Modelling Upholstered Furniture Frames Using the Finite Element Method

Abstract

This study employs the finite element method to propose a model-based design strategy for upholstered furniture frames. Three-dimensional discrete models of these frames were created, considering the orthotropic characteristics of pine (Pinus sylvestris L.) and spruce (Picea abies L.) wood, reinforced structurally with glue joints and upholstery staples. The modelling process utilised the CAE system Autodesk Inventor Nastran, applying the finite element method (FEM). Static analyses were performed by simulating standard loading conditions. The calculations incorporated the stiffness coefficients of the frame’s comb joint connections. The findings illustrate the stress distribution, displacements, and equivalent strains within the furniture frame models. The deformation and strength parameters of the frames introduce a novel perspective on designing upholstered furniture structures using the component-based FEM approach. These outcomes are applicable to the development of upholstered furniture designs.

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. The influence of dimensions on the pullout capacity of mortise and tenon joints can also be studied based on the finite element method (FEM). For a long time, the dimensions of pivot geometries were designed using empirical methods, which resulted in more wood waste. The optimal methodology of combining finite element analysis (FEA) allows us to investigate the effect of tenon geometric dimensions (length, width, and thickness) on and the bearing capacity of mortise and tenon joints. The use of FEA allows us to gain more knowledge about the joint and reduce the cost of materials and experimental time [52,53].

This study aims to validate a hypothesis regarding the feasibility of optimising upholstery frame designs using the CAE system Autodesk Inventor Nastran (In CAD, San Rafael, California). The analysis incorporated stiffness coefficients for frame elements joined with glue. The proposed evaluation and selection system is intended to streamline the design process and promote efficient wood resource management. The research scope included stiffness and strength analysis of the frame structure, focusing on displacement values under external loading. Pine and spruce timber from European regions, known for their diverse strength characteristics, were selected for testing.

2. Materials and Methods

The analysis of stiffness and strength for the selected frame structures, represented as 3D models, was conducted virtually using Autodesk Inventor Nastran 2020, employing the finite element method (FEM) [54,55]. This approach involves dividing the frame components into discrete sections, known as finite elements [56,57].

For the study, four groups of raw materials were distinguished based on their origin. Pine wood classified as EN 13556 PNSY (Pinus sylvestris L.) [58] was sourced from Poland (PL), Eastern Europe (EW), and Scandinavia (SC) [59,60,61]. Spruce wood, labelled PCAB (Picea abies L.) [58], was obtained from Scandinavia (SC). The material characteristics, including the modulus of elasticity and strength, were critical for modelling and aligned with furniture production requirements [61,62,63]. The strength classification of the wood followed EN 338 standards [64].

2.1. Building 3D Models of Bed Frames

In Autodesk Inventor Nastran, CAD software (In CAD Nastran 2019, San Rafael, CA, USA) was used to create continuous geometric models of the bed frame structures. Four distinct models were developed as part of the research:

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A1 W3 frame (80 mm wide elements) with a glue-free three-comb joint (Figure 1a);

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A1 W5 frame (80 mm wide elements) with a glue-free, five-comb joint (Figure 1b);

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A2 W5 frame (120 mm wide elements) with a glue-free, five-comb joint (Figure 1c);

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A2 W5 frame (120 mm wide elements) with a glue-free, seven-comb joint (Figure 1d).

To ensure realistic representation of the experimental (laboratory) conditions conducted concurrently, steel legs were integrated into each 3D model (Figure 2). Additionally, two tensors were included to simulate a four-point bending test of the long frame (bed edge) with a 600 mm span.

Following the principle of maximising geometric simplification to reduce the number of finite elements in the frame models, glue joints were not included as separate elements.

2.2. Structural Idealisation

The structural idealisation involved assigning specific physical properties of the raw material (material) to a given bed frame. The physical properties introduced into the calculations for the different types of raw material were derived from parallel experimental (laboratory) studies for A1 version long frames made of solid material, and their exact properties are shown in

Table 1. Material data used in the analysis of the A1 version of the numerical models made of solid material in strength classes according to EN 338 [64].

Species of Wood	Origin	Density		MOE
		kg/m3		MPa
Strength Classes		C24	C30	C24	C30
Pine	Poland	468	517	8100	10,300
	Scandinavia	500	504	10,000	10,100
	Eastern Europe	477	453	9100	8400
Spruce	Scandinavia	439	433	9100	8700

. Connections due to the longitudinal direction were modelled to simplify the action of bending forces in the longitudinal direction. The index was determined by the adopted modulus of elasticity (MOE) parameter for the longitudinal direction. The modelling assumed a tight fit and no friction between the key and tenon.

