## 2 Methodology  
The equivalent load in a one-dimensional task should have a distribution ensuring a solution corresponding to the actual behaviour of the structure and enabling the most accurate determination of all quantities necessary for the proper design of the LTP geosynthetic reinforcement. A membrane modelling the geosynthetic reinforcement, loaded with the equivalent load of a given general distribution, is shown in Figure 2. The parameter K is equivalent subgrade reaction value [kN/m\({}^{3}\)].  
In this paper, the loads in the form of a function (1) modelling all distributions from inverse triangular, through uniform, to triangular, were assumed:  
\[\overline{p}(x)=\frac{2q_{av}(x(1-2\delta)+\delta L/2)}{L/2}, \tag{1}\]  
where:  
\(\overline{p}(x)\)--function describing the equivalent load [kPa];  
\(q_{av}\)--equivalent load on the membrane [kPa];  
\(\delta\)--coefficient determining the load distribution [-];  
\(L\)--clear distance between adjacent columns [m];  
\(x\)--distance from the centre between the columns [m].  
Parameter \(\delta\) is the parameter in the range from 0 to 1. The load attains the inverse triangular distribution for \(\delta=0\), the uniform distribution--for \(\delta=0.5\), and the triangular distribution--for \(\delta=1\). Examples of load distributions depending on parameter \(\delta\) are shown in Figure 3.  
Figure 2: A membrane modelling the reinforcement of the LTP, boundary conditions and coordinate system.  
In further analyses, 41 load distributions from inverse triangular to triangular (\(\delta\in\langle 0,1\rangle\)) were taken into account with the \(\delta\) increment equal to 0.025. The deflection curve of the membrane was determined for each analysed load distribution. In order to determine the forces in the membrane and its deflection curve \(w(x)\), the method presented by the authors in [35] was applied. This method is based on expanding into the Taylor series the expression for the curve length used to determine the deformation in the membrane. The membrane tension force \(T_{h}\) can be determined from the condition for strain equality (see [35; 36]). Unfortunately, this equation is highly nonlinear and additionally the accuracy of the solution depends on the number of series terms in the expression for the curve length. Tension force is found using numerical methods and the Wolfram Mathematica program. In order to compare the deflection of the membrane under each of the analysed load distributions with the results of experimental tests (laboratory and in situ), the formula for the interpretation of the mapping error was introduced in the following form:  
\[ref=\sqrt{\frac{\int_{0}^{L/2}\left[w_{\text{exp}}(x)-w(x,T_{h})\right]^{2}dx} {\int_{0}^{L/2}\left[w_{\text{exp}}(x)\right]^{2}dx}} \tag{2}\]  
As 41 values of parameter \(\delta\) were introduced and the obtained solutions were compared with 5 cases (including 18 deflection curves resulting from the measurements), a program was written in Mathematica to automate the calculations. For example, Figure 4 shows the degree of accuracy of the deflection curve resulting from the measurements (blue line) for three load distributions represented by \(\delta=0\), \(\delta=0.5\) and \(\delta=1\).  
Figure 4: Comparison of the deflection curve of the reinforcement with the deflection curve of the membrane for three equivalent load distributions (inverse triangular, uniform and triangular), using Formula (2); (source of measurement data [10]).  
Figure 3: Examples of load distributions depending on parameter \(\delta\).