Wood Sci. TechnoL 19:353-361 (1985) S c ~

a n d T e c h n c d o g y

�9 Springer-Verlag 1985

Postbuckling behaviour of hardboard under shear

B. O~zarska-Bergandy and R. Ganowicz, Poznafl, Poland

Summary. The purpose of the study was to analyse the postbuckling behaviour of hard- board under shear both experimentally and theoretically. Two series of 400 mmx 400 mm x 3.2 mm and 400 mm x 400 mmx 4.0 mm hardboard were tested experimentally in the special apparatus which was supposed to satisfy the boundary and load conditions. During increasing shear load, the hardboard deformations were measured in order to determine the first and second critical loads and the failure load. The experimental results were compared with the numerical solution based on the yon K~rm~ln nonlinear theory of plates. A good agreement of experimental results with theoretical solution was achieved.

Introduction

The postbuckling behaviour o f different structures has been studied by a large number of workers (Aare 1965 b; Skaloud, Donea, Massonnet 1967; Sokolov 1952; Jakubowski 1983). The distinction between the buckling and postbuckling states results from a widely accepted recognition of the fact that the critical state of a structure does not imply its failure if the structure demonstrates a fully stable initial postbuckling behaviour. That is the case of flat plate stability.

The main interest of research dealing with the postbuckling behaviour of plates was directed at the theory itself, and, in the experimental part, at the steel plates (Aare 1965b; Skaloud 1964; Voronov, Kravchuk 1973; Walker 1969). Less attention, however, was paid to wooden material structures. The possibility of postbuckling state can be noted, for example, for webbed and box beams with webs of hardboard or plywood. Another example is the postbuckling state of hardboard or plywood working as rear side o f storage furniture (a wardrobe). According to the international standard ISO/TC 136/SC 1, each box structure is tested under the action o f a horizontal force causes possible the rear side of the structure to buckle.

The purpose of this paper is to study the postbuckling behaviour of a hardboard under pure shear, i.e., the loading which occurs in webs of beams and in rear side of storage furniture. Until now, the investigation of the postbuckling state of hardboard in shear has been unknown.

The problem of the postbuckling state of the plates may be presented mathematically on the basis of the yon I~rmfin large-deflection differential

354 B. Ozarska-Bergandy and R. Ganowicz

equations which form the nonlinear theory of plates. This theory is used when the plate deflections are large, and that is the case of postbuckling state (Huseyin 1975). The von Kfirmgn equations deal with homogeneous isotropic plates, but they do not concern the nonhomogeneous plates, such as a hardboard. There are some papers (Aare 1965a; Skaloud, Donea, Massonnet 1967) dealing with the postbuckling behaviour of plate stiffened with ribs on one side. This problem is to some extent similar to that of the nonhomogeneous plate, but it cannot be easily adapted to the hardboard. In view of these considerations, it has been decided to do some experiments and to compare the results with the numerical analysis based on the von I~rm~n theory.

Our experiments have been limited to square plates only. The following problems have been examined in detail:

1. Appearance of the first critical load. 2. Generation of half-waves on the buckled plate, 3. Appearance of the second critical load. 4. Determination of the ultimate load.

The paper is divided into two parts. In the first one experimental investigations are presented, the second one includes a comparison between the results based on the yon KArm~in theory and those achieved experimentally.

Experiments

The considered element is a thin, square hardboard simply supported along its edges. All the plate edges are subject to the action of uniformly distributed shear stresses of the constant intensity r (Fig. 1). In the experiments, the test pieces of 400 mm x 400 mm x 4.0 mm and 400 m m x 400 mm x 3.2 mm hardboard (a x a x g) were used. Hardboard density was 1,002 kg/m 3, and the mean moisture content about 7%. The test apparatus specially arranged to experimental investigations is shown in Fig. 2. The hardboard was placed in the pinned frame which was supposed to satisfy the appropriate boundary conditions and the pure shearing load (Fig. 2). This frame type has been used by many authors in the examinations of steel plates under pure shear (Ermeev, Aronchik 1982; Roman6w 1976).

In order to measure perpendicular deflections of the loaded plate, 36 dial indicators were placed on a special stiff frame. The programme of the experiments included the measurement of the hardboard deformation at each stress level when the shear load was increased subsequently from zero until failure was reached.

Thus we could observe the shape of the deformed plate as a function of in- creasing shear load. In order to determine the first and second critical loads and other characteristic points we decided to observe in details only each specimen central point deflection in relation to the increasing shear load. Fig. 3 presents examples of such deflections. These experiments were repeated for two sets of ten hard- boards 3.2 mm and 4.0 mm thick.

