Analysis of the lamella dimensions of the bed slat for the bariatric respondent: A modelling approach

ABSTRACT

This work analysed bed lamella dimensional characteristics for beds of bariatric users. Dimensions of the Slovak bariatric population were defined in the years 2020 to 2022 and used to design the length and thickness of bed lamellae. The thickness of lamellae made of spruce and beech wood (grade C24 and D50) was determined from the weight of the bariatric population while meeting the requirements of EN 1995. The bending deflection was calculated by a linear FEM model that was verified with the experimentally measured deflection of ungraded spruce and beech wood lamellae. The difference in deflection was 2.7% and 1.4% for spruce and beech, respectively. The minimal thickness of the lamella was always determined based on the allowable deflection limit criterion. The stress of the lamellae was below 70% of the allowable stresses of the spruce wood lamella (C24) and 37% in the case of beech wood lamella (D50). A non-linear dependence of the minimum lamella thickness on the weight of the bariatric respondent was presented. Designing a bed lamella loaded in bending is more effective by increasing the thickness of the spruce wood lamella than by using a beech wood lamella with a higher modulus of elasticity.

KEYWORDS:

• Obesity
• furniture design
• bariatric furniture
• bending
• solid wood bed

Introduction

The development of a mature society is associated with an increase in well-being. It closely relates to the rise in obesity that we have seen since the beginning of this millennium. The population's weight gain is becoming a serious medical, psychological, social, and economic problem. It primarily affects the people of economically developed countries in Europe and the USA. The incidence of obesity is on the rise and affects almost all socio-economic groups of the population. That is why the World Health Organization (WHO) has declared obesity an epidemic of the twenty-first century and a global epidemic because of its far-reaching health and social consequences (2020). The fight against it is one of the main priorities of WHO and the European Union. Substantial financial resources are spent in many fields of medicine to treat obesity itself and the health complications associated with it.

The percentage of obese people in the Slovak Republic is increasing, as shown in Figure 1, which contains data on the rate of obese people in the adult population.

Figure 1. Time series chart of the prevalence of obesity (BMI > 30) among adults, ages 18+, 1975–2020, in Slovakia. Source: Riley (2022).

These trends are similar worldwide (Bentham et al. 2017). Before the proper measures take place, the society must deal with the problems in various areas.

The bed, as an essential item of furniture, is an integral part of the interior for every person (Sydor et al. 2022). A person spends a third of their life in bed. The sunbed area is adapted for sleeping or resting. From the point of view of anthropometry, physiology, and hygiene, the bed size must be appropriate for the human body and changes in body position during sleep. The bed must have sufficient rigidity to maintain the body's position while preserving the spine's natural curvature. Modern solid wood beds also include bed lamellae that increase sleeping comfort and the breathability and hygienic maintenance of the mattress.

The bed is subjected to high demands. Its size and stiffness must correspond to the human body and changes in position during sleep. Strength characteristics in furniture construction can also be fundamentally affected by the occurrence of quality features in specific types of wood (Gejdoš et al. 2020). From an anthropometric point of view, the bed surface's length, width, and height are essential. Length and width affect the size of the bed surface, while the measurement is vital for comfortable standing and possibly sitting. Since the body height and weight of the population have been increasing in recent years, and the number of overweight and obese people is also increasing (Chuan et al. 2010, Chen et al. 2016), it is necessary to design beds with a perspective for people with a high BMI. Part of the strength characteristic of the bed is also the quality of the bed lamellae carrying the main load of the user. Bed lamellae are accessible in the bed frame or firmly anchored in the bed frame. They can also be part of a fixed frame. They are mostly made from spruce or beech wood but also laminated materials. Their main advantages are strength, breathability, and even distribution of weight, which prolongs the life of the mattress. It is not a single point stressed and does not bend.

