Beam theory

The concepts behind the analysis are not difficult. The piping system is simply a stmcture composed of numerous straight and curved sections of pipe. Although, for straight pipe, elementary beam theory is sufficient for the solution of the problem, it is not adequate for curved pipe. However, by the iatroduction of a flexibiUty factor, to account for iacreased flexibiUty of curved pipe over straight pipe, and a stress intensification factor, /, to account for... 

From basic beam theory, the moment-displacement and moment-force relationships for a guided cantilever are as follows ... 

At small strains the cell walls at first bend, like little beams of modulus E, built in at both ends. Figure 25.10 shows how a hexagonal array of cells is distorted by this bending. The deflection can be calculated from simple beam theory. From this we obtain the stiffness of a unit cell, and thus the modulus E of the foam, in terms of the length I and thickness t of the cell walls. But these are directly related to the relative density p/ps= t/lY for open-cell foams, the commonest kind. Using this gives the foam modulus as... 

The interpretive difficulty has been discussed in detail by Pawel and by Mortonin papers which appeared almost simultaneously. Both authors use arguments to show that the simple formula of bending-beam theory utilised by Stoney... 

The stresses in some folded structures can be determined with acceptable accuracy by applying elementary beam theory to the overall cross-sections of the plate assemblies. When assemblies are plates whose lengths are large relative to their cross-sectional dimensions (thin-wall beam sections, ribbed panels, and so on) and are in large plates... 

The flexural strength of the annealed polymers proved to be consistently about 30% higher than the strength of the quenched polymers as shown in Fig. 6.1. Tests were evaluated in accordance with ISO 178 [54]. As the samples yielded, they deformed plastically. Therefore, the assumptions of the simple beam theory were no longer justified and consequently the yield strength was overestimated. 

This effect might be interpreted by the Bethe dynamic potential approximation, which does not take into account the crystal orientation (as in the Blackman correction case) nor crystal thickness parameters. In terms of this approach, the effect of weak beams can be included in two-beam theory by replacing the potential coefficients, Vh, by ... 

In the case of thicknesses larger than mentioned above the intensities must be calculated according to the more general many-beam theory. The calculation should include summation over different groups of crystals having a certain distributions of thickness and orientation. A method based on the matrix formulation of the many-beam theory was developed for partly-oriented thin films and have been successfully applied samples [2]. The main problem in using direct many-beam calculation is to find the distribution functions for size and orientation of the microcrystals. However, it is not always possible to refine these functions in the process of intensity adjustment. Additional investigation of the micro-structure by electron microscopy is very helpful in such case. 

This test has an inherent problem associated with the stress concentration and the non-linear plastic deformation induced by the loading nose of small diameter. This is schematically illustrated in Fig 3.17, where the effects of stress concentration in a thin specimen are compared with those in a thick specimen. Both specimens have the same span-to-depth ratio (SDR). The stress state is much more complex than the pure shear stress state predicted by the simple beam theory (Berg et al., 1972 ... 

Sandorf, 1980 Whitney, 1985 Whitney and Browning, 1985). According to the classical beam theory, the shear stress distribution along the thickness of the specimen is a parabolic function that is symmetrical about the neutral axis where it is at its maximum and decreases toward zero at the compressive and tensile faces. In reality, however, the stress field is dominated by the stress concentration near the loading nose, which completely destroys the parabolic shear distribution used to calculate the apparent ILSS, as illustrated in Fig 3.18. The stress concentration is even more pronounced with a smaller radius of the loading nose (Cui and Wisnom, 1992) and for non-linear materials displaying substantial plastic deformation, such as Kevlar fiber-epoxy matrix composites (Davidovitz et al., 1984 Fisher et al., 1986), which require an elasto-plastic analysis (Fisher and Marom, 1984) to interpret the experimental results properly. 

As in the simple 3- or 4-point bending of a beam the vibrating reed device assumes the validity of the differential Eq. (2.2) which is due to Euler. Timoshenko25 included both rotary inertia and shear deformation deriving a more exact differential equation which reduces to the Euler equation as a special case. Use of the Timoshenko beam theory for anisotropic materials has been made by Ritchie et al.26 who derive a pair of equations for torsion-flexure coupling (which will always occur unless the axis of the beam coincides with the symmetry axis of the anisotropic material). 

L. Librescu and O. Song Thin-Walled Composite Beams. Theory and Application. 2005... 

As mentioned above, when the transverse dimensions of the beam are of the same order of magnitude as the length, the simple beam theory must be corrected to introduce the effects of the shear stresses, deformations, and rotary inertia. The theory becomes inadequate for the high frequency modes and for highly anisotropic materials, where large errors can be produced by neglecting shear deformations. This problem was addressed by Timoshenko et al. (7) for the elastic case starting from the balance equations of the respective moments and transverse forces on a beam element. Here the main lines of Timoshenko et al. s approach are followed to solve the viscoelastic counterpart problem. 

A similar equation can be written for Uyjj. These are the viscoelastic equations corresponding to the elastic ones in the elastic beam theory (7). 

Beam theory modelling of " L type adherend deflection... 

Transverse pull-off tests induce mainly mode 1 loading, provided the base panel is sufficiently rigid. Finite element analyses have been performed to look at this geometry in more detail, and will be reported elsewhere, but here a simple analytical beam theory expression is used to predict the pull-off failure load [21] ... 

Thus, corrected beam theory can be used to deduce the value of Ef from the mode II fracture test, and this allows an important cross-eheck on the results to be performed, as the value can be compared to the known, or independently measured value, , as used in eqn (1). Also the value of Ef so deduced should be independent of crack length, a. 

If one assumes that the measured crack length is likely to be only approximate in the presence of such extensive microcracking, the use of beam theory expressions requiring this parameter will also be subject to this uncertainty. However, corrected beam theory can be used to determine an effective crack length, a, via the measured compliance value and eqn (3) may be reananged to give an effective crack length thus fa)... 

Note G c values deduced via corrected beam theory 2,3, Cjic values deduced via corrected beam theory using eftective crack lengths, 0. ... 

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