7.3.1 Reliability of Case Furniture
The European Union Directive No. 2001/95/EC, relating to general product safety, includes also furniture. The appropriate European norms define safety requirements for the use of furniture; however, they do not apply to the evaluation of the
|
Fig. 7.79 Door of horizontal rotation axis with faultily working guides: a geometric scheme, b calculation scheme |
construction reliability at an extended period of time of their use. The use of traditional methods of designing and evaluating the strength requirements for furniture, based on subjectively accepted safety coefficients and certainty reserve, does not allow to judge the time and probability of damage to the furniture or its element. It is therefore obvious that a deterministic approach to furniture design is not reliable and makes it difficult for furniture manufacturers to establish warranty conditions beneficial for themselves and the customer, including the period of free warranty repairs. Therefore, it is appropriate to introduce new methods of designing, which would allow for the random nature of construction parameters, so that the reliability of furniture could be determined at the stage of construction. In most cases, durability of the furniture is determined by the strength of the structural nodes (Gozdecki and Smardzewski 2005; Smardzewski 2002a, b; Smardzewski and Gozdecki 2007). All construction parameters of systems needed to analyse the reliability of the furniture construction are determined by the relevant distribution of stresses and strength or loads and load-carrying capacity, above all in relation to the joints and elements. If both of these distributions are established, it will be possible to determine the probability of damage to the joint, and then the probability of damage to the furniture. The problem of reliability of the furniture construction has been addressed in a few publications (Smardzewski 2005; Smardzewski and Ozarska-Bergandy 2005), where the issue of testing stiffness of dowel or bolt joints in furniture for storage was discussed.
The basis for evaluation of reliability of furniture joints should be the probability of exceeding the border level of their strength. From the point of view of reliability, the calculation of strength amounts to determining the probability of exceeding a given level of border stresses Z, with the specified dispersion area, by random loading, which causes working load о at a given time. When using the furniture, one can observe complex cases, requiring to consider both the reduction of the connection strength, as well as the increase of internal stresses. Because the values Z and a are random variables, on the basis of their characteristics the probability of structural damage Ф(u) in the planned period of use should be determined. For random variables of strength Z and stresses a, characteristic for furniture joints, the form of distributions f(Z) and f(a) can be established, and then, on the basis of the size of the area of surface permeation (Fig. 7.80), determine the probability of damage:
F (Z <r).
According to Murzewski (1989), the probability that a certain value of strength Z is located in a narrow range dZ (Fig. 7.81) and that stress о does not exceed the strength Z0 is equal to
(7.267)
Fig. 7.81 Determining probability of failure-free work
Unreliability that is the probability of structural damage will therefore amount to
1 1
F = F(Z < r) = 1 – J /r(r)(1 – Fz(r))dr = J Fz(rfr(r)dr (7.268)
or for
z
R = j fz(Z) fr(r)dr dZ;
—1 —1 Z1 Z1
F = F(Z < r) = 1 — J fz(Z)Fa(Z)dZ = J (1 — Fr(Z))fz(Z)dZ. (7.270)
In order to assess the reliability of joints and case furniture, at the Department of Furniture of the University of Life Sciences in Poznan, a study on the strength of three populations of angular joints was carried out, 10 pieces each, in which two connectors were used: confirmat screw 05 x 50 mm, beech dowels with the dimensions of 06 x 32 mm and dowels 08 x 32 mm. The joints were made of unveneered particle board with the thickness of hp = 18 mm, density p = 660 kg m-3, bending strength kG =16 MPa, splitting strength kR = 0,35 MPa, absolute moisture content 8 %, as well as the shearing strength of glue kS =9 MPa and shearing strength of beech wood kB = 17 MPa. Compression loads were distributed by the strength machine ZWICK 1445, at the same time registering the value of the load P and displacement AP at the point of application of the force (Fig. 7.82). Based on these results, values of destructive force and the most important indicators of strength of the joints were determined.
The following were accepted as indicators of strength of the joints: shearing strength of the connector
shearing strength of the adhesive bond
splitting strength of the particle board
while the value of the bending moment Mi in the joint has been determined on the basis of the equation,
|
h As2 = 45° + arcsin — — —0-5 — u2, L — hp) 2+Щ |
(7.275) |
|
U2 = C3 — b2; |
(7.276) |
|
/ 2 — 0,5 1 (a — Sp L — hp) +h2 C3 = arccos – — , (L — hp) 2+hp ’ |
(7-277) |
|
1 a L — hp) 2+h2p b2 = arccos – —2 , |
(7.278) |
established on the basis of geometric dependencies in the deformed joint (Fig. 7.83).
