Effective width of stressed-skin panels

Sensitivity analysis of investigating the parameters effect on the effective width of stressed-skin panels with single-skin and double-rib is presented. Based on the results, a simple formula for predicting the effective width of stressed-skin panels was derived. With this formula, the effective width (ratio) of flange can be determined by multiplying the primary effective width ratio, which is given by an exponential function of the rib spacing ratio, by each modification factor for the flange thickness, the depth of rib, the moduli of elasticity in axial and shear, the load conditions and the location along the span, successively. INTRODUCTION Plywood stressed-skin panels are used for the floor, roof, or wall system in prefabricated wood houses. It has plywood skins (flanges) glued or nailed to the top and/or bottom of longitudinal lumber members (ribs). Because the flanges and ribs act as a single structural unit in carrying loads, this kind of panel is an efficient structural component. When the panel deflects under bending, direct stresses are induced in the plane of the flange; therefore the flange takes most of the bending resistance. The direct stresses developed in the plane of the flange are not uniformly distributed over the width of the flange. This non-uniform distribution is termed shear lag. If the spacing between ribs is large, not all of flange is effective in resisting bending. The non-uniform distribution of stresses in the flange means that a simple beam theory cannot be applied without modification. Thereby it is convenient, for design purposes, to consider a width of flange acting with the rib, which, if uniformly stressed, would contribute the same amount to the flexural resistance of the beam. This width of flange has been termed the Effective Width. The effective width can be determined analytically by using a plate theory, but it is very laborious in calculation. For practical design purposes, a simple and appropriate expression to determine such effective width is needed. However, as yet any formula for predicting the effective width has not been given in the Standard for Structural Design of Timber Structures (AIJ 1995)[4]. The study described in this paper addressed a prediction of the effective width of glued stressed-skin panels under bending. Several investigators [1][2][3] have indicated main parameters effecting on the effective width as follows: (1) The ratio of span length to rib spacing (rib spacing ratio), (2) the cross section of rib and its elastic moduli, (3) the thickness of flange and its elastic moduli, (4) the location along panel length, and (5) the type of loading. The sensitivity analysis of these parameters to the effective width was done for single-skin and double –rib panels made of plywood flanges and lumber ribs and with rigid flange-to-rib connections (glued joints) by making use of the mathematical model developed by Amana and Booth (1967). The sensitivity of each parameter was estimated as a relative value to the basic panel. NUMERICAL ANALYSIS The distributions of direct stresses in the plane of flange were calculated by use of the series solutions of plate flexure for Stressed-skin panels presented by Amana and Booth (1967), which is based on equal curvature of the flange and ribs and no-slip between flange and rib. The panel configuration used in the analysis was a single-flange double-rib type. Figure 1 and Table 1 show the dimensions and the material properties of the basic panel, respectively. 1 Professor, Dept. of Architecture, Nishinippon Institute of Technology, Fukuoka, Japan Fig. 1 Dimensions of basic panel Table 1 Material properties of basic panel Modulus of elasticity Modulus of rigidity Poisson’s ratio Ey (axial) Es (bending) Gxy Flange 45000 kgf/cm 4000 kgf/cm 0.07 (Plywood) [4413MPa] [392 MPa] Rib 80000 kgf/cm [7845 MPa] For simply supported panels, a two point load applied at the forth points of the span (F.P.L) was chosen as the basic loading type and besides it a uniformly distributed load (U.D.L) and a central point load (C.P.L) were considered. Effective width Be were defined as: b Be = 2∫σydx / σy|x=b 0 Figure 2 shows the concept of effective width. Fig.2 Concept of effective width Be PARAMETERS