2.3. Discretisation of Structures

The study of objects with a complex shape, properties, or complex conditions only makes it possible to obtain approximate solutions, which are arrived at by means of so-called discretisation, which involves transforming a field composed of an infinite number of parameters into a field expressed by a finite number of values and points. The result of the discretisation is the so-called discrete computational field. The discrete model is arrived at using the FEM approximation method [46,65,66]. It is a numerical tool designed to solve differential equations [67]. The discretisation of the design of the individual elements of the bed frames consisted of automatically dividing all the elements of the model into appropriately selected finite elements of defined shape and properties, connected to each other at specific points called nodes. All models were automatically subdivided into non-linear finite elements (parabolic type) with an assumed element size of 10 mm. The appearance of the selected mesh (computational) model is shown in Figure 3.

2.4. Entering Boundary Conditions

The next step was to define boundary conditions in the form of specifying the number and type of degrees of freedom at selected nodes, assigning interactions (contacts) between selected surfaces, thus simulating adhesive connections of elements or surface-to-surface interactions and introducing an external load on the structure. The boundary conditions describe the nature of the research being carried out and are intended to reflect the reality of the experimental tests carried out or the place of use of the products in question. For this reason, different boundary conditions have been introduced depending on the type of research being carried out.

2.5. Bed Edge Flexural Strength Test

The following boundary conditions were introduced in all 4 bed frame models:

• All four legs were stripped of their ability to translate in the vertical direction (ground simulation);
• Both thrusts were stripped of their ability to rotate and also to translate across the frame (bed edge);
• An external load was applied to the top surface of the thrust in the form of a force normal to the surface directed vertically downwards with a value of F = 1200 N (total load 2400 N) according to the standard EN 1725 [31];
• In the contacting surface of the long frame (the edge of the bed) with the thrusts, the interaction of the contacting surfaces was introduced on the principle of separation type surface contact;
• Interactions in the contacting surfaces of the frame elements were introduced on the basis of bonded bonding (without adhesive bonding) and also interactions on the basis of the surface contact of the separation type.

2.6. Testing the Bending–Torsional Strength of the Bed Frame

This study used the previously prepared bed frame models used in the bed edge bending strength test. Due to the test plan adopted, it was necessary to use all 4 bed frame models.

In order to carry out the tests correctly, the steel legs were removed from the frames and small corner roundings were made in order to precisely introduce the boundary conditions. A fragment of the selected 3D bed frame model is shown in Figure 4.

The following boundary conditions were introduced in all 4 bed frame models:

• Five out of six degrees of freedom were taken away from one corner of the chamfer surface—only the possibility of translation in the direction of the height of the frame was left (simulation of the corner of the frame sitting in the base bracket);
• An external load was applied to the opposite corner, diagonally on the frame, in the form of a force normal to the diagonally directed surface of the frame, with a value of F = 1147 N (force value derived from experimental tests on reference frames—experimental results are given in Appendix A);
• Interactions were introduced in the contacting surfaces of the frame elements on the basis of bonded bonding type (without adhesive bonding) and also interactions on the basis of separation type surface contact.

2.7. Assignment of Numerical Analysis Settings

The numerical analysis was carried out in the linear static FEM analysis mode. The measure of the stiffness of a given structure (k) is the ratio of the force acting on the product, which is an external load (Pz), to the displacement measured in the direction of this load (∆Pz), which is presented in the following equation:

$$\mathrm{k}={\displaystyle \frac{\mathrm{P}\mathrm{z}}{\mathsf{\Delta }\mathrm{P}\mathrm{z}}}{\displaystyle \frac{\mathrm{N}}{\mathrm{m}}}$$

(1)

where the components of the equation are as follows:

• k—measure of stiffness of a given structure;
• Pz—force acting on the product [N];
• ∆Pz—displacement [m].

3. Results and Discussion

The numerical analyses yielded results in the form of the following:

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Maps illustrating the displacement distribution of the frame’s structural elements, counter-braces, bed edges, and the entire bed frame in the direction of the applied load;

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Distribution maps of normal and reduced stresses occurring in the frame structure, counterweight slats, headboard, and bed frame components.

