The representativeness of a sample is always only partially verifiable. It is a relative notion. A sample can be representative of one, two, three or more variables, but is never completely identical to the total population. Even if representativeness is fully verifiable, it is verified at a given time.

We chose a wall of 99 tie rods, which we will call 100 tie rods to reduce the analyses to percentages. This wall where all the tie rods were tested is representative of what we found during the tests, whatever the type of tie rod, bar or cables. Families of tie rodsFirst of all, there are families of tie rods attached to the lines, i.e. to their altimetric position, in this example there are 5 lines of tie rods.

Family 1.

See Figure 16.1.

Fig. 16.1
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Family 1

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Family 2.

Line 1.

See Figure 16.2.

Fig. 16.2
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Family 2

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Family 3.

Lines 3 et 4.

See Figure 16.3.

Fig. 16.3
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Family 3

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Family 4.

Line 2.

See Figure 16.4.

Fig. 16.4
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Family 4

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This line of tie rods has lost most of its initial tension. This observation is not a particular case, but a general case of tie rods involved in the support of walls that receive rolling loads.

This observation concerning line 2 of our example is sometimes extended to several lines of tie rods. This is the case, for example, for a 16.5 m high wall with 11 tie lines (Fig. 16.5).

Fig. 16.5
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Tension forces of the tie rods

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Lines 3 and 4 have significantly lower force values than lines 2 and 5 (Fig. 16.6).

Fig. 16.6
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Force per line of tie rods

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Line 4 has significantly lower tension force values than lines 3 and 5.

Laurent Soyez's thesis is: Contribution to the study of the behavior of retaining structures subjected to railway operating loads (dynamic and cyclic loads) 2009 (Figs. 16.7 and 16.8).

Fig. 16.7
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Source Laurent Soyez

Cross section of the test embankment.

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Fig. 16.8
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Source Laurent Soyez

Diffusion of railway loads.

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Numerous tests have been carried out on a reinforced embankment at a scale of 1:1. In practice, the following curves show that the load applied to the embankment is maximum between 1- and 2-m depth and close to zero at 3.5 m depth. The parameter concerning the duration of load application is directly related to the depth of load application (Fig. 16.9).

Fig. 16.9
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Source Thesis de Laurent Soyez

Deformed as a function of load.

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We carried out a static tensile test on a tie bar receiving cyclic loads due to a convoy of ore wagons and measured deformations in the tie bar, which are close to those shown in the previous figure for a load of 250 kN.

Sampling should also consider the following.

  1. 1.

    Boundary zones of the retaining wall where tensile forces are lowest, in blue.

  2. 2.

    These boundary zones of the wall are locations where in-place soils are found and fill soils decrease, resulting in lower tension values in the tie rods than in tie rods of the same line or level.

  3. 3.

    Tie bars near construction joints blue and yellow arrows.

Construction joints between concrete wall panels have specific locations where rainwater or groundwater can flow, but in general, this flow or even seepage results in soil particles that create locally decompressed areas, hence the low-tension forces measured.

  1. 4.

    Zones of "normal" behavior, green frames

The central parts of the wall, avoiding the proximity of particular points, give a correct representation of the behavior of the wall, in general, two types of tie rods, tie rods that support only static loads and tie rods that receive, in addition, dynamic cyclic loads from heavy transport vehicles.

Analysis of the example and the sampling strategy (Fig. 16.10).

Fig. 16.10
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Examples of specific points

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For this wall used as a working example:

  • 52% of the tie rods are in "normal" situations

  • 11% are in the outer limit zones of the wall

  • 37% are close to the construction joints

The sampling strategy is an essential step in the design of scientific experiments, with or without particular experimental treatment, i.e. including measurements on an object.

Cochran formula.

16.1 N = t2xp(1-p)/m2

N: minimum sample size to obtain significant results for a given event and risk level.

t: confidence level (the standard value for the 95% confidence level is 1.96).

p: estimated proportion of the population with the characteristic.

m: margin of error (usually set at 5%).

For anchors called "normal".

With a 95% confidence level, see Student's law and the value of n equal to 1.96. (Fig. 122). The proportion of the population with these characteristics is 68%.

For a margin of error of 10%, the number of tie rods to be tested to reach these objectives is 43.

For the same population with lower assumptions, for example.

  • A confidence level of 90% (n = 1.64)

  • And a margin of error of 15%

  • The minimum number of tests is 16.

There are other more sophisticated methods of calculation, but this example shows the difficulty of choosing the items to be tested and the link between the number and its representativeness for a group (Fig. 16.11).

Fig. 16.11
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Student’s law

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