1 Study of a typical HWD test
1.3 Theoretical considerations
A basic semi-empirical analysis of the HWD force signal shape is presented here. This work presents a double aim. The first one is to better understand the macroscopic phenomena occurring during a test. The second application is operational. It is important during a field survey to be able to find rapidly a convenient test configuration to reach a target load. The objective is to determine and validate relationships between test parameters (fall height, mass, and buffer set) and resulting applied load, defined in terms of Fmax maximal strength and ∆t
pulse time values.
1.3.1Study of the buffer system
First, a specific study of the buffer system has been performed. The purpose is to 1- examine the homogeneity of the buffer system with the intent to interpret the aforementioned double peak, and 2- assess the buffer behaviour and especially stiffness for specific modellings (initial speed modelling in part 2). Two experiments were conducted. The first one was made
in static conditions (M Broutin/JC Deffieux collaboration). It was made twice (January and March 2006), since initially provided hard buffer set was highly heterogeneous. It has been replaced by a new one. Only the second experiment (with the new hard buffer set) is presented here. As the behaviour of the buffer material may be different under static and dynamic (HWD) load, the experiment has been reiterated under dynamic conditions, using external instrumentation on the HWD. Feasibility studies (M Broutin/JC Deffieux
collaboration) were performed in February 2008 and June 2008, and the final experiment in March 2009.
a - Static buffer analysis experiment
Details on the experiment are in (2006 internal publication)
Three buffer sets have been considered, which consist of height (8) hard buffers (“HB” in the following; SHORE 75), four (4) intermediate buffers (“IB” in the following; SHORE 55), and four (4) soft buffers (“SB” in the following; SHORE 45). Each buffer was placed between two metal plates (Fig. 1-6) and tested using a static press, by progressively increasing the applied load at constant low speed. Precision on the applied force was 1 kg. Precision on the buffer deformation, measured by three comparators placed at 120°, was better that 0,01 mm. The whole experiment was performed with a constant temperature of 15 °C.
Fig. 1-6 Test setup of the static buffer experiment
The three following parameters have successively been assessed: 1- Test repeatability
2- Influence of repositioning on test repeatability 3- Homogeneity of each buffer set
Hydraulic jack
Comparators Metal
plates
F
Studied buffer Force sensor
Vertical blocking of the inferior plate
- Test repeatability
The repeatability of the test has been assessed by repeating the measurement three times for each buffer, with a one (1) minute rest time between successive tests.
Fig. 1-7 displays the result for the first hard buffer (SHORE 75). Similar repeatability is observed for all buffers. Note that a perfect linearity is observed. Once more, this behaviour is common toall buffers.
Mean variance between the three measurements, calculated on all buffers and at all force levels, is 0,87 %.
Fig. 1-7 Repeatability between 3 consecutives measurements on a given buffer (first HB buffer on example)
- Influence of buffer repositioning
The same experiment has been led, but by removing and by repositioning the buffer between each of the three measurements. Mean variance between the three measurements is this time 0,91 %. It may be concluded that repositioning has no influence on the results.
- Homogeneity of each buffer set
Once it has been demonstrated that repositioning do not affect the results, homogeneity of each buffer set can be assessed (Fig. 1-8).
Fig. 1-8 Stiffness homogeneity of a buffer set (HB buffer on example)
Table 1-2 presents corresponding variances. Note that a 1 8 ratio for HB and 1 4ratio for IB and SB have been applied to theoretical results, as studied values correspond to a mean value on respectively 8, 4 and 4 buffers, considering for each buffer the mean value on the three measurements.
Actually, let us consider N measurement results (mi) of common U uncertainty, and name m
their mean value.
∑
= = N i i m N m 1
1 , so that sensitivity coefficients of m
iare λi= ∂m/∂mi = 1/N for
each ith measurement and
N U U U i i m =
∑
= = 3 1 2 2 λ .Table 1-2 Stiffness homogeneity of the buffer sets
Scatter of results is 1,5 to 3 times higher than for a single buffer, what implies that test precision is sufficient to judge homogeneity of the buffer sets. Precision seems very good. A translation in terms of precision on elastic moduli is given in the following.
HB IB SB
Standard deviation [mm] 0,0004 0,00035 0,0007
- Interpretation in terms of elastic moduli
An approximate calculation of elastic modulus of buffers consists in applying the (1-1) relationship: v S H F S F E= = ε (1-1)
Where H is the buffer height, S its section, and v the axial deformation imparted by the axial F applied load. According to Fig. 1-9, apparent moduli calculated by this method are 10,2; 4,1 and 2,8 MPa for the hard, intermediate, and soft buffers respectively.
