# Modelling Upholstered Furniture Frames Using the Finite Element Method
## 1 Introduction  
Upholstered wooden structures represent typical examples of box and frame furniture elements [1, 2]. These frames are constructed as sub-assemblies comprising several to a dozen or more components, joined together to create coherent structural units. The frames can be either solid or layered, with strips of material connected in parallel or crosswise configurations. Wood or wood-based panels of specified strength are commonly used as the primary materials in upholstered furniture. Sustainable practices highlight the preference for spruce, pine, birch, and beech wood species in furniture production [3, 4, 5], as they are renewable resources and meet high strength requirements. Pine and spruce, in particular, are well-suited for lightweight furniture applications due to their favourable properties [6, 7, 8, 9]. Their strength parameters comply with the design standards for furniture structures [10, 11, 12, 13]. A critical aspect of frame sub-assemblies is the joining of elements at frame nodes subjected to transverse forces. Proper stress distribution requires external loads to be applied directly to the frame plane, where the cross-sectional dimensions of elements are significantly smaller than their lengths. In these configurations, the influence of the frame structure's weight on internal forces is minimal. External loads induce shear forces and bending moments in frame components. Stress assumptions allow for the use of various techniques and algorithms to optimise the dimensions of these elements [14, 15]. Timber structures offer advantages such as durability, low weight, design flexibility, high stiffness, and excellent load-carrying capacity relative to the material used [16; 17; 18; 19; 20]. However, these benefits can be undermined by suboptimal component design. The ecological benefits of timber structures align with sustainable development goals, including reduced carbon footprints [21; 22; 23]. Frame structures must be designed to withstand anticipated loads while maintaining system rigidity [14; 24]. Joining frame components typically involves adhesive or mechanical fasteners (staples, screws, and nails) [25; 26]. Properly selected connectors ensure the strength of layered frame joints [27; 28; 29; 30]. Upholstered furniture frames are subjected to both static and dynamic loads during use [31]. Understanding the distribution of stress and deformation is crucial for optimising upholstered furniture design. The application of 3D modelling and FEM-(Autodesk Inventor Nastran) based software enhances the strength design of furniture structures. Widely used CAD tools enable the creation of wooden furniture models [32; 33; 34; 35; 36]. Numerical methods for analysing upholstered furniture frame structures [37; 38; 39], particularly finite element analysis [40; 41; 42; 43], facilitate material optimisation while ensuring structural strength. Optimising dimensions of primary structural components using orthotropic finite elements (e.g., pine and beech wood) involves varying thicknesses and cross-sections to evaluate deformation and stress states [44; 45]. Studies on the strength properties of frames joined with glue or mechanical fasteners [46; 47] have assumed isotropic material behaviour. Previous research demonstrated a high level of agreement (81%) between experimental and FEM results [48; 49]. The FEM method is a good tool for evaluating new joints and comparing them with a typical joint commonly used in wood and wood plastic furniture. The experimental results of one study showed that the stresses obtained by the FEM method were consistent, providing a reliable method for evaluating and optimising a new furniture structure [50]. Other authors experimentally studied the use of staples and connecting materials during the process of static lateral loading. Their experimental results showed that the load-carrying capacity of the members depends on the pullout force of the staples in the main members [51]. The influence of the carpentry joints used plays a role in stress transfer.