Beams maximum stress

In the deflection temperature under load test (heat distortion temperature test) the temperature is noted at which a bar of material subjected to a three-point bending stress is deformed a specified amount. The load (F) applied to the sample will vary with the thickness (t) and width (tv) of the samples and is determined by the maximum stress specified at the mid-point of the beam (P) which may be either 0.45 MPa (661bf/in ) or 1.82 MPa (264Ibf/in ). 

Fig. 3-2 Maximum stress and deflection equations for selected beams. ...

In simple beam-bending theory a number of assumptions must be made, namely that (1) the beam is initially straight, unstressed, and symmetrical (2) its proportional limit is not exceeded (3) Young s modulus for the material is the same in both tension and compression and (4) all deflections are small so that planar cross-sections remain planar before and after bending. The maximum stress... 

Observe that this is a geometric property, not to be confused with the modulus of the material, which is a material property. I, c, Z, and the cross-sectional areas of some common cross-sections are given in Fig. 3-1, and the mechanical engineering handbooks provide many more. The maximum stress and defection equations for some common beamloading and support geometries are given in Fig. 3-2. Note that for the T- and U-shaped sections in Fig. 3-1 the distance from the neutral surface is not the same for the top and bottom of the beam. It may occasionally be desirable to determine the maximum stress on the other nonneutral surface, particularly if it is in tension. For this reason, Z is provided for these two sections. 

The most common test is a minting beam type in which a carefully machined and polished sample is loaded as a beam while rotating in antifriction bearings. As any poitti in the periphery rotates from top to bortom to top position the stress changes Irani maximum compression to maximum tension and back to maximum compression From 4 to 8 or more individual tests at various maximum stress levels may he required to detemiine the endurance limit. 

First, the elastic stress distributions of the un-notched specimens are obtained from a finite element analysis. For the PI un-notched specimen, the discrepancy between the finite element and the analytical result is very small (about 0.01%), thus validating the finite element calculation in terms of accuracy through the meshing and the type of element used. Therefore a similar calculation is conducted on the G1 un-notched specimen where the span to height ratio is smaller. The mismatch on the maximum stresses at the bottom and at the top of the beam between the finite element calculations and the analytical solution is 0.74% in tension and 0.79% in compression (and remains constant upon further mesh refinement). This estimation of the stress distribution is then used for the following evaluation of the stress intensity factor. 

A maximum stress criterion (using the strength of a unidirectional beam as a reference) applies for the structure considered when taking into account the residual stress relaxation. 

The maximum stress on the baffles depended upon stirrer type. The following fmax/faverage values Were obtained 1.3 for cross-beam stirrer 1.6 for MIG 07 1.8 for turbine stirrer, and 2.5 for propeller stirrer. These values were independent of the tip speed of the stirrer. 

For a nniformly distributed load on a straight beam (elastically stressed) with its left and right ends simply supported [1], the maximum fiber stress, or flexural strength, can be expressed by the following formnla ... 

Make an elastic stress analysis of the product, which is an approximation. We calculate the maximum stress in the beam using Eq. (C.8) of Appendix C... 

The combination El of a material property and beam cross section geometry is referred to as the beam bending stiffness. It determines the curvature for a given applied moment. The maximum stress (t in a symmetric beam can therefore also be expressed by substituting in Eq. (C.3)... 

Bend-beam specimens provide a quantitative estimate of the maximum stress in the outer fibers of the bent beam when stressed below the yield stress. Experimental standards for bent beam specimen are given in ASTM G 39-99 [16]. In this test, both the specimen... 

It has been found by tests as well as by mathematical analysis that the torsional resistance of a section, made up of a number of rectangular parts, is approximately equal to the sum of the resistances of the separate parts. It is on this basis that nearly all the formulas for noncircular sections have been developed. For example, the torsional resistance of an Tbeam is approximately equal to the sum of the torsional resistances of the web and the outstanding flanges. In an I-beam in torsion the maximum shearing stress will occur at the middle of the side of the web, except where the flanges are thicker than the web, and then the maximum stress will be at the midpoint of the width of the flange. Reentrant angles, as those in I-beams and charmels, are always a source of weakness in members subjected to torsion. 

The maximum stress occurs at the surface of the beam farthest from the neutral surface, as given by the following equation ... 

The maximum stress in the beam can becalculated—/or r/iw case only—... 

A beam is being loaded in pure bending with a nominal applied maximum stress a and it contains a mode I edge crack. Using a Green s... 

Thus, the center deflection of the beam is a product of the maximum stress at the outer elements of the beam, and at the center a geometrical term (L / 6h)) divided by Young s modulus E(i), which is now time-dependent because of the viscoelastic relaxations in the beam, and decreases with time under stress as additional inelastic strains build up. However, the stresses in the viscoelastic beam continue to remain unaltered since they depend only on the applied forces and moments, which remain constant. 

A test specimen is held as a simply supported beam and is subjected to three-point bending as shown in Figure 1.26. Typically an Instron is used. Maximum stress and strain occurs at the underside of the test specimen, directly under the applied force. The preferred test specimen is 80 mm long, 10 mm wide, and 4 mm thick. Other specimens may be used if the length to thickness ratio is equal to 20. 

Section modulus Z) n. In a beam under load, the quotient of the moment of inertia of the beam s cross-section about its neutral axis divided by the distance from the neutral axis to the outermost surface of the beam He). The bending moment divided by the section modulus gives the maximum stress in the beam at any point along it. 

Fig. 6.80 Cavitation strain and total strain as a function of normalized beam height for creep a M = 0.165 N-m (80 MPa initial maximum stress), 1170 °C, 300 h b Af = 0.247 N-m (120 MPa initial maximum stress), 1170 °C, 142 h. The - - and — signs in this figure represent tensile and compressive stresses, respectively [5]. With kind permission of John Wiley and Sons...

The relationship in Eq. 1 is based on the elastic response of the beam material which would produce the stress relationship shown in Fig. 5-3A. The couple resisting the bending moment is made up of the elastic response of elements of the material alternately in tension and compression above and below the neutral plane. The material is assumed to have the same modulus in tension and in compression so that the neutral plane is in the center of the beam. The stress level is proportional to the distance from the neutral plane and is a maximum at the surface of the beam. 

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