3.1. Bed Edge Stiffness and Strength

The analyses revealed that the applied loading method and subsequent deformation of frame elements resulted in stress values consistent across all variants of frames within a given group, regardless of the material used. These stress levels, if nearing the material’s strength limits, could compromise the structural integrity of individual components and the entire frame. However, simulations indicated that normal stresses at critical points (e.g., near maximum deflection) remained below the material’s bending strength.

For these analyses, a constant load was applied throughout. Displacement (deflection) in the direction of the load was considered a direct measure of stiffness. It was observed that increased deflection corresponded to reduced stiffness of the structure.

Figure 5, Figure 6 and Figure 7 present displacement and normal stress maps for all analysed bed frame models constructed from various raw materials.

Table 2. Normal stresses and deflection (displacement) of the long frame (edge) of bed version A1 W3.

Frame Version	Species	Land (Habitat)	Strength Class	Normal Stress	Deflection
			EN 338 [64]	MPa	mm
A1 W3	Pine	Poland	C24	13.9	12.0
			C30	13.9	9.5
		Scandinavia	C24	13.9	9.8
			C30	13.9	9.7
		Eastern Europe	C24	13.9	9.8
			C30	13.9	9.7
	Spruce	Scandinavia	C24	13.8	10.7
			C30	13.8	11.1
A1 W5	Pine	Poland	C24	14.5	11.8
			C30	14.6	9.4
		Scandinavia	C24	14.6	9.6
			C30	14.6	9.5
		Eastern Europe	C24	14.6	10.5
			C30	14.6	11.4
	Spruce	Scandinavia	C24	14.6	10.6
			C30	14.6	11.0
A2 W5	Pine	Poland	C24	8.0	3.8
			C30	8.0	4.1
		Scandinavia	C24	8.1	3.4
			C30	8.1	3.2
		Eastern Europe	C24	8.0	4.8
			C30	8.4	4.4
	Spruce	Scandinavia	C24	8.0	4.2
			C30	8.0	4.2
A2 W7	Pine	Poland	C24	8.4	3.9
			C30	8.4	4.2
		Scandinavia	C24	8.5	3.5
			C30	8.5	3.3
		Eastern Europe	C24	8.3	4.9
			C30	8.4	4.4
	Spruce	Scandinavia	C24	8.4	4.3
			C30	8.4	4.3

summarises the exact values of normal stresses and displacements for all models. The highest stiffness was observed in frames made from Scandinavian pine wood.

In analysing the stiffness of all the frames constituting the bed edge, it should be stated that in the case of both the A1 and A2 frame constructions, the most rigid bed edge in most cases is an element made of pine wood from Scandinavia.

In analysing the stiffness of bed edges, the following was determined:

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For A1 frame versions, elements made from Eastern European and Polish pine wood exhibited the least rigidity.

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For A2 frame versions, the least rigid components were those constructed from Eastern European pine wood.

No significant differences in stiffness were noted based on the joint structures (W3, W5, and W7) of the frames (p < 0.005).

The analyses confirmed that in all cases, considering the applied loads, geometry, and materials, the bending strength of the timber remained within normative limits as per EN 1725 [31]. Stress levels were less than half of those observed in tests on simply supported beams.

Higher stresses were recorded in scenarios involving greater deflection, with geometry playing a significant role. For the A1 frame versions, stresses were approximately 40% higher than those in A2 frames. Specifically, the following results were found:

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In A1 frames, higher stresses occurred in W5 joints.

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In A2 frames, W7 joints exhibited higher stress levels.

Based on stiffness and strength considerations, the A2 W5 frame (or A1 W5) constructed from Scandinavian or Polish pine wood (strength class C30) is recommended for further bed frame development. This study makes it possible to confirm that the stresses obtained from the FEM analysis are close to empirical wattages. The actual stress ratios are associated with a higher material cost. As with other studies [50,51,52,53], the principle of replacing part of the experimental studies with FEM modelling can be confirmed. Discussions can be raised about the differences taking into account the friction forces for wood elements. The quality of the surface and the type of joint fit will play a role.

3.2. Frame Stiffness and Strength

Figure 8 and Figure 9 display the displacement and reduced stress maps for a selected bed frame model.

In these analyses, all simulations were conducted under uniform, constant loading. Displacement (deflection) in the direction of the load served as a simplified measure of stiffness. The results indicated that stiffness decreases as deflection increases.

