# Bending stiffness

Bending stiffness means the ability of a structure to resist the effect of forces that are out of its axial or planar direction. The problems on a paper machine include flutter and web wrinkling ^{1,2}. It is a basic quantity in engineering mechanics.

In paper and board, proper bending stiffness is needed for good runnability on the paper machine, printing presses and converting machines. In printed matter, sufficiently stiff products are easy to handle. In board packages, high bending stiffness gives rigidity and strength. In folding boxboard, high bending stiffness is needed for good runnability on a packaging machine, especially to ensure good performance of the creases. In short, problems may arise if the bending stiffness of paper is either too low or too high, while the stiffness of paperboard should almost always be as high as possible.

In practice, bending stiffness is most directly controlled by grammage; high grammage gives high bending stiffness. Grammage, though, is not the structural property that determines bending stiffness, it just reflects the changes in the direct control variables. From a practical point of view, grammage is also a poor control variable, because the goal is rather to decrease grammage and simultaneously keep bending stiffness constant.

## Basic relationships

Unlike the in-plane tensile properties, bending stiffness is a structural property and not a material property and it is non-trivially dependent on the macroscopic structure of paper or board, the different layers and thickness.

The parameter bending stiffness, *S _{b}*, is calculated from the standard expression for a uniform beam

(1)

${S}_{b}=\int E\left(z\right){z}^{2}dz$

where *z* is the coordinate in the thickness direction and *E* is elastic modulus. The elastic modulus can be a function of the *z*-coordinate, which reflects the effect of a layered structure. The equation can be reasoned as follows. When a beam or plate is bent, strains (*ε*) proportional to the distance from the neutral plane arise. The elastic modulus *E*, according to Hooke’s law, links stress (*σ*) and strain: *σ = E·ε*. Because the strain (and stress) are linearly proportional to the through-the-thickness coordinate (*z*), they have the form ε ≈ z. From this arises the term *E(z)·z*. The additional *z* arises from that the stresses in the various layers create a counter momentum to the bending moment. The moment arm is the distance from the neutral plane, that is the *z*-coordinate. Thus, we obtain the dependence on *z* squared, *E(z)·z ^{2}*. And finally, this is just integrated over the whole thickness (see Figure 1 ).

*Figure 1. A bent beam, the neutral plane and the strain distribution. The neutral plane is at z = 0.*

Above was mentioned the neutral plane, which is the zero of the z-coordinate. It is the plane where there is no strain. In a symmetric sheet, the plane is in the middle, in an unsymmetric sheet, it is the one more towards the stiffer side, the more unsymmetric the structure is. When the neutral plane is not in the middle, the tensile and compressive strains on the surface are not equal, but the strain is larger on the weaker side.

An important special case is constant elastic modulus, which leads to

(2)

where *d* is thickness. Because of the dependence on the cube of thickness, the result is sensitive to the correct value of thickness. For paper and board, the proper thickness to be used here is the effective thickness which is approximately in the middle of the roughness profile (for a more precise definition see Three-dimensional network).

It should be noted that the elastic modulus *E* (Gpa) is coupled with paper thickness *d* through density. Especially in calendaring, *E* and *d* change in inverse proportion to one another. Thus, calendering tends to decrease bending stiffness through d and increase it through *E*. The coupling can be removed by using the specific elastic modulus, *E/ρ = Ed/b*, where *ρ* is density and *b* grammage. *E/ρ* is then independent of calendering. Equation (2) can now be rewritten as

(3)

so that in taking the density dependence of elastic modulus into account, bending stiffness changes in proportion to thickness squared rather than cubed.

The bending stiffness index can be used to compare papers or boards of different grammage. The index, *S _{b,s}*, is obtained by dividing bending stiffness by the cube of grammage,

(4)

The units of the index are [Nm^{7}/kg^{3}].

Table 1 shows the bending stiffness index of some paper and board grades. The geometric average of MD and CD stiffness has been used in order to eliminate the effect of fibre orientation. The geometric average is also useful in comparing handsheets with machine-made papers. Structured papers are papers and boards which have some z-directional structure. LWC paper and art paper are coated, whereas folding boxboard has three layers of different pulps and corrugated board has a special structure to obtain a high thickness with a low grammage.

*Table 1. Typical bending stiffness index of (the geometric average of MD and CD) of some machine-made paper and board grades. Steel and aluminium are included for comparison.*

Paper grade | Bending stiffness index (10^{6} Nm^{7}/kg^{3}) |

SC paper | 0.23 |

LWC base paper | 0.68 |

Release paper (glassine) | 0.43 |

Fine paper (surface sized) | 0.53 |

Newsprint | 0.55 |

Structured papers | |

LWC paper | 0.33 |

Art paper (98 g/m2) | 0.36 |

Folding boxboard | 1.35 |

Corrugated board (C-flute) | 119 |

Other materials | |

Hard plastic (PET) | Up to 0.2 |

Steel | 0.032 |

Aluminium | 0.30 |

The furnish composition explains some of the differences seen in Table 1. Higher density, and thus low thickness, explains why SC paper and release paper have a lower bending stiffness index than newsprint and fine paper, respectively. In general, if mechanical pulp is replaced with chemical pulp, paper density increases and the effect cannot be compensated by an increase in specific elastic modulus.

Folding boxboard has a very high bending stiffness index because the chemical pulp is in the surface layers, where its positive effect on elastic modulus is highest without any effect on the density of the bulky middle layer made from mechanical pulp. Coated art paper and LWC paper have a lower bending stiffness index than the corresponding uncoated paper, fine paper and LWC base paper, respectively, because the coating layer has high density.