Fig. 1-9 Apparent stiffnesses of the different buffer sets
Precision on moduli is calculated in the following manner:
Let UX and λX be respectively the uncertainty on parameter X and sensitivity coefficients
relative to E. Parameters to be considered are the measured force, and the relative deformation (whose uncertainty is due to the scatter of buffer diameters and uncertainty on buffer
deformation measurement). Thus: 2 2 2 2 2 2 ε ε λ λ λ U U U
UE = F F + S S + with λi the sensitivity coefficient relative to the i parameter:
i E i ∂ ∂ = λ with ε λ S F E F 1 = ∂ ∂ = , ε λ F S S E S 2 1 − = ∂ ∂ = , S F E 2 1 ε ε λε =− ∂ ∂ = ,
and: 2 2 2 2 H H v vU U Uε = λ +λ ; 2 H v H =− λ ; H v 1 = λ D S S S U D U U = = × 2 π
λ , with UD the uncertainty on diameter.
Table 1-3 collects all results. Uncertainties on D and H have been obtained from Vernier caliper measurements.
UF (MN) UD (m) US (m2) UH (m) Uv (m) Uε (s.u)
HB 10-5 2,2.10-4 3,45.10-5 1,9.10-4 10-5 3,7.10-4
IB 10-5 1,4.10-4 2,20.10-5 4.10-4 10-5 1,5.10-3
SB 10-5 4,8.10-4 7,54.10-5 6,4.10-4 10-5 2,2.10-3
Table 1-3 Elementary uncertainties relative to the buffer moduli calculation
Sensitivity coefficients (Table 1-4) have been calculated using: - D = 100 mm = 0,1 m,
- H = 80 mm = 0,08 m,
- FHB = 1 200 kg = 0,012 MN, FIB = 0,010 MN and FSB = 0,006 MN,
- and the corresponding ε: εHB = 14,7 %, εIB = 29,9 % and εSB = 27,0 %.
λF λS λε
HB 867 1325 71
IB 426 543 14
SB 472 361 10
Table 1-4 Sensitivity coefficients relative to the parameters of the problem
Table 1-5 provides final precisions on moduli values. λFUF
(MPa) (MPa) λSUS (MPa) λεUε (MPa) UE
HB 9.10-3 4,6.10-2 2,6.10-2 0,0534
IB 4.10-3 1,2.10-2 2,1.10-2 0,0248
SB 5.10-3 2,7.10-2 2,3.10-2 0,0357
Table 1-5 Final precisions on moduli values
The previous approximate determination of elastic modulus does not take into account buffer radial deformation. An improved method is thus proposed using an identification procedure performed in CESAR-LCPC software [Humbert et al., 2005].
Interface element
• Perfect adherence
• Respect of the non
interpenetration condition Construction element p1 = F/STip Horizontal and vertical blocking "bound" Horizontal blocking Metal plate E = 200 000 MPa ; νννν = 0,15. Buffer Modulus ET ν ν ν ν = 0,51 "surf" (a) (b) 1
Fig. 1-10 Finite Elements modelling of the buffer behaviour under axial loading
Fig. 1-11 presents an example of buffer deformation under load.
Fig. 1-11 a - Vertical deformation of the buffer under axial load, b - Radial deformation of the buffer under axial load
Table 1-6 presents the results of the identification procedure for the three buffer sets. The values are slightly lower than the apparent moduli. This identification has been performed using axial deformation (1 equation and 1 unknown). The last two columns compare the radial numerical deformation corresponding to the identified modulus with the
experimentally-determined radial deformation. FEM calculated values are slightly lower than experimentally determined deformations, so that the model is considered to be consistent.
CESAR identified modulus (Εfitting)
FEM calculated radial relative deformation (εr) Experimental radial relative deformation (εr) HB 8,5 MPa 8,65% 10% IB 3,4 MPa 17,5% 20% SB 2,2 MPa 15,9% 18%
Table 1-6 Buffer moduli, identified from FEM calculation
Partial conclusions
This experiment has allowed demonstrating that all buffers present, at least under static action, an elastic linear behaviour, homogeneous for each buffer set.
An identification using a FEM model has allowed determining a static modulus value, which is in good agreement with the experimentally-determined apparent modulus. Note although that temperature may have a significant influence on rubber stiffnesses.
b - Dynamic analysis
The following results are taken from a more general experiment involving an accelerometer- based external instrumentation. The last part of this experiment will be presented in section 3.2.2.
The presented testing consists in positioning accelerometers on the falling mass and on the tray to which buffer bases are fixed. The experiment has beenperformed on the S3 structure.
