- Intro: Velkommen til præsentation af “ Vibrationer fra togtrafik: Femern Bælt-forbindelse”, 20 min …, … ,…
Introduction, Model setup, Convergence, BE, FE, Combined BE-FE, Conclusion
3 Introduction – Purpose/General aim/plot (Femern map)
The focus of this project lie around the evaluation of energy transmission/loss of waves emanating from the new planned RW
genes for the occupants in the existing buildings
4 Model setup
Full 3D model, with train passing a reference building. – The soil will be excited with a load at a given frequency/frequencies
wave propagation from the track towards the building.
The frequencies at one point/cut will depend on the train velocity, distance between wheel axles, which may vary, while the
frequency along the track also will depend on the distance between the sleepers and the radius of the wheel(imperfections) (and
the velocity) FREQUENCIES OF INTEREST 30-100Hz
In order to make a simplification of the model from a 3D model to a 2D model, the different contributions are assumed to act in
phase no response in the z-dir IF a homogeneous linear elastic isotropic material is selected and the load is applied in the xy-
plane only train load = line/surface load(in phase) – see next slide
5 Model setup
2D model layout
It is chosen to apply the load directly on the surface of the soil as an uniformly distributed load instead of on a stiffer sleeper,
which would have distributed load differently in order to keep a uniform displacement along the slepper(instead of a uniform
Soil model by BEM(TEA): BEM good for wave propagation problems – few DOFs as boundary discretized
Structure model by FEM: FEM well-suited for analysis of finite structures but requires refined mesh for dynamic problems
compared with a static problem, due to the time varying change of high-stress-gradient areas.
Uncoupled models/Coupled model – see next slide
6 Model setup
Uncoupled models Load applied in BE/TEA at different frequencies (between 30-100Hz). The response at 25, 33m FE
structure (no back coupling).
Coupled model Load applied in the BE model, where the FE structure is included back coupling/Soil-Structure Interaction
Comparison of: structural response, soil response
7 Convergence – Intro
Intro: Before the models are setup, a convergence analysis of the borrowed software TEA is carried out.
This analysis takes the element length, the diffraction around the end-point at the boundary and the reflection due to
impedance mismatch into account.
The BE model is using 3-noded quadratic shape function boundary elements, as seen on the FIGURE
8 Convergence – Element length
Converging results in the BE model require a reasonable representation of the waves (min. 4 nodes per wavelength depending
on the kind of elements being used)
9+10 Convergence – Element length
error at 33m < 2.5% at 100Hz ^ below 0.5% for the mean error(dist+freq), see next slide ( nele,wav = 4 is selected)
11 Convergence - Diffraction
Diffraction at the end-point(because of full-space outside) of the model (point), will occur, leading to a error near the boundary.
– The diffraction starts to occur for the last wave inside the model approaching the end-point the error is expected to
produce a certain error 1 wavelength away from the edge the appropriate fraction of a wavelength the model should extend
to provide results with a certain error is sought. (next slide)
12 Convergence - Diffraction
great model extents for converging results at low frequencies (long wavelengths)
13 Convergence - Diffraction
As explained, the error (at 50m) decrease for an increasing frequency followed by a stagnation
14 Convergence- Diffraction
Mean error (freq (at 50m)) is below 1% for at model extend of 15m. (with the 25m extend model as a reference)
15 Convergence - Multiple layers
Multiple layers are now introduced, leading to reflected waves at the interface more realistic representation of the soil
response at the surface. However some errors are introduced at the boundary, as reflection will occur, as the top material inside
the model, tend to feel the material outside at if it was a full-space consisting of the material occupying the half-space. this
will introduce reflection depending on the impedance mismatch (transmission), where a stiff top material will respond
unaffected by the softer material outside the model creating a free end(beam analog) and vice versa for a soft top material
fixed end…: In either case reflection will take place, increasing for an increasing impedance mismatch.
16 Convergence - Multiple layers
As reflection at the boundary occur for the reflected waves at the interface, error should decrease for every added top layer
thickness. The error is shown in the FIGURE, where the error drops relatively quickly, with one layer thickness to the right of the
reference point. Here the mean error (freq) drops below 10% (from 30Hz and above).
17 BEM – Cauchy, Navier, Somigliana
Cauchy equation = Strong formulation for elastodynamics
Cauchy equation reduce to the Navier equation for homogeneous isotropic linear elastic materials (with used of Hook’s law)
Det viser sig at være smart at bruge en fundamental løsning (Green’s function) til vægtningen i stedet for en arbitrær vægt
funktion (som ved FE) Somigliana’s indetity(i frekevens domæne) til at se sådan ud som kan sammenlignes med den svage
form i en FE formulering. – Det skal bemærkes at muligheden for at inkludere laster indeni domænet er ekskluderet her…(no
C = geometry constant; C=0.5 på randen
18 BEM - Greens
The Green’s function provide the response at point x at time t in direction i, for a unit magnitude concentrated force acting at
the point y at time tau in the l-idrection
By using the Green’s function for the weighting all element of the model is related.
