Stiffness of Eccentric Joints

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 results in the fact that in structural nodes that join the side walls with the bottom and top, the biggest loads on metal connectors of wall angular joints occur (Fig. 7.36). Such a state of loads makes it necessary to check the strength of the connection due to the shearing strength, splitting strength and compression strength of the particle board. Therefore, strength calculations should be carried out on models, for which initial data must derive from studies of elementary properties of elastic materials used to make joints and connectors.

The stiffness of joints is determined by experimental tests of models of corner joints (Fig. 7.37). On the basis of these studies, the coefficient of stiffness у is determined from the equations:

where

M bending moment and Ф rotation angle

Figure 7.38 illustrates the stiffness of joints established on the basis of clenching tests. Then, the stiffness coefficients were determined as the value of the derivative of function M = Дф) in a point (Table 7.5). The provided illustrations and tables

Fig. 7.36 The scheme of loading the core in a wall angular joint of the case furniture body

show that the tested joints belong to the group of semi-rigid or flexibility connection structural nodes of furniture.

For example, let us consider the horizontal partition loaded by the force qxy causing its bending (Fig. 7.39). In this case, as long as internal forces are balanced,

T = 12F = qxyydx, (7.96)

where

T friction force,

ц friction coefficient for particle boards,

F operational tension forces,

qxy operational forces of the partition,

Fig. 7.37 Models of eccentric joints: a trapezoid, b Rastex 15, c Rastex 15 with sleeve, d VB35, e VB35 with sleeve

y current coordinate (for the partition) and dx elementary depth of the partition,

then the pressures q’zx > 0 are distributed evenly (q’zx = q^). Increasing the value of operational load qxy will cause a change in the form of these pressures gradually to a trapezoid (0<qZx<qZX), triangular (0 = qZx<qZX) and finally a one-point.

As it can be seen from Fig. 7.39, the increase of the value of operational load qxy causes that the resultant Qn of pressures q’zx moves downwards, away from the axis of the core of the value z. The location from the resultant vector Qn of forces of mutual pressures of board elements can be calculated from the equation:

І= 1

where

Ai surfaces of pressures,

and after substituting values like in Fig. 7.39, we shall get

1

z = 6 d ttq,

where

q’^ surface pressures at the lower edge of the board, q’zx surface pressures at the top edge of the board and d thickness of the board.

There are such working conditions for which pressures qzx assume the form of even loads, growing linearly or concentrated forces. Because by assuming,

for Qu = T, that is q^ = q’zx, we obtain that z = 0,

for Qu > T, that is q’x > q’x, we obtain that

id 3 _ 2qZxlMx’

6 q’Zx + q’zx.

for Qu » T, that is q’^ > 0 and q’zx = 0, we obtain that z = 1/6 d, for Qu » T, that is q^ = 0 and 4zx = 0, we obtain that z = 1/2 d,

where

Qu resultant operational load.

When considering the cases,

Qu > Tiq’Zx > qZx, (7.101)

Qu » T and q’Zx > 0 and = 0, (7.102)

in the joint a balance of moments must occur deriving from external forces Qu and internal forces Qn in the form:

2 Quy = [2] qxyy2dx = QnZ, (7.103)

where

Qn = 2 (qZx + q’^ddx. (7.104)

By ensuring the connection made from particle boards and eccentric connectors sufficient stiffness and strength, it needs to be made sure that the size of stresses q^ and 4zx does not exceed the acceptable compression strength for particle boards k’W ~ 4 MPa. By using the scheme presented in Fig. 7.40, the value of these stresses can be determined, resulting from the equations in effect for passive forces RA and RB caused at stress points A and B by the resultant force Qn.

By writing the equations of balance for any 2-D system of forces relative to points A and B, we obtain that

Rb = 7Qn(3 – 2«q), Ra = Qn( 1 – (3 – 2«q)). (7.105)

While the balance of power in the entire joint shows that

qxyy2dx = 1 d2ftr(qZX + qZx) 3 – 2^” ; (7.106)

6 qzx ‘ qzx

which gives

1 d2

Assuming also that reactions RA and RB determined on the edges of the boards in points A and B, attributed to elementary sections of surfaces of pressures 3A = 3z9x, should correspond to the pressures q^ and q, the equations can be formulated

where

E linear elasticity module of the particle board in the direction of y axis,

£y£y relative normal strain of the board in point A and B,

3z3x elementary sections of the surface of pressure.

