Four-point flexural test: Difference between revisions

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The four-point flexural test provides values for the modulus of elasticity in bending $E_{f}$ , flexural stress $\sigma _{f}$ , flexural strain $\varepsilon _{f}$ and the flexural stress-strain response of the material. This test is very similar to the three-point bending flexural test. The major difference being that with the addition of a fourth bearing the portion of the beam between the two loading points is put under maximum stress, as opposed to only the material right under the central bearing in the case of three-point bending.

This difference is of prime importance when studying brittle materials, where the number and severity of flaws exposed to the maximum stress is directly related to the flexural strength and crack initiation. Compared to the three-point bending flexural test, there are no shear forces in the four-point bending flexural test in the area between the two loading pins.^[1] The four-point bending test is therefore particularly suitable for brittle materials that cannot withstand shear stresses very well.

It is one of the most widely used apparatus to characterize fatigue and flexural stiffness of asphalt mixtures.^[2]

Testing method

The test method for conducting the test usually involves a specified test fixture on a universal testing machine. Details of the test preparation, conditioning, and conduct affect the test results. The sample is placed on two supporting pins a set distance apart and two loading pins placed at an equal distance around the center. These two loadings are lowered from above at a constant rate until sample failure.

Calculation of the flexural stress $\sigma _{f}$

\sigma _{f}={\frac {3}{4}}{\frac {FL}{bd^{2}}}

^[3] for four-point bending test where the loading span is 1/2 of the support span (rectangular cross section)

\sigma _{f}={\frac {FL}{bd^{2}}}

^[4] for four-point bending test where the loading span is 1/3 of the support span (rectangular cross section)

\sigma _{f}={\frac {3}{2}}{\frac {FL}{bd^{2}}}

^[5] for three-point bending test (rectangular cross section)

in these formulas the following parameters are used:

$\sigma _{f}$ = Stress in outer fibers at midpoint, (MPa)
$F$ = load at a given point on the load deflection curve, (N)
$L$ = Support span, (mm)
$b$ = Width of test beam, (mm)
$d$ = Depth or thickness of tested beam, (mm)

Advantages and disadvantages

Advantages of three-point and four-point bending tests over uniaxial tensile tests include:

simpler sample geometries
minimum sample machining is required
simple test fixture
possibility to use as-fabricated materials^[6]

Disadvantages include:

more complex integral stress distributions through the sample

Application with different materials

Ceramics

Ceramics are usually very brittle, and their flexural strength depends on both their inherent toughness and the size and severity of flaws. Exposing a large volume of material to the maximum stress will reduce the measured flexural strength because it increases the likelihood of having cracks reaching critical length at a given applied load. Values for the flexural strength measured with four-point bending will be significantly lower than with three-point bending.,^[7] Compared with three-point bending test, this method is more suitable for strength evaluation of butt joint specimens. The advantage of four-point bending test is that a larger portion of the specimen between two inner loading pins is subjected to a constant bending moment, and therefore, positioning the joint region is more repeatable.^[8]

Composite materials

Plastics

Standards

ASTM C1161: Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
ASTM D6272: Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials by Four-Point Bending
ASTM C393: Standard Test Method for Core Shear Properties of Sandwich Constructions by Beam Flexure
ASTM D7249: Standard Test Method for Facing Properties of Sandwich Constructions by Long Beam Flexure
ASTM D7250: Standard Practice for Determining Sandwich Beam Flexural and Shear Stiffness

References

^ tec-science (2018-07-13). "Bending flexural test". tec-science. Retrieved 2019-11-09.
^ Pais & Harvey (Eds) (2012). Four Point Bending. Taylor & Francis Group. ISBN 978-0-415-64331-3.
^ ASTM C1161
^ ASTM D6272
^ ASTM C1161
^ Davis, Joseph R. (2004). Tensile testing (2nd ed.). ASM International. ISBN 978-0-87170-806-9.
^ ASTM C1161-13, section 4: http://www.astm.org/Standards/C1161.htm
^ Hasanabadi, M. Fakouri; Faghihi-Sani, M.A.; Kokabi, A.H.; Groß-Barsnick, S.M.; Malzbender, J. (September 2018). "Room- and high-temperature flexural strength of a stable solid oxide fuel/electrolysis cell sealing material". Ceramics International. 45: 733–739. doi:10.1016/j.ceramint.2018.09.236. ISSN 0272-8842. S2CID 139543143.

External links

ASTM C1161: Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
ASTM D6272: Standard Test Method for Flexural Properties of Unreinforced and Reinforced Plastics and Electrical Insulating Materials by Four-Point Bending
ASTM C393: Standard Test Method for Core Shear Properties of Sandwich Constructions by Beam Flexure
ASTM D7249: Standard Test Method for Facing Properties of Sandwich Constructions by Long Beam Flexure
ASTM D7250: Standard Practice for Determining Sandwich Beam Flexural and Shear Stiffness
ASTM C78: Standard Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading)

[1] tec-science (2018-07-13). "Bending flexural test". tec-science. Retrieved 2019-11-09.

