Flexure formula

TABLE 25.2 Summary of flexural formulas (after Schmitz and Brown (1965-1969), Vol. 2. p. 329)... 

Fiber-reinforced polymer systems, 38 Fickian diffusion, 665 Fick s law, 663,684 Field flow fractionation, 20 Filled polymers, 38 First normal stress coefficient, 545 difference, 640 First-order transition, 27,152 Flame-retardant additives, 861 Flammability, 847 Flashing, 804 Flash line region, 807 Flexibility of a chain molecule, 246 Flexible polymer molecules, 706 Flexural deformation under constant load, 825 Flexural formulas, 826 Flexural rigidity, 877 Floor temperature, 751 Flory-Huggins... 

Modulus of Rupture. The MOR is the ultimate bending strength of a material. Thus, MOR describes the load required to cause a wood beam to fail and can be thought of as the ultimate resistance or strength that can be expected (Figure 2, Point B Figure 6, Point B) from a wood beam exposed to bending-type stress. MOR is derived by the flexure formula... 

For a rectangular or square beam in bending under centerpoint loading, the flexure formula is varied to reflect loading conditions and beam geometry... 

Fiber Stress at Proportional Limit. The FSPL is the maximum bending stress a material can sustain under static conditions and still exhibit no permanent set or distortion. It is by definition the amount of unit stress on the y-coordinate at the proportional limit of the material (Figure 2, Point A Figure 6, Point A). FSPL is derived using the flexure formula... 

The flexural modulus is the ratio, within the elastic limit, of stress to corresponding strain. It is calculated by drawing a tangent to the steepest initial straight-line portion of the load-deflection curve and using an appropriate formula. 

For a beam of width b, the formula for flexural rigidity (D) can be expressed as ... 

It is obvious that flexure tests should be directly proportional to specimen width. Numerous tests on both types of machines have confirmed this. The modulus of rupture (MOR) formula predicts that flexural strength should be inversely proportional to the span. Combining these relationships we can calculate that the test on the 12 by 16 specimen should be 12/10 X 10/14 = 0.857 times the strength of the 10 by 12 specimen. This is reasonably close to the slope of 0.8 found for the correlation equation. Gypsum wallboard is not a homogeneous material as is assumed in the calculation of MOR. 

What about WPC deck boards Chapter 7, in this book covers this issue in detail. As a brief example, let us consider two WPC boards— Trex and GeoDeck. Trex has reported that flexural strength of their boards (solid boards of 5.5" width and 1.25" thickness) is 1423 psi. It means that a Trex board placed on two joists at 16" span would have a break load derived from the formula... 

It was shown above that the flexural stress at center-point load is described by the formula (7.20)... 

Flexural modulus for center-point load is given by the following formula ... 

Flexural modulus for uniformly distributed load on a straight beam (elastically stressed) with its left and right ends simply supported is given by the formula... 

Deflection of the loaded board can be predicted provided that the load and its location on the board is known, as well as the span, the moment of inertia, and the flexural modulus of the board, and assnming that the load and the deflection are within the linear relationship between each other. If the load is outside of this relationship (higher), a deflection wonld be higher than calculated using formulas... 

Formula ether ester cycloaliphatic epoxy groups curing agent Tensile Flexural Compression ... 

Load and support conditions for individual components depend on the complete structure (or system) analysis, and are unknown to be determined in that analysis. For example, if a plastic panel is mounted into a much more rigid structure, then its support conditions can be specified with acceptable accuracy. However, if the surrounding structure has comparable flexibility to the panel, then the interface conditions will depend on the flexural analysis of the complete structure. In a more localized context, structural stiffness may be achieved by ribbing and relevant analyses may be carried out using available design formulae (usually for elastic behavior) or finite element analysis, but necessary anisotropy or viscoelasticity complicate the analysis, often beyond the ability of the design analyst. 

The deflection of a beam as computed by the ordinary formulas is that due to flexural stresses only. The deflection in honeycomb (Chapter 7 Sandwiches) and short beams due to vertical shear can be high, and should always be checked. Because of the nonuniform distribution of the shear over the cross section of the beam, computing the deflection due to shear by exact methods is difficult. It may be approximated by ... 

The maximum deflection Y occurs at x=L, i.e., Y =FVI2> El nd this equation could be used to determine E from the beam deflection. If one inspects Fig. 4.5, the three-point geometry can be viewed as two attached cantilever beams. Replacing Fhy FI2 and L by Ul in the cantilever beam deflection formula, the maximum deflection in three-point bending can be determined, i.e., Tp=FZ,V48 /. The resistance of a beam to bending depends on El (Eq. (4.9)), which is termed the flexural rigidity. 

Definition Thermoplastic beads crosslinked during processing or by post-treatment offers better impact, stress cracking, and flexural props, than uncrosslinked grades Formula [CH2CH2]x... 

ASTM Formulas. For biaxial flexure tests, the ASTM formulas are available only for monolayered discs, and the guidelines for testing of discs have been provided by Salem and Powers [43]. In this case, the biaxial stress, a, in the disc is proportional to both the distance from the neutral surface and the moment but inversely proportional to the flexure rigidity, such that [44]... 

Hsueh et a/. s Formulas. Multilayered discs are considered in Hsueh etal. formulas [37-39]. In this case, the continuity conditions at the interfaces between layers are required in analyses, and the solutions are very difficult if not impossible to derive. Instead of solving directly the problem of multilayered discs subjected to biaxial flexure tests, the essence of Hsueh et a/. s analyses is to find the correlation between monolayered and multilayered discs subjected to biaxial moment. Then, utilizing this correlation, the existing solutions for monolayered discs can be converted to the solutions for multilayered discs. The analytical procedures are summarized as follows. 

The first method uses the standard formula for sold plastics, e.g. beam theory, but with a reduced flexural modulus E/e ... 

The compressive strength of the product should be determined experimentally because any prediction is not reliable enough as it concerns its increase with ageing. The flexural strength (modulus of rupture) may be approximately calculated from the formula proposed in FHA (1979) = 0.51 / / (in psi) or = 0.27 / / (in MPa). 

Results obtained by Ramakrishnan and Lokvik (1992) concerned concrete specimens subjected to flexure and reinforced with four types of fibres straight, corrugated and hooked steel-fibres, and polypropylene fibres up to 1 % volume. The relations between number of cycles N and maximum fatigue stress divided by the modulus of rupture are shown in Figure 11.11 as estimated regression lines for different kinds of fibres. The proposed formula for prediction of the fatigue behaviour is the following ... 

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