analysis was very stable and contact was sufficiently well captured by the good
though not perfectly fine mesh.
13
Figure 11: Roof crush analysis of fine model. Energy curves for internal energy and
hourglass energy with different velocities of the loading plate.
Figure 12: Roof crush analysis of fine model. Energy curves for kinetic energy with
different velocities of the loading plate.
The deformation plots in figure 13 and 14 also show the effect of a too high velocity.
The deformation is almost perfectly identical for all velocities up to 2000 mm/sec,
also for the fully integrated element, however, it is completely different for the high
velocity. The general physical behavior obtained from the simulation is such that the
slope of the force curve, see figure 15 is decreasing after 10 mm displacement,
stiffens (=increases) slightly at about 25 mm and flattens out at about 40 mm. The
events visible in the curve indicate a first local buckling at about 10 mm and finally at
40 mm a global buckling of the struts with major elasto-plastic deformations.
14
Figure 13: Roof crush analysis of fine model. Deformation plot at final state,
120 mm displacement of loading plate, velocity 2000 mm/s
Figure 14: Roof crush analysis of fine model. Deformation plot at final state,
120 mm displacement of loading plate, velocity 10000 mm/s
The force-displacement curve is the curve usually compared to the experimentally
obtained results. The comparison of the unfiltered curves in figure 15 (left) definitely
allow the conclusion that the high waviness for the high velocity is a clear indication
for an unrealistically chosen loading velocity. The waviness for the other two curves
15
at 2000 mm/sec is far less pronounced, though visible for both. This shows that the
structure folds and buckles locally and that the process itself is not smooth at all. The
filtered curves for all loading velocities and all variations concerning the elements,
see figure 15 (right), allow also the judgement that the highest velocity leads to a
completely different behavior, whereas the curves for the hourglass controlled elements show merely no difference for the velocities of 200 and 2000 mm/sec. However, there is a major difference in the final values to the curve for element type 16,
the full integrated element, though the curve is very similar to a far extent. Nevertheless, the deformation plots for a simulation with this element type do not show any
visible difference to the analyses with other element types and are not enclosed for
this reason.
Figure 15: Roof crush analysis of fine model. Contact force (scaled) at loading plate
for various element types and different velocities. Original plots taken directly from
analysis (left) and filtered plots using SAE 180 filter (right).
In conclusion the analysis demonstrates the capabilities of a dynamic analysis with
fairly low loading velocities concerning the capturing of the major effects needed for
design purposes using LS-DYNA. A quasi-static answer could be achieved for a
rather high loading velocity. As the analysis proceeds without many changes and
without any convergence problem, even the variations needed to judge the influence
of the dynamics onto the results can be performed with little effort. The CPU times on
a SGI R10000 computer with a clock of 250 MHz needed for the underintegrated
elements with hourglass control are about 2.5 h for 10000mm/s, 10 h for 2000 mm/s
and 100 h for the 200 mm/s runs. The analysis with the fully integrated element
needed about 24 h for the 2000 mm/s velocity. Obviously the higher loading velocities and the efficient hourglass elements have a remarkable effect to decrease the
computing time. The analysis can be easily performed over night on standard workstations. To judge the effect of the loading velocity in the simulation all variations
have not to be computed up to the final state. Mostly only a short duration of the
loading has to be checked, making the long lasting analysis with low velocities a rare
case.
16
Summary and Conclusions
The applicability of explicit programs as LS-DYNA to quasi-static analysis was shown
and the critical aspects were discussed. It is evident that the simulation models have
to be modified compared to the real situation in order to achieve the necessary efficiency. The engineers performing such simulations have to be able to judge their
results based on their knowledge of mechanics. The program LS-DYNA delivers full
information to allow such a judgement. Then reliable results with high predictability
can be obtained even for failure situations, where many implicit algorithms fail. Even
with rather coarse models results can be achieved for rather complex problems without much modeling effort which can be very well used for preliminary design purposes.
References
1. SCHWEIZERHOF, K., HAUPTMANN, R., ROTTNER, Th., RAABE, M. (1998)
"Silo Buckling Analysis Considering Nonuniform Filling - Dynamic versus Static
th
Analysis using LS-DYNA", 5 LS-DYNA User Conf., Southfield, Michigan.
2. ZIENKIEWICZ, O., TAYLOR, R.L. (1991) "The Finite Element Method, Vol. 1 and
th
2", 4 ed., McGraw-Hill, London, England.
3. ROTTNER, TH., SCHWEIZERHOF, K. (1999) "Transient FE-Analyses of Silo
Buckling Behavior Using Parallel Computers", Euroconference on "Parallel and
Distributed Computing for Computational Mechanics EURO-CM-PAR99, Weimar.
4. HALLQUIST, J.O., LIN, T., TSAY, C.S. (1993) "LS-DYNA Theoretical Manual",
"Nonlinear dynamic analysis of solids in three dimensions", Livermore Software
Technology Corp., Livermore.
5. BELYTSCHKO, T., TSAY, C.S. (1983) "A stabilization procedure for the quadrilateral plate element with one-point Quadrature", Int.J.Num.Meth.Engng. Vol. 19,
pp. 405-419.
6. BELYTSCHKO, T., WONG, B.L., CHIANG, H.-Y. (1989)" Improvements in loworder shell elements for explicit transient analysis", in: "Analytical and Computational Methods of Shells", ed. A.K.Noor, T.Belytschko and J.Simo; ASME, CEDVol. 3, pp. 383-398.
7. UNDERWOOD, P.G. (1983) "Dynamic Relaxation -- A Review", Ch. 5 in "Computational Methods for Transient Dynamic Analysis", T.Belytschko and T.J.R.
Hughes eds., North Holland, Amsterdam.
8. PARK, K.C., (1982), "A Family of Solution Algorithms for Nonlinear Structural
Analysis Based on the Relaxation Equations, Int.J.Num.Meth. Engng. Vol.18, pp.
1337-1347.
9. PAPADRAKAKIS, M., (1981), "A Method for the Automatic Evaluation of the
Dynamic Relaxation Parameters", Computer Meth. Appl. Mech. Eng. Vol.25 pp.
35 - 48
17
10. SCHWEIZERHOF, K., HAUPTMANN, R., KNEBEL, K., ROTTNER, TH., RAABE,
M. "Statische und dynamische FE-Stabilitätsuntersuchungen an Siloschalen mit
ungleichförmiger Schüttgutfüllung, Proc. Conf. "Finite Elemente in der Baupraxis
FEM'98", ed. Wriggers, Meißner, Stein, Wunderlich, Darmstadt (in German).
11. DIN 18800 part 4 (1990) "Stahlbauten, Stablitätsfälle, Schalenbeulen". Normenausschuß Bauwesen, Deutsches Institut für Normung e. V., Beuth Verlag GmbH,
Berlin, (in German).
12. ECCS (1988) "Buckling of Steel Shells", European Recommendations, Fourth
Edition, No. 56, European Convention for Structural Steelwork.
13. SCHWEIZERHOF, K. and COWORKERS (1999) "FEAP--MeKa", Mechanik
Karlsruhe, "A Finite Element Analysis Program". Basic documentation in: Zienkiewicz, O. C./ Taylor, R. L. (1989) "The Finite Element Method" Vol. 1, McGrawHill, London, England, 4. ed.
14. BATHE, J. (1998) "Crush simulation of cars with FEA", Mechanical Engineering,
pp. 82 - 83.
15. NCAC (1999) "TAURUS Model, Public Finite Element Model Archive", National
Crash and Analysis Center (NCAC), The George Washington University, Ashburn, Virginia.
18