The exact values of the reduced stresses as well as the displacement of the structure for all the analysed bed frame models using the tested raw material types are summarised in

Table 3. Reduced stresses and displacement of bed frame components in A1 W3 version subjected to diagonal force.

Species	Land (Habitat)	Strength Class	Normal Stress	Deflection
		EN 338 [64]	MPa	mm
Pine	Poland	C24	8.4	2.6
		C30	8.4	2.5
	Scandinavia	C24	8.3	2.3
		C30	8.3	2.3
	Eastern Europe	C24	8.4	2.3
		C30	8.4	2.5
Spruce	Scandinavia	C24	8.4	2.3
		C30	8.4	2.4

,

Table 4. Reduced stresses and displacement of bed frame components in A1 W5 version subjected to diagonal force.

Species	Land (Habitat)	Strength Class	Normal Stress	Deflection
		EN 338 [64]	MPa	mm
Pine	Poland	C24	5.9	3.7
		C30	5.9	2.9
	Scandinavia	C24	5.9	3.0
		C30	5.9	2.9
	Eastern Europe	C24	5.9	3.3
		C30	5.9	3.5
Spruce	Scandinavia	C24	5.9	3.3
		C30	5.9	3.4

,

Table 5. Reduced stresses and displacement of bed frame components in A2 W5 version subjected to diagonal force.

Species	Land (Habitat)	Strength Class	Normal Stress	Deflection
		EN 338 [64]	MPa	mm
Pine	Poland	C24	6.8	3.2
		C30	6.8	3.4
	Scandinavia	C24	6.8	2.7
		C30	6.8	2.8
	Eastern Europe	C24	6.8	3.6
		C30	6.8	4.0
Spruce	Scandinavia	C24	6.8	3.5
		C30	6.8	3.5

and

Table 6. Reduced stresses and displacement of bed frame components in A2 version W7 subjected to diagonal force.

Species	Land (Habitat)	Strength Class	Normal Stress	Deflection
		EN 338 [64]	MPa	mm
Pine	Poland	C24	5.5	3.9
		C30	5.5	3.2
	Scandinavia	C24	5.5	3.3
		C30	5.5	3.2
	Eastern Europe	C24	5.5	3.4
		C30	5.5	3.7
Spruce	Scandinavia	C24	5.5	3.3
		C30	5.5	3.2

.

The frames constructed from Scandinavian pine demonstrated the best performance in minimising deformation, while those made from Eastern European pine showed the highest deflection values. Increasing the width of frame elements significantly improved stiffness, reducing deflection.

Additional stress modelling suggested that incorporating heterogeneous raw materials did not compromise frame stiffness. Instead, the optimal simulations indicated enhanced stiffness and reduced deflection under normative loading conditions.

The strength of the analysed timber frames was directly influenced by the normal stress values and their distribution within the material and connectors. The results demonstrated that, considering the EN 1725 standard load [31], the geometry of the components, and the raw material properties, there was no risk of exceeding the bending strength or structural failure.

Previous studies [68,69,70,71,72] on larger furniture components support the conclusion that frames with constrained degrees of freedom, due to bonding with other elements, exhibit reduced deflection and lower normal stress values [73].

4. Conclusions

The highest stress values were observed at points of maximum deflection under the specified normal load conditions. Based on the modelling analyses of furniture frames, the following conclusions were drawn:

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The type of coniferous wood material used did not significantly affect the normal stress values in the elements.

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Modifications to the frame components also had no substantial impact on stress levels.

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Elements with larger cross-sectional dimensions demonstrated lower normal stress values, enhancing stiffness and strength. Therefore, the W5 frame version and Scandinavian or Polish pine wood with a strength class of C30 are recommended.

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Under diagonal force testing, all modelled frames exhibited torsional deformation.

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The W3 frame version with a triple-comb joint exhibited the highest stiffness, while the W7 version had the lowest stiffness and reduced stress values.

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In all analysed scenarios, considering load characteristics, element geometry, and material type, there was no risk of exceeding the wood’s bending strength, thereby ensuring structural integrity.

The design of the bed frame imposes certain limitations on achieving optimal stiffness and strength for individual elements. Using Polish or Scandinavian pine wood with a minimum strength class of 30 N/mm² requires precise sorting of lumber to meet these strength requirements [74,75,76,77]. Optimising structural selection processes to eliminate defects can significantly reduce material waste and enhance the strength properties of furniture frames.