Beams shear stresses

A beam is really two cantilever beams joined baek to baek (Figure 10.5a) and then turned upside down (Figure 10.5b). Thus, the largest eompressive stress oeeurs on the concave surfaee of the beam at the mid-span, while the largest tensile stress occurs on the convex surface at the mid-point. Although it is obvious that tensile and compressive forees are generated within a beam when it is bent, it may not be self-evident that shear forees also exist. In a loaded beam, shear stresses aet in both the horizontal and vertieal directions. 

We can reexamine the beam problem to determine the distribution of the transverse shearing stress -c z- know that the resultant of -c z is V which we obtain from Equation (D.7), i.e.,... 

Accordingly, we find it difficult to determine the distribution of the transverse shearing stress in a beam, much less in a plate. Procedures for determining the approximate transverse shear stress distribution in plates are described in Section 6.5.2. 

The beam is also subject to a shear stress that varies over the beam cross-section. 

Shear Stress. For a rectangular cross-section, the maximum value of Q occurs at the neutral axis, and, because the width b of the beam is a constant 3 in., the maximum value of the shear stress occurs at the neutral axis. 

Apart from the short beam shear test, which measures the interlaminar shear properties, many different specimen geometry and loading configurations are available in the literature for the translaminar or in-plane strength measurements. These include the losipescu shear test, the 45°]5 tensile test, the [10°] off-axis tensile test, the rail-shear tests, the cross-beam sandwich test and the thin-walled tube torsion test. Since the state of shear stress in the test areas of the specimens is seldom pure or uniform in most of these techniques, the results obtained are likely to be inconsistent. In addition to the above shear tests, the transverse tension test is another simple popular method to assess the bond quality of bulk composites. Some of these methods are more widely used than others due to their simplicity in specimen preparation and data reduction methodology. 

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 ... 

Fig. 3.17. Effect of stress concentrations on short beam shear specimens (a) thin specimen (b) thick...

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. 

Fig. 3.18. (a) Shear stress eontours and (b) shear stress distributions aeross the thickness of a three-point bending specimen in a short beam shear test. After Cui and Wisnom (1992). Reproduced by permission of... 

Equations (17.20) are Laplace transforms of the equations of viscoelastic beams and can be considered a direct consequence of the elastic-viscoelastic correspondence principle. The second, third, and fourth derivatives of the deflection, respectively, determine the forces moment, the shear stresses, and the external forces per unit length. The sign on the right-hand side of Eqs. (17.20) depends on the sense in which the direction of the strain is taken. 

It is well known that the elementary theory of beams described above becomes inadequate for beams with transverse dimensions of the same order of magnitude as their length. This section deals with the theory to be applied to thick non-slender beams. This theory appears to be relevant in the context of dynamic mechanical analysis. The first fact to be considered is that when the beam is flexed it experiences a shear stress that provokes a relative sliding of the adjacent transverse sections. As a consequence, the larger the transverse section, the higher is this shear strain. The final effect is an increase in the total deflection of the beam (Fig. 17.5). 

Owing to the nonuniformity of the shear stress in the section, an effective value A for the area of this section, instead of the geometric value A = bd should be used. This is a consequence of considering only the deformation of the neutral fiber of the beam. In this way, A /A would be the relationship between the mean shear effects and the shear effects in the neutral fiber. Its value is always less than 1. The controversial question is now to decide the real meaning of the word mean in the present context. If the mean value corresponding to the neutral fiber is taken, then according to Eqs. (17.61) and (17.62), A = 2A/3>. However, this is a naive assumption. A much more convenient approach to... 

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. 

The horizontal shear stress is a maximum at the mid-depth of the beam (known as the neutral plane beeause here the bending stress is zero) and the shear stress falls to zero at the upper and lower surfaces. The shear stress in the neutral plane is ... 

The largest stresses are observed as shear stresses at the corners of the die at the lowest temperature. Three commercially available epoxy-based molding compounds were studied. Two of these materials are standard packaging formulations for smaller devices. Both strain gauge and beam bending experiments showed comparable stress levels with these two materials. The third material is a rubber modified, low stress material. As expected, stress levels in devices packaged with this material, as well as stresses observed in the beam bending apparatus, were considerably lower than those for the other two materials. 

Standard methods use rectangular beam test pieces. The geometry of the beam is chosen to make shear stresses and flexure across the width unimportant. For three-point loading a span-to-depth ratio of 16 is generally satisfactory but does vary with the material characteristics. The quite different situation of deliberately introducing shear forces to measure interlaminar strength was discussed in Section 5. 

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