Topics in Dynamic Analyses
1. LO LOAD ADIN ING G and and BOUN UNDA DARY RY CO COND NDIT ITIO ION NS 2. DA DAM MPI PIN NG and MATE TER RIA IAL L RES ESPO PONS NSE E
This talk is for current users users of FLAC – the objective is to address some common difficulties; thus, some level of experience with dynamic simulations is assumed.
1 LOADING and BOUNDARY CONDITIONS Seismic input to FLAC Dynamic input may come from within the grid (e.g., train vibrations in a tunnel) or from outside of the grid (e.g., earthquake waves coming from a distant source).
To model sources within the grid the dynamic excitation is simply applied directly directly to the the appropria appropriate te gridpoin gridpoints ts – for example example pressu pressure re loading loading from from an explosion or oscillating forces due to a vibrating machine. Quiet boundaries are normally required to reduce reflections at artificial boundaries, but a free field boundary is not required required . The rest of this part of the talk is concerned with sources located outside of the grid, so that dynamic excitation must be applied to part of the model boundary. In this case, free field f ield boundaries are normally used, so that the simulation reproduces the effect of a plane wave propagating into the grid -
Free field boundaries are used to ensure that incoming plane waves remain plane
(No spurious absorption of incoming wave by side quiet boundaries)
The bottom boundary may be either fixed (driven by a velocity or acceleration history) or “free” (supported by static forces and driven dynamically by a stress history)
Mejia & Dawson (see proceedings of this Symposium – paper 04-10) present a very clear description of the ways in which seismic input may be applied to a model. (The figures in this section are reproduced from their paper, with permission).
There are two main options 1. 2.
Rigid base (velocity or acceleration history applied directly) Flexible base (velocity history converted to applied stress history)
If the target motion is provided for any location except for the base of the model, then deconvolution is necessary, to develop a time history to be applied at the model base such that the simulation would reproduce the target motion at the specified location, under free field conditions (e.g., no structures). Normally the program SHAKE is used for deconvolution. SHAKE is an equivalent-linear program, and is thus unable to follow nonlinearity directly; it adjusts the secant shear modulus and damping of each layer iteratively to obtain the approximate effect of nonlinearity, averaged over the whole time
SHAKE works in the frequency domain, using the sum of the upward- and downward-propagating waves. At each interface between layers, there is an analytical solution for the reflected & transmitted portions of each wave. By solving the resulting system of equations, transmission between any two locations (e.g., between base & surface) may be computed. (After Mejia & Dawson, 2006)
SHAKE input & output is available either 1. 2.
at the boundary between two layers – termed within motion, which is a superposition of upward- and downward-propagating waves; or at a notional free surface of the same depth as the requested layer boundary – the motion that would occur at an outcrop free surface. Thus, the outcrop motion is simply twice the upward-propagating wave.
For a rigid FLAC model base, the following example illustrates the procedure.
(After Mejia & Dawson, 2006)
The use of SHAKE to compute the required input motion for the rigid base of a FLAC model leads to a good match between the target surface motion and the surface motion computed by FLAC , for a model that exhibits with a low level of nonlinearity. (The input motion already contains the effect of all the layers above the base, because it contains the downward-propagating wave). A different approach must be taken if FLAC is to model more realistic systems, such as 1.
sites that exhibit strong nonlinearity; or
2.
the effect of a surface or embedded structure.
In the first case, the real nonlinear response is not accounted for by SHAKE in its estimate of the base motion. In the second case, secondary waves from the structure will be reflected from the rigid base, causing artificial resonance effects.
For most sites encountered in practice (except those where the existence of a very stiff bedrock justifies a rigid base) a flexible base to the FLAC model should be used. In this case, the quiet base condition is selected, and the upward-propagating wave only from SHAKE used to compute the input stress history. (This is derived as the outcrop velocity history, converted to a stress history by using the formula τ = −CS ρ v ).
By using the upward-propagating wave only at a quiet FLAC base, no assumptions need to be made about secondary waves generated by internal nonlinearities or structures within the grid, because the incoming wave is unaffected by these; the outgoing wave is absorbed by the compliant base. As an example, consider the dam shown here. A rigid base leads to non physical oscillations. The inputs in both cases (rigid & flexible) were derived by deconvoluting the same surface motion).
