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in these formulas the following parameters are used: = Stress in outer fibers at midpoint, = load at a given point on the load deflection curve, = Support span, (mm) = Width of test beam, (mm) = Depth or thickness of tested beam, (mm)
L is the length of the support (outer) span; b is width; d is thickness; For the 4 pt bend setup, if the loading span is 1/2 of the support span (i.e. L i = 1/2 L in Fig. 4): = If the loading span is neither 1/3 nor 1/2 the support span for the 4 pt bend setup (Fig. 4): Fig. 4 - Beam under 4 point bending
1940s flexural test machinery working on a sample of concrete Test fixture on universal testing machine for three-point flex test. 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.
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
Figure 1 - Equilibration of a deflected sandwich beam under temperature load and burden in comparison with the undeflected cross section. The stress resultants and the corresponding deformations of the beam and of the cross section can be seen in Figure 1. The following relationships can be derived using the theory of linear elasticity: [3] [4]
Euler–Bernoulli beam theory (also known as engineer's beam theory or classical beam theory) [1] is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams. It covers the case corresponding to small deflections of a beam that is subjected to lateral ...
Simply supported beam with a single eccentric concentrated load. An illustration of the Macaulay method considers a simply supported beam with a single eccentric concentrated load as shown in the adjacent figure. The first step is to find . The reactions at the supports A and C are determined from the balance of forces and moments as
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]