BENDING STIFFNESS OF PAPER AND PAPERBOARD
Introduction
A force applied parallel to the axis of a paper strip is called a tensile force; it elongates the strip. A force applied perpendicular to the plane of the strip is called a bending force; it bends the strip, see Figure 1. The bending stiffness of paper and paperboard characterizes the resistance of the material to such a bending force. Accordingly, to measure bending stiffness one typically subjects a strip of the material to a known bending force and observes the resulting bending deformation. In Figure 1, the applied bending force produces a displacement δ of the strip's upper edge. With a given bending force, a paper of low bending stiffness will have large δ , and a paper of high bending stiffness a small δ .
The paper strip in Figure 2 bends under its own weight, an alternative arrangement that can also be used to estimate the bending stiffness. Many different instruments have been devised to measure the bending stiffness of paper and paperboard. We will encounter two additional schemes for measuring bending stiffness further below.
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The two instruments probably most widely used in determining the bending stiffness of paper and paperboard are introduced here: the Taber bending stiffness tester and the Lhomargy bending stiffness tester. As we will see, these two instruments evaluate bending stiffness in completely different ways. They differ not only with respect to the measurement scheme employed, but also in the actual quantity measured. Details follow farther below under “Tests”.
Significance
Long ago, scientists have discovered that the best measure of bending stiffness of a material is a quantity generally called “flexural rigidity”. It has a sound physical basis and is related to more fundamental quantities as follows:
where b is the width (breadth) of a paper strip, E is Young’s modulus of the paper, and t is the caliper. A closely related quantity is the specific flexural rigidity, given by
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It is the flexural rigidity per unit width. As we shall see below, the Lhomargy bending stiffness tester directly measures the specific flexural rigidity of a sample.
Notice that caliper has a very strong effect on bending stiffness, entering both equations (1) and (2) as t3. Doubling the caliper means an eightfold increase in bending stiffness. This is the main reason why thick paperboard is so much stiffer than paper.
Papers of the same thickness are stiffer when made from highly beaten stock than when made from lightly beaten stock. This is because paper from highly beaten stock has higher density and therefore greater Young's modulus E. It is this increase in E that according to equations (1) and (2) makes for stiffer paper.
Pulps high in hemicellulose content make stiffer paper because they have higher E. For the same reason, the addition of starch, sodium silicate, and other dry-strength agents increases bending stiffness. Fillers and loading materials, on the other hand, generally decrease bending stiffness because they lower E.
Increased moisture content also lowers bending stiffness because it lowers E.
The addition of groundwood to a chemical pulp is very effective in raising the bending stiffness of papers of a given basis weight. Although E generally drops, tending to lower the bending stiffness, this effect is overcompensated by an increase in bulk and therefore thickness.
Stiffness is the most important property in folding box boards since the utility of the box depends upon its resistance to bulging when filled. This property is also important in index bristols, typing papers, and playing cards, where the paper must stand upright during use. A certain amount of stiffness is desirable in bond papers where it is a factor in the handle or feel of the paper. Stiffness is one of the most significant properties of liner board. The higher the stiffness the more rigid is the container made from the board, and the greater the resistance of the container to loading or crushing forces.
Stiffness is undesirable in some papers such as tissues, toweling, and labels. Plasticizers are added to glassine to lower the stiffness while other papers are often pebbled or embossed to reduce the stiffness.
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Tests
First Method: Taber Stiffness Test
The Taber instrument allows one to determine only an empirical measure of bending stiffness; it does not measure flexural rigidity. It is particularly useful with paperboard.
