**4. IFEM**

According to the concept of CM, the residual stress existing in welded specimens can be calculated through the measured out-of-plane displacement normal to a plane of interest. Several assumptions are made in the linear elastic finite element analysis: (1) small displacements or deformations of the welded components or structures, (2) linear elastic behavior during the material removal, and (3) unchanged boundary conditions during loading process. Therefore, it can be defined as an inverse finite element method (IFEM). In general, a set of linear equation below describes the welded components or structures in the linear elastic finite element:

$$[K][\mu] = [F] \tag{4}$$

ensuring computation accuracy when carrying out transient welding temperature

*Simulation model and weld sequence of butt welded joint: (a) without considering back-gouging; (b) with*

*Residual Stress Evaluation with Contour Method for Thick Butt Welded Joint*

*DOI: http://dx.doi.org/10.5772/intechopen.90409*

In principle, plastic strain is the generation source of welding residual stress, which is mainly determined by the maximum welding temperature and restraint condition. Here, the restraint condition is not considered, and the maximum welding temperature distribution is used to describe the features of welding temperature field of the butt welded joint. During the thermal analysis of the butt welded joint, the initial temperature is assumed to be 20°C. To simulate the heat input of moving welding arc during welding, the volumetric heat source with uniform density distribution is employed. For this heat source, the welding arc energy is dependent on the welding current, voltage, speed, and arc efficiency. When the welding process is SMAW, the arc heat efficiency is 0.6 and the welding arc length is 30 mm. And the heat source volume denoted the considered weld pool volume and can be obtained by calculating the volume fraction of the elements in currently being welded zone. The nonlinear isotropic Fourier heat flux was also employed for heat conduction. The temperature-dependent thermal properties such as thermal conductivity, specific heat, and density are considered. **Figure 9** presents the fusion zone in middle cross-section of the butt welded joint with or without considering back-gouging. The simulated fusion zone is slightly greater than the Xgroove area, while the maximum temperature is approximately 2300°C. Comparing to the macrostructure of the butt welded joint, the fusion zone with considering back-gouging agrees better than that without considering back-gouging. Therefore,

nonlinear heat transfer computation.

*5.1.1 Root weld*

**115**

*considering back-gouging.*

**Figure 8.**

$$\begin{array}{c} |K| = \\\\ \frac{E}{(1-2\nu)(1+\nu)} \begin{bmatrix} \frac{\partial^{2}(1-\nu)}{\partial x^{2}} + \left(\frac{\partial^{2}}{\partial y^{2}} + \frac{\partial^{2}}{\partial x^{2}}\right) \frac{(1-2\nu)}{2} & \frac{\partial^{2}}{2\partial x\partial y} & \frac{\partial^{2}}{2\partial x\partial x} \\\\ \frac{\partial^{2}}{2\partial x\partial y} & \frac{\partial^{2}(1-\nu)}{\partial y^{2}} + \left(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial x^{2}}\right) \frac{(1-2\nu)}{2} & \frac{\partial^{2}}{2\partial y\partial x} \\\\ \frac{\partial^{2}}{2\partial x\partial x} & \frac{\partial^{2}}{2\partial y\partial x} & \frac{\partial^{2}(1-\nu)}{\partial x^{2}} + \left(\frac{\partial^{2}}{\partial x^{2}} + \frac{\partial^{2}}{\partial y^{2}}\right) \frac{(1-2\nu)}{2} \end{array}$$

where *E* is the Young's modulus, 210 GPa, *υ* is Poisson's ratio, 0.3, ½ � *K* is the stiffness matrix of the welded component or structure, ½ � *u* is the matrix of nodal displacements, and ½ � *F* is the matrix of external nodal force.