Values of the coefficients of the strength of joints have been presented in Table 7.11. According to them, only low splitting strength of particle boards could pose a serious threat to the failure-free work of the furniture. For these reasons,
|
Type of joint |
Symbol |
Strength (MPa) |
|||||
|
Shearing strength of the connector Zb |
Shearing strength of the adhesive bond Zs |
Splitting strength of the board ZR |
|||||
|
Average |
Std. deviation |
Average |
Std. deviation |
Average |
Std. deviation |
||
|
Dowel 6 |
Z6 |
7.03 |
0.59 |
8.64 |
0.72 |
0.501 |
0.044 |
|
Dowel 8 |
Z8 |
5.22 |
0.27 |
8.87 |
0.41 |
0.662 |
0.033 |
|
Confirmat screw |
ZK |
19.86 |
1.84 |
0.272 |
0.025 |
|
Table 7.11 Indicators of the strength of joints |
for further research on the nature of the random variable of strength f(Z) and stresses f(a), only the results of the determination of splitting strength of particle boards were selected.
Bearing in mind the distribution of internal forces acting on nodes of the furniture body (Fig. 7.84a, b), and also taking into account only the criterion of splitting strength of particle boards, stresses caused by operational load were calculated from the equation:
|
Fig. 7.84 Distribution of internal forces: a in the furniture body, b in the joint |
examined joints. Greater surface area of the charts imposition suggests a higher probability of damage to the joint. The lack of imposition, however, informs that the joint is fail-safe within a given range of given loads.
|
Type of joint |
Symbol |
Stresses aR (MPa) |
|
|
Average |
Std. deviation |
||
|
Dowel 6 |
N6 |
0.368 |
0.088 |
|
Dowel 8 |
N8 |
0.368 |
0.088 |
|
Confirmat screw |
NK |
0.105 |
0.024 |
|
Table 7.12 Values of stresses in joints according to the criterion of splitting strength of boards |
In order to determine numeric values of the probability of damage to the joint, a new random variable should be considered:
Y = Z – r. (7.284)
The condition for the security of the construction is then the assumption that
R = F(Y > 0); (7.285)
hence, the probability of damage is
0 0 і
F = J fY(Y)dY =J J fz(Y + r)fr(r)drdY (7.286)
—і —і —Y
For the discussed case furniture joints, it has been assumed that both random variables Z and a, as is shown in Fig. 7.85, are described by a normal distribution:
and
where
r = N and z = z average values of stress and strength and Sn and Sz standard deviations of stress and strength;
Hence, the standard deviation of the new random variable Y is described by the following equation:
Delamination strength [MPa]
Fig. 7.85 Distribution of stress f (N) and strength f (Z) of joints with a a dowel with a diameter of 6 mm, b a dowel with a diameter of 8 mm, c a confirmat screw
By assuming the marking:
with
thereby
(7.295)
The value of Ф(и) corresponds to the value of probability of the damage occurring and amounts to
і 2.46
Ф(и) = F(u) = 0.5(2+1) dla u > 0. (7.296)
On this basis, the probabilities of damage to the various types of joints have been determined (Table 7.13). This table shows that the most unreliable joints are those, where dowels with a diameter of 6 mm have been used as a connector. In this case, the probability of fail-safe work in the given load conditions amounts to 0.927149. For dowel joints with a diameter of 8 mm, this probability amounts to 0.999178, while for confirmat screw joints—0.999999.
Reliability of furniture as a system consisting of many unreliable joints can be calculated on the basis of the component reliabilities of the joints. In this case, case furniture should be treated as a system of serially connected structural nodes (Fig. 7.87). It is characterised by the fact that damage to one structural node leads to the damage to the whole construction.
|
Table 7.13 Values of the probability of damage to the various types of joints
|
Probability of fail-safe operation of this construction is expressed by the product of the probabilities of independent events of the 8 component items:
8
R(0=IJ Ш (7.297)
і
In the discussed case Rj(t) = R2(t) =.. .R8(t); therefore, the probability of fail-safe work of a furniture body with dowel joints with a diameter of 6 mm amounts to
R(t) = Rf(t) = 0.9271498 = 0.546004. (7.298)
Probability of fail-safe work of a furniture body with a construction using the other joints has been presented in Table 7.14.
From summaries given in Table 7.14, it appears that the reliability of the furniture construction is the lower, the more it contains elements joined with unreliable construction nodes. Systems built with many structural elements must contain a
|
Table 7.14 Probability of fail-safe work of a furniture body
|
very high reliability joints, such as confirmat screw joints. If a high reliability of the system consisting of a large number of elements is required, then reserves should be used, which allow to greatly reduce the chance of damage, and at the same time connections with a lower probability of fail-safe operation should be used. In constructions of case furniture, and in particular in multi-chamber systems, double or triple reinforcement systems should therefore be used.