To see clearly what is being measured by the instrument, we focus attention on the paper specimen and ignore most of the mechanical parts of the apparatus. At the left of Figure 3 is shown how the sample is clamped. The top clamp, fixed to the pendulum of the instrument and rotatable around the point where the paper exits the clamp, grips the specimen firmly. The bottom clamp consists of two rollers that must not grip the sample but be left slightly open to permit free slippage. In a test, the sample is forced to bend by rotating the upper clamp 15o with respect to the straight line connecting the two clamps; this is shown on the right of Figure 3. The roller clamp will exert an unknown load P on the sample. This load causes a bending moment that varies along the paper strip. The quantity actually measured in the test is the bending moment at the center of rotation of the upper clamp. This bending moment is measured under an arbitrary
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set of conditions and in this sense represents an empirical measure of bending stiffness. (It is possible although fairly difficult to calculate the specific flexural rigidity from the measured Taber stiffness and vice versa.)
Test Specimens: Cut at least five test specimens free from scores or blemishes 1.50 in wide by 2.75 in long, parallel to, and at right angles to, the machine direction. The 1.5 in width is critical and should be cut precisely; the length is not critical.
Method of Test:
1. Description of the Taber instrument and definition of terms
1.1. The instrument rests on three stand rods of adjustable length.
1.2. Notice the large stationary disk with the circular scale of stiffness units. The zero point of the scale is at the top, and two identical scales, one to the left and one to the right, range up to 100.
1.3. The rotatable driving disk is concentrically mounted in front of the stationary disk. Notice the “degree deflection” marks at the top of the driving disk, corresponding to 0o, 7.5o, and 15o deflection.
1.4. The operating switch is the long, black plastic switch located below the two disks and pointing downward. When you push the operating switch to the left side, the driving disk rotates counterclockwise; when you push the operating switch to the right side, the driving disk rotates clockwise. Notice that the driving disk stops moving immediately you release the operating switch.
1.5. The pendulum is mounted in front of the driving disk. The pendulum rotates on a low-friction bearing that is located in the center of the driving disk. Notice the upper stud, the lower stud, and the upper specimen clamp, all fixed to the pendulum. Also notice the pendulum mark, a fine line etched into the upper end of the pendulum.
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1.6. The roller clamp is mounted on the driving disk and functions as the lower specimen clamp. The roller clamp consists of two roller units that can be independently pulled off the driving disk. If you pull a roller unit towards you, it will slide out, and you can observe that it is held by a stud on the driving disk. In addition, each of the roller units has a projecting pin at the rear. This pin fits into the hole below the large studs on the driving disk.
1.7. When you pull both roller units off the driving disk, you will notice that they are different in that only one of the two bears the sliding bottom gauge.
1.8. To enable the sliding bottom gauge to slide freely up and down past the roller, turn the adjustment knob fully clockwise. When you now turn the roller unit upside down, the sliding bottom gauge will indeed slide past the roller. Under no circumstances should the sliding bottom gauge be bent out of the way of the roller by brute force!
1.9. Testing under standard conditions means that the roller unit with the sliding bot-tom gauge is mounted on the right (rollers down); the effective sample length between clamps is then 5 cm.
1.10. Testing under sensitive conditions means that the roller unit with the sliding bottom gauge is mounted on the left (rollers up); the effective sample length between clamps is then 1 cm.
1.11. Find the small grey container usually kept near the instrument. It contains two types of special weights. First, there is a series of range weights inscribed “500 units”, “1000 units”, “2000 units”, “3000 units”, and “5000 units”, respectively. These range weights are attached to the lower stud on the pendulum to extend the range of the instrument so that relatively stiff paper and board can be tested. When using these range weights, the rollers must be mounted downward for standard testing conditions (see 1.9.).
1.12. Second, the small grey box contains the ten-unit compensator; it is attached to the upper stud on the pendulum to extend the range of the instrument so that relatively limp paper can be tested. The ten-unit compensator is used only with the roller units mounted upward for sensitive conditions (see 1.10.). Never use the ten-unit compensator under standard conditions or when a range weight is being used.
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1.13. Given that the roller units can be either up or down, that one of the five available range weights can be used or none at all, and that the deflection angle can be either 7.5o or 15o, there are a total of eight testing conditions; these are listed in Table 1. For each sample, depending on its Taber stiffness, one of these testing conditions is optimal. Accordingly, the range of Taber stiffness values appropriate with each of the eight testing conditions is listed in column 2 of Table 1.
