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On the difference between flexural moduli obtained by three-point and four-point bending tests
Polymer Testing 25 (2006) 214–220 www.elsevier.com/locate/polytest
Test Method
On the difference between flexural moduli obtained by three-point and four-point bending tests F. Mujika Department of Mechanical Engineering, Polytechnical University College, University of the Basque Country, Plaza de Europa, 1, 20018 San Sebastia´n, Spain Received 12 September 2005; accepted 21 October 2005
Abstract It has been experimentally seen that flexural moduli obtained by three-point and four-point bending tests are different for the same specimen. The slope of the load-displacement curve increases as load increases in both tests, showing an apparent stiffening of the specimen. The present work analyses the effect of the variation of the support span and the load span caused by the variation of the contact zone between the specimen and support and load rollers. These effects have been analysed by classical beam theory without taking into account shear effects. Experimental differences greater than 5% for the bending modulus have been obtained for the same specimen tested in three-point and four-point bending, using two specimens of different carbon/epoxy composite materials. After corrections based on the analysis developed in this work, the relative differences between three-point and fourpoint moduli for the same specimen were under 1% for both specimens. q 2005 Elsevier Ltd. All rights reserved. Keywords: Three-point bending; Four-point bending; Bending modulus; Unidirectional composite; Carbon/epoxy
1. Introduction Bending tests are used for determining mechanical properties of unidirectional composite materials. Due to the important influence of shear effects in the displacements, great span-to-depth ratios are used in order to eliminate these effects. Three-point and four-point test configurations are used in order to obtain flexural strength and flexural modulus. The rotation of the cross sections in the deformation process leads to the contact zone between specimen and cylindrical supports changing in a three-point bending test. Furthermore, in a four-point bending test the contact between specimen and cylindrical loading noses also changes. Timoshenko [1] included the effect of the variation of the support span in a the three-point test E-mail address: [email protected]. 0142-9418/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2005.10.006
configuration due to the bending rotation at supports as a problem where the superposition principle is not applicable. Theobald et al. [2] carried out an experimental analysis in order to analyze the influence of load and geometric configuration in bending tests. The distance between loading noses or load span in a four-point bending test was varied in order to study the influence of this factor on the flexural strength and modulus. Experimental results showed that bending modulus varied in a significant manner when load span varied. Theocaris et al. [3] investigated the three-point bending test at large deflections including friction forces at the supports, axial forces along the beam and the effect of span shortening due to roller supports. Hayat and Suliman [4] performed tensile, three-point bending and four-point bending tests on glass reinforced phenolic laminates. The specimens used in four-point bending were wider than the ones in three
F. Mujika / Polymer Testing 25 (2006) 214–220
215
Nomenclature a0,
b0,
L 0 dimensions in the undeformed configuration a, b, L dimensions in the deformed configuration qA, qB, qC, qD bending angles in the basic system dC, dD displacements in the basic system I moment of inertia with respect to the middle plane P applied load in three-point bending QZP/2 applied load at each nose in four-point bending w, h width and thickness of the specimen, respectively q3p bending angle at supports in a three-point bending test d3p displacement of the middle point in a threepoint bending test h3p term related to support span reduction in a three-point bending test
point bending. Modulus obtained by three-point and four-point bending tests were different, but the higher values did not correspond always to the same method. Brancheriau et al. [5] analyzed the influence of the shear force, supports and loading head indentation in three-point and four-point bending tests on wooden samples. The same specimens were tested by both kind of test with maximum loads much lower than those corresponding to failure. They stated that a three-point bending test under-estimates the modulus of elasticity value in relation to a four-point bending test. In the present work, the chord slope of the loaddisplacement curve between 0.1 and 0.3% strain points has been used for modulus calculation. In three-point bending and four point bending, the displacement used has been that corresponding to the loading nose. Furthermore, quarter point loading has been used in four-point bending. As long as the span-to-depth ratio used has been greater than 40, shear effects have not been considered. The hypothesis of small displacements has been assumed. Thus, the rotated angles in bending have been considered small quantities and second and higher order terms related to these angles have been neglected.
RA, RC radius at support and at loading noses, respectively Ef flexure modulus k3p, k4p constants related to error in three-point and four-point bending tests, respectively 33p, 34p maximum strains in three-point and fourpoint bending tests, respectively 4p q4p ; q A C bending angles at supports and at load application points in four-point bending, respectively hA, hC, h1, h2 terms related to support span and load span reduction in a four-point bending test x1, x2, x small quantities d4p displacement of the load application points C in a four-point bending test r, D geometric factors related to support and load application radii
In the case of three-point bending, solutions can be obtained by replacing a0Zb0ZL0/2. For four-point bending, displacements and angles can be obtained using the principle of superposition. The results needed for the basic system can be obtained, for instance, using the conjugate beam method without considering shear effects. The angles and displacements necessary for analyzing the cases of three-point and four-point bending are: Rotated angles at supports A and B qA Z
Pa0 b0 ðL0 C b0 Þ 6L0 Ef I
qB Z
Pa0 b0 ðL0 C a0 Þ 6L0 Ef I
(1)
Rotated angles at points C and D qC Z
Pa0 b0 ðb0 Ka0 Þ 3L0 Ef I
(2)
2. Displacements and angles in the basic system The system shown in Fig. 1 has been used as the basic system for displacement and angles calculation.
