Bending of beams

Consider the stresses that arise in a regular beam, of thickness h, subjected to bending such that the beam has a curvature R, as shown in Fig. 4.8. The central 

Each layer also exerts a force d/ and, hence, a moment yd/about an axis in the neutral surface. The moments from successive layers cooperate in rotating the cross-sections about this axis. The total bending moment Mis given by 

For a rectangular beam, b(y) is a constant (=6) and the integral in Eq. (4.5) can be solved to give I=bhVl2. It is useful to note that Young s modulus can be determined from Eq. (4.5) provided a technique is available to measure the radius of curvature (e.g., by optical interference or with a profilometer). 

The stress state in a bent beam is usually more complex than that encountered in pure bending, with non-uniform bending moments and shearing forces. In these cases, the normal stresses produced by the bending moment are still given by Eq. (4.6) but, in addition, shearing stresses t may be present such that 

Consider the cantilever beam shown in Fig. 4.9 in which it can be shown that M=F L-x) and Vis constant along the beam length. Using Eqs. (4.7) and (4.8) for a beam of rectangular cross-section, one finds the shear stresses are parabolic, being zero at the free surfaces of the beam and rising to a maximum of ZFUbh at the neutral surface. 

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An analogous line of reasoning shows that at a strain of 1.25% the stress intensity in the glass is 125,000 psi (862 MPa) and in plastic B and C at 12,600 and 4,500 psi (87 and 31 MPa), respectively. The corresponding loads on rods made with plastics B and C are 34,400 lb (15,600 kg) and 32,375 lb (14,700 MPa), respectively. Additional detailed information is available concerning this analysis as well as developing data for plain RP plates, composite plates, bending of beams and plates, etc. (10). 

From elastic analysis of bending of beam, the apparent interlaminar shear strength Xjlss is given by ... 

These are well-known formulae from the technical theory of bending of beams and bars (Housner and Vreeland 1966). They may be used for the ealeulation of maximum local strains in Uners, for instance in the case of settlement in compacted elay liners (Scherbeek and Jessberger 1992 Seherbeek 1993). In applying these equations it is assiuned that the neutral... 

The above solution is based on the shear lag model and ignores bending in the adherends. Another situation of considerable importance is the bending of beams and plates made with two or more materials. For example, consider the case of... 

Beams are often subject to different boundary conditions and forces in MEMS design. A comprehensive reference for the bending of beams in different situations is Roark s formulas for stress and strain [4]. We considered a fixed-free cantilever beam that is subjected to a point load at its free end in the spring analysis above. The fixed-free boundary condition means that one end is fixed, so that both its displacement and slope do not change under the applied force. The other end of the beam is free to both move and change its slope in response to a point load F, as shown in Figure 2.7. In that case the deflection of the beam y(j ) as a function of position x is given by... 

The basis functions so obtained can be used for the solutions of many two dimensional problems in multiply connected circular and semicircular regions. Among these problems are those for clamped perforated circular plates, two dimensional thermal stresses in a multi hole circular cylinder whose outer boundary is free from traction. Other problems involve bending and tosion of prismatic bars, and two dimensional heat conduction. A few examples will be given for clamped plates, bending of beams and heat conduction. 

If the constitutive law and the evolution of damage D (e) Is given, the process Is deterministic and a unique response is possible for a set of data. But when one proceeds to make experiments on structures ( compression or tension tests, bending of beams), then always appears a certain scatter In the experimental results. Care taken in the experimental procedure may reduce it, but this scatter always exists and Its magnitude depends on many parameters material characteristics (porosity, density and size of the heterogeneities...), geometry and size of the structure, experimental equipment, etc... 

Why, then, are bicycles not made of wood (There was a time when they were.) That is because metals, and polymers, too, can readily be made in tubes with wood it is more difficult. The formula for the bending of a tube depends on the mass of the tube in a different way than does that of a solid beam, and the optimisation we have just performed - which is easy enough to redo - favours the tube. 

M, = bending moment to cause rupture b = width of beam d = depth of beam. Compressive strength... 

From an FMEA of the system design, a Severity Rating S) = 1 was allocated, relating to a safety critical failure in service. It is required to find the optimum unequal angle section size from the standard sizes available. It is assumed that the load is carried at the section s centre of gravity, G, and only stresses due to bending of the section are considered, that is, the torsional effects are minimal. The combined weight of the beam and tie rod are not to be taken into account. 

Myklestead, N. O., A New Method of Calculating Natural Modes of Uncoupled Bending Vibration of Airplane Wings and Other Types of Beams, Journal of the Aeronautical Sciences, Vol. 11, No. 9. April 1944, pp. 153-162. 

However, the foregoing derivation is valid only for isotropic beams of rectangular cross section. For beams of nonrectangular cross section, the parabolic stress distribution is not correct. Also, for laminated beams, the parabolic distribution is most assuredly incorrect because of layer inhomogeneity. In fact, for laminated beams, we must expect different shapes of stress distribution in each layer as seen in Figure 6-19 for wide beams (there interpreted as cylindrical bending of a long strip, i.e., a special plate). 

Insertion devices are placed in the electron path of a synchrotron. They increase the photon flux by several orders of magnitude. Similar to the FEL principle they operate by forcing the electrons on a wavy path. At each bend of the path synchrotron light is emitted. In contrast to the FEL device there is no coherence. Instead, the light intensity sums up to form the effective beam. Two kind of insertion devices are used. In wigglers the curvature of the electron path is high. In undulators it is relatively low. 

Refractive bending of light beams by sea salts- refractive index blank [3]... 

Figure 20. Elastic-Plastic Solution for Bending of Blast Loaded Beams. (Reprinted with permission from ref. 15. Copyright 1983 Elsevier Science.)...

An instrument known as a refractometer has been used for many years to measure the refractive index of liquids and liquid solutions for the purpose of both quantitative and qualitative analysis (see Chapter 15). A refractometer measures the degree of refraction (or bending) of a light beam passing through a thin film of the liquid. This refraction occurs when the speed of light in the sample is different from a reference liquid or air. The refractometer measures the position of the light beam relative to the reference and is calibrated directly in refractive index values. It is rare for any two liquids to have the same refractive index, and thus this instrument has been used successfully for qualitative analyses. 

Figure 2.10 Data from the bending of a simply supported cylindrical beam consisting of 20% polyvinyl alcohol in water. In this case L = 110 mm d = 7 mm...

Berg, C.A., Tirosh, J. and Israeli, M. (1972). Analysis of short beam bending of fiber reinforced composites. In Composite Materials Testing and Design (2nd Conf), ASTM STP 497 (C.E. Browning ed.), ASTM, Philadelphia, PA, pp. 206-218. 

Cui, W.C. and Wisnom, M.R. (1992). Contact finite element analysis of three- and four-point short beam bending of unidirectional composites. Composites Sci. Technol. 45, 323-334. 

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