Quantifying mesh resistance to opening of netting panels: experimental method, regression models, and parameter estimation strategies

Abstract

The increased mesh resistance to opening of netting panels manufactured with thick and stiff twines has a notable impact in the structural response and selective performance of the fishing gears. The only available method to quantify the mesh resistance to opening of netting panels was described in Sala et al. (in Experimental method for quantifying resistance to the opening of netting panels. ICES Journal of Marine Science, 64: 1573–1578, 2007b). We present an alternative method with a similar methodology: we attempt to estimate the mechanical and geometrical properties of a netting material that best fits the experimental measurements of a netting panel. We introduce three major contributions: (i) a considerably simpler uniaxial experimental set-up, which stretches a netting sample in the normal direction of the meshes while leaving free its deformation in the transverse direction; (ii) more accurate theoretical models for mesh resistance to opening; and (iii) new strategies to estimate the parameters of the models. We present the results of the analysis of polyethylene, compacted polyethylene, single-twine, and double-twine netting. Some of the assessed combinations of estimation strategies and theoretical models have an excellent goodness of fit with experimental data. The method proved to be a simple yet accurate way to quantify the mesh resistance to opening of netting panels.

bending stiffness, flexural rigidity, mesh resistance to opening, regression, twine

Introduction

In recent years, there is a tendency in some sectors of the fishing industry towards the use of thicker and stiffer twines in the manufacture of netting materials for the codend of trawls. The increased mesh resistance to opening of such materials has a notable impact in the structural response and performance of the fishing gears. For example, an increased mesh resistance to opening hinders mesh opening in the codend (O’Neill, 2004), which affects the escapement of small fish. Theoretical and experimental studies demonstrate that mesh resistance to opening plays a major role in the reduction of selective performance of trawls (Lowry and Robertson, 1996; Herrmann and O’Neill, 2006; Sala et al., 2007a; Herrmann et al., 2013). Theoretical models for mesh resistance to opening are generally based on the beam theory of solid mechanics. In diamond mesh panels, the predominant netting in towed fishing gears, the resistance to opening is mainly characterized by the bending stiffness of the netting twine. An increased twine bending stiffness also changes the overall shape of the fishing gear during fishing operations (Priour, 2001; O’Neill, 2004). Therefore, methods to quantify the mesh resistance to opening and to incorporate this property in theoretical models of netting materials are necessary to accurately predict the selective performance of fishing gears by simulation. Despite this, research about this topic is still scarce (Priour and Cognard, 2011).

The only available method to quantify mesh resistance to opening of netting panels is described in Sala et al. (2007b). The method uses a specially designed instrument that applies normal and transversal displacements to a netting sample and measures the generated reaction forces. Then, twine bending stiffness and geometric parameters of the netting are estimated through non-linear regression analysis of the obtained experimental data. The asymptotic solution for a bending twine proposed in O’Neill (2002) is used as a model in the regression analysis. Although the method proved to be robust and useful to estimate mesh resistance to opening, the authors reported several problems: inconsistencies between normal and transversal forces and displacements, occasional unrealistic estimates of geometrical parameter values, and systematic lack of fit of the model to the experimental data. Another concern is that the authors carried out the regression analysis using the force as an independent variable and the displacement as a dependent variable, despite the force was an effect caused by an applied displacement in the experimental set-up. The complexity of the experimental set-up required by this method is another important drawback.

A method to estimate the twine bending stiffness was proposed by Priour and Cognard (2011). The method measures the out-of-plane bending deformation of a netting sample and then adjusts a theoretical model of a cantilever beam to the experimental data, in order to estimate the twine bending stiffness EI. The method is very simple, but it has some drawbacks. It does not take into account the knot size, which can have an important effect on the shape of the codends (PREMECS: Development of predictive model of cod-end selectivity, 2000). In addition, it cannot estimate the slope angle between twines and knots at the insertion points.

The goal of this article is to describe a simple but accurate experimental method to quantify the mesh resistance to opening of netting panels. We follow a methodology similar to Sala et al. (2007b), that is, we attempt to estimate the mechanical and geometrical properties of a netting material that best fits the experimental measurements of a netting panel. This research introduces three original contributions:

• The biaxial experimental set-up used in Sala et al. (2007b) requires a very complex measurement instrument that is not commercially available. In contrast, this work proposes a new uniaxial experimental set-up that notably simplifies the required measurement instrument.

• The regression model used in Sala et al. (2007b) is the asymptotic model for a bending twine described in O’Neill (2002). This model is an approximate solution. This work uses more accurate models to describe mesh resistance to opening: the exact model described in O’Neill (2002) and two recently developed models based on finite element analysis (de la Prada and González, 2014a).

• The parameter estimation strategy used in Sala et al. (2007b) fixed one of the geometrical properties of the netting (the slope angle between the twine and the knots) and leaved the remaining parameters unconstrained. As a result, the estimates were sometimes out of physical limits. This work assesses other estimation strategies that avoid that problem.

Material and methods

Theoretical models for mesh resistance to opening

Priour (2001) proposed a theoretical model for mesh resistance to opening based on the assumption that the couple created by mesh twines on the knot varies linearly with the angle between twines. Although this model can easily be introduced in numerical formulations for netting structures, it does not involve parameters with specific physical interpretation, and therefore, it is not suited to identify the mechanical properties of the netting (e.g. twine bending stiffness) from experimental data.