(After Mejia & Dawson, 2006)
Anomalies encountered when motion at depth is specified
If a specific ground-motion history is required at a certain depth in a layered site, then deconvolution is used (as discussed) to determine what motion must be applied at the flexible (quiet) base of the model.
Even with this (apparently) simple system, there may be surprising results.
The following example illustrates one effect that is often overlooked.
Required motion at 300m depth
Layered site Ground surface
acceleration
Target motion
Deconvoluted incoming motion to be applied at base
Why are there extra oscillations?
about 1 Hz
To check, we run FLAC with the specified base input, and obtain the following acceleration response: JOB TITLE : Observed Acceleration at B in FLAC Model
FLAC (Version 5.00) LEGEND 3-Jun-05 12:00 step 114707 Dynamic Time 4.0000E+01
4.000
HISTORY PLOT Y-axis : 2 X acc eleration( 1, 91)
2.000
X-axis : 101 Dynamic time
0.000
-2.000
-4.000
5
10
15
20
25
30
35
Itasca Consulting Group, Inc. Minneapolis, Minnesota USA
This history is very similar to the required response at 300 m depth … the oscillations in the input wave have disappeared.
Why are there “spurious” oscillations in the in
e?
If we propagate a single cycle of a sinusoidal wave into a layer, it is reflected without change of sign from the free surface, as illustrated by the following simulations: Run FLAC3D (file qb1c.dat) Now, we propagate a continuous wave … Run FLAC3D (file qb.dat)
The velocity component in an upward-traveling wave is reflected at the free surface, and the resulting downward-traveling wave is superimposed on it, causing reinforcement at some points and cancellation at others. For the original site model, we send in single-frequency waves, and plot maximum velocity observed at each depth -
5 Hz
1.04 Hz
1.200
1.000
0.800
1.200
1.000
300 m depth
0.800
0.600
0.600
0.400
0.400
0.200
0.200
0.000
0.000
These are standing waves
Thus, if we insist that the history at 300 m contains all frequencies (including 1.04 Hz), then the base incoming wave must be given a “boost” at 1.04 Hz to overcome
To illustrate the effect in another way, we perform many tests at different frequencies, and plot the maximum steady-state velocity at 300 m depth versus frequency –
There is an infinite series of specific frequencies at which the transfer function between the base and the given depth is zero.
y 1.600 t i c o 1.400 l e v 1.200
m u m 1.000 i x a 0.800 m d e 0.600 z i l a 0.400 m r o N0.200 2
4
6
8
10
Frequency - Hz
12
14
2 DAMPING and MATERIAL RESPONSE The new Hysteretic Damping (HD) option of FLAC and FLAC3D allows users to represent energy loss more realistically than with Rayleigh damping, and avoid the latter’s small-timestep penalty. HD is described and illustrated in two papers in the Proceedings: 04-02 and 07-04. In summary, HD adjusts the tangent shear modulus M T as a continuous function of shear strain. M T is derived from the secant modulus M S . For example M S =
a
1 + exp {−( L − xo ) / b} where L = log10 γ
This is the sigmoidal curve “sig3” with 3 fitting parameters The provided functions allow a good fit to cyclic laboratory tests
1.2 r 1 o t c a f n 0.8 o i t c u d 0.6 e r s u 0.4 l u d o M 0.2
0 0.0001
0.001
0.01
0.1
strain-%
1
10
In addition, the following “rules” are used (& illustrated on the figure):
4.0E+05
1.
Upon reversal, the complete state (stress, strain, modulus, etc) is pushed onto a (Last In, First Out) stack. When the strain again passes through the same point, the previously-saved state is restored.
2.
A new & identical (but inverted) curve is started upon reversal;
3.
