# Analysis of the Impact of Forces in Hanger Rods on Power Boiler Operation
## 2 Materials and Methods  
The numerical analysis of a unit as large as a power boiler requires the model to include all the elements that interact with each other. This approach is mainly imposed by the fact that the boiler elements, such as the furnace chamber or separators, are connected to each other by shrouds that stiffen the membrane walls to protect them from the effect of pressure from combustion gases. Membrane walls, being orthotropic elements, are characterized by good stiffness and tensile strength, but they demonstrate much worse bending strength, especially when the pressure of combustion gases acts on them in a direction perpendicular to the wall. The boiler assemblies, such as the furnace chamber and the separators, work together, and this co-operation also affects the tension of the forces in the hanger rods. Building detailed numerical models for boilers is not economical, and it takes a great deal of time and processor power. Therefore, it is justified to build models that replace the real models of membrane walls using equivalent elements with a less complex structure, such as a flat plate, but with the same stiffness (Figure 1). The figure below shows the studied boiler's typical repetitive fragment, which is where high stresses can occur. For such a solution, the model constructed from tubes and fins (Item 1 and 2 in the figure--made of 3D elements such as SOLID 186) can be replaced with a flat plate model (3), which considers the equivalent stiffness of a model made of 3D elements. A model with equivalent stiffness (3) can be made of 2D elements like SHELL 181. Figure 1 shows a fragment of the boiler wall with 13 tubes only, which resulted in 196,325 nodes, while the wall made as a flat plate contained only 27,749 nodes. Considering that boilers are constructed using thousands of such tubes and fins, one can see the time and computer memory savingsthat can be achieved with the presented model. For the presented simplified model, the computation time is more than five times shorter. Such a model is referred to as an equivalent orthotropic model, and it has so far been used in smaller units with support from the bottom [5; 6], i.e., at different boundary conditions compared to boilers suspended by means of hanger rods. In cases of bending the membrane walls in a direction perpendicular to the y-axis (as in Figure 2), the infinite stiffness of the tubes is often assumed [8; 9]. In most cases, this approach is correct. However, bearing in mind that the diameter of the tubes and their pitch may vary depending on the boiler size, it seems justified to take account the stiffness of the entire system in the equations of the equivalent orthotropic plate (tubes and fins), which was taken into consideration in the boiler numerical model under analysis. The use of an equivalent orthotropic plate in the MES analyses was possible thanks to the use of SHELL elements. For this purpose, the ANSYS Workbench R2, 2023 commercial package [14] was used. The element that met the conditions of the thin-walled orthotropic plate was SHELL181. It is an element with four nodes and six degrees of freedom in each node. SHELL181 takes into account the linear cross-shear effect. The transverse stiffness matrix for the equivalent orthotropic plate can be presented as follows [14]:  
\[[E]= \begin{bmatrix}E_{11}&0\\ 0&E_{22}\end{bmatrix} . \tag{1}\]  
Figure 1: Fragment of the boiler wall modelled using 3D elements (1—\(\phi\)63.5 \(\times\) 6.3 mm tube and 2—fins t = 6 mm); a flat plate (3)—equivalent stiffness was adopted. Figure 2: (**a**) The internal forces and stiffness of the membrane wall Am and the equivalent orthotropic plate Ap for the membrane state, and (**b**) the internal moments and stiffness of the membrane wall Dm and equivalent orthotropic plate Dp for the flexible state. For an isotropic material it can be written that \(E_{11}=k\cdot G\cdot h\) is expressed in [N/mm], where \(k\) is the correction factor, \(G\) is the shear modulus [MPa], and \(h\) is the thickness of the analysed coating [mm].