On the Calculation of Deflection of a Semitrailer Chassis under Various Loading Conditions: an Experimental and Numerical Investigation

In this paper, a simple approach is presented for the calculation of deflection of a semi trailer chassis since the less deflection become a unique selling point of a semi trailer. First of all, by using the 3D data of the chassis a function for the moment of inertia of the cross section is created and then the chassis is modelled as a Euler Bernoulli Beam. Different loading conditions coming from the semi trailer test procedures are applied. The deflections of the semi trailer chassis are numerically calculated by using three different meshless methods. Strong Form Meshless Implementation of Taylor Series Method I (SMITSM I), Strong Form Meshless Implementation of Taylor Series Method III (SMITSM III) and Symmetric Smoothed Particle Hydrodynamics (SSPH) method are studied for the calculation of deflection of a semi trailer chassis then, comparisons are made with the results of experiments. We also compared the effectiveness of the SMITSM I, SMITSM III and SSPH method by considering various particle distributions. It is observed that the SMITSM I and the SMITSM III have the conventional convergence properties yields smaller L2 error norms than SSPH method for different loading cases. Finally, the approach presented here may be used and found reliable for the calculation of deflection of the semi trailer chassis before the release of detail design. Introduction Meshless Smoothed Particle Hydrodynamics (SPH) method, proposed by Lucy (1977) to study three-dimensional (3D) astrophysics problems, has been successfully applied to analyze transient fluid and solid mechanics problems. However, it has two shortcomings such as inaccuracy at particles on the boundary and tensile instability. Many techniques have been developed to alleviate these two deficiencies among which are Corrected Smoothed Particle Method (CSPM) (Chen, Beraun, Jin, 1999a; 1999b), Reproducing Kernel Particle Method (RKPM) (Liu, Jun, Zhang, 1995; Liu, et al., 1995; Chen, et al. ,1996) and Modified Smoothed Particle Hydrodynamics (MSPH) method (Zhang and Batra, 2004a; 2004b; 2007a; 2007b]. The MSPH method has been successfully applied to study wave propagation in functionally graded materials (Zhang and Batra, 2007a), can capture the stress field near a crack-tip, and simulates the propagation of multiple cracks in a linear elastic body (Zhang and Batra, 2007b). The SSPH method has been applied to 2D homogeneous elastic problem successfully (Zhang and Batra, 2009). On the other hand, the SSPH method (Zhang and Batra, 2008; Zhang and Batra, 2009; Tsai, et al., 2013; Karamanli and Mugan, 2012) is more suitable for homogeneous boundary value problems, cannot be easily applicable to nonlinear problems, requires at least fourth order terms in basis functions for the buckling problems which increases the CPU time. Due to the SSPH method may not yield accurate results for solving nonhomogeneous problems with its underlying formulation, an alternative approach is investigated especially for nonhomegenous problems (Karamanli and Mugan, 2013). Three different implementations of MITSM including the approach presented in (Karamanli and Mugan, 2013), called Meshless Implementation of Taylor Series Method I, II and III (MITSM) are presented in (Karamanli, 2015). Although the SSPH method and MITSM depend on TSEs, the main difference between these two approaches is as follows: the SSPH method calculates the value of the solution at a node by using the values of the solution at the other nodes and then substitute it into the governing differential equation; thus, nonhomogeneous terms in the governing differential equation are also evaluated pointwise at the nodes. This approach results in approximation errors especially in the existence of nonsmooth nonhomogeneous terms. On the other hand, the SMITSM I, II and III methods approach substitute the TSEs of the solution and nonhomogeneous term into the governing differential equation and then utilize exact recursive relations between the coefficients of the expansions of the solution and nonhomogeneous term; it yields improvement in accuracy that is verified in (Karamanli and Mugan, 2013). In section 2, the formulation of the meshless methods used in this paper are presented for 1D application. In section 3, The chassis of the semi trailer is modelled as a beam based on Euler Bernoulli beam theory. The moment of inertia of the Euler Bernoulli beam is defined as a function by using the moment of inertia values of totally 23 sections due to the non-uniform structure of the semi trailer chassis. In Section 4, three types of loading conditions are investigated. The performance of the Strong Form Meshless Implementation of Taylor Series Methods (SMITSM I and SMITSM III) and Symmetric Smoothed Particle Hydrodynamics (SSPH) method are compared with the experimental results. It is found that the SMITSM III and SSPH methods have less convergence in numerical examples than the SMITSM I. Formulation In this section, two different basis function formulations based on the DTM Differential Transform Method (DTM) are given for 1D dimensional case. These methods are named as followings;  Strong form meshless implementation of Taylor series method I (SMITSM I) and,  Strong form meshless implementation of Taylor series method III (SMITSM III) The detail explanation and the formulations of the mentioned meshless methods can be found in [Karamanli and Mugan, 2012; Karamanli and Mugan, 2013; Karamanli, 2015). a. Strong form meshless implementation of Taylor series method I (SMITSM I) The formulation of the SMITSM III for a 1D problem can be written as follows