Fig. 1. Basic configuration for angles and displacements calculation.
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F. Mujika / Polymer Testing 25 (2006) 214–220
qD Z
Pa0 b20 C 2a0 b0 K3a20 6L0 Ef I
where RA is the support radius. The actual span L during the test is L Z L0 K2h3p
Displacements at points C and D dC Z
Pa20 b20 3L0 Ef I
Pa20 b20 C 2a0 b0 Ka20 dD Z 6L0 Ef I
(3)
where Ef is the flexure modulus; I the moment of inertia with respect to the middle plane: IZwh3/12; w the width of the specimen; and h the thickness of the specimen. 3. Three-point bending Fig. 2 shows a three-point bending test in the undeformed and deformed configurations. A reduction in the support span occurs in the deformed configuration, due to the rotation at supports. The rotated angle at supports in a three-point specimen can be obtained from either qA or qB in Eq. (1) replacing a0Zb0ZL0/2, obtaining 3PL20 q3p Z 4Ef wh3
(4)
As long as the rotation angle is small, the span corresponding to the undeformed configuration is used in Eq. (4). According to Fig. 2, the error in each support due to the variation of the contact zone between the specimen and the support is h3p Z q3p RA
(5)
(6)
The displacement of the middle point in a threepoint bending test when shear effects are not considered can be obtained from either dC or dD of Eq. (3) for the actual support span, being d3p Z
PL3 4Ef wh3
(7)
Replacing Eqs. (6) and (7) and neglecting small terms related to bending angles of higher order than 1, results in h3p PL30 d3p Z 1K6 (8) L0 4Ef wh3 Taking into account that the unique variable in the parenthesis of Eq. (8) is the load P, this equation can be written as d3p Z
PL30 ð1Kk3p PÞ 4Ef wh3
(9)
where k3p Z
9L0 RA 2Ef wh3
(10)
From Eq. (9), the difference of displacements between 2 points is Dd3p Z
DPL30 ½1Kk3p ðP1 C P2 Þ 4Ef wh3
(11)
From Eq. (11) the chord flexure modulus is Ef Z
m3p L30 ½1Kk3p ðP1 C P2 Þ 4wh3
(12)
where m3pZDp/Dd3p is the chord slope between the considered points. Eq. (12) can be written as Ef Z E3p ½1Kk3p ðP1 C P2 Þ
(13)
where E3p Z
Fig. 2. Undeformed and deformed three-point bending test configuration.
m3p L30 4wh3
(14)
is the modulus usually calculated in three-point flexure. Considering that it is a small quantity, the maximum strain in a three-point bending test is calculated using the initial span as
F. Mujika / Polymer Testing 25 (2006) 214–220
33p Z
3PL0 2Ef wh2
(15)
Extracting P from Eq. (15) and replacing in Eq. (13) taking into account Eq. (10) results in 3R 3p Ef Z E3p 1K A 33p C 3 (16) 1 2 h Eq. (16) shows that the error due to span variation only depends on the strain range adopted for modulus calculation, the radius of the supports and the thickness of the specimen. 4. Four-point bending There are two sources of error in the case of fourpoint bending: the contact between specimen and loading noses and the contact between the specimen and supports. Fig. 3 shows a four-point bending test in the undeformed and deformed configurations. The dimensions after deformation and before deformation are related as L Z L0 K2hA
(17)
a Z a0 KhA KhC Z a0 Kh1
h2 Z hA KhC According to Eq. (17) and Fig. 3, supports span decreases and load span increases in a four-point bending test. The error terms at supports and at load applications points are hA Z q4p A RA
(18)
hC Z q4p C RC where RC is the radius of the load noses. The angle at supports qA and the angle at load application points qC are obtained from Eqs. (1) and (2) using the principle of superposition as q4p A Z
Qa0 b0 2Ef I
q4p C Z
Qa0 ðb0 Ka0 Þ 2Ef I
(19)
where QZP/2 is the load applied at each nose. The displacement at load application points is obtained from Eq. (3) using the principle of superposition as Qa2 ð3b2 C 2abKa2 Þ 6LEf I
d4p C Z
h1 Z hA C hC b Z b0 KhA C hC Z b0 Kh2
217
(20)
Replacing a, b and L from Eq. (17) results in Qða0 Kh1 Þ2 ½3ðb0 Kh2 Þ2 C 2ða0 Kh1 Þ 6ðL0 K2hA ÞEf I
d4p C Z
!ðb0 Kh2 ÞKða0 Kh1 Þ2
(21)
Neglecting terms of higher order than 1 in the parentheses of Eq. (21) results in d4p C Z
Q a20 ð1Kx1 Þ 2 3b0 C 2a0 b0 Ka20 K2h1 6Ef I L0 ð1Kx2 Þ !ðb0 Ka0 ÞK2h2 ð3b0 C a0 Þ
(22)
where x1Z2(h1/a0) and x2Z2(hA/L0) Neglecting terms of higher order than 1 in Eq. (22) and taking into account that when x/0, 1/1KxZ1C xCx2C. results in 1Kx1 Z 1 C x2 Kx1 Z 1Kx 1Kx2 Fig. 3. Undeformed and deformed four-point bending test configuration.