The first quarter-cycle of loading is scaled by one-half relative to all other cycles;
3.0E+05 2.0E+05
s s e 1.0E+05 r t s r a 0.0E+00 e h S-1.0E+05
-2.0E+05 -3.0E+05 -0.1%
0.0%
0.1%
Shear strain %
0.2%
Hysteretic Damping (HD) not only adds energy loss to dynamic straining, it also causes the mean shear modulus to decrease , for large cyclic strains. This may lead to unexpected results – e.g., an increased response amplitude, due to a shift in resonant frequency closer to the predominant frequency of input waves. Before running a dynamic model with HD, an elastic simulation should be made without damping, to observe the maximum levels of cyclic strain that occur. If the cyclic strains are large enough to cause excessive reductions in shear modulus, then the use of HD is questionable – it will be performing outside of its intended range of application. The model properties and input amplitude should be checked. If properties and input are reasonable, and the large cyclic strains are limited to a small regions, then consider the possibility of excluding HD from these regions and using a yield model in the regions (since the large strains imply that yielding should occur). Even if cyclic strains under elastic conditions are small, the use of a yield model may increase the strains …
The Hysteretic Damping formulation is not intended to be a substitute for a yielding constitutive model . It may be used in conjunction with a yield model (e.g., Mohr Coulomb) but conflicts in the domain of application should be avoided, for meaningful results. In particular, large plastic strains may cause HD to “take over,” and dominate the behavior. As an illustration, consider the following slope model, subject to rigid-base excitation (an actual user example): 20 m high slope of 30O angle
Cohesion = 104 Pa, friction angle = 30o (close to failure)
(10
-01
1.500
1.000
0.500
Horizontal velocity history applied to base:
0.000
-0.500
-1.000
-1.500
)
Max. shear strain increment 0.00E+00 2.00E-02 4.00E-02 6.00E-02 8.00E-02 1.00E-01 10%
With NO damping, but with plasticity:
Contour interval= 2.00E-02 shear_mod Contour interval= 1.00E+08 Minimum: 1.00E+08 Maximum: 3.00E+08 (10 (10
-01
05
)
0.200
y 2.000 t i c o 1.000 l e v l a 0.000 t n o z-1.000 i r o H -2.000
-3.000
s 0.000 s e r t s -0.200 r a e h -0.400 S -0.600
-0.800
)
When Hysteretic Damping is added, the 10% strains in the slope region lead to (essentially) zero shear modulus there. Without any constraint, a block of material becomes detached, and moves as a free body, after input ceases.
Max velocity = 3.5 m/s
Shear strain – max = 160%
Velocity
0.000
0.200
-0.200
0.000 -0.200
-0.400 -0.600 -0.800 -1.000 -1.200 -1.400 -1.600
c o n t i n u i n g m o t i o n
-0.400 -0.600 -0.800 -1.000 -1.200 -1.400 -1.600
1.6 m/s
If Hysteretic Damping is excluded from the yielding region, the response stabilizes.
Regular HD
Shear strain – max = 2% (10
-02
)
Velocity (10
-01
0.000
2.000
-0.400
1.000
-0.800
0.000
-1.200
-1.000
-1.600
-2.000
-2.000
-3.000
)
0.2 m/s
1
2
3
4
5
6
7
8
9
Thus, if there is extensive yielding, it should be represented by a yielding model (which performs its own energy dissipation) , and not by hysteretic damping.
We plan to make this action automatic in future code releases – i.e., during yielding, hysteretic damping will be “switched off” (as well as the associated strain accumulation for the HD calculation). Note that Rayleigh damping is already switched of during plastic flow.
Finally, with an initial shear stress present in a model, care must be taken when using HD. In this example, a column is loading with the Loma Prieta horizontal record
With no initial shear stress
With initial stress, simply set with INI command
4.0E+05
4.0E+05
3.0E+05
3.0E+05 2.0E+05
2.0E+05 s s e r t s r a e h S
s s e r t s r a e h S
1.0E+05 0.0E+00 -1.0E+05
1.0E+05 0.0E+00 -1.0E+05
-2.0E+05
-2.0E+05
-3.0E+05
-3.0E+05
-0.2%
-0.1%
0.0%
0.1%
0.2%
-0.2%
0.3%
By using HD to obtain the static state, stresses and strains in the HD logic are compatible. (See paper 07-04 for
0.0%
0.1%
Shear s tr ain %
Shear s tr ain %
With shear stress installed by simulating static loading with HD active
-0.1%
4.0E+05 3.0E+05 2.0E+05 s s e r t s r a e h S
1.0E+05 0.0E+00 -1.0E+05 -2.0E+05
0.2%
0.3%
CONCLUSIONS There are many potential pitfalls when setting up and performing dynamic simulations. If the results look strange, it is worthwhile spending time to understand exactly what is going on. If necessary, perform some simplified runs to reveal the effects of boundaries, loading method, damping, nonlinear materials, zone size and embedded structures.
