The load-carrying capacity of eccentric joints with eccentric connectors to a great extent determines the strength of structural nodes of case furniture. The distribution of internal forces generated while forcing torsional deformations of the furniture body (Dzi^gielewski and Smardzewski 1995) results in the fact that in structural nodes that join the side walls with the bottom, the biggest loads on wooden or metal connectors of wall angular joints occur (Fig. 7.67).
Such a state of loads makes it necessary to check in the joint the shearing strength, splitting strength and compression strength of the particle board. Therefore, strength calculations should be carried out on in situ models, for which initial data must derive from elementary studies of elastic properties of materials used to make joints. Most often, bolt connections in eccentric joints are mounted directly in the particle board, which in this case constitutes a type of nut. Its elastic features have been given in the literature of the subject and in the vast majority these results can be used directly in engineering studies. However, in order to get the most reliable results of engineering calculations for particle board, an orthotropic model of the elastic properties is proposed, taking into account various properties of the surface layers (with microchip layer no. 1) and middle layer of the board (layer no. 2).
For orthotropic material, characterised by the appropriate dependencies between strains el, et, er and stresses aL, aT, aR, we use known dependencies for stress tensors:
|
Г = Ei • £i ; |
(7.215) |
|
Ty = Gj • уц, |
(7.216) |
vxz _ vzx ____ vxy vyx ф vzy vyz
Ex Ez ’ Ex Ey ’ Ez Ey ‘
For a three-layer particle board with the thickness of 18 mm, where the thickness of individual layers amounts to h1 = 3 mm, h2 = 12 mm, it can be assumed that (Kociszewski et al. 2002) Ex1 = 3850 MPa, Ex2 = 1030 MPa, and calculate substitute Young’s modulus, using the equations:
|
n
|
|
Physical quantity |
Layer h1 = 3 mm |
Layer h2 = 12 mm |
|
Ex (MPa) |
3850 |
1030 |
|
Ey (MPa) |
3301 |
883 |
|
Ez (MPa) |
123.9 |
33.2 |
|
v« |
0.368 |
0.098 |
|
Vxy |
0.270 |
0.072 |
|
vzy |
0.059 |
0.016 |
|
Table 7.10 Elastic properties of a three-layer particle board |
For the most of the examined particle boards £x-subst usually amounts to 2950 MPa, and the remaining modules, respectively, Ey-subst = 2530 MPa, and Ez-subst = 95 MPa, whereas Poisson’s ratios vxz-subst = 0.282, vxy-subst = 0.207 and
vzy- subst = °.°45.
Bearing in mind the assumed proportions and determined values of substitute linear elasticity modules and Poisson’s ratios, other elastic values for individual layers of the particle board can be specified (Table 7.10).
For practical reasons, a particle board is mostly treated as a homogeneous isotropic material, assuming that the connection of the core with the particle board should shift only post-axial forces Pb (Fig. 7.67) and rotational moments Ms caused by mounting operations. Acceptable loads for the examined joint are transferred through the adhesion surface of the thread and nut according to the scheme shown in Fig. 7.68.
At a given point of the loaded threaded surface, unit forces at of normal impact act, with the versor compatible with the normal at the given point i to the surface of the thread. Also unit forces т will occur, originating from the friction impact of
tangentials to the bolt line and the line creating the loaded side of the thread outline. Counterparts of these stresses are elementary forces dNi, dTi and dT/.
When solving the load scheme of one thread coil (Fig. 7.69), we check the conditions of self-suppression of the thread and values of resultant forces.
|
N—1—,g |
(7.221) |
|
H — Fzi tg(y + p), |
(7:222) |
|
1 F’ — F ■ |
(7.223) |
|
cos a |
|
|
n. — F’ u — F 1 |
(7:224) |
|
cos a |
|
|
u _ tg p 1 . cos a cos a |
(7:225) |
For the examined connector of the eccentric joint, geometric characteristics in accordance with the values indicated in Fig. 7.70 can be assumed.
In order to describe the distribution of stresses along the length of the thread core or in the nut body (particle board) of a semi-cross-wall joint (Fig. 7.71), the distribution of loads of the thread has to be described, whereas
Fig. 7.71 Model of a semi-cross-wall joint
where
Es linear elasticity module of the bolt core,
Ep linear elasticity module of the board and En linear elasticity module of the nut,
Detailed model of the load of the thread has been given in the work (Dietrich 2008), explaining reasons and character of various loads of the thread coils (Fig. 7.72).