[pais1-2] Pais & Harvey (Eds) (2012). Four Point Bending. Taylor & Francis Group. ISBN 978-0-415-64331-3.

[3] ASTM C1161

[4] ASTM D6272

[5] ASTM C1161

[davis1-6] Davis, Joseph R. (2004). Tensile testing (2nd ed.). ASM International. ISBN 978-0-87170-806-9.

[7] ASTM C1161-13, section 4: http://www.astm.org/Standards/C1161.htm

[8] Hasanabadi, M. Fakouri; Faghihi-Sani, M.A.; Kokabi, A.H.; Groß-Barsnick, S.M.; Malzbender, J. (September 2018). "Room- and high-temperature flexural strength of a stable solid oxide fuel/electrolysis cell sealing material". Ceramics International. 45: 733–739. doi:10.1016/j.ceramint.2018.09.236. ISSN 0272-8842. S2CID 139543143.

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[2]

[3]

[4]

[5]

[6]

[7]

[8]

@@ Line 1: / Line 1: @@
+{{Short description|Mechanical test for materials}}
-The '''four-point bending [[flexure|flexural]] test''' provides values for the [[Flexural modulus|modulus of elasticity
-in bending]] <math>E_f</math>, [[flexural stress]] <math>\sigma_f</math>, flexural strain <math>\varepsilon_f</math> and the flexural stress-strain response of the material. This test is very similar to the [[Three point flexural test|three-point bending flexural test]]. The major difference being that the addition of a fourth bearing brings a much larger portion of the beam to the maximum stress, as opposed to only the material right under the central bearing.
+The '''four-point flexural test''' provides values for the [[Flexural modulus|modulus of elasticity in bending]] <math>E_f</math>, [[flexural stress]] <math>\sigma_f</math>, flexural strain <math>\varepsilon_f</math> and the flexural stress-strain response of the material. This test is very similar to the [[three-point flexural test|three-point bending flexural test]]. The major difference being that with the addition of a fourth bearing the portion of the beam between the two loading points is put under maximum stress, as opposed to only the material right under the central bearing in the case of three-point bending.
-This difference is of prime importance when studying [[Brittleness|brittle]] materials, where the number and severity of flaws exposed to the maximum stress is directly related to the [[flexural strength]] and [[Fracture mechanics|crack initiation]].
+This difference is of prime importance when studying [[Brittleness|brittle]] materials, where the number and severity of flaws exposed to the maximum stress is directly related to the [[flexural strength]] and [[Fracture mechanics|crack initiation]]. Compared to the three-point bending flexural test, there are no shear forces in the four-point bending flexural test in the area between the two loading pins.<ref>{{Cite web|url=https://www.tec-science.com/material-science/material-testing/bending-flexural-test/|title=Bending flexural test|last=tec-science|date=2018-07-13|website=tec-science|language=en-US|access-date=2019-11-09}}</ref> The four-point bending test is therefore particularly suitable for brittle materials that cannot withstand [[shear stress]]es very well.
-It is one of the most widely used apparatus to characterize [[Fatigue (material)|fatigue]] and [[Moment distribution method#Flexural stiffness|flexural stiffness]] of asphalt mixtures.<ref name="pais1">{{cite book|last = Pais & Harvey (Eds)|title = Four Point Bending|publisher = Taylor & Francis Group|year = 2012|url = https://www.google.com/search?tbs=bks:1&q=isbn:9780415643313&gws_rd=ssl|isbn = 978-0-415-64331-3}}</ref>
+It is one of the most widely used apparatus to characterize [[Fatigue (material)|fatigue]] and [[Moment distribution method#Flexural stiffness|flexural stiffness]] of asphalt mixtures.<ref name="pais1">{{cite book|last = Pais & Harvey (Eds)|title = Four Point Bending|publisher = Taylor & Francis Group|year = 2012|url = https://www.google.com/search?q=isbn:9780415643313|isbn = 978-0-415-64331-3}}</ref>
  <!-- https://books.google.ch/books?id=WXvbfyY76GYC&lpg=PR7&ots=EYbm7pE__i&dq=CEN%20standard%20four-point&pg=PP1#v=onepage&q=CEN%20standard%20four-point&f=false -->
@@ Line 13: / Line 13: @@
 Calculation of the flexural stress <math>\sigma_f</math>
-:<math>\sigma_f = \frac{3}{4}\frac{F L}{b d^2}</math><ref>ASTM C1161</ref>  for four point bending test where the loading span is 1/2 of the support span (rectangular cross section)
+:[[File:4-point bend simplified.svg|thumb|4-point bend loading]]<math>\sigma_f = \frac{3}{4}\frac{F L}{b d^2}</math><ref>ASTM C1161</ref>  for four-point bending test where the loading span is 1/2 of the support span (rectangular cross section)
-:<math>\sigma_f = \frac{F L}{b d^2}</math><ref>ASTM D6272</ref>  for four point bending test where the loading span is 1/3 of the support span (rectangular cross section)
+:<math>\sigma_f = \frac{F L}{b d^2}</math><ref>ASTM D6272</ref>  for four-point bending test where the loading span is 1/3 of the support span (rectangular cross section)