According to the Southwell theory (Skan, Southwell 1954) the critical buckling load H~ may be determined as the horizontal asymptote ordinate to which the curve load-deflection approaches (Fig. 4), The Southwell theory deals with ideal

Postbuckling behaviour of hardboard under shear

H =-X

0

I N

m ,.~ n

H-T.0.g

Fig. 1. Scheme of hardboard loading. H = ~ ' a ' g

355

I L _ j I

Hellicel mechdnism

./Pinned freme

JHardboard

. / /Spr ing dynamometer

Fig. 2. Scheme of testing arrangement

5 Hu . . . . .

Her 4

2 cz~

0 5

H~

He9 - -

H~r~ 4 o

3

/ 7

a

f O ~

/

/ Q "

0 1 2 3 4 5 6 7 8 9 10 11 rnm 12 geflection w

Fig. 3a and b. Load-deflection curves for hardboard under sheai=, a 3.2 mm thick; b 4.0 turn thick; H~ first critical load; H~ second critical load; Hu failure load; w deflection of the specimen central point

elastic structures without imperfections and stable postbuckling state. In this case, when the load increases gradually from zero, the structure does no t deflect until the load reaches the critical value. In fact, the plate is initially imperfect and not absolutely flat. Therefore, the buckling behaviour of such structure is dif- ferent from the perfect one described by Southwell. This problem has been discussed by severaI authors: (Noale I975; Tereszkowski 1980). Fig. 3 shows that

356

i Asymptote

Deflection (wi)""

B. O~rska-Bergandy and R. Ganowicz

Fig. 4. Theoretical curve: load-deflec- tion for the ideal elastic structure with- out stable postbuckling state

Table 1. Values of characteristic loads of tested hardboards under shear

Hard- Thick- Shear Modulus of First critical load H~ Second Failure board ness modulus G elasticity E kN critical load Hu No. mm MPa MPa load I ~ kN

theoretical experimental kN

1 3.3 2,493 3,910 2.75 2.80 4.00 4.75 2 3.2 2,495 4,200 2.69 3.10 3.95 4.60 3 3.4 1,993 3,850 2.90 2.55 4.20 4.40 4 3.2 2,065 4,340 2.78 2.92 4.00 4.25 5 3.2 2,475 4,500 2.87 3.05 3.55 4.40 6 3.5 2,794 4,420 3.71 4.20 5.00 5.60 7 3.6 2,182 4,070 3.72 3.85 4.55 5.00 8 3.3 2,268 3,810 2.68 2.80 3.55 5.00 9 3.6 2,156 3,640 3.32 2.90 3.55 3.85

l0 3.6 2,032 3,830 3.50 3.50 5.00 5.50 11 3.7 1,953 3,840 3.81 4,00 5.10 5.60 12 3.8 2,024 3,670 3.94 3,45 3.90 4,30 t3 3.8 1,987 3,890 4.18 3.50 4.50 5.00 14 3.9 1,952 3,810 4.43 3.50 4.00 4.55 15 3.9 2,005 3,910 4.54 3.90 4.75 5.60 16 4.0 1,860 3,500 4.39 3.90 5.00 5.85 17 4.0 1,885 3,300 4.13 2.60 4.20 4.50 18 4.0 2,043 3,750 4.69 4.50 5.70 6.10 19 3.9 2,162 3,850 4.48 4.15 5.35 5.50 20 3.9 2,204 3,920 4.56 4.20 5.00 5.50

the process of ha rdboard buckling is composed of various stages. Before the stability loss, while the shear load increases from zero to the critical value Hc~, the deflections are small and propor t ional to the load. Near the critical load the hardboard deflections increase significantly, while the load increases slightly. It is the horizontal part of the curve load-def lect ion in Fig. 3. According to the Southwell theory we can assume that it is just the horizontal asymptote of the load- deflection relation, and then we can consider this load as the first critical buckling load I-I~.

As shown in Fig. 3, this load does not imply plate failure. When the process is continued, a further increase of both load and deflections can be observed. We

Postbuckling behaviour of hardboard under shear 357

have considered this stage as the postbuckling range limited by the second critical load I ~ . This load magnitude can be determined in the same way as the first critical load H~. It means that the hardboard deflections increase significantly while the load is almost constant (the second horizontal part of the curve load- deflection). Above the second critical load, the deflections increase rapidly until the hardboard failure Hu.

Table 1 presents experimental and analytical values of the first critical load, experimental values of the second critical and failure loads for each specimen tested. The first critical load can be calculated on the basis of the linear theory (Timoshenko, Gere 1961). The result is the same as the one received from the nonlinear theory. For the rectangular isotropic plate the first critical load is given in the form:

7z2D l-I~r = k - - (1)

a

where

E g 3 D Flexural rigidity of a plate

12(1 - v 2)

g Plate thickness E Young's modulus (values see Table 1) v Poisson's ratio (v = 0.15 for the hardboard) a Plate length and width b k Parameter depending on the ratio - - (length to width) - for the square plate

k = 9.34. a

The average critical load depends on E. For the hardboard 3 .2ram thick: I-~ = 2.54 kN, and for 4.0 mm thick: I-~ = 4.87 kN. Average modulus of elasticity E was measured in the statical bending for each tested specimen. The obtained results are presented in Table 1.

