## 2 Materials & Methods
### The Numerical Model
For the calculations of these environmental forces, it is necessary to calculate the exposed area of the cargo in m\({}^{2}\) (longitudinal and transverse), knowing that the influence of both effects is 1 kN per m\({}^{2}\). Although the influence of wind pressure is produced along the entire height of the cargo, in the case of green water (sea sloshing), the exposed area is calculated only until a maximum height of 2 m above the weather deck. Therefore, knowing the cargo mass (m), the corrected accelerations longitudinal (ax\({}_{\text{(C)}}\)), transverse (ay\({}_{\text{(C)}}\)), and vertical (az\({}_{\text{(C)}}\)), and the environmental forces of wind pressure and sea sloshing (longitudinal and transverse), the total forces in the three axes sustained by the cargo can be calculated as follows:  
Figure 2.2  
Figure 2.3  
Figure 2.4  
In the balance of forces and moments of a cargo item stowed on board, Annex 13 of the CSS Code includes the named \"advanced calculation method\" and the \"alternative calculation method\". The \"alternative method\" is considered the most accurate for the  
Figure 2.5 Correction factor for transverse acceleration (ay) when B/GM < 13.  
calculation of the suitability of securing devices because both angles formed by the lashing line with the horizontal and transverse plane are taken into account. The first one is the vertical angle (\(\alpha\)) formed between the lashing line and horizontal plane where it is fixed (deck, hatch cover, double bottom), and the second one is the horizontal angle (\(\beta\)) formed by the lashing line with the transverse plane (port-starboard), which is known as the angle of deviation from the transverse. This is because, in practice, the lashing lines do not usually have a purely longitudinal (fore-aft) or transverse direction (port-starboard). Thus, a lashing line with a non-zero \(\alpha\) and \(\beta\) angles will produce securing effects in both longitudinal (fore-aft) and transverse (port-starboard) directions. The securing capacity of the lashing lines is determined by multiplying their calculated strength (CS) by the values of fx or fy, depending on the effects in the longitudinal or transverse direction. The values of fx and fy, which are functions of the \(\alpha\) and \(\beta\) values, but also on the friction coefficient (\(\mu\)), can be calculated according to the following Equations:  
Figure 2.6  
Figure 2.7  
Using Equations (9) and (10), once the optimal fx and fy values are known using the proposed methodology, the corresponding a and b angles can be calculated by clearing these equations. In Figure 5 is represented the fx and fy values corresponding to a friction coefficient (\(\mu\)) of 0.3. In the alternative calculation method, the safety factor applied between the maximum securing load (MSL) and the calculated strength (CS) of each lashing device is 1.35, i.e., CS = MSL/1.35. Therefore, the balance of external forces and the securing capacity to transverse and longitudinal sliding is expressed as follows:  
Figure 2.8  
Figure 2.9  
where n is the number of lashing lines and fz is a correction factor for the vertical force (Fz), which depends on the friction coefficient (\(\mu\)) and is represented in Figure 3.0  
It is relevant to mention that the term fz\(\cdot\)Fz was only applied to Equation (12) because it is assumed the worst-case condition, where the maximum longitudinal acceleration (Fx)  
Figure 3.1 The values of fy (a) and fx (b) corresponding to a friction coefficient (\(\mu\)) of 0.3.  
coincides with the maximum vertical acceleration (Fz) upwards during the pitching motion. This ship's behavior is not produced during the rolling motion. In Appendix A are included all abbreviations and symbols.