Stiffness of Screw Joints

Screw joints of case furniture belong to the group of susceptible connections, for which the total stiffness of the system depends on the stiffness of all its components. In engineering calculations, using models of these connections as perfectly stiff nodes it seems too optimistic, while articulated models are too unfavourable. Only the susceptible models allow the correct mathematical description and numerical modelling of the stiffness of construction of case furniture assembled with screws.

Previous studies on the load-carrying capacity of confirmat-type connectors (Bachmann 1983) show that the average strength of the screw, set at a depth of 30 mm, for pulling in the parallel direction to the wide surfaces of the board amounts to 1250 N (Fig. 7.43). By adopting this value as force in the core of the screw, and

Fig. 7.43 Strength of the screw for pulling in the parallel direction to the width of the board surface (own development based on Bachmann 1983)

Fig. 7.44 The dimensions of the screw and angular confirmat-type joint

choosing the geometric characteristics of the screw and material constants of a three-layer particle board (Table 7.6), in order to determine the stiffness of angular joints, as shown in Fig. 7.44, the cones of impact must be determined, taking into account only stresses of the screw head (Fig. 7.45) and stresses from the screw cone (Fig. 7.46). Then, the following needs to be calculated in the order:

• stiffness coefficient of screw cs,

• stiffness coefficient of board (sleeve) ck,

• load coefficient £

• initial stress of joint Qo,

• maximum stress of joint Qmax,

• residual stress of joint Qr,

• tightening moment of screw M,

• acceptable compression of the board caused by initial stress of the screw Ahp and

• maximum load Pr of the screw causing a contraction in the board equal to Ahp.

For the scheme from Fig. 7.45, we get

_______ 4h________ 4(hp – h1 – yp)

Ck nE1 ( (D0 + h1)2-D2 (Do + h1 + Xp)2-D0

_______ 4( yp – hQ___________________ 4h_______

nE2 ^ ^D£+2ip+h£+2h1^ 2 – d2 pE^ (hp + h1)2 – D2^)

= 4,803; 085 x 10-^mm) ,

c

f = s = 0.913,

cs + ck

Qo = Q(1 – f)k = 129.36 N,

Qmax = Qo + f Q = 1270.6N;

Qr = Qmax – Q = 20.6 N.

The value of shortening Ahp of the vertical connection element, caused by maximum force Qmax deriving from initial installation, was obtained from the equation:

This result can be easily verified using numerical modelling (Smardzewski and Ozarska-Bergandy 2005). As the calculation scheme, the model shown in Fig. 7.46 was selected with a force value Qmax = 1262.89 N. Additionally, taking into account the friction force on the cone and thread of the screw (Fig. 7.47), the tightening moment M has been determined from the equation:

M = MT + MG, (7.129)

where

Building a numerical model of a susceptible confirmat-type joint, the effect of compression of a vertical board was simulated by the compression of the

Fig. 7.47 Internal forces for a screw

non-threaded part of the screw, set on the section hp with the force Pr = 24,714.76 N, causing a contraction of the core Ahp = 0.0885 mm.

The distribution of reduced stresses according to Mises demonstrated that the biggest strains of the particle board occur in the part tightened by the screw head and horizontal element. Along the threaded part of the screw, the stresses are minor and do not have an impact on damaging the material of the middle layer of the board. In addition, exerting installation stresses only with the cone part of the screw head worsens the conditions of work of the joint through the increase of linear shortenings in the direction of the force of initial stress. The maximum tightening moment of the screw does not cause stresses that are destructive to the board at the length of the thread; however, it does cause stresses that are destructive in the board tightened by the head of the confirmat.

The solutions presented above included the case of loading a connector with axial installation forces Qmax. In practice, in the construction of case furniture, more complex states of loads act on the nodes. The most dangerous include the bending moments, which cause mutual stresses of connection elements. Deformations of the connector and deformations of board elements cause that the stiffness of the structural node depends on the geometry of elements after deformation and material susceptibility. The deformation of a joint loaded by the bending moment takes place gradually. Along with the growth of the value of the bending moment M, the values of stress force P, the character of surface stresses qyz and the value of the angle ф change (Fig. 7.48).

Assuming the indicators as in Table 7.6, it can be written that for ф = 0 and P = 0 (Figs. 7.48 and 7.49):

where

Qo < k nDosn

or

For ф = 0 and P >0 (Fig. 7.49), the stresses change the value from qyz to q’yz. The stiffness of the joint and value of the bending moment is written as follows:

for the part of screw with the thread,

1 _ 4 [Lz – hp)

Dl

= T ’

(7.137)

ck E2P hp – dO

M = 1 hpSoEon^ h°

– d02

(7.138)

for the part of screw with the head,

As it can be seen, the stiffness of the structural node depends on the suscepti­bility of the connector and type of board materials used.