Stress theories maximum

Subsection A This subsection contains the general requirements applicable to all materials and methods of construction. Design temperature and pressure are defined here, and the loadings to be considered in design are specified. For stress failure and yielding, this section of the code uses the maximum-stress theory of failure as its criterion. 

Since its inception, the design requirements of the code have been based on the maximum-stress theory of failure. Over the past 50 years, it has been established that yielding under pressure correlates better with the maximum-shear-stress theory. Therefore, both Division 2 and Section III, Nuclear Vessels, are based on this latter theory, resulting in a more precise evaluation of the stresses in the various p s of a vessel. 

They are the maximum stress theory and the maximum shear stress theory. ... 

This theory is the oldest, most widely used and simplest to apply. Both ASME Code, Section VIII, Division 1, and Section I use the maximum stress theory as a basis for design. This theory simply asserts that the breakdown of... 

Figure 1-1. Graph of maximum stress theory. Quadrant i biaxiai tension Quadrant II tension Quadrant III biaxial compression Quadrant IV compression.

For simple analysis upon which the thickness formulas for ASME Code, Section I or Section VIII, Division I, are based, it makes little difference whether the maximum stress theory or maximum shear stress theory is used. For example, according to the maximum stress theory, the controlling stress governing the thickness of a cylinder is circumferential stress, since it is the largest of the three principal stresses. According to the maximum shear stress theory, the controlling stress would be one-half the algebraic difference between the maximum and minimum stress ... 

Division 2 stress analysis considers all stresses in a triaxial state combined in acc-ordance with the maximum shear stress theory. Division 1 and the procedures outlined in this bcx)k c-onsider a biaxial state of stress combined in accordance with the maximum stress theory. Just as you would not design a nuclear reactor to the rules of Division 1, you would not design an air receiver by the techniques of Division 2. Each has its place and applications. The following discussion on categories of stress and allowables will utilize information from Dicision 2, which can be applied in general to all vessels. 

Figure 9.17 graphically shows a comparison of the four methods of analysis for a two-dimensional stress system (Xr = 0). The upper right quadrant represents tension for both fjc and fy. The upper left quadrant represents fy in tension and/r in compression, and the lower right quadrant represents in tension and fy in compression. The wlnl lines in the figure represent the locus of the conditions at which yield is assumed to begin according to the four theories. The square a-b-c-d represents the maximum stress theory. Point a represents equal tension in both the X and y perpendicular directions, both of which are considered to be equal to the yield-point stress obtained from a simple tensile lest. 

A macroscopic theory of strength is based on a phenomenological approach. No direct reference to the mode of deformation and fracture is made. Essentially, this approach employs the mathematical theories of elasticity and tries to establish a yield or failure criterion. Among the most popular strength theories are those based on maximum stress, maximum strain, and maximum work. The maximum stress theory states that, relative to the material symmetry axes x-y, failure of the RP will occur if one of three ultimate strengths is reached. There are three inequalities, as follows ... 

The maximum strain theory is similar to the maximum stress theory. Associated with each strain component, relative to the material symmetry axes, e, Cy, or there is an ultimate strain or an arbitrary proportional limit, Xg, T or S, respectively. The maximum strain theory can be expressed in terms of the following inequalities ... 

The maximum stress theory is shown as solid lines in Fig. 2.28. On the right-hand side of the figure is the uniaxial strength of directional RPs with fiber orientation 6 from 0° to 90° on the left-hand side, laminated RPs with helical angle a from 0° to 90°. Both tensile and compressive loadings are shown. The tensile data are the solid circles and the com-... 

For over three decades, there has been a continuous effort to develop a more universal failure criterion for unidirectional fiber composites and their laminates. A recent FAA publication lists 21 of these theories. The simplest choices for failure criteria are maximum stress or maximum strain. With the maximum stress theory, the ply stresses, in-plane tensile, out-of-plane tensile, and shear are calculated for each individual ply using lamination theory and compared with the allowables. When one of these stresses equals the allowable stress, the ply is considered to have failed. Other theories use more complicated (e.g., quadratic) parameters, which allow for interaction of these stresses in the failure process. 

Engineers have known for some time that the maximum shear stress theory and the distortion energy theory predict yielding and fatigue failure in ductile materials better than does the maximum stress theory. However, the maximum stress theory is easier to apply, and with an adequate safety factor it gives satisfactory designs. But where a more exact analysis is desired, the maximum shear stress theory is used. 

Two basic theories of strength are used in the ASME Boiler and Pressure Vessel Code. Section I, Section IV, the ASME Code, VIII-1, and Section m. Division 1, Subsections NC, ND, and ME use the maximum stress theory. Section DB, Division 1, Subsection NB and the optional part of NC, and the ASME Code, VHI-2, use the maximum shear stress theory. 

In the two sections of the ASME/ANSI Code for Pressure Piping B3I that ate used primarily with the ASME Boiler and Pressure Vessel Code, both ANSI B31.1 and B31.3 use the maximum stress theory. B31.3 is unique in that it uses the maximum stress theory but permits allowable stresses to be established on the same basis as the ASME Code, VIII-2, which requires use of the maximum shear stress theory. The other sections of B31 also use the maximum stress theory. They require that in addition to the stresses caused by internal and external pressures, those stresses caused by thermal expansion of the piping are to he considered. 

The required thicknesses and other design requirements vary somewhat depending upon the design theory chosen. The maximum stress theory is used for the design of most tall vessels. This theory is used in the ASME Code, VIII-1, and the API 620 and 650 design rules. The effects of using other theories are discussed later. 

One of the simplest failure theories for orthotropic lamina is the maximum stress theory of failure. This theory states,... 

Determine the allowable stress for uniaxial off-axis loading of the unidirectional fiber-reinforced composite lamina, shown in Figure 8.17. Assume that the lamina strength properties Xl, Xf Xp, Xt, and S are known. Use the maximum stress theory of failure. 

The maximum stress theory is illustrated in Figure 9.2a in which Equation 9.11 are plotted as a function of 0 for an E-glass/epoxy composite with properties Xl = X =150 ksi, Xt=4 ksi, Xy = 20 ksi, and 5 = 6 ksi. Experimental data are superimposed on the theoretical strength plots, with tension data denoted by solid circles and compression data by sohd squares. The theoretical cusps predicted by theory are not home out by the experimental results, nor is there exact agreement in variation of stress with 0 between theory and experiment. 

One of the shortcomings of the maximum stress theory of failure is that there are no terms which account for interaction between stress components for the case of biaxial (or off-axis) loading. Another is that five independent equations must be satisfied. The Tsai-Hill theory of failure for anisotropic materials overcomes both of the above mentioned shortcomings. This theory can be expressed in terms of principal material stress components as follows ... 

FIG U RE 9.2 Theoretical curves for uniaxial strength of unidirectional E glass/epoxy composite lamina versus off-axis loading angle 0. Experimental tension (solid circles) and compression (solid squares) data are superposed on theoretical curves, (a) Maximum stress theory of failure and (b) Tsai-Hill theory of failure. (Adapted from Jones, R. M., Mechanics of Composite Materials, Taylor Francis, London, 1998.)... 

Maximum stress theory. Includes unloading of cracked lamina as well as geometric nonlinearity due to changes in the ply angle. 

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