The stiffness indices of other materials, plastic, steel and aluminium are lower than for paper. This is one of the advantages of using paper and board as packaging materials; they are stiff in relation to their weight. On the other hand, paper and board have a relatively low compressive strength.

Even if the structure of paper or board is unsymmetric (e.g., coating only on one side), bending stiffness is the same in bending towards the top or bottom side, provided that during the measurement, in-plane strains everywhere in the sheet stay in the linearly elastic region. This is because the elastic modulus is the same in tension and compression. However, in measurements, bending stiffness often appears to be different when the bending is towards the top or the bottom side. This must arise from some non-linearities, and all bending stiffness measurement standards require that an equal number of measurements be made in both directions and just the average given as the result.

At large bending, the local strains within the sheet may increase so much that the bending direction becomes significant. The concave side in compression goes into the non-linear region of the stress-strain curve first. Therefore, an unsymmetric sheet exhibits lower stiffness when bent excessively towards its weaker side. The non-linearity in compression can sometimes be reversible and therefore no permanent change in curvature needs to occur, even if the specimen is taken into the non-linear region.

The bending radius r determines when the in-plane compressive strain becomes inelastic. This happens at

(5)

Font 16:

} (5)

where *ε _{y,c}* is the yield strain in compression. Since compressive failure usually occurs at strains of less than 1%, it can be estimated that

*ε*= 0.2%. Thus, the bending radius must typically be larger than

_{y,c}*r*> 250

*d*in order that the specimen remains in the elastic region and bending stiffness is symmetric. For example, with a

*d*= 100 μm thick paper, the bending radius must be

*r*> 25 mm. This is a fairly small radius (similar to paper rolled around three fingers) and thus most measurement methods are on the safe side. With paperboard

*d*= 500 μm thick, the corresponding radius is

*r*> 125 mm.

## Practical challenges related to bending stiffness

Printed matter of low bending stiffness has poor appearance and is difficult to handle. For example, a magazine may not stay flat on a sales stand, its pages may droop annoyingly during reading and it may be difficult to browse. On rare occasions, too high bending stiffness can also cause handling problems. In addition, handling problems are influenced by sheet size, the orientation of the paper in the magazine and possibly curl of the paper. The problems are biggest with wood-containing magazine papers, because of their low grammage and high density.

Inappropriate bending stiffness — either too low (which is more common) or too high — also causes runnability problems. There may be a relatively narrow stiffness range in which paper performs properly. The problems on a paper machine include flutter and web wrinkling ^{1,2}. On printing presses there can be problems in folding and stapling. Some of the problems arise from changes in paper stiffness with ambient conditions, as humid paper is less stiff than dry paper.

In packaging boards, bending stiffness is one of the most important properties. High bending stiffness is needed for good runnability on packaging machines and for rigid and strong packages. High stiffness is pursued by using three-layer structures where the middle layer gives thickness and a small contribution to strength, and the surface layers give a high elastic modulus.

When low bending stiffness causes problems, the cause is usually the CD stiffness, which is typically two to four times lower than the MD stiffness. This anisotropy arises from fibre orientation and CD drying shrinkage that affects the elastic modulus.

Bending stiffness problems often arise when one tries to lower grammage, as this usually leads to lower stiffness. Although some of the negative effects of reduced grammage *b*, such as lower strength and opacity, can be combated, less can be done with bending stiffness. This is because bending stiffness is proportional to *b ^{3}*, while tensile strength is related to

*b*and opacity to 1 – 1/

*S · b*(because

*Sb*>> 1, Table 2).

*Table 2. Approximate dependence of some paper properties on grammage b; S is the light scattering coefficient and E elastic modulus. In the approximation with opacity it is assumed that R _{ꝏ} > 0.9.*

Property | Dependence on grammage |

Tensile strength | T ≈ T_{0} · b |

Opacity | R_{0}/R_{ꝏ} ≈ S · b/(S · b + 1) |

Bending stiffness | S_{b} ≈ E · b^{3} |

In practice, there are three ways to improve bending stiffness:

- Thicker or bulkier paper: the main problem is to achieve a smooth printing surface with high bulk.
- Higher elastic modulus: this cannot be done by measures that reduce paper thickness.
- Higher elastic modulus on the surfaces than in the middle of the sheet: if each surface layer equals 5% of paper thickness, doubling their elastic modulus raises bending stiffness by 27%.

The last one of these, a layered structure, is the most effective way to obtain both high bending stiffness and good surface properties with minimum grammage. This is exploited in multi-ply boards, which are usually made with three headboxes and couched together in the press section. A layered structure can also be achieved by promoting the enrichment of fines on paper surfaces during drainage, multi-layer (stratified) forming with one headbox, surface sizing (when the size does not penetrate the sheet), coating and gradient calendering. Figure 2 illustrates the advantage gained in bending stiffness if a homogeneous sheet made from a blend of chemical and mechanical pulp is replaced with a two or three-ply structure. This example assumes that the elastic modulus and density of the blend is equal to the weighted average of the pure components. This assumption is often inaccurate.

*Figure 2. Calculated bending stiffness vs. the blending ratio of a chemical and mechanical pulp; homogeneous sheet and a two or three-ply sheet where the mechanical pulp is placed in one layer only. Chemical pulp ^{3} : E/ρ = 7.69·10^{6} Nm/kg, ρ = 826 kg/m^{3}; mechanical pulp: E/ρ = 3.02·10^{6} Nm/kg, ρ = 400 kg/m^{3}.*