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Effective width of stressed-skin panels

Kikuchi, S1

ABSTRACT

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 plywoodskins (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, asyet 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

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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/cm2 4000 kgf/cm2 0.07

(Plywood) [4413MPa] [392 MPa]

Rib 80000 kgf/cm2

[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

B p (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.

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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-e1e −−= L/SS

BBBB(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 K 1(=Be/Be(d=14cm)) of 

effective width ratio by rib depth d

As shown in Figure 5, the relation of Be/Be(d=14cm)(=K 1) to d14/d can be expressed by the following expression K 1,

where K 1 gives the modification factor of effective width ratio by varying the rib depth d.

1/γ14

1 ÷ ø

 öçè 

æ =

d

dK  (2)

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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 K 2(=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)

K 2 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 K 3(=Be/Be(Ey=45000kgf/cm2) and Ey45/Ey can be approximated by the following expressions, in

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Fig. 8  Relationship between effective width ratio

  Be/Be(t=12mm) and flange thickness t

Fig. 9 Modification factor K 2(=Be/Be(t=12mm)) of 

effective width ratio by flange thickness t

which Ey45 is the basic axial elastic modulus (Ey=45000kgf/cm2) (see Figure 11).

where Ey45=45000kgf/cm2 [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 K 3

(=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

K 4(=Be/Be(Gxy=4000kgf/cm2)) and Gxy/Gxy40 can be approximated by the following expressions (see Figure 13):

where Gxy40=4000kgf/cm2

[392Mpa] (basic shear elastic modulus of flange)

K 3 and K 4 give the modification factors of effective width ratio Be/S by the flange elastic moduli Ey and Gxy,respectively.

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Fig. 12 Relationship between effective width ratio

  Be/Be(Gxy=4000kgf/cm2) and flange shear elastic

modulus Gxy

Fig. 13 Modification factors

  K 4(=Be/Be(Gxy=4000kgf/cm2)) of effective width

ratio by flange shear elastic modulus Gxy

Figure 14 shows the effect of Poisson’s ratio υυυυxy of flange. It can be seen that Poisson’s ratio does not affect the

effective width.

Fig. 14 Effect of Poisson’s ratio υυυυxy 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

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Observation from Figure 15, it can be seen that almost the same value of effective width ratio has been given at thelocation 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 K 5(=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:

  K 5 = 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. K 5 gives

the modification factor of the effective width ratio by loading type and location along the span.

Fig. 17 Modification factors K 5(=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 (K 5=1) 0.01067 0.01811 0.02226 0.00998 -0.01000 0.00972 0.01119 0.02035

b (K 5=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 K 1: modification factor by rib depth d (Eq. 2)

  K 2: modification factor by flange thickness t (Eq. 3)

  K 3: modification factor by axial elastic modulus Ey of flange (Eq.4)

  K 4: modification factor by shear elastic modulus Gxy of flange (Eq.5)

  K 5: 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/cm2 [4413MPa] and Gxy=4000kgf/cm2 [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 

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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, 1- 21.

[4] Architectural Institute of Japan, (1995). “Standard for Structural Design of Timber Structures.”