T J ~ . T

m m f ~ 2o 3-,-3.83k,

'=' -10

a

1 - H - ? . 9 2 kN mmf 2-H-3,37kN

= 20 1 2 3 3-H-3.83kN o \ / /

-1 2 C? 04 . . . . . . . . ---- Oiogono[ [ - ] I u ~

' b

Fig. 5a and b. The stages of half-waves generation in the hardboard in shear. 8 3.2 mm thick; b 4.0 mm thick

358 B. O2arska-Bergandy and R. Ganowicz

The second purpose of our research was the investigation of the half-waves generation in the hardboard in shear. The hardboard deformation along the diagonal is presented in Fig. 5. The analysis of the deformed hardboard (Fig. 5) indicates that the diagonal of the hardboard buckles first in the form of one half- wave until the first critical load H~ is exceeded (curve No. 1). In the post- buckling stage a significant jump from one into three half-waves has been ob- served, q'he ordinates of half-waves gradually grow during the load increase (curves No. 2 and 3). The deformation of the buckled hardboard due to the second critical load is presented in the Fig. 6. This kind of deformation results from the half- waves superposition along "X" and "'Y" coordinates of the plate (Fig. 6). Along the compressed diagonal I - I I three half-waves are formed, however only one along the tensile diagonal HI-IV.

Experiments presented above point out that there is a very simple, experimental method to determine the critical load and to describe the postbuckling state of the hardboard in shear. It seems necessary to analyse this problem theoretically and compare the solution with the experimental results.

T h e o r e t i c a l C o n s i d e r a t i o n s

The problem of the postbuckling behaviour can be solved also on the basis of the nonlinear theory of plates. This theory provides the two governing nonlinear, partial differential K~rm~n's equations:

__D V 2 V 2 (w) - L (w, (o) = 0 (2) g

and

_ _ l L (w, w ) = 0 (3) 1 V2V 2 (~9) ~_ _f E

where

~0 = (p (x, y) w = w (x, y)

L (w, ~0)

~2W ~2~9 ~2 W ~2~0 ~2W ~2~0 L (w, ~0) ~x 2 ~y2 + - - 2 - - - - ~y2 ~X 2 ~X 8y ~x ~y

82 ~2 V2 = ~5x2 + --By 2 (V 2 The Laplace's operator)

Eg 3 D 12(1 - v 2) Flexural rigidity of a plate

v Poisson's ratio

Airy's stress function Deflection function describing the shape of the middle surface of the buckled plate Nonlinear differential operator defined as follows:

(4)

g Thickness of the plate.

Postbuckling behaviour of hardboard under shear 359

Y 1F g

&

I 1 2 3 4 5 6 lg x I 1 Z 3 4 5 6 Ig

Fig. 6. The buckled hardboard deformations due to the second critical load. a 3.2 rnm thick; b 4.0 mm thick

__ \ ~ / X \

. f

- f

o I / "--.~<..

2

H-2.85kN "\

i!

. . , , , - . - - - - - _ . , [

o

- ( . . . _ _ - _ _ _ _ ~ .

1:, -07 / '

-77"_.4~

�9 ~ - ' r

\ i,,s.: XL~. "~..-,x.d o - -

Fig. 7 a - f t . Comparison of the theoretical and experimental contour maps of the hardboard deflections: 8, b = experimental results; e, d = numerical results

360 B. Ozarska-Bergandy and IL Ganowicz

mm ~ Plate thickness - 3,2turn 20~- Shearing load H-3.5kN

'~- 101- T A1 B2 ~ 1 ~ - - - ~ 1 1 ~ ~ E s F6 ][ ~ ~ i....,~ ~ r , i i ~ ~2~"~,,~ _- v

"s ~20~-a ~ (score 1:2) 10}- ~3 u~ UIO(~JROI I-~L

m~o& ~ Rote thickness -4.Omm She~ring toad H-3.SkN ][

--'~'- ...... [3 Ol, -'-- ...... - Diagonot I-~-

-20 b

Fig. & Comparison of theoretical and experimental deformation of hardboard under shear (deflections along the compressed diagonal I-ll). - - - Numerical results, - - experimental results

To solve the problem numerically the deflection function w (x, y) was assumed in the form of the double Fourier series which satisfied the boundary conditions of simply supported plate edges:

~zx 2~zx 2~ty 7zx 3 ~ y + w ( x , y ) = g fll s i n - - sin rty + f22sin sin +f13sin sin a a a a a

where

a

+ f3t sin 3 n_____XXa sin na y + f33 sin 3 ga x sin 3 ga Y)~ (5)

g Thickness of the plate.

fll, fzz, finn are coefficients of the plate deflection to be determined. These equa- tions were then used for detailed computations of the hardboard under shear. The values of the material parameters of the hardboard were taken from the experi- mental examinations (Ozarska-Bergandy 1983). The numerical computations were made using the method described by Jakubowski (1983) for the computer ODRA-1305 in FORTRAN language. As a result of the computations, the values for plate deflections in relation to the shear loading were obtained. On the basis of the numerical results, the contour maps of the hardboard deflec- tions were presented (Fig. 7 c, d) and then compared with the experimental results (Fig. 7 a, b). Figure 8 introduces the theoretical and experimental deformation of the hardboard along the compressed diagonal. The comparison shows a very good agreement between the theoretical and experimental results.

Thus the assumption that the hardboard can be treated as a homogeneous plate has been justified. Therefore equations and formulae of the isotropic, homo- geneous plate may be adopted in the case of the hardboard postbuckling behaviour. In fact, the hardboard may be regarded as the isotropic plate in its plane and non.homogeneous transwersally. This nonhomogeneity has an influence on the direction of the hardboard buckling. In each experiment we observed buckling of the hardboard in the direction of the porous side.

Postbuckling behaviour of hardboard under shear

Conclusions

361

The results of this study just i fy the following conclusions:

1. The nonlinear theory o f the isotropic plates may be used for the descript ion of the postbuckl ing state of hardboard .

2. There is a very simple exper imental method for investigation of the post- buckling behaviour of hardboard .

References

Aare, J. J. 1965a: Calculations of the elastic plates working in shear with stiffness of contour ribs in compression taken into account. Trudy Tallinskogo Politekhnicheskogo instituta, Nr. 229. Serija A: 3-15 (in Russian)

Aare, J. J. 1965b: Postbuckling behaviour of plates in shear. Izvestiya Akademii Nauk SSSR, Seriya Fiziko-i Mekhanicheskikh-Matematicheskikh Nauk. Moscou. Vol. XIV: 82-88 (in Russian)

Huseyin, K. 1975: Nonlinear theory of elastic stability. Leyden: Noordhoff Intemat. Publishers.

Jakubowski, S. 1983: Postbuckling analysis of simply supported square disk under shear. Applied Mechanics Institute, Techni~l University of IxScl~ (in Polish)

Ermeev, P. G.; Aronchik, A. S. 1982: Investigations of thin, steel plate under shear. Stroiteinaya Mekhanika i Raschot Soorujenii 12: 75-81 (in Russian)

Neale, K. W. 1975: Effect of imperfections on the elastic buckling of rectangular plates. ASME J. Appl. Mechan. 42:115-120

O~'arska-Bergandy, B. 1983: Postbuckling investigations of the hardboard and plywood under shear. Ph.D. Thesis. Agriculture University of Pozna~ (in Polish)

Roman6w, F. 1976: Critical stresses of simply supported sandwich plates in shear. Theoretical and Applied Mechanics 16:199-213 (in Polish)

Skan, S.; Southwell, R. 1954: On the stability under shearing forces of a fiat elastic strip. Proceed. Royal Soc. of London, Series A, 105:252-274

Sokolov, P. 1953: On postbuckling strength of the plate in compression. Trudy, Nr. 7, NISS, Moscou: 724-750 (in Russian)

Skaloud, M.; Donea, J.; Massonnet, Ch. 1967: Comportement postcritique d'une plaque carrre raidie cisaillre uniformrmenL Pattie I and IL IABSE Publications 27:187-210; 28: 137-156

Tereszkowski, Z. 1980: An experimental method for determination of critical loads of plates. The Archive of Mechanical Engineering, 23:485 - 492 (in Polish)

Timoshenko, S. P.; Gere, J. M. 1961: Theory of elastic stability. 2nd Edition. New York: McGraw-Hill

Walker, A. C. 1969: The postbuckling behaviour of simply supported square plates. The Aeron~ Quart. 20:205-222

Voronov, K.; Kravchuk, U 1975: Postbuckling investigations of thin-walled beams. Sbornik "Isledovanie metallicheskikh konstrukcii s profilirovanymi elementami secheniya", Habarovsk: 421-446 (in Russian)

International Organization of Standardization ISO/TC 136/SC 1: Furniture test methods

(Received June 6, 1984)

Prof. Dr. in~. t ( Ganowicz and Dr. in~. B. Ozarska-Bergandy Akademia Roinicza Katedra Mechaniki ul. Wojska Polskiego 38/42 Poznan, Poland