Knowing the value of the linear elasticity module E for the board and assuming acceptable relative normal strain of the board EyE! y at points A and B, the values qxy in the function of strains can be estimated as follows: 1

Acceptable reactions at points of pressure should not exceed the values calcu­lated from the following equation:

Rb = 24Eddx(ey – 4), Ra = 24Eddx(5ey + 194). (7.110)

If acceptable values of operational loads are exceeded, for which Qu » T and q’zx = 0 and q’zx = 0, we obtain that z = 1/2 d. Therefore, at the junction of board elements surface pressures do not appear, but edge pressures do (Fig. 7.41).

From the balance of forces in this node, it results that

1qxyy2dx = 2F"d + F'(l – t), (7.111)

where

F’ and F" resultant forces of pressures and l arm or the force F’

and that the value of squeezes at the point of forces F" and F’ operating amount to, respectively:

F F

Al" = —, Dl’ = —, (7.112)

к"’ k’ ’ ( )

where a = l or 0.5 d.

Fig. 7.42 Reduced stresses according to Mises caused by the load on an eccentric joint with the stress qxy = 711 N/m2

Assuming a controlled size of compression for the particle board, resulting from its strength to compression, the acceptable operational load should be as follows:

qxy = 1 [Al"k"d + 2Al’k'(l – t)}. (7.114)

y2dx

The analytical determination of acceptable values of operational loads requires the use of the stiffness coefficient y, which values for specific types of connectors have been provided in Table 7.5. Adopting from this table the stiffness of the joint Rastex 15 for the elastic range у = 568.8 N m/rad, and using the above formula, for the horizontal partition with dimensions 0.4 x 0.8 m, we can determine the acceptable operational load qxy at the level 711 N/m2. For the trapezoid joint, in which у = 192.4 N m/rad, qxy cannot exceed the value 240.5 N/m2. The results of these analyses can be verified by numerical calculations. Figure 7.42 shows the model of a semi-cross-joint built from a mesh of finite elements. 20-node ortho­tropic block elements and gap-type contact elements were used to create it. Together with this, elastic characteristics of the board were used according to data provided in Table 7.6.

As it can be seen in Fig. 7.42, as a result of the rotation of the core, contact stresses appear on the particle board which contribute to decalibrating the diameter of the hole, in which an outline of the nut thread was made. The form of defor­mations of the node determined on the basis of numerical calculations indicates that the assumptions as to the location of the rotating points of the core for the math­ematical model are correct.

Table 7.6 Mechanical properties of particle board (based on Bachmann 1983; Smardzewski 2004b, c)

Property

Unit

Value

E1—Young’s modulus of external layers of the board

MPa

4656

E2—Young’s modulus of the internal layer of the board

1080

Es—Young’s modulus of steel

200,000

v—Poisson’s coefficient for all materials

0.3

kW—delamination resistance of the board

MPa

0.71

kW—shearing strength of the board

5.35

Eg—stiffness of the gap-type element

1080

hp—thickness of the board

mm

18

h1—thickness of the external layer of the board

3

h2—thickness of the internal layer of the board

12

Lz—length of the screw

50

l,—height of the ith surface of the cone of impact

Do = d—external diameter of the thread

7

D,—diameter of the ith surface of the cone of impact

Dz—external diameter of the screw head

10

do—screw core diameter

4

в—angle of elementary section of the friction surface

deg

у—angle of cone of the screw head

21

A,—surface of the ith part of the cone of impact

mm2

A—friction surface of cone of the screw head

S—skip of the thread

mm

3

dz—width of elementary section of the friction surface

MT—friction moment

N mm

MG—moment on the thread

N—pressure force

N

T—friction force

i—coefficient of friction of the board against metal