(23)
where xZx1Kx2 Replacing Eqs. (22) and (23) and neglecting second order terms, results in
218
d4p C Z
F. Mujika / Polymer Testing 25 (2006) 214–220
Q a20 ð1KxÞ 3b20 C 2a0 b0 Ka20 6Ef I L0
K2h1 ðb0 Ka0 ÞK2h2 ð3b0 C a0 Þ
From Eq. (13) the chord flexure modulus is
(24)
Eq. (24) is valid for any load span in a four-point bending. In the experimental part of the present work quarter point loading has been used, being a0ZL0/4 and b0Z3L0/4. Replacing these values and taking into account that QZP/2 results in PL30 3hA 2hC d4p Z 1K3 C (25) C L0 L0 8Ef wh3 In order to calculate hA and hC, replacing the mentioned values of a0 and b0 in Eq. (19), the angles at points A and C are q4p A Z
9PL20 16Ef wh3
q4p C Z
3PL20 8Ef wh3
(26)
(27)
where rZRC/RA is the ratio between the loading noses and the support radius. From Eqs. (18), (25) and (27) the displacement at load application points is PL30 hA d4p Z 1K3 D (28) C L0 8Ef wh3 where DZ(9C4r)/3 Taking into account that the unique variable in the second term of the parenthesis in Eq. (28) is P, this equation can be written as PL30 d4p ð1Kk4p PÞ C Z 8Ef wh3
(29)
where k4p Z
27L0 DRA 16Ef wh3
(30)
According to Eq. (29) the increment of displacement between two points is Dd4p C Z
DPL30 ½1Kk4p ðP1 C P2 Þ 8Ef wh3
m4p L30 ½1Kk4p ðP1 C P2 Þ 8wh3
(31)
(32)
where m4p Z DP=Dd4p C is the chord slope between the considered points. Eq. (32) can be written as Ef Z E4p ½1Kk4p ðP1 C P2 Þ
(33)
where E4p Z
m4p L30 8wh3
(34)
is the modulus usually calculated in four-point bending when the displacement of the load application points is used. The maximum strain in a four-point bending test considering that it is a small quantity is calculated using the initial span as 34p Z
From Eqs. (18) and (26) the ratio between the terms hC and hA is hC 2 Z r 3 hA
Ef Z
3PL0 4Ef wh2
(35)
Extracting P from Eq. (35), and replacing in Eq. (33) taking into account Eq. (30) results in 9DRA 4p 4p 31 C 32 Ef Z E4p 1K (36) 4h Eq. (36) shows that the error due to the load span and support span variations depends on the strain range adopted for modulus calculation, on the radius of the supports and the loading noses and on the thickness of the specimen. When the same strain range is used in four-point and three-point bending tests, the ratio between error terms in Eqs. (16) and (36) is: 9DRA =4h 3 Z D 3RA =h 4
(37)
The ratio in Eq. (37) only depends on the ratio between loading noses and supports radii. When the load and supports have the same radius, i.e. rZ1 the ratio in Eq. (37) is 3.25. Even in the limit case that r/0 the ratio in Eq. (37) is 2.25. Therefore, the error in modulus calculation in four-point bending is much greater than the corresponding error in three-point bending. It is worth underlining that this fact is independent of material properties, support span and specimen dimensions; it only depends on the parameter D, related to the radii of loading noses and supports. The error corresponding to third point loading in four-point bending can be determined in a similar
F. Mujika / Polymer Testing 25 (2006) 214–220
manner than the followed in this section replacing the values a0ZL0/3 and b0Z2L0/3 in Eqs. (19) and (24). 5. Experimental procedure 5.1. Materials, tests and apparatus AS4/8552 and IM7/8552 carbon/epoxy unidirectional composite materials from Hexcel Composites have been used in order to check the analytic predictions. One specimen of each material with fibres oriented longitudinally was tested in three-point and four-point bending in a universal testing machine, INSTRON 4206, at a displacement rate of 1 mm/min. The data acquisition system was Testworks 4 from MTS. Five tests were performed for each specimen. The radius of supports and loading noses were: Supports RA Z 2 mm Loading noses
RC Z 5 mm
The length of both specimens was 150 mm, the support span was 100 mm and in the case of four-point bending quarter loading was used. The dimensions of the specimens were: AS4=8552
w Z 16:6 mm
h Z 1:81 mm
IM7=8552
w Z 15:2 mm
h Z 2:37 mm
5.2. Small displacements condition ISO 14125:1998 [6] specifies that the ratio of maximum displacement to span must be less than 10% for considering small displacements. From Eqs. (4) and (7) without considering the change in span, results in d3p q3p ! 0:1 Z L0 3
(38)
Therefore, the maximum bending angle must be less than 0.3 rad (178). This angle limitation can be considered also for four-point bending. In the case of three-point bending, the relation between the angle at supports and the maximum deformation can be obtained from Eqs. (4) and (15) as q3p Z
L0 3p 3 2h
(39)
In the case of four-point bending, the relation between the angle at supports and the maximum deformation can be obtained from the first of Eqs.