The greater the number of lamellae in the bed frame, the more durable the grid is, and the better the weight is distributed. Twenty-eight lamellae are considered the standard number. However, a higher number of lamellae is permissible. The distance between the lamellae should reach a maximum of 4–6 cm. The current load capacity of the grates is usually up to 150 kg which is not sufficient for overweight people. The number of lamellae, their width, thickness, and the gaps between the individual members also greatly affect the load-carrying capacity of the bed. The material of the lamella, processing, and montage quality also influence the load capacity of the bed and its reliability.

Wiggermann et al. (2017) argue that BMI is a good predictor of a person's space to go from supine to sideways. This study showed that a patient with a BMI greater than 35, lying in the middle of a standard 91 cm wide bed, would have insufficient room to turn in any direction without lateral displacement. Patients with a BMI greater than 45 would not have enough space to turn at all, even if they were moved to the edge of the bed surface. The psychological problem of obese patients related to limited movement is also significant. The bed size must be adapted to involuntary changes in body position during sleep (Adami et al. 2011).

The purpose of the study was to analyse and define the dimensional characteristics of bed lamellae of solid wood beds for obese users made of renewable materials, namely spruce and beech wood.

Materials and methods

Two types of material – ungraded spruce lamellae (cross-section of 20 × 80 mm) and ungraded beech lamellae (cross-section of 15 × 80 mm) were tested in bending. The density of lamellae was determined in the entire volume of a lamella. After testing, a sample was taken from the lamella for determination of the moisture content. The result of the test was used for verification of the FEM model.

The dimensions of the lamellae reflected the requirements of bariatric patients. The bed width dimension should include the space necessary for the bariatric respondent to turn in both directions without lateral repositioning. Figure 2 shows the amount of space required to allow an end-user centred in the bed to comfortably turn in one or both directions without lateral repositioning.

Figure 2. Minimal space required to change the position of the body in the bed comfortably from a centred position: 1TD – to turn in one side, 2TD – to turn in two sides (Wiggermann et al. 2017).

When determining the width of the lying surface, the width of the male shoulders (1) is the most critical dimension. Because the hips of bariatric respondents are wider than the shoulders, the maximum body width is replaced by the seat width (Sw). Because each person changes their position several times during the night, increasing the minimum width of the bed surface by another 25% is necessary. The bed's width of the lying surface (b₁ or b₂) was determined according to the formula (Prokopec 1998). The width of the bed's lying width was calculated using the modified formula (2). (1) $b_1=1.5(\mathrm{Sw}+2s_x)+0.25\left[1.5\right(\mathrm{Sw}+2s_x\left)\right]$(1) (2) $b_2=2(\mathrm{Sw}+2s_x)+0.25\left[2\right(\mathrm{Sw}+2s_x\left)\right]$(2) where Sw is the seat width and s_x is standard deviation.

A finite element method model of the lamella

The finite element method (FEM) of a tested bed lamella was carried out using the ANSYS (Ansys, Inc., Canonsburg, PA, USA). Beech and spruce wood are used to manufacture lamella; therefore, a 3D FEM model that included the orthotropic properties of wood was created. Table 1 summarises the physical properties used in a calculation model of bed lamellae.

Table 1. Mechanical properties of lamellae (Réh et al. 2019).

Mechanical properties (at the moisture content = 12%)			Spruce wood	Beech wood
Density	(kg/m^3)	ρ	440	720
Modulus of elasticity	(GPa)	EL	11.429*	11.994*
		ER	0.83	1.13
		ET	0.65	0.63
Shear modulus	(GPa)	GLR	0.64	1.2
		GRT	0.04	0.19
		GLT	0.87	0.93
Poisson's ratio	(–)	μLR	0.041	0.044
		μRT	0.35	0.33
		μLT	0.033	0.027
Ultimate tensile strength	(MPa)	ft,||	84.0	130.0
		ft,⊥	1.5	3.5
Ultimate compression strength	(MPa)	fc,||	30.0	46.0
		fc,⊥	4.1	7.9
Ultimate shear strength	(MPa)	fs	5.3	12.3
Ultimate wood bending strength	(MPa)	σMOR	60.0	104.0
Bending modulus of elasticity of wood	(GPa)	EMOE	9.6	13.1

*Modulus of elasticity E_L was taken from results of verifying experiments.

