Numerical Modelling of Human-Bed Systems

Studies confirm that pressure in healthy blood vessels of the skin amounts to 32 mmHg (4.3 kPa) and enables proper blood circulation (Krutul 2004). By examining the stresses on muscle tissue, it was found that stresses up to 34.6 kPa lasting for 35 min cause its stiffening, which leads to pressure pain (Gefen et al. 2005). Adverse loads on soft tissues of the human body, caused by lying down on a base that is too hard, can be reduced by proper support of the user’s torso using a soft and flexible material.

The conditions of the effect of the base on the human body in a lying down position can be illustrated using numerical calculations, using the algorithm of the finite elements method. Based on the transverse cross-sectional model of the human body, established at the height of the chest (Fig. 8.59), from the atlas of anthro­pometric characteristics measurements (Gedliczka 2001) the mass and dimensions of a person constituting actual load of the mattress have been determined. For the anatomical model selected in this way, using scanning, a two-dimensional mesh of finite elements was applied. During scanning, it was ensured that individual parts of the human body were covered by various grids, with varying degrees of density, depending on the type of tissue and skeletal system. In the static system, the symmetrical half of the analysed object was assumed for calculations (Fig. 8.60).

Fig. 8.59 Transverse cross section of the chest at shoulder-height (http://www. meddean. luc. edu)

Fig. 8.60 Mesh model reflecting: a the cross section of the human body on a flexible mattress, b the state of stresses in the human-bed system

The created cross-sectional model of the human body was propped up on a flexible mattress, and then it was assigned support bonds enabling vertical shifts. By defining the contact between the mattress and the human body, contact points were identified between the outer surface of the selected cross section of the body and the upper plane of the mattress. It was also assumed that the external load will be caused only by forces of gravity. For individual parts of the human body, the elastic properties provided by Gefen et al. (2005) were assumed and summarised in Table 8.6.

By analysing the contact between the human body and the mattress and the impact of flexibility of the mattress on the values of stresses in the human body,

Table 8.6 Elastic properties of human body parts (Gefen et al. 2005)

Type of tissue

Poisson’s ratio

Young’s modulus Et (kPa)

Shear modulus Gt (kPa)

Bone tissue


22.5 x 106

865 x 106

Muscle tissue




Fat tissue




a series of calculations were carried out, respectively, for a representative female, with anthropometric characteristics constituting the scale of the 50th centile (weight 65 kg), as well as the 95th centile (weight 87.8 kg). The results of the numerical calculations were presented in the system az = f(Et/Em), where Et—module of tissue flexibility, Em—module of mattress flexibility, for:

• point A located inside the mattress,

• point B on the contact surface of the body with the mattress, and

• point C inside the human body.

They were also illustrated in Figs. 8.61, 8.62 and 8.63.

It can be concluded from Fig. 8.61 that the stresses inside the mattress decrease together with the reduction of its stiffness. For a user from the 95th centile, stresses inside the mattress are about 22-25 % greater in relation to the stresses caused, in the same places, by a user of the 50th centile.

By analysing the contact stresses in point B (Fig. 8.62) it can be concluded that their value decreases, along with the reduction of the stiffness of the mattress to the value corresponding to the stiffness of human tissue. Then increases together with a clear decrease in the stiffness of the mattress in relation to the stiffness of the tissue.

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0


The smallest stress value, 4.2 kPa for the 50th centile and 5.0 kPa for the 95th centile, was obtained using the proportion E/Em equal to 1.

The changes of stresses shown in Fig. 8.63 in the function of the coefficient EJ Em set out in point C have demonstrated that while lying down on the back, stresses inside the human body increase slightly along with a decrease in stiffness of the mattress. With a 20-fold decrease in the stiffness of the mattress, the stresses inside the human body increased linearly by 6.7 %, thus by 0.34 % for every 20 % in the reduction of stiffness of the mattress. The calculations carried out prove that we obtain the correct stiffness of the spring layer of the mattress if we use materials characteristic of Young’s modulus similar or lower than the linear elastic modulus of the human body’s soft tissue.

[1] Rab

w = x/ = 8 Ea?

whereas for a board evenly loaded along its edge, this deflection is expressed by the formula:

[2] d2


W = Cw(h – 3) + C01(h – 3) + D (Jel – 1)2′ (8.15)

Mooney-Rivlin model with three parameters:


W = Сш(/t – 3) + C0t(/2 – 3) + Cn(h – 3)(h – 3) + ъ(Jel – 1)2′ (8.16)

Mooney-Rivlin model with five parameters, where N =2:

[4] _ і

W =Y, Cij(h – 3)i(/2 – 3)j – (Jel -1)2.

i+j=1 D