TABLE 1 : The eight test conditions available with the Taber instrument.
Each works well over the limited range of Taber stiffness values
given in column 2.
1 / 2 / 3 / 4 / 5 / 6 / 7 / 8 / 9Test
Condi-
tion
No.
No. / Test Range,
gf´cm / Test Length,
cm / Rollers Mount-
ed / Specimen
Size,
inch
inch / Range
Weight / 10 Unit Compen-
sator / Angle
of
Deflec-
tion
tion / Multi-
plier
1 / 0 - 10 / 1 / Up / 1½ ´ 1½ / 0 / Use / 15.00 / 0.1
2 / 10 - 100 / 5 / Down / 1½ ´ 2¾ / 0 / 0 / 15.00 / 1
3 / 50 - 500 / 5 / Down / 1½ ´ 2¾ / 500 / 0 / 15.00 / 5
4 / 100 - 1000 / 5 / Down / 1½ ´ 2¾ / 1000 / 0 / 15.00 / 10
5 / 200 - 2000 / 5 / Down / 1½ ´ 2¾ / 2000 / 0 / 15.00 / 20
6 / 300 - 3000 / 5 / Down / 1½ ´ 2¾ / 3000 / 0 / 15.00 / 30
7 / 500 - 5000 / 5 / Down / 1½ ´ 2¾ / 5000 / 0 / 15.00 / 50
8 / 1000 - 10,000 / 5 / Down / 1½ ´ 2¾ / 5000 / 0 / 7.50 / 100
2. Preparing the instrument for testing
2.1. Make sure the tip of the rear stand rod is screwed on tight.
2.2. By pushing the operating switch, set the driving disk to zero. That is to say, line up the zero mark on the driving disk with the zero mark of the scale on the stationary disk.
2.3. Close the jaws of the upper clamp on the pendulum by adjusting the clamp screws. Make sure the line along which jaws meet is perfectly aligned with the central mark scribed on the pendulum.
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2.4. Level the instrument by adjusting the tips of the two front stand rods so that the pendulum mark is directly in line with the zero mark on the driving disk. The pendulum is now in perfect balance.
3. Preliminary test of a new sample
3.1. First we must obtain a rough estimate of the Taber stiffness of our sample, for only then can we decide what are the optimum test conditions for this sample, using Table 1. This first preliminary test is done under standard conditions, listed in row 2 of Table 1.
3.2. Cut a specimen to size 1½ in ´ 2¾ in. The width of 1½ in is critical and should be cut precisely. If you wish to test stiffness in the machine direction (MD), then MD must be parallel to the length direction (2¾ in) of your specimen.
3.3. Mount the rollers down for standard conditions, see 1.9. The sliding bottom gauge must always be below, not above, the rollers.
3.4. Insert your test strip between the jaws of the upper clamp and the rollers of the roller clamp, with the specimen resting lightly on the sliding bottom gauge. As you center the specimen in the upper clamp by adjusting the clamp screws, make sure it lines up with the central mark on the pendulum.
3.5. Next, center the specimen between the rollers. Move the left hand roller toward the specimen until the roller contacts it without deflecting the pendulum. Then bring the right hand roller into light contact with the specimen. On the head of each of the adjustment knobs is scribed a black line. Observe the position of this line on the right hand adjustment knob and then back off ¼ turn. This procedure ensures not only constant clearance in all tests but also enough clearance so the specimen can slide between the rollers. If this condition is not satisfied, your measurements are meaningless.
3.6. Push the operating switch, see 1.4., to the left side. This sets the driving disk in counterclockwise rotation, and the specimen is deflected. The end point is indicated when the pendulum mark, see 1.5., is aligned with the 15o mark on the driving disk. These two marks, forming one line, point to your reading on the scale. This is your left hand reading.