The construction of a furniture piece is done by creating appropriate bonds between its particular elements, subassemblages and assemblages. Choosing the right kind of joints for the designed furniture piece depends mainly on the type and form of the construction, but it should always lead to ensure its high stiffness and strength, and ease of realisation technologically.
A fragment of the structure in which parts are joined using connectors, interfaces and/or glue is called a joint. An element of a joint for connecting two parts is called a connector, while an interface refers to properly formed fragments of connected parts (Fig. 4.21). The quality of furniture joints is usually determined by assigning
reliability, strength and stiffness characteristics. The reliability of joints is characterised by a number of measurable indicators. One of the most important indicators of reliability is the probability of failure-free work, i. e. work without damages within a given time period, or the probability of realising a given measurable work, e. g. the number of hours of usage of the furniture piece, the number of cycles of dynamic loads, the number of damages and abrasions or scratches. (Smardzewski 2005, 2009). If the problem were to be restricted to the analysis of the strength of a given element of the furniture piece, then the designer should answer the question: What is the probability of not exceeding the permissible level by the workload, or in other words—the probability of not exceeding the appropriate level of permissible stress in a given time by stress.
Practice shows that furniture manufacturers do not assume any probability of overwork by the produced structure in a given time. Therefore, they do not know the criteria of reliability of the produced furniture and cannot properly assess the time of their correct exploitation until damages occur. Usually, the time period of warranty validity on a product is determined on the basis of the designer’s intuition rather than a pragmatic statistical analysis.
Let us consider the results of studies of a large number of angular wall joints in time t (Smardzewski 2005, 2009). At the end of the test, let n(t) of undamaged and m(t) of damaged joints remain. In this case,
R(t) = —, (4.40)
n
is the probability of non-damage, that is failure-free work, while the probability of damage will be equal to
p(t) = m«. (4.41)
n
Because the probability of damage and non-damage are events that are mutually exclusive, then the sum of them will amount to
nW + = R(t) + P(,) = 1. (4.42)
n m
The density of the probability of damages f(t) (frequency of damages) of joints in the unit of time is a derivative of the function P(t) in relation to time or other units:
f(t) = dP() – 1dm() _ dR()
dt n dt dt
thereby
t
P(t) = j f (t)dt,
0
therefore
The integral of the probability density in the probability theory, in the general sense, is called the distribution function of a given random variable. The average failure-free operating time of an element is determined on the basis of a known distribution of probability density f(t) or on the basis of the results of statistical studies. In the first case, the expected operating time E(t) = T amounts to
1
E(t) = T = J tf(t)dt.
By using this relation, it can be written as follows:
T = – tR'(t)dt,
0
whereby after integration by parts:
It is obvious that at t = 0 and t = ro, the first part of that equation will be equal to zero, thus
In the case of the examined statistical sets of furniture joints (Table 4.8):
n
ti
n where
ti operating time of the i-th connection until damage.
The behaviour of individual parts of the furniture piece influenced by operational loads depends not only on the fundamental laws of Newtonian mechanics, but also on the physical characteristics of materials used to make the construction. Joints of
|
Joint type |
Probability of failure-free work |
|
|
Unclenching |
Clenching |
|
|
Dowel d =6 mm |
0.81 |
0.85 |
|
Dowel d =8 mm |
0.94 |
0.59 |
|
Confirmat screw d = 5 mm |
0.99 |
0.99 |
|
Eccentric without sleeve |
0.84 |
0.81 |
|
Eccentric with sleeve |
0.23 |
0.69 |
|
Trapezoidal |
0.62 |
0.21 |
|
VB35 without sleeve |
0.70 |
0.49 |
|
VB35 with sleeve |
0.92 |
0.69 |
|
Table 4.8 The probability of failure-free operation of selected furniture joints |
furniture, like other components, are characterised by a limited resistance to loads causing both stresses and strains. For the designer, an important premise for choosing a specific connector or interface is the carrying capacity of the joint. The carrying capacity is the ability of taking up external loads by a material, joint or construction. The maximum load that can be transferred by the designed system is called the strength limit (Table 4.9).