-:<math>\sigma_f = \frac{3}{2}\frac{F L}{b d^2}</math><ref>ASTM C1161</ref>  for three point bending test (rectangular cross section)
+:<math>\sigma_f = \frac{3}{2}\frac{F L}{b d^2}</math><ref>ASTM C1161</ref>  for three-point bending test (rectangular cross section)
 in these formulas the following parameters are used:
@@ Line 26: / Line 26: @@
 * <math>d</math>   = Depth or thickness of tested beam, (mm)
-== Advantages and disdvantages ==
+== Advantages and disadvantages ==
 '''Advantages''' of three-point and four-point bending tests over uniaxial tensile tests include:
 * simpler sample geometries
@@ Line 34: / Line 34: @@
 '''Disadvantages''' include:
-* more complex stress distributions through the sample
+* more complex integral stress distributions through the sample
 == Application with different materials ==
 === Ceramics ===
-Ceramics are usually very brittle, and their flexural strength depends on both their inherent [[toughness]] and the size and severity of flaws. Exposing a large volume of material to the maximum stress will reduce the measured flexural strength because it increases the likelihood of having cracks reaching [[Fracture mechanics|critical length]] at a given applied load. Values for the flexural strength measured with four-point bending will be significantly lower than with three-point bending.<ref>ASTM C1161-13, section 4: http://www.astm.org/Standards/C1161.htm</ref> , Compared with [[Three-point flexural test|three-point bending test]], this method is more suitable for strength evaluation of butt joint specimens. The advantage of four-point bending test is that a larger portion of the specimen between two inner loading pins is subjected to a constant bending moment, and therefore, positioning the joint region is more repeatable.<ref>{{Cite journal|last=Hasanabadi|first=M. Fakouri|last2=Faghihi-Sani|first2=M.A.|last3=Kokabi|first3=A.H.|last4=Groß-Barsnick|first4=S.M.|last5=Malzbender|first5=J.|date=2018-09|title=Room- and high-temperature flexural strength of a stable solid oxide fuel/electrolysis cell sealing material|url=https://linkinghub.elsevier.com/retrieve/pii/S0272884218327019|journal=Ceramics International|doi=10.1016/j.ceramint.2018.09.236|issn=0272-8842}}</ref>
+Ceramics are usually very brittle, and their flexural strength depends on both their inherent [[toughness]] and the size and severity of flaws. Exposing a large volume of material to the maximum stress will reduce the measured flexural strength because it increases the likelihood of having cracks reaching [[Fracture mechanics|critical length]] at a given applied load. Values for the flexural strength measured with four-point bending will be significantly lower than with three-point bending.,<ref>ASTM C1161-13, section 4: http://www.astm.org/Standards/C1161.htm</ref> Compared with [[Three-point flexural test|three-point bending test]], this method is more suitable for strength evaluation of butt joint specimens. The advantage of four-point bending test is that a larger portion of the specimen between two inner loading pins is subjected to a constant bending moment, and therefore, positioning the joint region is more repeatable.<ref>{{Cite journal|last1=Hasanabadi|first1=M. Fakouri|last2=Faghihi-Sani|first2=M.A.|last3=Kokabi|first3=A.H.|last4=Groß-Barsnick|first4=S.M.|last5=Malzbender|first5=J.|date=September 2018|title=Room- and high-temperature flexural strength of a stable solid oxide fuel/electrolysis cell sealing material|journal=Ceramics International|volume=45|pages=733–739|doi=10.1016/j.ceramint.2018.09.236|s2cid=139543143 |issn=0272-8842}}</ref>
 === Composite materials ===
-* [[DIN]] 53293: Testing of sandwiches - Bending test
 <!-- From ASTM D7250. Section 5 -->
-<!-- To insure that simple sandwich beam theory is valid, a good
+<!-- To ensure that simple sandwich beam theory is valid, a good
 rule of thumb for a four-point bending test is the span length divided by
 the sandwich thickness should be greater than 20 (L1/d > 20) with the ratio
@@ Line 51: / Line 51: @@
 === Plastics ===
-==Standards==
+== Standards ==
 <!-- for advanced ceramics-->
 * [[ASTM]] C1161: Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature
@@ Line 61: / Line 61: @@
 * ASTM D7250: Standard Practice for Determining Sandwich Beam Flexural and Shear Stiffness
-==See also==
+== See also ==
 *[[Bending]]
 *[[Euler–Bernoulli beam equation]]
@@ Line 81: / Line 81: @@
 * [http://www.astm.org/Standards/D7249.htm ASTM D7249]: Standard Test Method for Facing Properties of Sandwich Constructions by Long Beam Flexure
 * [http://www.astm.org/Standards/D7250.htm ASTM D7250]: Standard Practice for Determining Sandwich Beam Flexural and Shear Stiffness
+* [http://www.astm.org/Standards/C78.htm ASTM C78]: Standard Test Method for Flexural Strength of Concrete (Using Simple Beam with Third-Point Loading)
 [[Category:Materials testing]]