Wood Sci. Technol. 28:35-44 (1993) W o o d S c i e n c e a n d T e c h n o l o g y

�9 Sprtnger-Verlag 1993

Stability of cabinet furniture backing boards

J. Smardzewski and St. Dziggielewski, Pozndn, Poland

Summary. Rigidity of cabinet furniture and load carrying ability of different classes of sur- face constructions depend on stability of sheets used as sheathing or back walls of these structures. The process of toss of stability by large sheets of wood boards (fiber and plywood boards) has not been sufficiently studied. The aim of this study was to carry out laboratory investigations on the phenomenon of the loss of stability by fiber and pine plywood boards used as back walls differently mounted to bodies of cabinet furniture. Ex- perimental results were verified by theoretical calculations which showed satisfactory con- sistency. They also confirmed possibility of checking dimensions of these boards in finished constructions,

Introduction

Studies on rigidity of cabinet furniture are carried out using two different methods of support of the construction examined, i.e. tightening three or four quoins to the base (Fig. 1). Much research has been conducted (Dziegielewski, Smardzewski 1988, 1990; Ganowicz, Dziuba, OZarska-Bergandy 1978; Ganowicz, Kwiatkowski 1978) on the character of spatial work of furniture supported in three quoins (Fig. 1 a). It is evident from these investigations that permanently torsional defor- mation character of these constructions depended only on torsional deformations of individual furniture components. However, no similar data is available on the response of such construction supported in four quoins. Moreover, nothing is known on the character of work of the back wall in such conditions.

The objective of this paper was to study the stability of the back wall mounted in different ways to the body of the furniture supported in four quoins. It was expected that the results will help assess stability of the main stiffener of real dimensions made of fiber board and plywood subjected to tangential loads.

~ C --4

a b

Fig, l a and b, Methods of testing cabinet furniture rigidity; a by supporting in three quoins; b by supporting in four quoins

36

Experimental methods

According to Korolev (1973), if a furniture body supported in four quoins is load- ed as shown in Fig. lb, a state of pure shearing stress will occur in its back wall (Fig. 2). If a critical value of these stresses is exceeded, the back wall will deflect too much leading to greater furniture body deformation. Consequently, deter- mination of critical values of operational loads will result in the rational utiliza- tion of materials and assembly methods for wood boards used as back walls in cabinet furniture and all kinds of construction elements (girders, I-bars, latticed skin etc.).

Figure 3 shows furniture bodies used in the experiment. They were made of three-layer chip board with back walls of 3 mm thick fiber boards and 5 mm thick pine plywood. The walls were mounted to the frames into notches without gluing (Fig. 4a), into rabets and to butt using staples every 150mm (Fig. 4b), (Fig. 4c) and by gluing them to notches using polyvinyl acetate glue (Fig. 4d).

Elastic properties of the materials used in the experiment are presented in Table 1. Bodies with differently mounted back walls were subjected to stresses schematically shown in Fig. 2. Since determination of the critical values of opera- tional loads acting on the examined piece of furniture is correlated with determina- tion of the critical shearing stresses in back walls manufactured from fiber boards and plywood, the key task in the experiment was to study the process of loss of stability in back walls loaded by shearing forces.

The most common method used to determine critical load values in boards is Southwell's method (Skan, Southwell 1954). This method was also used by O~ar- ska-Bergandy, Ganowicz (1985) in their experiments on stability of small sheets of wood boards. It consisted of measuring the board deflection in relation to

i a

b

Nxy

Fig. 2. Distribution of stresses in the back wall mounted to the body supported in four quoins

1! i Fig. 3. Dimensions of the experimental piece of furniture sub- jected to laboratory tests

6 :, t ~ l

00 ~ a ; d

o

,~ K ~ - - 1 F - T%.~ , ii

ij

b 4 1600 ~ - -

37

Fig. 4 a - d . Methods of mounting plywood and fiber boards to the body of furniture: a jointed in the notch; b partially elastic, butted to the body of furniture (staples); c partially elastic, in the rabbet of the body of furniture (staples); d glued to the body of furniture

Table 1. Properties of the wood boards examined

Property Unit Fiber of board measure

Type of material

Plywood Chip board

Along Across

Thickness mm 2.9

E module MPa 4530.1

G module MPa 2064.7

Poisson coeff. - 0.281

Bending strength MPa 35.02

Humidity % 4.7

5.1 18

9 179.5 691.1 -

822.0 1 752.2

0.439 0.031 -

122.4 10.7 -

6.4 6.7

Deformation Fig. 5. Method of checking sheet stability acc.to skan and Southwell (1954)

38

loads, making a graph of dependencies F = f(1) (load-deformation) for middle points of the shield surface. The value F~r was read from the graph as the asymptote ordinate to which the curve of deflection approaches (Fig. 5). Ex- periments of this kind are difficult and cumbersome (OZarska-Bergandy 1983). It was, therefore, decided that the occurrence of the critical state shall be determined by theoretical analysis. This allowed initiation of the external loads in the vicinity of the critical point.