219
(26) and (35) as q4p A Z
3L0 4p 3 4h
(40)
Having fixed maximum strain at 0.3% in both cases, the angle for the thinnest specimen in Eq. (39) is 0.08 rad and in Eq. (40) is 0.12 rad. Thus, the hypothesis of small displacements is satisfied in three-point bending and in four-point bending for both specimens. 5.3. Strains, loads and displacements When two strain points are used for the chord modulus calculation, loads and displacements corresponding to both points are necessary. As strain gauges are not usually used in bending tests, the value of strain can be obtained approximately from the displacement according to Eqs. (9) and (15) in three-point bending and Eqs. (29) and (35) in four-point bending, without considering span variations. The relation between strain and displacement for both cases is 3Z
6dh L20
(41)
where 3 is the maximum strain in three-point and fourpoint bending (33p or 34p) and d is the displacement of the loading nose in both configurations (d3p or d4p C ). In spite of the displacements measured by the machine having errors related to the driving system and possible indentation of the specimen, considering that these errors do not vary in a significant manner between the two points adopted, they are partially eliminated when the difference between displacements is used for chord slope calculation. Nevertheless, Eqs. (16) and (36) are not recommended for modulus correction, as long as the error in strain caused by the mentioned errors of displacement data are additive. Eqs. (13) and (33) are preferred for modulus correction as long as they are based on load data, in spite of an iterative procedure being necessary due to Ef not being known, as explained below. 6. Results and discussion Replacing the initial modulus E3p and E4p of Eqs. (14) and (34) obtained from tests in Eqs. (10) and (30) as flexural modulus Ef, the constants k3p and k4p, respectively, were obtained. Replacing these constants in eqs. (13) and (33), the corrected modulus corresponding to three-point and four-point were obtained, respectively. The process was repeated until constant values of flexural modulus were obtained.
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F. Mujika / Polymer Testing 25 (2006) 214–220
Table 1 Uncorrected and corrected flexure modulus obtained by three-point and four-point bending tests Experimental
AS4/8552 (E4pKE3p)/ E3p!100 IM7/8552 (E4pKE3p)/ E3p!100
Corrected
E3p (MPa)
E4p (MPa)
Ef3p (MPa)
Ef4p (MPa)
131,077 (G128) 5.1%
137,802 (G211)
129,428
129,551
150,813 (G227) 5.3%
158,866 (G151)
0.1% 149,116
150,355
0.8%
Table 1 shows the initial modulus E3p and E4p, and the constant values Ef3p and Ef4p after the iterative process applied to the mean values. For both specimens the constant value was obtained after two iterations in three-point bending and after four iterations in fourpoint bending. After the first iteration step the values obtained were very close to the end constant value for both test configurations. According to Table 1, the initial difference between modulus obtained by three-point and four-point bending is above 5% in both cases and is reduced below 1% after the correction process. The final difference is greater for the thicker specimen (0.8%), probably due to the greater influence of shear effects in the displacements. 7. Conclusions Due to the bending rotation at supports, support span decreases in both, three-point and four-point bending tests. Furthermore, due to the rotation at loading noses in a four-point bending test the load span increases in such a test. The effect of the mentioned span variations is much greater in four-point bending than in three-point
bending. Having fixed the strain range for modulus determination, error terms do not depend on material properties or load span used. They only depend on the thickness of the specimen and on the radius of supports and loading noses. Differences in experimental modulus calculated without correction of greater than 5% for the same specimen tested in three-point and four-point bending decrease to below 1% after corrections considered in the present work. If modulus calculation is carried out without corrections, both test methods, but particularly fourpoint bending, overestimate the flexure modulus. Acknowledgements The author wish to thank the Ministry of Education and Science of Spain for its financial support on the research project DPI 2004-02642, ‘ Flexure and Interlaminar Fracture Behaviour of Carbon/Epoxy Multidirectional Laminates’. References [1] S. Thimoshenko, Strength of Materials, Part 2 (Spanish edition), Espasa-Calpe, Madrid, 1978. [2] D. Theobald, J. McClurg, J.G. Vaughan. Comparison of ThreePoint and Four Point Flexural Bending Tests, International Composites Expo 1997, Washington, 1997, pp. 1–9. [3] P.S. Theocaris, S.A. Paipetis, S. Paolinelis, Three-point bending at large deflections, Journal of Testing and Evaluation 5 (6) (1977) 427–436. [4] M.A. Hayat, S.M.A. Suliman, Mechanical and structural properties of glass reinforced phenolic laminates, Polymer Testing 17 (1998) 79–97. [5] L. Brancheriau, H. Bailleres, D. Guitard, Comparison between modulus of elasticity values calculated using 3 and 4 point bending tests on wooden samples, Wood Science and Technology 36 (2002) 367–383. [6] ISO 14125:1998, Fibre-reinforced Plastic Composites, Determination of Flexural Properties, 1998.