The dimensions of the FEM model of the lamellas and the method of loading are shown in Figure 3.

Figure 3. Shape and dimensions of lamella in bariatric beds.

The model was verified by comparing the force-deflection diagram of bed lamellae tested by a three-point bending test (Figure 4).

Figure 4. Scheme of the three-point bending test of the lamella (CEN 1993).

Bending test of bed lamellae for verifying FEM model

The bending strength f_m and modulus of elasticity of lamellae E were calculated using the formula of 3-point bending and data tested according to CEN (1993): (3) $f_m={\displaystyle \frac{F_{\max }l}{b{h}^3}\mathrm{and}E={\displaystyle \frac{l^3}{4b{h}^3}{\displaystyle \frac{\mathrm{\Delta F}}{\mathrm{\Delta w}}}}}$(3) where F_max is the force at the failure of lamella in bending, b and h is the thickness and the width of the sample and a ratio ${\displaystyle \frac{\mathrm{\Delta }F}{\mathrm{\Delta }w}}$ is the slope of the linear part of the force-deflection diagram. The loading speed was 30 mm/min, and the failure of samples happened approximately three minutes after the beginning of loading.

The characteristic strength of lamellae $f_{m,k}$ (the 5th percentile) for individual tree species was calculated using the formula: (4) $f_{m,k}=f_m-1.645s-{\displaystyle \frac{s}{\sqrt{N}}}$(4) where N is the number of samples and s is standard deviation.

Assessment of the thickness of bed lamellae

Regarding strength assessment, wooden structures must comply with conditions specified in the standards (CEN 1995, 2016). The analysis of the load-carrying capacity of the wooden bed structure for the newly considered loads is carried out according to the rules stated in these standards. To verify the structural design of the bed in the ultimate limit state and the serviceability limit state, the limits set by the EN 1995 standard must be satisfy. In general, to design and assess the reliability of wooden structural elements in the limit state, the following condition must be met: (5) $S_d\le R_d$(5) where S_d means the effect of the design load value (internal forces, stress, strain, etc.) and R_d is the corresponding value of resistance or prescribe value (i.e. strength, limit strain) of wooden construction.

The design value of the wood-based material resistance for the ultimate limit state is defined following the formula: (6) $f_d=k_{\mathrm{mod}}{\displaystyle \frac{f_k}{{\gamma }_M}}$(6) where $f_k$ is a characteristic value of strength, $k_{mod}$ is a modifying factor considering the loading and moisture content, and ${\gamma }_M$ is a partial factor of reliability for material properties. The stress in the bed lamella caused by bariatric respondents depends on the weight of a respondent and the toughness of the material used.

The design value of the effect of the load in the case of the serviceability limit defines the lamella deflection as follows: (7) $w_d=w_{\mathrm{inst}}\left(1+k_{\mathrm{def}}\right)$(7) where $w_{\mathrm{inst}}$ is an instant bending deflection due to the acting force and k_def is a factor of deformation (0.6 in the case of the short load duration, service class 1). Instant deformation of simply supported bed lamellae was calculated by the verified FEM model. The weight of a bariatric respondent m acts, in a critical case, in the centre of a lamella. The patient's critical mass was distributed over two adjacent lamellae and the loading force is calculated as follows: (8) $F={\displaystyle \frac{\mathrm{mg}}{2}}$(8)

For designing the bed lamella, the high-quality spruce lamellae, included in the C24 grade, and beech lamellae from the D50 grade were selected. The properties of wood for these grades were used to calculate the minimum allowable lamella thickness for bariatric patients. The parameter values used for designing the lamella thickness are shown in Table 2.

Table 2. Parameters to design the lamella thickness.