Line theory is applied to solve critical loads and stresses. Fiber board, in this case, should be treated as an isotropic sheet subjected to shearing forces (Fig. 6) for which the differential equation of deflection surface takes the form (Timoshenko, Woynowsky-Krieger 1940).

D \3x4 4- 2 ~ + 34y.] Oy

where

N Edge tangential forces, w Function describing the surface of board buckling

Eh 3 D - Flexural rigidity of a plate,

~2(1 -- v) 2

E Young's modulus, v Poisson's ratio, h Thickness of the plate, b Width of the plate, a Height of the plate.

The critical value of forces should be determined by integrating the Eq. (1) substituting " w " by the appropriate form of the function describing the shape of the deformed board.

For a board mounted into a notch without gluing (Fig. 4 a) Bergmann, Reissner (1929); Huber (1922, 1956); Timoshenko, Woynowsky-Krieger (1940) and Timoshenko, Goodier (1951) recommend that function " w " be substituted by a dual trigonometric series

mzrx n~zy w = ~ ~ Amn sin s i n - - (2)

m=l n= , a b ? Ny//X //,~ r/~

/ Fig. 6. Isotropic sheet subjected to shearing loads

39

For a board with a partially elastic mount (Fig. 4b, c) Huber (1956) suggests the following function:

w = A s i n n X s i n n Y s i n ~ x - y (3) b b a b

Complete mounting of the back wall(Fig. 4d) which corresponds to the case where the isotropic sheet is fastened along the entire edge is described by Timoshenko and Goodier (1951) by the following equation:

A ( 2 : x ) ( 2bY ) w = - - l - c o s 1 - c o s (4) 4

For ortotropic sheets, i.e. boards with perpendicular fibers (Fig. 7), the biharmonic equation takes the following form:

0 4W 0 4W 0 4W 0 2W DxZ~ox' + 2H~ax_oy_ + - - = 2N• 0x (5) Dy 0Y 4 0y

where

Nxy, w as before, and:

Gh 3 1 1 H = - - 6 + 2 (DxVy + DyVx),

D• = Exh3/(12(1 - VxVy)) 1

Dy = E y h 3 / ( 1 2 ( 1 - VxVy))

Ex; Ey Vx; Vy G h a, b

Torsional rigidity of board acc.to (12)

Flexural rigidity of board for both directions of orthotrophy X and Y acc.to (12)

Appropriate modulus of linear elasticity, Appropriate Poisson's ratio, Shear modulus, Thickness of the plate, Size of the plate.

where

D• : DyV x.

Determination of the critical value consisted of integrating Eq. (5), substituting function "w" by the equation describing buckled plywood as an orthotropic sheet.

~I ~

7 Y �9 .~ a =// Fig. 7. Orthotr0Pic sheet subjected to shearing

loads

40

For individual methods of plywood mounting, the following forms of " w " func- tion were assumed:

- dual trigonometric series (2) for orthotropic sheet joint-mounted along the edge (Fig. 4 a),

- d u a l trigonometric series (2) which, according to Bergman and Reissner (1929), describes the surface of the ortotropic sheet mounted along the edge (Fig. 4 d),

- function (6),

w = A (1 - cos 2 (m~nx + n b x ) ) (6)

which describes the case of the orthotropic sheet mounted not completely elastical- ly along the edge (Korolev 1973).

Theoretical determination of critical load values Fcr for the wood boards ex- amined considerably simplified the laboratory investigations. Measurements were limited to recording deflections (deformations) of the back wall in the middle and to measuring the corresponding load values. Figure 8 shows the device which was specially constructed for this purpose. Table 2 and Figs. 9 and l0 present investiga- tion results on the loss of stability of back walls, while Figures 11 and 12 illustrate buckling of back walls along the shorter symmetry axis of the sheet.

o

Fig. 8. Equipment for measuring board deformation and determining loss of stability: 1)columns with pegs; 2)horizontal slat; 3)measurement factors

41

F (daN)

140 120 100 8O 6O 40 2O 0

Fkr 78 daN

1 2 3 /~

120 lOO / [ 80 r I 60 ~o 1 I 20 / 0

1 2 3 4

120! 1001-- / ~ - ~ . . . .

. . . . . . . 80 1 60

40 2 0-1 j

Wma x (mm) 1 2 3 4 wrnax (ram) b

Fkr = 126 daN 14.,0 120 - ~ - - - ' " ' ~ - 1001 / 80 60

3 40 4

20 1 0

Wma x (mm) 1 2 3 4 Wmax (mm} d

Fig. 9 a - d . Maximal dislocation of middle points of the fiber board in relation to mounting method: a jointed in the notch; b partly elastic to butt; c partly elastic in the rabbet; d fixed by gluing

F (daN)

140 120 100 8oi 60 4O 2O 0

Fkr = 113 dan

J ' - - - - L . . . - . - -

/ l

1 2 3

120 loo- ,,,.N 60

20 r I 0

1 2 3

- -- -- 120' 100 8O 1 60 4O 2O

0 4 Wmax (mm)

b

160 140 120 100 8O

3 60 40 20 0

/., Wma x (mm) d

/ i

1 2 3 4 Wma x (mm)

! !