Test Method
On the difference between flexural moduli obtained by three-point and four-point bending tests F. Mujika Department of Mechanical Engineering, Polytechnical University College, University of the Basque Country, Plaza de Europa, 1, 20018 San Sebastia´n, Spain Received 12 September 2005; accepted 21 October 2005
Abstract It has been experimentally seen that flexural moduli obtained by three-point and four-point bending tests are different for the same specimen. The slope of the load-displacement curve increases as load increases in both tests, showing an apparent stiffening of the specimen. The present work analyses the effect of the variation of the support span and the load span caused by the variation of the contact zone between the specimen and support and load rollers. These effects have been analysed by classical beam theory without taking into account shear effects. Experimental differences greater than 5% for the bending modulus have been obtained for the same specimen tested in three-point and four-point bending, using two specimens of different carbon/epoxy composite materials. After corrections based on the analysis developed in this work, the relative differences between three-point and fourpoint moduli for the same specimen were under 1% for both specimens. q 2005 Elsevier Ltd. All rights reserved. Keywords: Three-point bending; Four-point bending; Bending modulus; Unidirectional composite; Carbon/epoxy
1. Introduction Bending tests are used for determining mechanical properties of unidirectional composite materials. Due to the important influence of shear effects in the displacements, great span-to-depth ratios are used in order to eliminate these effects. Three-point and four-point test configurations are used in order to obtain flexural strength and flexural modulus. The rotation of the cross sections in the deformation process leads to the contact zone between specimen and cylindrical supports changing in a three-point bending test. Furthermore, in a four-point bending test the contact between specimen and cylindrical loading noses also changes. Timoshenko [1] included the effect of the variation of the support span in a the three-point test E-mail address: [email protected]. 0142-9418/$ - see front matter q 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.polymertesting.2005.10.006
configuration due to the bending rotation at supports as a problem where the superposition principle is not applicable. Theobald et al. [2] carried out an experimental analysis in order to analyze the influence of load and geometric configuration in bending tests. The distance between loading noses or load span in a four-point bending test was varied in order to study the influence of this factor on the flexural strength and modulus. Experimental results showed that bending modulus varied in a significant manner when load span varied. Theocaris et al. [3] investigated the three-point bending test at large deflections including friction forces at the supports, axial forces along the beam and the effect of span shortening due to roller supports. Hayat and Suliman [4] performed tensile, three-point bending and four-point bending tests on glass reinforced phenolic laminates. The specimens used in four-point bending were wider than the ones in three
F. Mujika / Polymer Testing 25 (2006) 214–220
215
Nomenclature a0,
b0,
L 0 dimensions in the undeformed configuration a, b, L dimensions in the deformed configuration qA, qB, qC, qD bending angles in the basic system dC, dD displacements in the basic system I moment of inertia with respect to the middle plane P applied load in three-point bending QZP/2 applied load at each nose in four-point bending w, h width and thickness of the specimen, respectively q3p bending angle at supports in a three-point bending test d3p displacement of the middle point in a threepoint bending test h3p term related to support span reduction in a three-point bending test
point bending. Modulus obtained by three-point and four-point bending tests were different, but the higher values did not correspond always to the same method. Brancheriau et al. [5] analyzed the influence of the shear force, supports and loading head indentation in three-point and four-point bending tests on wooden samples. The same specimens were tested by both kind of test with maximum loads much lower than those corresponding to failure. They stated that a three-point bending test under-estimates the modulus of elasticity value in relation to a four-point bending test. In the present work, the chord slope of the loaddisplacement curve between 0.1 and 0.3% strain points has been used for modulus calculation. In three-point bending and four point bending, the displacement used has been that corresponding to the loading nose. Furthermore, quarter point loading has been used in four-point bending. As long as the span-to-depth ratio used has been greater than 40, shear effects have not been considered. The hypothesis of small displacements has been assumed. Thus, the rotated angles in bending have been considered small quantities and second and higher order terms related to these angles have been neglected.
RA, RC radius at support and at loading noses, respectively Ef flexure modulus k3p, k4p constants related to error in three-point and four-point bending tests, respectively 33p, 34p maximum strains in three-point and fourpoint bending tests, respectively 4p q4p ; q A C bending angles at supports and at load application points in four-point bending, respectively hA, hC, h1, h2 terms related to support span and load span reduction in a four-point bending test x1, x2, x small quantities d4p displacement of the load application points C in a four-point bending test r, D geometric factors related to support and load application radii
In the case of three-point bending, solutions can be obtained by replacing a0Zb0ZL0/2. For four-point bending, displacements and angles can be obtained using the principle of superposition. The results needed for the basic system can be obtained, for instance, using the conjugate beam method without considering shear effects. The angles and displacements necessary for analyzing the cases of three-point and four-point bending are: Rotated angles at supports A and B qA Z
Pa0 b0 ðL0 C b0 Þ 6L0 Ef I
qB Z
Pa0 b0 ðL0 C a0 Þ 6L0 Ef I
(1)
Rotated angles at points C and D qC Z
Pa0 b0 ðb0 Ka0 Þ 3L0 Ef I
(2)
2. Displacements and angles in the basic system The system shown in Fig. 1 has been used as the basic system for displacement and angles calculation.
