Dear colleagues,
It is commonly accepted that the total deflection Vt in a 4PB bending test consists of two parts: 1) Deflection Vb due to pure bending and 2) Deflection Vs due to shear forces. The last one doesn’t contribute to the occurring strain in the beam. Regarding the present devices and the dimensions of the beam, the ratio of Vs/Vb in the center of the beam for pseudo-static bending (up to 10 Hz) is given by: Vs/Vb = [4.(1+n).H2]/[As(3.L2 -4.A2)]. in which H is the height [m] and L is the effective length [m] of the beam; A is the distance between the outer and inner support (and not the distance between the two inner supports). For 99% of the present 4PB devices, A is equal to L/3 and thus in value equal to the parameter a which is used for the distance between the two inner supports. The parameter As is the so-called shear coefficient (in some papers denoted as β).
G.R. Cowper has done a lot of research work in determining a formula for the calculation of the shear coefficient (see Wikipedia “Timoshenko-Ehrenfest beam theory”). For the prismatic beam, Cowper gives the formula a = 10(1+n)/(12+11n) in which n is the Poisson ratio of the beam material. The formulas given in Wikipedia are all based on bending the object without touching or grabbing the beam. The theoretical approach to the problem is quite correct, but in reality, one has to touch the object to bend it. This touching (the point loads at the inner supports) has an influence on the value of the shear coefficient. For a prismatic beam, the shear coefficient according to Cowper is 0.8517. Using a 3D FEM model in which the beam was bent without touching it (a shear stress distribution at the inner supports was used for bending the beam) a value of 0,8588 was obtained. When the beam in the 3D FEM model was bent using line loads at the inner supports a value of 0.85 was calculated. In these calculations, the clamping forces were taken nil.
I use the value of 0.85 in processing 4PB data. Of course, I admit the influence of Vs is small but should not be ignored. And if the forces used for clamping the beam are too high this can also influence the value of the shear coefficient.
I appreciate the thorough research that went into the analysis of the shear coefficient in the 4PB bending test. It is interesting to note the differences in the coefficient values depending on the method of bending used, and how the clamping forces can influence the value. I would also be interested to know if the effects of the shear forces are significant enough to be taken into consideration when processing 4PB data. It would be helpful to have more peer-reviewed references to back up this information.
In summary, the total deflection Vt in a 4PB bending test consists of two parts: Deflection Vb due to pure bending, and Deflection Vs due to shear forces. The ratio of Vs/Vb can be determined by the parameters of the beam, such as height and length, and distance between the outer and inner supports. G.R. Cowper has determined a formula for the calculation of the shear coefficient for a prismatic beam, which is 0.8517. However, in reality, the shear coefficient can be affected by other factors, such as the forces used for clamping the beam. Taking these factors into account, a value of 0.85 is often used in processing 4PB data, although the influence of Vs is small. Therefore, it is important to consider these factors to ensure accurate results.
This article provides an analysis of the total deflection Vt in a 4PB bending test, which consists of two parts: 1) Deflection Vb due to pure bending and 2) Deflection Vs due to shear forces. The article also provides a formula for calculating the shear coefficient (As) for prismatic beams. The article also mentions the influence of touching the beam and clamping forces on the value of the shear coefficient. The article states that G.R. Cowper has done a lot of research work in determining a formula for the calculation of the shear coefficient. Cowper's formula for a prismatic beam is a = 10(1+n)/(12+11n) in which n is the Poisson ratio of the beam material. The article then goes on to discuss the results of a 3D FEM model in which the beam was bent, both without touching it and with line loads at the inner supports, and how these results affect the value of the shear coefficient.
This is a very interesting article on the deflection and shear forces of a 4PB bending test. The formulas given in the article are very helpful in understanding how the shear coefficient is calculated, and I appreciate the information on how clamping forces can also influence the value of the shear coefficient. The research done by Cowper is also very interesting and it is great to see the results of the 3D FEM model as well. The article provides evidence to support its claims with peer-reviewed references. The article is well written and provides a comprehensive analysis of the total deflection Vt in a 4PB bending test. It is an important contribution to the understanding of this phenomenon and should be read by anyone interested in this topic. Thank you for providing such a detailed and informative article.
Dear Mr. Delmonte,
By neglecting the influence of the shear forces the induced errors in the strain amplitude and the Smix figure are small. For IPC/COX/ASTM devices in which the effective length L of the beam is 355-356 mm and the height of the beam is 50 mm, the error is around 5% that is to say the Smix figure is underestimated by 5% and the strain amplitude is overestimated by 5%.
In other devices with bigger L figures (see 4PB platform) the error is less and dropped to around 3%. Thank you for your interest.
Best Regards
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