19 BEM – global system of equations
Somigliana’s identity discretized – As seen, the field quantities is weighting via the Green’s function and interpolated with the
shape functions (peg!)
The global system of equations for the BE model is summarized in HU = GP.
20 BEM – Results soil 1
Two soil profiles selected – Number 1, see FIGURE. – The response will appear random, with locally increasing response for an
increasing distance to the source. This happens due to locally constructive/destructive interference, see FIGURE(next slide)
21 BEM – results soil 1
where reflection of 1, 2, 3….. wavelengths, will create constructive/destructive interference along x(axis). The change increase
for an increasing frequency, as the wave length is decreasing, creating more reflections close to the source(see FIGURE on
22 BEM – results soil 1
The response in the vertical direction, provide larger amplitudes, but with the same randomness……
23 BEM – results soil 1 (x-dir)
The response as function of the frequency at a the supports of the structure in 25 and 33m (as well as a reference point 8m from
the source), is shown in FIGURE (x-dir). – It is clear that the first reflection at the interface is observed corresponding the a P-
wave reflected at the interface providing constructive response 33m from the source. The same is seen for the 25m point, with a
slightly higher frequency, corresponding to the smaller reflection length.
The response at the reference point 8m from the source is a bit more blurry, due to the relatively wide point of application, near
the reference point.
24 BEM – results soil 1 (y-dir)
The response in the vertical direction is more similar at the building site and at the reference point. It is however noted that the
response at the building site is actually increasing for frequencies above 40Hz
2D model shown again – explain uncoupled model soil response in 25 and 33m used for forced displacements in the FE
Mention supports = pinned
Global system of eq’s – Dynamic stiffness matrix – Hysteretic damping
27 FEM – Results structure 1
Maximum response of beams and columns - in x- and y-dir
Depend on mode shape and if the load is applied in phase/counter phase
28 FEM – results structure 1
Normalized response x and y-dir dynamic amplification
29 FEM – results structure 1
Størst gener ved største acc (= kraft (f = ma)). Acceleration response x and y-dir eigenfrequency at 70 and 98Hz provide the
most “mærkbare” response, due to a high frequency and a high dynamic amplification of the structure.
30 Combined BE-FE
Results from the coupled model. Explain Coupled model SSI
31 Combined BE-FE
BE-FE structure: eight-noded quadrilateral elements, element division = element division in FE program
32 Combined BE-FE
A certain amount of wavelengths around the building will influence and hereby interact with the building structure influenced
by smaller soil mass for high frequencies the soil is easiest affected by the presence of the building at high freq’s && the
building will apart from this provide the largest reaction force, when the building is excited at its natural frequencies providing
the largest reaction forces (f = m·(-omega²·U)), see FIGURE next slide
33 Combined BE-FE
Reaction forces…………….show on FIGURE
34 Combined BE-FE
Soil response y-dir in the UNcoupled model.
35 Combined BE-FE
Soil response y-dir in the Coupled model.– It is clear to see the supports of the structure connection points, as the response near
the natural frequencies of the structure change noticeable
36 Combined BE-FE
Uncoupled/Coupled soil response y-dir great influence of the response near the eigen freq’s(e.g. 40 and 70Hz)……
37 Combined BE-FE
Soil response y-dir as fct of freq
38 Combined BE-FE
The response of the soil tend to increase in the Coupled model just below the natural freq’s, where the soil and the structure is
acting in phase. At the eigenfrequency, the soil and the structure act in counter phase, minimizing the soil response, as the
structure act as a tuned-mass-damper.
Reference point affected by reflections!...
39 Combined BE-FE
The response of the coupled and the uncoupled structure is now shown. It is seen, that the eigenfrequencies of the uncoupled
structure, is below the natural freq’s of the structure in the coupled model, as it is pinned, while the coupled structure have
eigen freq’s similar to a fixed structure.
40 Combined BE-FE
By fixing the uncoupled model, the eigen freq’s decrease and is now comparable with the freq’s in the coupled model. The
response of the uncoupled model is also decreased and is now in the same order of magnitude.
41 Combined BE-FE
It is also possible to make a pinned/simple supported structure in the coupled model done by reducing the intersection point
in TEA. The natural freq’s now drop to a level comparable with the pinned uncoupled model, but the response remain in the
same order of magnitude, as for the fixed structure in the coupled model = low.
• The response of the soil in the coupled model is greatly influenced for:
• An increasing load application frequency
• Near the natural frequencies of the structure
• Response increase for load application frequency natural frequencies
• Response drastically reduced at the natural frequencies
• The wave field will also change around the structure
• Reflections from supports will change wave field in front of building
• The amplitudes in the wave field behind the structure are reduced (shielding)
• The responses of the uncoupled structure are larger than the responses of the coupled structure
Field measurements amplitude/vibration measurements = highly dependent on location
43 Train riddle