Fig. 1. Basic configuration for angles and displacements calculation.
216
F. Mujika / Polymer Testing 25 (2006) 214–220
qD Z
Pa0 b20 C 2a0 b0 K3a20 6L0 Ef I
where RA is the support radius. The actual span L during the test is L Z L0 K2h3p
Displacements at points C and D dC Z
Pa20 b20 3L0 Ef I
Pa20 b20 C 2a0 b0 Ka20 dD Z 6L0 Ef I
(3)
where Ef is the flexure modulus; I the moment of inertia with respect to the middle plane: IZwh3/12; w the width of the specimen; and h the thickness of the specimen. 3. Three-point bending Fig. 2 shows a three-point bending test in the undeformed and deformed configurations. A reduction in the support span occurs in the deformed configuration, due to the rotation at supports. The rotated angle at supports in a three-point specimen can be obtained from either qA or qB in Eq. (1) replacing a0Zb0ZL0/2, obtaining 3PL20 q3p Z 4Ef wh3
(4)
As long as the rotation angle is small, the span corresponding to the undeformed configuration is used in Eq. (4). According to Fig. 2, the error in each support due to the variation of the contact zone between the specimen and the support is h3p Z q3p RA
(5)
(6)
The displacement of the middle point in a threepoint bending test when shear effects are not considered can be obtained from either dC or dD of Eq. (3) for the actual support span, being d3p Z
PL3 4Ef wh3
(7)
Replacing Eqs. (6) and (7) and neglecting small terms related to bending angles of higher order than 1, results in h3p PL30 d3p Z 1K6 (8) L0 4Ef wh3 Taking into account that the unique variable in the parenthesis of Eq. (8) is the load P, this equation can be written as d3p Z
PL30 ð1Kk3p PÞ 4Ef wh3
(9)
where k3p Z
9L0 RA 2Ef wh3
(10)
From Eq. (9), the difference of displacements between 2 points is Dd3p Z
DPL30 ½1Kk3p ðP1 C P2 Þ 4Ef wh3
(11)
From Eq. (11) the chord flexure modulus is Ef Z
m3p L30 ½1Kk3p ðP1 C P2 Þ 4wh3
(12)
where m3pZDp/Dd3p is the chord slope between the considered points. Eq. (12) can be written as Ef Z E3p ½1Kk3p ðP1 C P2 Þ
(13)
where E3p Z
Fig. 2. Undeformed and deformed three-point bending test configuration.
m3p L30 4wh3
(14)
is the modulus usually calculated in three-point flexure. Considering that it is a small quantity, the maximum strain in a three-point bending test is calculated using the initial span as
F. Mujika / Polymer Testing 25 (2006) 214–220
33p Z
3PL0 2Ef wh2
(15)
Extracting P from Eq. (15) and replacing in Eq. (13) taking into account Eq. (10) results in 3R 3p Ef Z E3p 1K A 33p C 3 (16) 1 2 h Eq. (16) shows that the error due to span variation only depends on the strain range adopted for modulus calculation, the radius of the supports and the thickness of the specimen. 4. Four-point bending There are two sources of error in the case of fourpoint bending: the contact between specimen and loading noses and the contact between the specimen and supports. Fig. 3 shows a four-point bending test in the undeformed and deformed configurations. The dimensions after deformation and before deformation are related as L Z L0 K2hA
(17)
a Z a0 KhA KhC Z a0 Kh1
h2 Z hA KhC According to Eq. (17) and Fig. 3, supports span decreases and load span increases in a four-point bending test. The error terms at supports and at load applications points are hA Z q4p A RA
(18)
hC Z q4p C RC where RC is the radius of the load noses. The angle at supports qA and the angle at load application points qC are obtained from Eqs. (1) and (2) using the principle of superposition as q4p A Z
Qa0 b0 2Ef I
q4p C Z
Qa0 ðb0 Ka0 Þ 2Ef I
(19)
where QZP/2 is the load applied at each nose. The displacement at load application points is obtained from Eq. (3) using the principle of superposition as Qa2 ð3b2 C 2abKa2 Þ 6LEf I
d4p C Z
h1 Z hA C hC b Z b0 KhA C hC Z b0 Kh2
217
(20)
Replacing a, b and L from Eq. (17) results in Qða0 Kh1 Þ2 ½3ðb0 Kh2 Þ2 C 2ða0 Kh1 Þ 6ðL0 K2hA ÞEf I
d4p C Z
!ðb0 Kh2 ÞKða0 Kh1 Þ2
(21)
Neglecting terms of higher order than 1 in the parentheses of Eq. (21) results in d4p C Z
Q a20 ð1Kx1 Þ 2 3b0 C 2a0 b0 Ka20 K2h1 6Ef I L0 ð1Kx2 Þ !ðb0 Ka0 ÞK2h2 ð3b0 C a0 Þ
(22)
where x1Z2(h1/a0) and x2Z2(hA/L0) Neglecting terms of higher order than 1 in Eq. (22) and taking into account that when x/0, 1/1KxZ1C xCx2C. results in 1Kx1 Z 1 C x2 Kx1 Z 1Kx 1Kx2 Fig. 3. Undeformed and deformed four-point bending test configuration.