When considering total relative displacements of chosen coils i and j distant from each other by Az in the post-axial direction, we will find identity connection of elongations or shortenings of the bolt and nut body with the deflections of observed coils,
Dus Dun (vis ^ vin) (vjs + vjn); (7-227)
where
Aus change of distance of the bolt thread coils as a result of elastic
elongation or shortening of the bolt core,
Aun change of distance of the nut thread coils as a result of elastic
shortening of the nut core,
vis and Vjs elastic deflections of the ith or jth coil of the bolt thread, respectively, measured on the average thread diameter, vin, Vjn elastic deflections of the ith or jth coil of the nut thread, respectively,
cooperating with the bolt coils.
Degree of differentiation (concentration) of the distribution of expenditures or pressures for loaded parts of the thread can be written by the equations:
q(z) = 1 [q'(m) cosh(kz) — q'(0) cosh(k(m – z))], (7.228)
k sinh(km)
r(z) = 1 [r(m) cosh(kz) — r(0) cosh(k(m — z))], (7.229)
k sinh(km)
where
We will consider cases of operational load of the core with forces caused during mounting or during exploitation of the furniture body.
The initial stress caused by screwing in the core or loading the thread with the force from the eccentric joint (Fig. 7.73) allows to assume the following operating conditions of the joint: the bolt core is extended and the particle board (nut body) is compressed. For such assumptions, border conditions can be written in the following form:
for z = 0
|
Fig. 7.73 Distribution of loads of thread coils, caused by screwing in of the thread with the force from the eccentric joint |
core or loading of the |
|
|
for z = m |
К = К = о, |
(7.237) |
|
q'(0) = – k2F, |
(7.238) |
|
|
q'(m) = 0, |
(7.239) |
|
|
therefore, |
kF q(z) = c°sh(k(m z)). sinh (km) |
(7:240) |
The force necessary for the initial stress (before mounting the joint) is determined from the condition of compression strength of the particle board kW =4 MPa and by receiving a value equal to
p(D – Dr)2 w F<n-b r^~ kW
Therefore, F < 36.30 N.
The moment M on the clutch screwing the core in the particle board should therefore amount to
therefore M = 88.42 N mm.
More unfavourable, however, is the work of the joint associated with the forces caused by mounting of connections and mounting of the furniture body. Stress of the core caused by the eccentric joint, however, should not be greater than the value of the force possible to be shifted, due to the shearing strength of the particle board. Using this condition, the value of forces can be determined:
of the operating stress from the condition of compression strength of the particle board k^ =3.5 MPa:
F < nDmkW,
therefore, F < 560.5 N,
the moment M on the clutch of the screwdriver:
equal to M = 1365.38 N mm.
For such conditions of use of the furniture, both the bolt core (shaft) and particle board (nut body) shall be subject to stretching. Operational conditions for this joint are as follows:
for z = 0
F"= F
1 s 1 ’
F0 = 0
1 n
|
FS =0; |
(7.247) |
|
|
Fl = F; |
(7.248) |
|
|
4(0) |
1 F = CESAS; |
(7.249) |
|
q'(m) |
1 F = C F A ’ nn |
(7.250) |
|
F 1 |
cosh (kz) cosh(k (m — z)) |
, (7.251) |
|
Ck sinh (km) |
EnAn EsAs _ |
where
On this basis, assuming the following numerical data:
M 8.5 mm,
P 2 mm,
A0r n(D – Dr)2/4,
En 1800 MPa,
Es 200,000 MPa,
An n(Rp)2,
As n(Dr)2/4,
D2 5.15 mm,
Dr 4.3 mm, and D 6 mm,
expenditures can be calculated, which have been presented in Fig. 7.74.
On the basis of values of the expenditures illustrated in Fig. 7.74, it can be noted that the initial mounting stresses related to mounting of the connector in the board do not contribute to a significant loading of the thread. Only operational loads dependent on the nature of the work of the core and nut significantly increase the level of load up to a value of 75 N/mm.