EFFECTING ON EFFECTIVE WIDTH Rib spacing For the basic panels with the dimensions and material properties shown in Figure 1 and Table 1 respectively, the effective widths Be at the center of the span for the forth point loads were calculated from the plate theory varying the panel width Bp (25cm~136.5cm) and the panel length L (91cm~546cm). The results are plotted in Figure 3 in relation of the effective width ratio Be/S,and the rib spacing ratio L/S. Fig. 3 Relationship between effective width ratio Be/S and rib spacing ratio L/S As shown in Figure 3, the rib spacing ratio L/S affects largely the effective width ratio Be/S, and Be/S can be expressed in terms of L/S as follows: β) ( α e 1 e − − = L/S S B (1) where α=0.3838 β=0.4687 Rib depth Owing to the calculated results by the plate theory for the panels with the forth point loads, when the rib width Bs, the flange thickness t and the span length L are kept constant, the effective width Be increases with increase in the rib depth ratio d14/d, in which d14 is the basic rib depth (d=14cm), and also smaller the rib spacing ratio L/S, the effect of d on Be becomes larger. Above fact is shown in Figure 4. Fig. 4 Relationship between effective width ratio Be/S and rib depth d Fig. 5 Modification factor K1(=Be/Be(d=14cm)) of effective width ratio by rib depth d As shown in Figure 5, the relation of Be/Be(d=14cm)(=K1) to d14/d can be expressed by the following expression K1, where K1 gives the modification factor of effective width ratio by varying the rib depth d. 1/γ 14 1 = d d K (2) where γ= 2(L/S) + 12.5 for L/S<4 γ=11(L/S) 23.5 for L/S≧4 Rib width Figure 6 shows the effective width ratio Be/Be(Bs=3.8cm) versus rib width Bs for the basic panels. Observation from this figure indicates that the variation of effective width ratio is less than a few percent for a practical range of rib width and the effect of varying rib width is small. Elastic modulus of rib As shown in Figure 7, it can be said that Young’s modulus Es of rib does not affect the effective width of flange. Fig. 6 Effect of rib width Bs on effective width Be Fig. 7 Effect of rib Young’s modulus Es of rib on effective width Be Flange thickness Keeping the rib dimensions, the span length and the elastic moduli of flange constant, the ratio of the effective width to that of the basic flange thickness K2(=Be/Be(t=1.2cm)) increases with increase in the flange thickness ratio t/t12 except when the rib spacing ratio is large (L/S≧10) (see Figure 8). This relationship can be expressed as follows (see Figure 9): where t12=12mm (basic flange thickness) K2 gives the modification factor of effective width ratio by the flange thickness t. Elastic moduli of flange Figure 10 shows the calculated results of effective width of varying the flange axial elastic modulus Ey, when the rib dimensions, the span length and the flange thickness have been kept constant. It can be recognized from this figure that the smaller the rib spacing ratio L/S is, it has the larger effect of Ey. The relationship between K3(=Be/Be(Ey=45000kgf/cm2) and Ey45/Ey can be approximated by the following expressions, in Fig. 8 Relationship between effective width ratio Be/Be(t=12mm) and flange thickness t Fig. 9 Modification factor K2(=Be/Be(t=12mm)) of effective width ratio by flange thickness t which Ey45 is the basic axial elastic modulus (Ey=45000kgf/cm) (see Figure 11). where Ey45=45000kgf/cm [4413MPa] (basic axial elastic modulus of flange) Fig. 10 Relationship between effective width ratio Be/Be(Ey=45000kgf/cm2) and flange axial elastic modulus Ey Fig. 11 Modification factor K3 (=Be/Be(Ey=45000kgf/cm2)) of effective width ratio by flange elastic modulus Ey Figure 12 shows the calculated results of effective width of varying the flange shear elastic modulus Gxy. It also can be seen that the smaller the rib spacing ratio L/S is, the effect of shear modulus becomes larger. The relationship between K4(=Be/Be(Gxy=4000kgf/cm2)) and Gxy/Gxy40 can be approximated by the following expressions (see Figure 13): where Gxy40=4000kgf/cm [392Mpa] (basic shear elastic modulus of flange) K3 and K4 give the modification factors of effective width ratio Be/S by the flange elastic moduli Ey and Gxy, respectively. Fig. 12 Relationship between effective width ratio