Bending stiffness of multilayer productsThere are many paper and board products whose elastic properties vary more or less discontinuously in the z-direction. These include coated papers and multilayer boards. The bending stiffness of multilayer structures cannot be calculated directly from the simplified equation for a beam (Bending stiffness, Eq 2). If each layer in the structure is homogeneous, its contribution to stiffness is given by
The symbols are defined in Figure 1; particularly |hi – z0| is the distance between the neutral plane and the ith layer. The stiffness Sb of the whole sheet is obtained by summing the contributions of all the N layers,
Figure 1. Bending stiffness of a multilayer sheet where di is the thickness of layer i, hi the coordinate of its mid-plane from the mid-plane of the whole sheet and z0 the coordinate of the neutral plane from the mid-plane of the sheet. Note that the coordinate can get either a negative or a positive value depending on which side of the mid-plane it is.
Equation (1) tells that the contribution of a layer to bending stiffness is large if the layer is far from the neutral plane.
In Eq. (1), one has to know where the neutral plane is relative to the mid-plane of the sheet, z = 0. A shortcut by assuming that z0 = 0 leads to wrong results. The assumption is valid only with symmetric products. In other cases, the neutral plane can be calculated by requiring that in bending there is no net stress on a perpendicular cross-section of the sheet and that the cross-section remains perpendicular to the sheet tangent. The first condition leads to
where ε(z) is the in-plane strain; and the second to
where ε0 is a constant. From Eqs. (3) and (4) it follows
An alternative way to Eqs. 1–5 to calculate the bending stiffness of a multilayer structure is to use the following equations 1.
and the coordinates zi are defined in Figure 2.
Figure 2. Notation of Eqs. (6)–(9).
The simplest and most important special case of multilayer structures is a symmetric three-layer structure, whose bending stiffness is
where E1 and E2 are the elastic moduli of the middle layer and surface layers, respectively, and d1 and d are the thicknesses of the middle layer and the whole sheet. This equation can be used for coated papers, because they are usually symmetric. For paperboard Eq. 10 can be used only as a quick estimate, because the structure is seldom symmetric (in folding boxboard the back side usually has lower grammage and stronger pulp than the front side).