Ultimate limit state	kmod (–)	kh (–)	fm,k (MPa)	Lamella thickness (mm)	Distance of supports (mm)
	0.9	1.3	spruce 24 beech 50	80	700
Serviceability limit	kdef (–)	ylim, (mm/m)	Emean (MPa)
	0.6	l/150	spruce 11,000 beech 14,000	80	700

The lamella thickness was determined by meeting both the serviceability limit and ultimate limit state (Equation 5(5) $S_d\le R_d$(5) ).

Results and discussions

Determining the lamella dimensions

To determine the width of the bed, we used the width of the seat determined according to the hip dimension (Table 3) and Equation 2(2) $b_2=2(\mathrm{Sw}+2s_x)+0.25\left[2\right(\mathrm{Sw}+2s_x\left)\right]$(2) . The width of the bed matching the distribution of seat width is 140 cm. The specified width dimension will ensure sleep for the bariatric respondent with the possibility of turning in both directions without lateral repositioning. However, the stated width of the bed can cause problems when laying the bedclothes, which is less important when considering a comfortable sleep.

Table 3. Descriptive statistics of 225 bariatric respondents.

Dimensions	Descriptive statistics
	N	Average	Min.	Max.	Percentiles					Std. Deviation
					1st	5th	50th	95th	99th
Body mass (kg)	225	143	93	242	97.00	105.00	139.00	191.00	233.00	27.36
Waist circumference (cm)	225	137	100	188	106.00	113.00	136.00	169.00	184.00	16.81
Hip circumference (cm)	225	146	108	192	110.00	121.00	145.00	173.00	191.00	15.71
Seat width (waist based) (cm)	225	43.69	31.85	59.87	33.76	35.99	43.31	53.82	58.60	5.36
Seat width (hip based) (cm)	225	46.74	34.39	61.15	35.03	38.54	46.18	55.10	60.83	5.00
BMI (kg.m^−2)	225	49	36	90	36.36	39.04	47.96	60.94	76.38	7.89

To ensure a sufficient load-bearing capacity for the bed, the centre will be supported (Figure 5). It will provide stability, and there will be an increase in the overall bearing capacity of the bed. The length of the lamella is then derived from the width of the bed, and it will be 70 cm.

Figure 5. Example of a bed construction.

The results of bending characteristics of the tested material

The test results are shown in Table 4. The average bending strength values are comparable to the properties of ideal wood. The modulus of elasticity of the beech wood was rather low. It was caused by the deviation of the fibres in the lamella, which was observed in the failure zone. On the other hand, the mechanical characteristics of beech wood in bending showed significantly lower variability (Figure 6).

Figure 6. Force and bending diagrams: (a) spruce wood (h = 20 mm), (b) beech wood (h = 15 mm). The dashed line shows the result of the FEM simulation.

Table 4. Bending characteristics of bed lamellae.

	Spruce			Beech
	Strength	Modulus	Density	Strength	Modulus	Density
Average	75.6	11,426	429	108.4	11,994	708
Std. Dev.	16.11	2498	35	10.47	995	39
5th percentile	49.09	7316	372	91.18	10,357	644
Var	21%	22%	8%	10%	8%	5%

The linear dependence of the strength on the modulus of elasticity was confirmed (Figure 7). There was a significant dependence of strength on density only in the case of spruce wood. The result indicates that it is more reliable to estimate the strength of spruce bed lamellae based on their elasticity.

Figure 7. Correlation between strength, modulus of elasticity, and density.

Comparison of bend from the FEM and experiments

The deflection of wooden bed lamellae was investigated using numerical simulations of the three-point bending test of boards using the finite element method (Figure 8) of the ANSYS program. For verification, the thickness of modelled beech and spruce wood lamellae was the same as the thickness of the tested material (20 and 15 mm).

Figure 8. Finite element calculation model for three-point board bending.