O’Neill (2002) proposed a physically oriented approach to the problem by modelling the twine as a bi-dimensional double-clamped beam (Figure 1) and describing the equations governing its bending assuming that: (i) the slope angle θ between the twines and the knots at the insertion points remains fixed, (ii) the bending moment is proportional to the curvature of the twine, and (iii) there is no twine extension. He found two analytical solutions: the exact solution and an asymptotic solution. The exact solution expresses the coordinates of the endpoint of the twine p₁ (Figure 1) as a set of implicit non-linear equations. To overcome the high complexity of the exact solution, the asymptotic solution expresses the coordinates of p₁ as an explicit function of the tensile forces acting on it:

$$x_{\mathrm{twine}}=L_{\mathrm{twine}}\cos \beta +4\sqrt{\frac{EI}{F}}\left\{\cos \left(\frac{\theta +\beta }{2}\right)-\cos \beta \right\},$$

(1)

$$y_{\mathrm{twine}}=L_{\mathrm{twine}}\sin \beta +4\sqrt{\frac{EI}{F}}\left\{\sin \left(\frac{\theta +\beta }{2}\right)-\sin \beta \right\},$$

(2)

where θ is the slope angle between the twines and the knots at the insertion points, L_twine is the twine length, EI is the twine bending stiffness, $F=(F_x^2+F_y^2{)}^{0.5},$$\beta ={\tan }^{-1}(F_y/F_x),$ and F_x and F_y are the tensile forces that act at the end of the twine. This approximation is very close to the exact solution when ${\epsilon }^2=EI/(F{L}_{\mathrm{twine}}^2)<0.04$⁠. Both analytical solutions were used in O’Neill (2003) to study the factors influencing the measurement of netting mesh size. The asymptotic solution was used in Sala et al. (2007b) to develop an experimental method for quantifying mesh resistance to opening. It has been demonstrated that the model proposed in Priour (2001) can be modified to approximate the asymptotic solution for small values of ɛ² (O’Neill and Priour, 2009).

Figure 1.

Double-clamped beam model of a mesh twine.

In de la Prada and González (2014a), a different approach was followed aiming at the development of models that enable a fast and accurate evaluation of elastic forces in the twine as a function of its deformation. The mesh twine was also modelled as a double-clamped beam (Figure 1), but its force–displacement response was calculated by finite element analysis (Zienkiewicz, 2000). Then, fitting techniques were used to develop two dimensionless models. The first model, named the polynomial model, fits a polynomial of the radial and tangential force components F_r and F_φ that act at the end of the twine. This results in a polynomial degree (m, n) of (2, 3) for F_r and (1, 4) for F_φ:

$$F_r(r,\phi )=\left(\frac{EI}{L_{\mathrm{twine}}^2}\right)\sum _{0<i+j<m+n}{c_{ij}{r}^i(\cos \phi )}^j,$$

(3)

$$F_{\phi }(r,\phi )=\left(\frac{EI}{L_{\mathrm{twine}}^2}\right)\sum _{0<i+j<m+n}{c_{ij}{r}^i(\cos \phi )}^j,$$

(4)

where $r=(x_{\mathrm{twine}}^2+y_{\mathrm{twine}}^2{)}^{0.5}/L_{\mathrm{twine}},$$\psi ={\tan }^{-1}(y_{\mathrm{twine}}/x_{t\mathrm{wine}}),$ and φ = ψ− θ.

The second model, called the spline model, calculates F_r and F_φ as the gradient of the potential elastic energy V of the twine. V was interpolated with (N_r− 1) × (N_φ− 1) bicubic spline patches, where each patch (i, j) spans the rectangular region [r_i, r_i+1] × [φ_j, φ_{j +1}] and expresses V as:

$$V_{ij}(r,\phi )=\left(\frac{EI}{L_{\mathrm{twine}}}\right)\sum _{k=0}^3\sum _{l=0}^3{c}_{kl}^{ij}(r-r_i{)}^k(\phi -{\phi }_j{)}^l.$$

(5)

A comprehensive description of the polynomial and spline models is provided in de la Prada and González (2014a), with comparison to the abovementioned exact and asymptotic models described in O’Neill (2002). The spline model is very accurate and it is well suited for simulation methods based on energy minimization (de la Prada and González, 2014b). The polynomial model has a simpler mathematical form at the cost of a slightly lower accuracy (5–10% of deviation), as reported in de la Prada and González (2014a).

Experimental set-up

The experimental set-up is portrayed in Figure 2. A rectangular netting sample is attached between an upper fixed bar and a bottom free bar, with the normal direction of the meshes aligned to the vertical axis of the figure (i.e. perpendicular to the bars). The free bar can move in the normal direction of the netting while keeping its orientation parallel to the fixed bar. The attachment system between the netting sample and the bars allows the netting knots to freely move in the transverse direction of the netting when the sample is stretched. In this way, the force applied to meshes and twines in the transverse direction of the netting is zero. The panel has m_n and m_t netting meshes in normal and transverse directions, respectively. During the experiment, the sample is stretched in the normal direction of the netting to open the meshes.

Figure 2.

Design of the experimental set-up and general view of a netting sample during the test.

The sample is stretched by applying a force F_panel to the free bar. The normal length of the panel L_panel, defined as the distance between centres of the upper and bottom knots, is calculated as L_panel = D₀− (D_R + D_L)/2 − D₁− D₂, where distances D₀, D₁, and D₂ are measured at the beginning of the experiment and distances D_R and D_L are measured for each value of F_panel (both distances should be equal in theory, but in practice slight differences can appear due to misalignment of the free bar). F_panel is generated by applying calibrated weights to the free bar. The weight of the free bar (0.68 N) and mounting hooks (0.04 N each) is also included in F_panel. Distances D₁ and D₂ are measured with a Vernier caliper, while distances D₀, D_R, and D_L are measured with digital laser rangefinders with an accuracy of ±0.5 mm. Other measuring procedures are also compatible with the experimental set-up described in Figure 2: for example, L_panel could be prescribed and the required F_panel could be measured.

The objective of this work is not to quantify the mesh resistance to opening for a wide range of netting materials, but rather to investigate different combinations of regression models and parameter estimation strategies for the proposed uniaxial experimental set-up. Hence, a number of new and unused netting samples was tested. All of them are used in commercial North Sea trawls. Their main characteristics are given in Table 1: materials are sorted according to the perceived mesh resistance to opening from manual manipulation of the netting, from low [polyethylene (PE) 80 × 2.5] to high [double-twine compacted polyethylene (CPE2) 80 × 4] stiffness.

Table 1.

Main characteristics of the netting samples: nominal stretched mesh size (L_mesh), nominal twine diameter (D_twine), external knot width (a_ext) and height (b_ext), number of meshes in transverse (m_t) and normal (m_n) directions, and linear density (Rtex).

Netting	Lmesh (mm)	Dtwine (mm)	aext (mm)	bext (mm)	mt × mn	Rtex (g/1 000 m)
PE 80 × 2.5	80	2.5	8.5	5.5	4 × 8	2 540
PE 100 × 2.5	100	2.5	8.5	5.5	3 × 10	2 870
PE 80 × 3	80	3	11.0	6.0	3 × 12	4 225
PE 80 × 4	80	4	12.5	8.5	3 × 10	5 623
PE 100 × 4	100	4	12.5	8.5	3 × 8	6 474
CPE 80 × 5	80	5	16.0	12.0	3 × 8	11 423
CPE2 80 × 4	80	4	25.0	14.0	2 × 5	12 310

Material codes: PE is traditional single-twine greed-braid polyethylene, CPE is single-twine compacted polyethylene, and CPE2 is double-twine compacted polyethylene.

The following steps were performed for each netting sample:

• The netting is attached to the bars and distances D₁ and D₂ are measured. At this moment, F_panel is equal to the weight of the bottom free bar and hooks (0.7 N).

• F_panel is increased. Load increments start with 0.5 N and increase up to a maximum value of 9.8 N as L_panel increases.

• Distances D_R and D_L are measured at every minute to monitor netting twine relaxation (Sala et al., 2007b); when they get stabilized, the final values are recorded.

• Steps (ii) and (iii) are repeated until L_panel reaches 80% of m_n·L_mesh, where L_mesh is the nominal mesh size. Above this value, the main characteristic contributing to mesh resistance to opening is twine axial stiffness EA rather than twine bending stiffness EI (de la Prada and González, 2014a).

Netting materials used in fishing gears may experience high tensile forces able to generate permanent plastic deformations in twines and knots. To simulate such situation in the experimental set-up, the maximum value of F_panel reached in step (iv) is applied for 1h to the netting. Then, steps (ii) and (iii) are repeated with decreasing values of F_panel in step (ii). In this way, two sets of data are obtained: a loading cycle and an unloading cycle.

Data analysis

To analyse the experimental data, it is necessary to make the same assumptions as in Sala et al. (2007b), which define the idealized panel represented in Figure 3: (i) the netting is homogeneous, so all the meshes experience the same deformation, (ii) knots can be represented as rectangles of size a × b, and (iii) twines emerge from knots at the corners of the rectangles. Note that the knot size a × b is smaller than the measured outer knot size a_ext × b_ext given in Table 1. The mechanical and geometrical parameters of this idealized netting panel are estimated by fitting theoretical models for mesh resistance to opening to experimental data. The observed variable in the experiment is the distance between twine knots in the normal direction of the netting

$$y_{\mathrm{knots}}=\frac{L_{\mathrm{panel}}}{2m_{\mathrm{n}}}$$

(6)

and the explanatory variable is the vertical force applied to the twines

$$F_y=\frac{F_{\mathrm{panel}}}{2m_{\mathrm{t}}}.$$

(7)

The predicted values for the distance between knots can be expressed as follows:

$${\hat{y}}_{\mathrm{knots}}=b+y_{\mathrm{twine}}(EI,L_{\mathrm{twine}},\theta ),$$

(8)

where y_twine can be calculated with any theoretical model for mesh resistance to opening. Therefore, four parameters can be estimated: EI, L_twine, b, and θ. Notice that the knot dimension a cannot be estimated with the proposed experimental set-up, because the transversal length of the panel is not measured. Preliminary analysis shows that there are no outlier points in the experimental data. For simplicity, the observed and the explanatory variables are assumed to be distributed with constant variance. Therefore, non-linear least-squares regression is used (Seber, 1989).

Figure 3.

Idealized netting where mesh twines are modelled as beams emerging from the corners of rectangular knots.

With respect to the regression model $y_{\mathrm{twine}}(EI,L_{\mathrm{twine}},\theta )$ in Equation (8), the four models for mesh resistance to opening described in the section Material and methods were assessed: the exact model, the asymptotic model, the polynomial model, and spline model. The use of the asymptotic model is straightforward, since it provides y_twine as an explicit function of the explanatory variable F_y. The other models need to be numerically solved for every evaluation in the regression analysis.

A remark must be made regarding the use of the asymptotic model in this experimental set-up, where $F_x=0\Rightarrow \beta ={\tan }^{-1}(F_y/F_x)=\mathrm{\Pi }/2$ in Equations (1) and (2). The asymptotic approximation is very close to the exact solution when ɛ² < 0.04 (O’Neill, 2002), but ɛ² usually ranges from 0.4 and 0.1 in most part of the loading and unloading cycles of our experiments. This is explained in Figure 4, which shows the dimensionless vertical displacement of the twine calculated with Equation (2) for β = Π/2, as a function of the parameter ${\epsilon }^2=EI/(F{L}_{\mathrm{twine}}^2)$ and the slope angle θ. Since F_x = 0, the mesh can be opened in the normal direction with a relatively small force. Hence, the condition ɛ² < 0.04 can only be achieved if θ is small and the mesh is nearly completely opened (y_twine/L_twine ≥ 0.75). The only way to achieve ɛ² < 0.04 without completely opening the mesh is to apply a transversal force F_x > 0 to reduce β as in the experimental set-up used in Sala et al. (2007b). Nevertheless, de la Prada and González (2014a) show that the vertical position of the twine y_twine predicted by the asymptotic model is close to the exact model even if β = Π/2 (relative error below 10% in most of the range of displacement), because most of the error in this model is in the horizontal position x_twine. As a result, the asymptotic solution can still be used as a model in this experimental set-up, but caution should be taken to interpret its goodness of fit.

Figure 4.

Dimensionless vertical displacement of the twine (y_twine/L_twine) calculated with the asymptotic solution as a function of ${\epsilon }^2=EI/(F{L}_{\mathrm{twine}}^2)$ for different slope angles θ.

Parameter estimation strategies

Regression analysis can generate parameter estimates that are out of physical limits. For example, Sala et al. (2007b) reported that the estimates for θ were often negative, which is physically impossible. To circumvent this problem, constraints can be applied to the parameters. Table 2 summarizes the two types of parameter constraints considered in this work.

Table 2.

Constraints that can be applied to the parameters in the regression analysis.

Constraint	EI	Ltwine	b	θ (°)
Fixed	–	$L_{\mathrm{mesh}}/2-2D_{\mathrm{twine}}^{\ast }$	$D_{\mathrm{twine}}^{\ast }$	0
Min	0	Lmesh/2 − aext	0	5
Max	∞	Lmesh/2	bext	90

$D_{\mathrm{twine}}^{\ast }=D_{\mathrm{twine}}$ for single-twine netting and $D_{\mathrm{twine}}^{\ast }=2D_{\mathrm{twine}}$ for double-twine netting.

A fixed constraint means to remove the parameter from the regression analysis, which greatly reduces the required computational effort. For example, Sala et al. (2007b) fixed θ to 0° to avoid negative estimates, and this constraint was also considered in this work. Fixed constraints for L_twine and b are obtained by assuming that the knot size a × b matches $2D_{\mathrm{twine}}^{\ast }\times D_{t\mathrm{wine}}^{\ast },$ with $D_{\mathrm{twine}}^{\ast }=D_{\mathrm{twine}}$ for single-twine netting and $D_{\mathrm{twine}}^{\ast }=2D_{\mathrm{twine}}$ for double-twine netting. This is a plausible assumption after a visual inspection of several netting samples (Figure 3).

Another approach is to constrain parameters between minimum and maximum physical limits. The limits for L_twine and b are obtained by assuming that the knot size a × b can be neither negative nor greater than a_ext × b_ext. A minimum value of 5° for θ seems reasonable to avoid knot overlapping when no forces are applied to the netting, and it is also consistent with visual observations in the netting samples.

Finally, the different constraints listed in Table 2 were combined to form the four parameter estimation strategies summarized in Table 3. All of them were assessed in this work.

Table 3.

Description of the different parameter estimation strategies used in the regression analysis.

Estimation strategy	Constraint applied on parameter
	Ltwine	b	θ
1	–	–	–
2	Min/max	Min/max	Min/max
3	Fixed	Fixed	Min/max
4	–	–	Fixed

Results

Figure 5 shows an example of the datasets obtained in the loading cycle and in the unloading cycle. The number of recorded measurements in every netting panel and cycle ranged between 17 and 21. The two datasets are quite different. The length of the panel $L_{\mathrm{panel}}^{\mathrm{end}}$at the end of the unloading cycle is higher than the length $L_{\mathrm{panel}}^{\mathrm{start}}$ at the beginning of the loading cycle. The ratio $L_{\mathrm{panel}}^{\mathrm{end}}:L_{\mathrm{panel}}^{\mathrm{start}}$ is higher in PE netting samples (1.52 on average) than in the CPE sample (1.23) or the CPE2 sample (1.37). The two datasets were analysed separately to quantify the mesh resistance to opening.

Figure 5.

Experimental data obtained for PE 80 × 3, showing the difference between loading and unloading cycles.

Loading cycle

Table 4 summarizes the results of the non-linear regression with unconstrained parameters (parameter estimation strategy No. 1). Confidence intervals for estimates of EI are represented as a percentage; confidence intervals for other parameters are omitted because they are of the same order of magnitude as confidence intervals for estimates of EI. The goodness of fit is measured with the coefficient of determination R². Results indicate that all models can fit a diverse variety of experimental datasets, as represented by their ability to achieve very high R² values at the cost of providing estimates of L_twine, b and θ that are often out of their physical limits. Estimates of EI are inconsistent in some cases. For example, the estimates for PE 80 × 4 and PE 100 × 4 are very different, despite their stiffness seems very similar when they are manipulated by hand.

Table 4.

Results of the analysis with unconstrained parameters (parameter estimation strategy No. 1), loading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	θ (°)	R2
PE 80 × 2.5	Exact	23 ± 6	28	9.9	0	0.9995
	Polynomial	58 ± 23	53	−16.8	22	0.9995
	Spline	10 ± 3	20	17.9	−21	0.9996
PE 100 × 2.5	Exact	21 ± 22	37	9.3	−2	0.9956
	Polynomial	22 ± 39	32	13.6	4	0.9973
	Spline	10 ± 34	24	21.8	−17	0.9930
PE 80 × 3	Exact	28 ± 60	27	11.4	−1	0.9965
	Polynomial	33 ± 36	32	5.1	6	0.9984
	Spline	13 ± 9	19	18.8	−21	0.9969
PE 80 × 4	Exact	92 ± 66	32	3	19	0.9995
	Polynomial	107 ± 31	43	−10.0	25	0.9994
	Spline	50 ± 86	25	9.7	9	0.9996
PE 100 × 4	Exact	33 ± 9	21	23	0	0.9991
	Polynomial	85 ± 38	41	2.4	22	0.9987
	Spline	19 ± 8	17	26.7	−12	0.9993
CPE 80 × 5	Exact	153 ± 7	22	15.6	1	0.9998
	Polynomial	140 ± 31	25	10.2	7	0.9974
	Spline	181 ± 6	23	14.5	4	0.9997
CPE2 80 × 4	Exact	1001 ± 70	34	1.7	25	0.9996
	Polynomial	289 ± 32	24	9.7	10	0.9986
	Spline	238 ± 10	19	16.3	0	0.9992

This unconstrained regression analysis exposed the degree of correlation in the parameters of two models. The exact model generates several solutions with R² > 0.99, showing some degree of correlation between parameters: Figure 6 reveals a non-linear correlation between EI and θ, and a linear correlation between b and L_twine. Nevertheless, it is easy to select the best solution, since most of them have unrealistic parameter estimates: among the solutions with plausible estimates, the one with highest R² is listed in Table 4. The asymptotic model exhibits a very strong correlation between EI and θ, and between b and L_twine (Figure 6). It generates multiple solutions with identical R², most of them with plausible estimates, so it is not possible to identify the best solution. Hence, results for this model are not listed in Table 4. An analysis of Equations (2) and (8) reveals that parameters are not identifiable for this regression model because its basis functions are not orthogonal when β is constant (Seber, 1989).

Figure 6.

Multiple solutions of the regression with unconstrained parameters for PE 100 × 4, showing the correlation between parameters in the exact and the asymptotic models: (a) bending stiffness EI vs. slope angle θ and (b) knot height b vs. twine length L_twine.

Table 5 presents the results of the regression with min/max constraints on all the parameters (parameter estimation strategy No. 2). Despite the constraints on the parameters, R² values are almost as high as in Table 4. Confidence intervals for the spline model are unusually high. Table 6 reports the results for the parameter estimation strategy No. 3, which applies fixed constraints on L_twine and b and min/max constraints on θ. In this strategy, the fixed values for L_twine and b correlate with visual observations of the netting. Furthermore, the computational effort of the regression analysis is reduced by about one order of magnitude with respect to the previous estimation strategies (e.g. from 10 min to 20 s for the polynomial and the spline models). R² values are still very high, confidence intervals are reduced, and the estimates of EI obtained with the three models are closer to each other than in Tables 4 and 5. Note that the constraint on θ was activated in only one netting sample (PE 80 × 3). Regarding the asymptotic model, the strong correlation between EI and θ is still present in this analysis.

Table 5.

Results of the analysis with min/max constraints on all the parameters (parameter estimation strategy No. 2), loading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	θ (°)	R2
PE 80 × 2.5	Exact	27 ± 2	35.2	2.0	7	0.9928
	Polynomial	33 ± 37	36.8	0.0	11	0.9979
	Spline	32 ± 214	33.7	3.8	8	0.9991
PE 100 × 2.5	Exact	27 ± 1	45.4	0.9	5	0.9926
	Polynomial	31 ± 44	41.5	3.7	12	0.9956
	Spline	30 ± 261	41.5	3.8	10	0.9872
PE 80 × 3	Exact	57 ± 1	35.5	2.8	15	0.9931
	Polynomial	34 ± 36	32.8	4.6	7	0.9984
	Spline	39 ± 395	32.4	5.3	8	0.9946
PE 80 × 4	Exact	119 ± 2	32.0	3.8	19	0.9949
	Polynomial	66 ± 8	33.4	0.0	17	0.9987
	Spline	67 ± 98	27.9	6.3	15	0.9996
PE 100 × 4	Exact	159 ± 1	42.0	2.8	29	0.9983
	Polynomial	85 ± 38	41.0	2.4	23	0.9987
	Spline	111 ± 220	37.6	6.3	25	0.9986
CPE 80 × 5	Exact	445 ± 1	32.4	6.1	18	0.9993
	Polynomial	159 ± 36	26.5	8.8	9	0.9973
	Spline	244 ± 71	25.8	11.5	10	0.9997
CPE2 80 × 4	Exact	683 ± 1	28.5	7.4	18	0.9993
	Polynomial	294 ± 33	24.2	9.5	10	0.9986
	Spline	373 ± 111	22.5	12.4	9	0.9993

Table 6.

Results of the analysis with fixed constraints on L_twine and b and min/max constraints on θ (parameter estimation strategy No. 3), loading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	θ (°)	R2
PE 80 × 2.5	Exact	34 ± 1	35	2.5	10	0.9955
	Polynomial	33 ± 6	35	2.5	10	0.9938
	Spline	35 ± 6	35	2.5	10	0.9990
PE 100 × 2.5	Exact	39 ± 2	45	2.5	9	0.9814
	Polynomial	56 ± 24	45	2.5	17	0.9182
	Spline	77 ± 39	45	2.5	23	0.9464
PE 80 × 3	Exact	29 ± 2	34	3	5	0.9765
	Polynomial	35 ± 6	34	3	9	0.9974
	Spline	37 ± 25	34	3	9	0.9914
PE 80 × 4	Exact	126 ± 3	32	4	19	0.9920
	Polynomial	99 ± 12	32	4	15	0.9780
	Spline	126 ± 11	32	4	19	0.9904
PE 100 × 4	Exact	219 ± 2	42	4	30	0.9873
	Polynomial	164 ± 14	42	4	24	0.9759
	Spline	218 ± 13	42	4	29	0.9852
CPE 80 × 5	Exact	248 ± 2	30	5	19	0.9914
	Polynomial	198 ± 5	30	5	15	0.9958
	Spline	273 ± 6	30	5	20	0.9946
CPE2 80 × 4	Exact	310 ± 3	24	8	18	0.9852
	Polynomial	233 ± 7	24	8	13	0.9938
	Spline	314 ± 10	24	8	18	0.9881

Finally, Table 7 summarizes the results with a fixed constraint on θ and leaving the remaining parameters unconstrained (parameter estimation strategy No. 4), as in the analysis carried out in Sala et al. (2007b). With this estimation strategy, the asymptotic model can only estimate EI, and therefore L_twine and b are not listed for this model. R² values are high except when the asymptotic model is applied to very stiff materials (CPE 80 × 5 and CPE2 80 × 4). However, estimates of EI are inconsistent for all models: estimates for stiff materials (PE 80 × 4 and PE 100 × 4) are very similar to that for very soft materials (PE 80 × 2.5 and PE 100 × 2.5), which seems abnormal. In addition, estimates of L_twine and b are often out of physical limits.

Table 7.

Results of the analysis with fixed constraint θ = 0° (parameter estimation strategy No. 4), loading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	R2
PE 80 × 2.5	Exact	24 ± 3	28.4	9.7	0.9995
	Polynomial	20 ± 9	27.5	9.4	0.9930
	Spline	22 ± 4	28.1	9.4	0.9992
	Asymptotic	41 ± 10	–	–	0.9936
PE 100 × 2.5	Exact	23 ± 17	37.9	8.7	0.9956
	Polynomial	18 ± 6	27.9	17.6	0.9969
	Spline	19 ± 13	32.3	13.2	0.9896
	Asymptotic	37 ± 21	–	–	0.9812
PE 80 × 3	Exact	29 ± 8	27.7	10.7	0.9964
	Polynomial	26 ± 6	27.4	10.1	0.9979
	Spline	27 ± 10	27.2	10.6	0.9952
	Asymptotic	49 ± 20	–	–	0.9876
PE 80 × 4	Exact	35 ± 4	20.6	13.9	0.9995
	Polynomial	28 ± 12	21.1	12.2	0.9929
	Spline	33 ± 4	20.3	13.8	0.9996
	Asymptotic	39 ± 31	–	–	0.9574
PE 100 × 4	Exact	33 ± 5	21.2	23.2	0.9991
	Polynomial	28 ± 10	21.8	21.7	0.9934
	Spline	31 ± 5	20.9	23.1	0.9991
	Asymptotic	41 ± 31	–	–	0.9656
CPE 80 × 5	Exact	148 ± 5	21.2	15.9	0.9998
	Polynomial	97 ± 14	21.0	13.9	0.9960
	Spline	144 ± 5	21.0	15.9	0.9998
	Asymptotic	30 ± 56	–	–	0.7612
CPE2 80 × 4	Exact	240 ± 11	18.8	16.1	0.9988
	Polynomial	164 ± 13	18.8	14.3	0.9968
	Spline	233 ± 11	18.6	16.1	0.9989
	Asymptotic	32 ± 64	–	–	0.7161

Low R² values are marked in bold font.

A visual inspection of the residual plots of all the analysis confirms that the models have a very good fit when R² > 0.98. R² values between 0.9 and 0.98 often under-predict the experimental data, and values below 0.9 correspond to inaccurate fits. Results from the parameter estimation strategies No. 2 (Table 5) and No. 3 (Table 6) are shown in Figures 7 and 8. Figure 7 shows a box plot of the R² values for different combinations of estimation strategies and regression models. Figure 8 plots the estimates of EI against the linear density of the netting (kRtex).

Figure 7.

Box plot of the R² values from the parameter estimation strategies No. 2 and No. 3, loading cycle.

Figure 8.

Estimates of EI against the linear density of the netting obtained with parameter estimation strategies No. 2 (Table 5) and No. 3 (Table 6).

Unloading cycle

Parameter estimation strategies No. 1 and 4 have not been used for the unloading cycle due to the disadvantages exposed in the loading cycle. Table 8 summarizes the results with the strategy No. 2: min/max constraints on all the parameters (compare with the loading cycle in Table 5). R² values are very good for stiff materials and acceptable for soft materials. Estimates of θ are considerably increased in all materials compared with the loading cycle. Visual observations of the netting samples after the experiment also show an increased θ, but it is not as high as the estimated values. Average estimates of EI are slightly lower than in the loading cycle. The polynomial and spline models exhibit very wide confidence intervals in some cases.

Table 8.

Results of the analysis with min/max constraints on all the parameters (parameter estimation strategy No. 2), unloading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	θ (°)	R2
PE 80 × 2.5	Exact	23 ± 0.4	34.0	0.1	49	0.9897
	Polynomial	17 ± 195	31.5	2.2	46	0.9798
	Spline	24 ± 219	31.5	2.3	51	0.9873
PE 100 × 2.5	Exact	21 ± 1	41.7	1.7	41	0.9871
	Polynomial	15 ± 85	41.5	0.3	46	0.9617
	Spline	49 ± 255	41.6	1.1	56	0.9724
PE 80 × 3	Exact	31 ± 0.3	35.1	0.2	50	0.9903
	Polynomial	12 ± 108	30.6	3.8	44	0.9635
	Spline	12 ± 92	31.2	3.2	44	0.9637
PE 80 × 4	Exact	41 ± 0.4	30.6	1.4	48	0.9860
	Polynomial	22 ± 78	29.2	2.1	44	0.9907
	Spline	42 ± 137	27.6	4.0	50	0.9930
PE 100 × 4	Exact	107 ± 1	40.0	2.1	53	0.9937
	Polynomial	24 ± 53	38.0	2.7	41	0.9917
	Spline	65 ± 132	37.5	3.9	50	0.9947
CPE 80 × 5	Exact	80 ± 1	24.3	8.8	35	0.9931
	Polynomial	52 ± 117	24.2	8.2	32	0.9971
	Spline	78 ± 600	24.0	8.6	37	0.9963
CPE2 80 × 4	Exact	234 ± 1	29.0	2.8	40	0.9987
	Polynomial	93 ± 44	23.4	7.8	30	0.9993
	Spline	249 ± 254	31.2	0.1	43	0.9989

Table 9 presents the results with fixed constraints on L_twine and b, and min/max constraints on θ (compare with the loading cycle in Table 6). Fittings for PE netting samples have R² < 0.9, which indicates a poor fit. This fact is confirmed by a visual inspection of the residual plots. Conversely, CPE 80 × 5 shows acceptable R² values and CPE2 80 × 4 shows very good fits. For these stiff materials, EI estimates are of the same order of magnitude than those calculated for the loading cycle.

Table 9.

Results of the analysis with fixed constraints on L_twine and b and min/max constraints on θ (parameter estimation strategy No. 3), unloading cycle.

Netting	Model	EI (N/mm2) (%)	Ltwine (mm)	b (mm)	θ (°)	R2
PE 80 × 2.5	Exact	264 ± 30	35	2.5	52	0.8343
	Polynomial	433 ± 18	35	2.5	50	0.8563
	Spline	279 ± 31	35	2.5	52	0.8321
PE 100 × 2.5	Exact	752 ± 37	45	2.5	55	0.7635
	Polynomial	853 ± 48	45	2.5	52	0.8490
	Spline	787 ± 37	45	2.5	55	0.7678
PE 80 × 3	Exact	239 ± 42	34	3	56	0.7947
	Polynomial	348 ± 22	34	3	54	0.8261
	Spline	351 ± 0	34	3	59	0.8561
PE 80 × 4	Exact	535 ± 33	32	4	48	0.8086
	Polynomial	973 ± 26	32	4	47	0.8482
	Spline	567 ± 35	32	4	48	0.8054
PE 100 × 4	Exact	712 ± 27	42	4	51	0.8513
	Polynomial	1373 ± 22	42	4	50	0.8415
	Spline	753 ± 28	42	4	51	0.8485
CPE 80 × 5	Exact	263 ± 15	30	5	40	0.9661
	Polynomial	214 ± 15	30	5	35	0.9764
	Spline	277 ± 16	30	5	40	0.9614
CPE2 80 × 4	Exact	171 ± 6	24	8	35	0.9975
	Polynomial	133 ± 8	24	8	30	0.9948
	Spline	182 ± 7	24	8	35	0.9949

Discussion

The proposed uniaxial experimental set-up has a major advantage over the biaxial set-up used in the ROD-m prototype instrument developed in Sala et al. (2007b): it can be carried out in standard uniaxial universal testing machines, provided special fixtures are mounted to allow the attached knots to freely move in the transverse direction of the netting. Even if such a machine is not available, the experiment can easily be carried out by manual means with a simple and inexpensive set-up, as in this work. On the other hand, the uniaxial experimental set-up does not provide measurements of transverse data and the knot width a cannot be directly estimated from the experiment. However, the good results obtained with the parameter estimation strategy No. 3, based on simple assumptions about the values of L_twine and b, suggest that the knot width a could also be estimated as a = (L_mesh− 2L_twine)/2. Note that better approximations could be obtained using the results from O’Neill (2003).

The data analysis assumed that all the meshes experience the same deformation. This assumption is not completely true because the panel was hold in the vertical position during the experiment. Twines at the top of the panel support more weight than those at the bottom due to the mass of the netting. The ratio of the weight of the netting to the weight of the free bar ranged from 0.34 (PE 100 × 2.5) to 1.51 (CPE2 80 × 4). The ratio of the weight of the netting to the maximum applied force ranged from 1.7 to 2.5%. Hence, the assumption about uniform mesh deformation is plausible except at the beginning of the loading cycle and at the end of the unloading cycle. The assumption would be correct during all the cycle, provided the panel was hold in the horizontal position.

Regarding the parameter estimation strategies, strategies No. 1 and No. 4 do not offer advantages over the others and, in fact, have important disadvantages: estimates of EI are inconsistent and estimates of geometrical parameters are often out of physical limits. The other two strategies have their own advantages and disadvantages. Strategy No. 2 (min/max constraints on all parameters) provides slightly better fits, but strategy No. 3 (fixed constraints on L_twine and b and min/max constraints on θ) simplifies the analysis, narrows the confidence intervals, and provides EI estimates that are closer across different models. Moreover, the min/max constraint on θ is hardly activated and could be removed, resulting in an even simpler unconstrained regression analysis. The lower computational effort required by strategy No. 3 can be an advantage when the number of experimental data points is high (e.g. hundreds of points). In these cases, strategy No. 2 can take hours to analyse the experimental data while strategy No. 3 can take minutes. The main disadvantage of strategy No. 3 is that it cannot provide good fits for the unloading cycle, except for the very stiff materials. It seems that the large mesh opening applied to PE netting samples before the unloading cycle has introduced permanent deformations in the twines which cause it to not match the idealized netting material described in Figures 1 and 3. On the contrary, the permanent deformations in CPE netting samples seem to be small due to their increased stiffness and strength, and therefore, analysis results for their unloading cycles are still quite acceptable.

Regarding the regression models, the asymptotic model exhibits identifiability problems in the proposed uniaxial experimental set-up. Similar problems of correlation between parameters have already been reported in Sala et al. (2007b). All three of the other models provide similar and very good fits. The exact model seems more reliable due to its narrower confidence intervals. However, this comes at the cost of a very complex computer implementation. The polynomial and spline models are easier to implement, but they provide wider confidence intervals. The variations in estimates provided by different models may seem surprising. However, the estimates are not absolute measurements of the netting properties, but rather calibration parameters for different theoretical models, as observed in Sala et al. (2007b). Therefore, the model used to analyse the experimental data should be the same model that will be used to make predictions of the netting behaviour, for example, to simulate gear behaviour (Priour, 1999; Takagi et al., 2004; Lee et al., 2005; Li et al., 2006; Priour et al., 2009) or codend selectivity (Herrmann, 2005; O’Neill and Herrmann, 2007).

A detailed analysis of the fits revealed that the estimates given by the exact and spline models are interchangeable. See, for example, the estimates for CPE 80 × 5 in Table 5. They are very different, but when the exact model is evaluated with the parameters estimated with the spline model, it provides a R² value of 0.9987. Conversely, when the spline model is evaluated with the parameters estimated with the exact model, it provides a R² value of 0.9988. Visual inspections of the residual plots confirm that both fits are extremely good and virtually identical. On the contrary, estimates from the polynomial model are not interchangeable with other models. This behaviour agrees with the numerical experiments in de la Prada and González (2014a), which showed that the exact and spline models provide almost identical results. In fact, this also suggests that the narrow confidence intervals of the exact model are not realistic, since very similar goodness of fits can be obtained with very different parameter values. In this sense, the wide confidence intervals of the spline model seem more realistic.

The EI estimates obtained with our best estimation strategies (Figure 8) are smaller than those obtained in Sala et al. (2007b) for similar materials. It is difficult to compare the results from both works, because (Sala et al., 2007b) does not provide qualitative or quantitative indicators of the goodness of fit.

The difference between the loading and the unloading experimental datasets seems a result of the plastic deformations in the netting due to long-term exposure to high stress. Such plastic deformations may be related to visco-elastic creep, which can occur in polymers at room temperatures (McCrum et al., 1997). The available theoretical models for mesh resistance to opening assume a lineal material, and therefore they can only predict the behaviour of netting that operates in the linear range. They cannot be used to predict plastic deformations. For this reason, the loading and the unloading cycles need to be analysed separately, and the data analysis gives different parameter estimates in both cycles. In fact, Figure 5 shows that the mesh resistance to opening is different in both cycles. This result suggests that further research is required to investigate how the loading history affects the mesh resistance to opening of netting during the lifespan of a fishing gear.

Some objections can be made to the experiments presented in this work. In Sala et al. (2007b), a series of pretension cycles were applied to the netting samples to remove the irreversible part of the elongation and to safeguard against knot slippage (Sala et al., 2004). We applied such pretension by manual means, which obviously cannot achieve the high tensile loads applied in Sala et al. (2007b). However, we believe that this does not invalidate the excellent results obtained with the proposed combination of experimental set-up, regression models, and estimation strategies. Another concern is that several samples of the same material should be tested to obtain average estimates of mesh resistance to opening, as in Sala et al. (2007b). Due to resource limitations, we only tested one sample per material, which means that the obtained estimates may be affected by irregularities or defects in the sample. Nonetheless, as stated in the Introduction, the goal of this work is not to quantify the mesh resistance to opening for a range of netting materials, but rather to investigate the soundness of the presented method.

Conclusions

The method we have presented proved to be a simple yet effective method to quantify the mesh resistance to opening of netting panels. Its main advantage over the method described in Sala et al. (2007b) is the simplicity of our uniaxial experimental set-up, which does not require complex and specially designed measuring instruments. In fact, our experiment can be carried out by manual means or in standard universal testing machines with inexpensive modifications of clamps and fixtures. The advantage over the method described in Priour and Cognard (2011) is that the presented method takes into account the knot size and can estimate the angle θ.

We recommend starting the data analysis assuming that L_twine = L_mesh/2 −2D_twine and b = D_twine, in order to estimate EI and θ with an unconstrained non-linear regression. This kind of analysis is simple and fast, and often provides excellent results when the netting material has not suffered permanent plastic deformations due to large mesh opening. If this analysis fails to provide a good fit, a second analysis should be carried out to estimate the four parameters EI, L_twine, b, and θ, with a min/max constrained non-linear regression using the parameter limits listed in Table 2. This approach always provides good fits and parameter estimates within physical limits. Our method does not allow one to estimate the knot width a from experimental data, although it could be estimated as a = (L_mesh− 2L_twine)/2. Better approximations might be obtained using the results from O’Neill (2003).

The three theoretical models—exact, polynomial, and spline—for mesh resistance to opening provide very similar goodness of fit. We recommend analysing the experimental data with the same model that will be used to predict netting deformations. It was found that the loading history can modify the mesh resistance to opening of a netting panel. Further research is required to investigate this issue.

Author notes

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Handling editor: Finbarr (Barry) O’Neill