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The three-point bending flexural test provides values for the modulus of elasticity in bending, flexural stress, flexural strain and the flexural stress–strain response of the material. This test is performed on a universal testing machine (tensile testing machine or tensile tester) with a three-point or four-point bend fixture.
Here, is the distance from the neutral axis to a point of interest; and is the bending moment. Note that this equation implies that pure bending (of positive sign) will cause zero stress at the neutral axis, positive (tensile) stress at the "top" of the beam, and negative (compressive) stress at the bottom of the beam; and also implies that the ...
For a 3-point test of a rectangular beam behaving as an isotropic linear material, where w and h are the width and height of the beam, I is the second moment of area of the beam's cross-section, L is the distance between the two outer supports, and d is the deflection due to the load F applied at the middle of the beam, the flexural modulus: [1]
The deflection at any point, , along the span of a center loaded simply supported beam can be calculated using: [1] = for The special case of elastic deflection at the midpoint C of a beam, loaded at its center, supported by two simple supports is then given by: [ 1 ] δ C = F L 3 48 E I {\displaystyle \delta _{C}={\frac {FL^{3}}{48EI}}} where
Fig. 3 - Beam under 3 point bending. For a rectangular sample under a load in a three-point bending setup (Fig. 3), starting with the classical form of maximum bending stress: = M is the moment in the beam; c is the maximum distance from the neutral axis to the outermost fiber in the bending plane
Using the free body diagram in the right side of figure 3, and making a summation of moments about point x: = + = where w is the lateral deflection. According to Euler–Bernoulli beam theory , the deflection of a beam is related with its bending moment by: M = − E I d 2 w d x 2 . {\displaystyle M=-EI{\frac {d^{2}w}{dx^{2}}}.}
Values for the flexural strength measured with four-point bending will be significantly lower than with three-point bending., [8] 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 ...
The starting point is the relation from Euler-Bernoulli beam theory = Where is the deflection and is the bending moment. This equation [7] is simpler than the fourth-order beam equation and can be integrated twice to find if the value of as a function of is known.