Be/Be(Gxy=4000kgf/cm2) and flange shear elastic modulus Gxy Fig. 13 Modification factors K4(=Be/Be(Gxy=4000kgf/cm2)) of effective width ratio by flange shear elastic modulus Gxy Figure 14 shows the effect of Poisson’s ratio υxy of flange. It can be seen that Poisson’s ratio does not affect the effective width. Fig. 14 Effect of Poisson’s ratio υxy of flange on effective width Types of loading In the above discussions, the type of loading was a two point load at the forth points of the span (F.P.L), and the effective width was determined at the center of span. The effective width Be varies with loading type and location along the span length, as shown in Figure 15. Fig. 15 Variations of effective width Be by types of loading Fig. 16 Ratios of effective width Be to that of two point load at fourth points (FPL) and at center of span Observation from Figure 15, it can be seen that almost the same value of effective width ratio has been given at the location of y=0.35L regardless of the type of loading. Figure 16 has been rewritten Figure 15 as the ratio of the effective width Be to that of a two point load at the fourth points (F.P.L) and at the center of span. Based on Figure 16, variation of the effective width ratio K5(=Be/Be(FPL, y=L/2)) at each location along the span length can be expressed by a linier relation to the rib spacing ratio L/S as follows: K5 = a (L/S) + b (6) where a, b = coefficients Figure 17 shows the results at y=0.5L, 0.35L and 0.1L. The values of coefficients a and b are listed in Table 2. K5 gives the modification factor of the effective width ratio by loading type and location along the span. Fig. 17 Modification factors K5(=Be/Be(FPL, y=L/2)) of effective width ratio by types of loading and location along span Table 2 Values of coefficients a and b in Eq. 6 Fourth point loads Central point load Uniformly distributed load (F.P.L) (C.P.L) (U.D.L) y=0.5L 0.35L 0.1L y=0.5L 0.35L 0.1L y=0.5L 0.35L 0.1L a (K5=1) 0.01067 0.01811 0.02226 0.00998 -0.01000 0.00972 0.01119 0.02035 b (K5=1) 0.8530 0.7247 0.6040 0.8754 0.1031 0.8554 0.8350 0.6533 FORMULA FOR PREDICTION OF EFFECTIVE WIDTH Based on the above discussions, the following formula for predicting the effective width ratio Be/S was proposed. where K1: modification factor by rib depth d (Eq. 2) K2: modification factor by flange thickness t (Eq. 3) K3: modification factor by axial elastic modulus Ey of flange (Eq.4) K4: modification factor by shear elastic modulus Gxy of flange (Eq.5) K5: modification factor by type of loading (F.P.L, U.D.L, C.P.L) and location y along the span (Eq.6) Equation 7 has been derived based on the basic panel which has the flange thickness t=1.2 cm, the flange elastic moduli Ey=45000kgf/cm [4413MPa] and Gxy=4000kgf/cm [392MPa], and is subjected to a two point load at the forth points of the span. CONCLUSIONS The followings were showed from the sensitivity analysis of stressed-skin panels. (1) The effective width of flange is governed primarily by the ratio of span length to rib spacing (the rib spacing ratio), and the effective width ratio (the ratio of effective width to rib spacing) can be expressed by an exponential function of the rib spacing ratio, (2) the variation in the thickness of flange, the moduli of elasticity in axial and shear of flange, the depth of rib, the type of loading and the location along the span length have effect on the effective width, and (3) meanwhile, Poisson’s ratio of flange and the width of rib and its elastic modulus have not significant effect on the effective width in the range of practical use. Based on the sensitivity analysis, Equation 7 was derived for predicting the effective width ratio. REFERENCES[1] Amana, E.J., and Booth, L.G. (1967). “Theoretical and experimental studies on mailed and glued plywood stressed-skin components: Part 1 Theoretical study.” J. Inst. Wood Science, 4, 1, 43 69. [2] Amana, E.J., and Booth, L.G. (1968). “Theoretical and experimental studies on nailed and glued plywood stressed-skin components: Part 2 Experimental study.” J. Inst. Wood Science, 4, 2, 19 34.[3] Hirashima, Y., (1973). “Study on bending of wooden stressed-skin panels.” Bull. Cov. Exp. Sta., No.255, 121. [4] Architectural Institute of Japan, (1995). “Standard for Structural Design of Timber Structures.”