Along with the appearance of external loads, constructions of furniture face strains, the size of which depends on the stiffness of joints used (Fig. 4.22).
The stiffness of the joint is determined by the coefficient k. It marks the strains caused by the external load. The best way of expressing joint stiffness is by the ratio of the value of the bending moment M to the value of the rotation angle of the node Ф (Fig. 4.23):
In the literature, however, many other ways of defining the stiffness coefficient are encountered, for example, by measuring the displacement of dp of point p on the direction of the force P (Fig. 4.24). By conducting the experiment in such a way, the authors define the joint stiffness as follows:
P
– [N/m].
dp
Both expressions determine joint stiffness, and a comparison of the obtained results and the assessment of the quality of structural nodes are possible only if identical test methods or mathematical transformations are applied, which enable to express stiffness in the form of a quotient of the bending moment and rotation angle of the joint. The next page demonstrates the transformations of equations expressing linear displacements on expressions describing shear strains.
|
Table 4.9 Stiffness and strength of selected furniture joints
|
For the load scheme causing unclenching of the joint (Fig. 4.25), we obtain
Mr
Ur
where for markings like in Fig. 4.25:
|
Fig. 4.25 Load scheme causing unclenching of the joint |
|
C = C1 + C2; |
(4.62) |
|
L2 + hp p (L + hp) – C1 = arccos L2 + h2 > |
(4.63) |
|
/ 0,5 L L2 + h2p C2 = arccos L2 + hi |
(4.64) |
|
For the load causing clenching of the joint (Fig. 4.26), the stiffness coefficient can be calculated from the equation: |
|
|
Mz k = tgaz = —, Uz |
(4.65) |
|
where for markings like in Fig. 4.26: |
|
|
Mz = Pcos(As2)((L – hp)2 + ’ , |
(4.66) |
|
Uz = U1 + U2; |
(4.67) |
therefore
Uz = 2U2. (4.74)
Table 4.9 provides example stiffnesses and strengths of selected furniture joints.
Tables 4.8 and 4.9 show that furniture joints are characterised not only by different reliability, but also diverse stiffness and strength. Due to the type of joint and type of joined materials, the stiffness of structural nodes can vary from very small to matching the stiffness of joined elements or exceeding it many times. Small stiffness of joints k3 = М3/ф3 (Fig. 4.27) causes that in the idealisation of the actual object, they should be treated as articulated joints. Stiffness determined by the quotient k1 = М1/ф1 exceeds the stiffness of joined elements, which is why in analytical models, joints of such characteristics are considered to be perfectly stiff. Between the curves k1 and k3, there is a huge set of furniture joints showing characteristics of susceptible connections (semi-stiff). Calculating susceptible joints requires a detailed specification of the distribution of all the forces in the structural node and determining places of mutual effect of contact surfaces.
The joints can be divided into two main groups: with a mechanical connector, and shaped and shape-adhesive (Fig. 4.28).
Fig. 4.27 Characteristics of the stiffness of furniture joints
Fig. 4.28 Division of furniture joints
Joints with mechanical connectors form a large group of metal and plastic separable and inseparable structural nodes. Currently, the most representative can be considered joints with connectors such as staples, nails, bolts, screws, hooks and eccentric joints.
Shaped and shape-adhesive joints contain shaped interfaces in specific parts of furniture elements, which ensure their independent connection without or with the use of glue as a connector. Of course, shape-adhesive joints prevail in this group. Formed and perfected for generations, they provide the inseparability of the construction, therefore, a satisfactory stiffness and strength. Due to the mutual system of joined elements, these joints are applied in the design of skeletal furniture, case furniture and bearing structures of upholstered furniture.