Temperatures during the tests are provided in appendix 1.2. The buffer set used is composed of the 8 hard buffers.
50G accelerometer fixed on the falling mass
25G accelerometer fixed on the buffer tray
(50 G)
Fig. 1-12 Test setup related to the accelerometer-based mass motion study
An approximate calculation is needed to determine a priori the necessary measurement ranges. Assuming a free fall, the initial mass velocity at the impact point at buffer top is respectively V0 = 2,8 m.s-1 or 1,4 m.s-1 at H0 = 400 mm and H0 = 100 mm,. Velocity varies
from V0 to zero between occurrence of the signal and peak value (occurring 15 ms later), so
that the mean acceleration on this time frame is about 19 G and 9,5 G for the two H0 values.
In the case of a sinusoidal F(t) force signal (and consequently the γ(t) vertical acceleration
signal as the fundamental principle applied to falling mass implies γ(t) = F(t) / M0), relation
between maximal value and mean value presents a ratio of 2, so that the expected maximal acceleration values of the falling mass is about 38 G for H0 = 400 mm (19 G for H = 100). A
50 G accelerometer is thus chosen for following the falling mass motion.
The same reasoning is conducted for the second accelerometer, when supposing that
displacement corresponds to the plate displacement, so that displacement varies from zero to maximal deflection in a half pulse time. Maximal expected acceleration is in this case about 9 G for H0 = 400 and the half for H0 =100 mm. A 25 G accelerometer is used, what gives a
good security margin.
As seen before, the frequency spectrum of the HWD force-signal and ensuing deflections occupies the 0 – 80 Hz range. Accelerometer transfer function is of low-pass type, with a cut- off frequency around 250 Hz (see 50 G accelerometer response in Fig. 1-13 for instance), so that no undesirable filtering is applied to the signal.
Fig. 1-13 Transfer function of the accelerometer fixed on the falling mass
Collection of the accelerometer responses is performed using a spider acquisition unit (http://www.hbm.com/). Acquisition rate is 3 200 Hz. Raw data have been filtered with a 300 Hz Bessel filter what does not affect the results.
Fig. 1-14 and Fig. 1-15 display the recorded raw acceleration signals for a 100 mm fall height. Acceleration is taken positive when directed downwards. Zero base is made before dropping of the mass. Then the free fall is characterized by a 1 G constant acceleration. A negative acceleration (upwards sense) is then observed corresponding to the first impact. Note that maximal measured mass acceleration is about 15 G (against 19 G in the approximated
calculation). Several free fall and rebounds succeed then. The elapsed time between principal impact and first rebound is about 200 ms. Note that this value depends on the fall height (see for instance Fig. 1-44 in 2.2.2. hereafter, corresponding to a 400 mm fall height).
The second accelerometer presents accelerations higher than predicted, what may be due to rapid vibrations of the thin metal tray under impact. As acceleration has an important
magnitude but short pulse time, corresponding displacement, obtained by double integration, is less than 1 mm. It will be neglected in the following.
Free fall Principal impact First rebound Mass dropped
Fig. 1-14 Accelerations measured on the falling mass and buffer tray during a HWD test (1/2)
Fig. 1-15 Accelerations measured on the falling mass and buffer tray during a HWD test (2/2)
Fig. 1-16 presents the displacement of the falling mass obtained from a double integration of the 50 G accelerometer signal. As displacement of the base of the buffers is neglected, negative parts of the curve reflect deformation of the buffer under the successive rebounds until stabilization around the final position. It appears that final deformation under static weight (6,8 kN) is negligible in comparison with the one under the dynamic load (120 kN).
Fig. 1-16 Buffer axial deformation during a HWD test, from double integration of acceleration measurements
Buffer modulus is obtained by studying the relationship between force and deformation, considering an equivalent buffer (of same height and modulus as the 8 buffers, but a
0
8D diameter, withD0 the diameter of each buffer). Deformation under the 120 kN applied force is around 12 mm. This corresponds to a 10,5 MPa modulus, when performing the same identification procedure as previously.
Partial conclusions
This paragraph has shown that all buffers of a given set present, at least under static
conditions, an identical behaviour. As their geometry is also identical, the force signal double peak can not be explained by buffers heterogeneity.
Static and dynamic moduli have been calculated.
1.3.2Theoretical prediction of maximal force and pulse time
The experimental study presented in § 1.3.1 has shown that the rubber buffers have a linear- elastic behaviour and present similar stiffness values. The 8 buffers system may be modelled as a unique equivalent spring of k spring value.
Under this assumption, the applied strength over the pavement surface can be expressed as: