# Large-Amplitude Vibrations of Spider Web Structures
## 2 Materials and Methods
### Modelling of Spider Web  
Many researchers have investigated the mechanical properties of spider silk. In this model, radial and spiral captured threads are made of dragline silk of the _Argiope aurantia_ spider with a circular fibre diameter of 4 microns. Generally, spiral threads are made of viscid silk which has lower strength but higher extensibility than dragline silk and this combination optimise the function of a spider web. However, in this research, we focussed on the structure behaviour itself, therefore, the same material properties were applied on both radial and spiral thread. The material properties of spider web used are listed in Table 1[4]. A large-deformable finite element model of spider web structures with geometric nonlinearities was developed using Abaqus based on the typical spider web structure. It is noted that there are two main thread elements in the spider web structure: spiral thread and radial thread as shown in Figure 1[9]. The structural performance is dominated by the radial thread while the spiral tread acts as a non-structural member so that higher forces are required to break the radial thread. Based on the finite element method, both relatively simple linear problems and nonlinear analysis can be solved. The spider web model was developed on a 2-D plane (X and Y axis), but it is a 3-D model because the free vibration of the spider web would happen on the third axis, Z. The analysis is divided into two parts: linear analysis and nonlinear analysis. For linear analysis, five different patterns of spider web were built to compare the effects of different parameters and structural patterns on natural frequencies and corresponding mode shapes. For nonlinear analysis, natural frequencies extracted from large amplitude free vibration and corresponding mode shapes were investigated, taking geometric nonlinearities into account. In this study, six patterns of spider web structure were studied, as presented in Figure 2. The spider web structure is constructed based on the actual dimensions in previous study [14]. The radii of spiral threads are 0.3 m and the length of each radial thread is 0.4 m. The overall size of the model is 0.8 m \(\times\) 0.8 m. The spacing between adjacent spiral threads is 0.03 m for all patterns except patterns 5 and 6 which have the spacing of 0.015 and 0.01 m, respectively. The ratio of the spiral to radial thread diameter is 1 and the angle between the radial thread is 45, 30 and 22.5 degrees in patterns 1, 2 and 3, \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Material** & **Density (kg/m\({}^{3}\))** & **Tensile Modulus (GPa)** & **Poisson’s Ratio** & **Strength (GPa)** & **Strain (\%)** \\ \hline A. aurantia silk & 1098 & 34 & 0.49 & 1.75 & 26 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Engineering properties of the _A. aurantia_ silk [4]. Figure 1: Typical spider web geometry. respectively. Patterns 4 and 5 have the same angle as pattern 1 and its capture threads are series of concentric circles instead of spiral threads. At first, the spider web was modelled as a truss structure, which means the member only carries tensile and compressive axial loads without rotation and torque. In addition, truss elements only need to set the cross-sectional area to define element properties in ABAQUS. As mentioned before, the spider web was modelled on a 2-D plane while vibrating in a third axis (Z axis), but ABAQUS could not analyse the frequencies without mass and stiffness matrix on the third axis. Hence, beam elements were applied instead. Beam element is usually defined by stiffness (K), torque (J), and moment of inertia (I), but for spider silk, a sort of cable-type element, the bending, torsional and shear rigidities should be as small as possible. The boundary conditions exhibited the characteristic of spider web anchorages. From a structural engineering view mentioned in [16], a spider web is a cable structure whose segments only sustain tension.