Fkr = 136 daN

1 2 3 4 Wrnax (mm)

Fig. 1 0 a - d . Maximal dislocation of middle points of pine plywood board in relation to mounting method: a jointed in the notch; bpa r t ly elastic to butt; c partly elastic in the rabbet; d fixed by gluing

42

Table 2. Calculated values 800 • 1 600 mm dimensions

of critical loads for back walls of single cupboards of

Material Method of mounting Critical force [daN]

Fiber jointed 81.3 board partly elastically 135.5

fixed 127.9 Pine jointed 120.1 plywood partly elastically 121.1

fixed 143.6

Results and discussion

It is evident from Figs. 9 and 10 as well as from Table2 that the critical values of shearing loads in back walls were determined only for plywood and fiber boards mounted in notches along the edge without gluing (Fig. 4a, d). Experiments were interrupted in the case of the remaining methods of mounting (Fig. 4b, c) because staples were torn out of the bot tom part of the body as a result of back wall buckling. It can also be seen from Table 2 that theoretical investigations were 93~ consistent with labora tory tests only within the tested range of stability of back walls. It is also worth noticing that theoretical values were higher. This could be a t t r ibuted to a significant simplification of the mathematical model and to the integration of equation (1) and (5). Moreover, analytical considerations presented here apply to ideally flat sheets, while the experimental plywood and fiber boards showed slight initial curvatures (especially conspicuous in the case of plywood). This is easily noticed in the type of surface deformations of the back wall as il- lustrated in Fig. 11 and 12.

w(mm) A B C D E F G

a

-5 0 5

b (mm)

d

I 8OO 4

Fig. l l a - d . Deformation of fiber boards sub- jected to shearing stress applied along the shorter axis of symmetry for different mounting methods: a jointed in the notch; b partly elastic to butt; c partly elastic in the rabbet; d fixed by gluing

w(mm)

d

A B C D E F

0 5"

b (mr'n)

[ -SL i~.L_ L I [ I 5 ~ t ] i ~ ' - - - - ~ l ~ l m ' ~ d

43

Fig. 12 a - d . Deformation of pine plywood - 5 ~ ~_ ~ , ! boards subjected to shearing stress applied 0 ~ . ~ q along the shorter axis of symmetry for different 5 - mounting methods: a jointed in the notch;

800 b partly elastic to butt; e partly elastic in the 4 rabbet; d fixed by gluing

Conclusions

On the basis of the analysis of the results obtained the following conclusions can be drawn:

1. Values of critical shearing loads in back walls determined theoretically were 93~ consistent with labora tory tests. This creates possibilities for checking dimen- sions of back walls.

2. Heterogeneity of the plywood in its cross-section as well as initial curvatures result in varying forms of buckling of the wood boards examined.

3. Loads should not exceed 110daN, otherwise staples will be torn out.

References

Bergmann, S.; Reissner, H. 1929: Zeitschrift for Flugtechnik und Motorluftschiffahrt. 20: 475

Dziggielewski, St.; Smardzewski, J. 1988: Die Steifigkeit yon geschlossenen Kastenm6beln. Maschinenskript, Lehrstuhl fi~r M6belbau, Landwirtschaftsakademie Poznafi

Dzi~gielewski, St.; Smardzewski, J. 1990: Der EinfluB der Befestigungsart der hinteren Wand und der Ttir auf die Steifheit der KastenmObel. Holztechnologie 3 :136-139

Ganowicz, R.; Dziuba, T.; Ozarska-Bergandy, B. 1978: Theorie der Verformungen yon Schrankkonstruktionen. Holztechnologie 19:100-107

Ganowicz, R.; Kwiatkowski, K. 1978: Experimentelle PrUfung der Theorie der Verformungen yon Schrankkonstruktionen. Holztechnologie 19:202-206

Huber, M.T. 1956: Files. Vol. 2 Warszawa: PWN Huber, M.T. 1922: Theory of rectangular anisotropic plates with technical application for

concrete plates, beam grids etc. Lwow: Archive of Scientific Society in Lwow Korolev, V. 1973: Grundlagen der rationellen MObelkonstruktion Moskow: Nanka OZarska-Bergandy, B. 1983: Postbuckling investigations of the hardboard and plywood under

shear. Ph. D. Thesis. Agriculture University of Poznari (in Polish) OZarska-Bergandy, B.; Ganowicz, R. 1985: Postbuckling behaviour of hardboard under shear.

Wood Sci. Technol. 4 :353-361

44

Skan, S.; Southwell, R. 1954: On the stability under shearing forces of flat elastic strip. Proc. Roy. Soc. of London, Series A, 105:252-274

Timoshenko, S.; Goodier, J.N. 1951: Theory of elasticity. New York: McGraw-Hill Book Comp., Inc.

Timoshenko, S.; Woynowsky-Krieger, S. 1940: Theory of plates and shells. New-York: McGraw-Hill Book Comp., Inc.

(Received July 3, 1992)

Prof. Dr. inZ. St. Dzi~gielewski and Dr. inz. J. Smardzewski Akademia Rolnicza Katedra Meblarstwa ul. Wojska Polskiego 38/42 Poznafi, Poland