(23)
where xZx1Kx2 Replacing Eqs. (22) and (23) and neglecting second order terms, results in
218
d4p C Z
F. Mujika / Polymer Testing 25 (2006) 214–220
Q a20 ð1KxÞ 3b20 C 2a0 b0 Ka20 6Ef I L0
K2h1 ðb0 Ka0 ÞK2h2 ð3b0 C a0 Þ
From Eq. (13) the chord flexure modulus is
(24)
Eq. (24) is valid for any load span in a four-point bending. In the experimental part of the present work quarter point loading has been used, being a0ZL0/4 and b0Z3L0/4. Replacing these values and taking into account that QZP/2 results in PL30 3hA 2hC d4p Z 1K3 C (25) C L0 L0 8Ef wh3 In order to calculate hA and hC, replacing the mentioned values of a0 and b0 in Eq. (19), the angles at points A and C are q4p A Z
9PL20 16Ef wh3
q4p C Z
3PL20 8Ef wh3
(26)
(27)
where rZRC/RA is the ratio between the loading noses and the support radius. From Eqs. (18), (25) and (27) the displacement at load application points is PL30 hA d4p Z 1K3 D (28) C L0 8Ef wh3 where DZ(9C4r)/3 Taking into account that the unique variable in the second term of the parenthesis in Eq. (28) is P, this equation can be written as PL30 d4p ð1Kk4p PÞ C Z 8Ef wh3
(29)
where k4p Z
27L0 DRA 16Ef wh3
(30)
According to Eq. (29) the increment of displacement between two points is Dd4p C Z
DPL30 ½1Kk4p ðP1 C P2 Þ 8Ef wh3
m4p L30 ½1Kk4p ðP1 C P2 Þ 8wh3
(31)
(32)
where m4p Z DP=Dd4p C is the chord slope between the considered points. Eq. (32) can be written as Ef Z E4p ½1Kk4p ðP1 C P2 Þ
(33)
where E4p Z
m4p L30 8wh3
(34)
is the modulus usually calculated in four-point bending when the displacement of the load application points is used. The maximum strain in a four-point bending test considering that it is a small quantity is calculated using the initial span as 34p Z
From Eqs. (18) and (26) the ratio between the terms hC and hA is hC 2 Z r 3 hA
Ef Z
3PL0 4Ef wh2
(35)
Extracting P from Eq. (35), and replacing in Eq. (33) taking into account Eq. (30) results in 9DRA 4p 4p 31 C 32 Ef Z E4p 1K (36) 4h Eq. (36) shows that the error due to the load span and support span variations depends on the strain range adopted for modulus calculation, on the radius of the supports and the loading noses and on the thickness of the specimen. When the same strain range is used in four-point and three-point bending tests, the ratio between error terms in Eqs. (16) and (36) is: 9DRA =4h 3 Z D 3RA =h 4
(37)
The ratio in Eq. (37) only depends on the ratio between loading noses and supports radii. When the load and supports have the same radius, i.e. rZ1 the ratio in Eq. (37) is 3.25. Even in the limit case that r/0 the ratio in Eq. (37) is 2.25. Therefore, the error in modulus calculation in four-point bending is much greater than the corresponding error in three-point bending. It is worth underlining that this fact is independent of material properties, support span and specimen dimensions; it only depends on the parameter D, related to the radii of loading noses and supports. The error corresponding to third point loading in four-point bending can be determined in a similar
F. Mujika / Polymer Testing 25 (2006) 214–220
manner than the followed in this section replacing the values a0ZL0/3 and b0Z2L0/3 in Eqs. (19) and (24). 5. Experimental procedure 5.1. Materials, tests and apparatus AS4/8552 and IM7/8552 carbon/epoxy unidirectional composite materials from Hexcel Composites have been used in order to check the analytic predictions. One specimen of each material with fibres oriented longitudinally was tested in three-point and four-point bending in a universal testing machine, INSTRON 4206, at a displacement rate of 1 mm/min. The data acquisition system was Testworks 4 from MTS. Five tests were performed for each specimen. The radius of supports and loading noses were: Supports RA Z 2 mm Loading noses
RC Z 5 mm
The length of both specimens was 150 mm, the support span was 100 mm and in the case of four-point bending quarter loading was used. The dimensions of the specimens were: AS4=8552
w Z 16:6 mm
h Z 1:81 mm
IM7=8552
w Z 15:2 mm
h Z 2:37 mm
5.2. Small displacements condition ISO 14125:1998 [6] specifies that the ratio of maximum displacement to span must be less than 10% for considering small displacements. From Eqs. (4) and (7) without considering the change in span, results in d3p q3p ! 0:1 Z L0 3
(38)
Therefore, the maximum bending angle must be less than 0.3 rad (178). This angle limitation can be considered also for four-point bending. In the case of three-point bending, the relation between the angle at supports and the maximum deformation can be obtained from Eqs. (4) and (15) as q3p Z
L0 3p 3 2h
(39)
In the case of four-point bending, the relation between the angle at supports and the maximum deformation can be obtained from the first of Eqs.