The mechanical properties listed in Table 1 (Réh et al. 2019) were used in the simulations. For the sake of comparison, the modulus of elasticity in the longitudinal direction was identical to the average modulus of the measured wood species. Following the computer simulations, the strain, i.e. the deflection of the board (Figure 9) and the tensile/compressive stress σ_|| (Figure 10) arising in the direction of the fibres during the bending of the loaded board, were determined for different values of the loading force.

Figure 9. Deflection in three-point bending of beech lamella (board) (board thickness h = 15.3 mm, loading force F = 2500 N).

Figure 10. Tensile stress in the fibre direction in the three-point bending of beech lamella (board) (board thickness h = 15.3 mm, loading force F = 2500 N).

The FEM calculated deflection of the average specimens, which depends linearly on the magnitude of the applied force, is shown in Figure 6. The difference between average sample deflection and FEM deflection was 2.6% for spruce and 1.4% for beech wood. It could be concluded that the numerical model could reliably determine the instant elastic deflection of the lamella. Depending on the used wood species and its quality, the FEM model and limit values can be used for designing the dimensions of the lamellae for bariatric respondents.

The thickness of the bed lamellae for the bariatric population

The calculated lamella thicknesses meet the load bearing and serviceability limits. The results show that the factor for determining the optimal dimensions was the allowable deflection (4.67 mm). The maximal stress in the spruce lamella ranged from 10 to 14 MPa (11 to 17 MPa in the case of the beech lamella) (Figure 11) depending on the weight of the observable bariatric population. A person weighing 300 kg loaded up the spruce lamella to 67% of its allowable stress (38% in the case of a beech lamella). Moreover, the maximal stress below the proportional limit meets the criteria of the linear force-deflection relationship of the proposed linear FEM model.

Figure 11. Maximum lamella stress after meeting the allowable deflection limit criteria.

The minimum thickness of the lamella is affected by the wood species, but in terms of practice, these differences are negligible (Figure 12). For example, a 30 mm lamella made of beech wood is sufficient for a 200 kg person. A thickness of 32 mm will suffice from spruce wood.

Figure 12. The minimum thickness of bed lamellae for bariatric respondents.

Conclusions

The anthropometric analysis of Slovakia's bariatric population indicates that more than 20% of the adult population has an increased body mass index above 35. The bed lamellae for adult users are designed for a weight of up to 150 kg, and their length does not consider the increased body mass and increased body dimensions of people with a higher BMI. The study includes the increased weight, waist, hip circumference and seat width values and enables the proper design of bed for bariatric users, especially elements made of renewable materials, such as spruce and beech wood. Summing out the literature review, numerical analysis, and experimental verification enable the formulation of the following conclusions and observations:

1. The linear FEM model is suitable for reliable calculation of bed lamella deflection for a bariatric respondent. The model was verified by comparing it to experimentally measured spruce and beech wood lamella bending deflection.

2. The minimal lamella thickness that meets the dimensioning criteria was always determined based on the serviceability limit criterion. The load of the lamella from the proposed bed was below 70% of the allowable stress in the case of a spruce wood (C24 grade) or 37% in the case of a beech wood (D50 grade).

3. The minimum lamella thickness dependence on the bariatric respondent's weight was non-linear. It is more advantageous to increase the stiffness of the lamella by changing the dimensions of the spruce wood lamella than by using a beech wood lamella with a higher modulus of elasticity.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This research was supported by the Slovak Research and Development Agency [contract No. APVV-20-0004] The Effect of an Increase in the Anthropometric Measurement of the Slovak Population on the Functional Properties of Furniture and the Business Processes, [APVV-21-0049] Processing of the beech raw material into the dimension timber and glued boards with significant dimensional stability, [APVV-21-0015] Utilisation and transfer of biomimetic mechanisms of wood into the design of a new form and properties of furniture, interior, and housing and [KEGA 009STU-4/2021] Innovations in the teaching processes of technical subjects by implementing augmented and virtual reality.