219
(26) and (35) as q4p A Z
3L0 4p 3 4h
(40)
Having fixed maximum strain at 0.3% in both cases, the angle for the thinnest specimen in Eq. (39) is 0.08 rad and in Eq. (40) is 0.12 rad. Thus, the hypothesis of small displacements is satisfied in three-point bending and in four-point bending for both specimens. 5.3. Strains, loads and displacements When two strain points are used for the chord modulus calculation, loads and displacements corresponding to both points are necessary. As strain gauges are not usually used in bending tests, the value of strain can be obtained approximately from the displacement according to Eqs. (9) and (15) in three-point bending and Eqs. (29) and (35) in four-point bending, without considering span variations. The relation between strain and displacement for both cases is 3Z
6dh L20
(41)
where 3 is the maximum strain in three-point and fourpoint bending (33p or 34p) and d is the displacement of the loading nose in both configurations (d3p or d4p C ). In spite of the displacements measured by the machine having errors related to the driving system and possible indentation of the specimen, considering that these errors do not vary in a significant manner between the two points adopted, they are partially eliminated when the difference between displacements is used for chord slope calculation. Nevertheless, Eqs. (16) and (36) are not recommended for modulus correction, as long as the error in strain caused by the mentioned errors of displacement data are additive. Eqs. (13) and (33) are preferred for modulus correction as long as they are based on load data, in spite of an iterative procedure being necessary due to Ef not being known, as explained below. 6. Results and discussion Replacing the initial modulus E3p and E4p of Eqs. (14) and (34) obtained from tests in Eqs. (10) and (30) as flexural modulus Ef, the constants k3p and k4p, respectively, were obtained. Replacing these constants in eqs. (13) and (33), the corrected modulus corresponding to three-point and four-point were obtained, respectively. The process was repeated until constant values of flexural modulus were obtained.
220
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Table 1 Uncorrected and corrected flexure modulus obtained by three-point and four-point bending tests Experimental
AS4/8552 (E4pKE3p)/ E3p!100 IM7/8552 (E4pKE3p)/ E3p!100
Corrected
E3p (MPa)
E4p (MPa)
Ef3p (MPa)
Ef4p (MPa)
131,077 (G128) 5.1%
137,802 (G211)
129,428
129,551
150,813 (G227) 5.3%
158,866 (G151)
0.1% 149,116
150,355
0.8%
Table 1 shows the initial modulus E3p and E4p, and the constant values Ef3p and Ef4p after the iterative process applied to the mean values. For both specimens the constant value was obtained after two iterations in three-point bending and after four iterations in fourpoint bending. After the first iteration step the values obtained were very close to the end constant value for both test configurations. According to Table 1, the initial difference between modulus obtained by three-point and four-point bending is above 5% in both cases and is reduced below 1% after the correction process. The final difference is greater for the thicker specimen (0.8%), probably due to the greater influence of shear effects in the displacements. 7. Conclusions Due to the bending rotation at supports, support span decreases in both, three-point and four-point bending tests. Furthermore, due to the rotation at loading noses in a four-point bending test the load span increases in such a test. The effect of the mentioned span variations is much greater in four-point bending than in three-point
bending. Having fixed the strain range for modulus determination, error terms do not depend on material properties or load span used. They only depend on the thickness of the specimen and on the radius of supports and loading noses. Differences in experimental modulus calculated without correction of greater than 5% for the same specimen tested in three-point and four-point bending decrease to below 1% after corrections considered in the present work. If modulus calculation is carried out without corrections, both test methods, but particularly fourpoint bending, overestimate the flexure modulus. Acknowledgements The author wish to thank the Ministry of Education and Science of Spain for its financial support on the research project DPI 2004-02642, ‘ Flexure and Interlaminar Fracture Behaviour of Carbon/Epoxy Multidirectional Laminates’. References [1] S. Thimoshenko, Strength of Materials, Part 2 (Spanish edition), Espasa-Calpe, Madrid, 1978. [2] D. Theobald, J. McClurg, J.G. Vaughan. Comparison of ThreePoint and Four Point Flexural Bending Tests, International Composites Expo 1997, Washington, 1997, pp. 1–9. [3] P.S. Theocaris, S.A. Paipetis, S. Paolinelis, Three-point bending at large deflections, Journal of Testing and Evaluation 5 (6) (1977) 427–436. [4] M.A. Hayat, S.M.A. Suliman, Mechanical and structural properties of glass reinforced phenolic laminates, Polymer Testing 17 (1998) 79–97. [5] L. Brancheriau, H. Bailleres, D. Guitard, Comparison between modulus of elasticity values calculated using 3 and 4 point bending tests on wooden samples, Wood Science and Technology 36 (2002) 367–383. [6] ISO 14125:1998, Fibre-reinforced Plastic Composites, Determination of Flexural Properties, 1998.