Stress trajectory

Figure 7.1. Early diagrams showing the relationship between stresses created by forces on bones and the internal architecture of the skeleton (a) Culmann s calculation of the stress trajectories in a crane, (b) Wolff s drawing of the trabecular orientation in the upper part of the femur, and (c) a photograph of the cross-section of the upper part of the femur.

Similar method was used in [3] with heat flow and in [4, 5] with stress trajectories. The purpose of this work is to develop efficient method for identification of current densities within a domain where some data on electric potential are provided. 

Potential for detecting fault sealing characteristics based on the rotation of the stress trajectories... 

In either case, whether faults occur in homogeneous sandstone or with clay in the fault, the principal stresses are markedly rotated outside the fault zone. This demonstrates that a rotation of the stress trajectories cannot uniquely characterise open and nonsealing faults. 

The inclined cracks that arise in shear zones under the main stress influence are distributed to approximately 2/3 of a beam height. In this case, the ends of a crack settle at some distance from the load and a support. Such cracks are characteristic for some beams with longitudinal reinforcement Rn = 6400 kg/cm2. For the same beams, and for all beams with reinforcement Ra = 3400 kg/cm2, crack formation occurs in most cases on the bottom side of the beam. An increase of loading on the crack is distributed along the main compression stress trajectory and its top is situated from a load center on approximately 14 of the beam s height. 

The stress distribution in a component can be visualised using so-caUed stress trajectories. These trajectories always run in the direction of the maximum principal stress. Their distance is inversely proportional to the stress so that the stress trajectory density is a measure of the locally acting stress. Each abrupt change in cross section deflects the stress trajectories which then move closer together. Thus, a local stress concentration arises. 

The term stress trajectory is due to the fact that the stress distribution in a component is analogous to the velocity distribution of a laminar, frictionless fluid. The stress concentration at changes in the geometry corresponds to the disturbed flow of the fluid at similar geometries. Different from fluid flows, stress trajectories cannot become turbulent. 

Figure 4.1 shows sketches of the stress trajectories near differently shaped notches. If we look at the stress trajectories at a cross section at the notch root, we see that they are not evenly distributed, but become more narrow at the notch root. Thus, there is a local stress concentration, with a maximum stress Crnax in the notch root as shown in figure 4.2. The shape and size of the... 

Fig- 4.1. Stress trajectories in notched components. The stress trajectories are aligned with the maximum principal stress, their density is a measure of the stress level. At the notch root, there is a stress concentration in both geometries... 

Initially, the response is within the elastic range of behavior. In this range, the plane strain constraint e z = 0 is sufficient to determine the stress trajectory. From Hooke s law. 

Integration of the differential equations from first yield for an elapsed time of t = 1 yields the result shown in Figure 7.34. The dash curve is the initial yield locus. Note that the stress trajectory deviates sharply from linearity once yielding begins, even though the state of deformation is always uniaxial plane strain extension. 

As the magnitude of plastic strain becomes large in the material compared to the elastic strain, the stress trajectory approaches the dot-dash line with slope 0.5. This is precisely the trajectory that would be predicted if elastic strains would be neglected entirely from the outset. For comparison, the corresponding results for kinematic hardening the shown in the graph as the dotted curve, barely visible under the solid curve. 

In certain zones such as embedded items areas would create where stresses in concrete are unacceptable reinforcement may be arranged to reduce these stresses. Due to temperature hot spots may occur in concrete zones which could be unacceptable. The stresses could be high. By providing some of these local reinforcement will reduce stresses to acceptable values in concrete. The most appropriate technique is to use advance analytical means such as finite element. Stress trajectories with and without reinforcement could be developed. These stress trajectories would determine the sizes and the zones to which these reinforcement could form shapes. The extent of such reinforcement will have bond lengths between 24 and 48 diameters of the bar. Obviously these reinforcement would be evaluated under prestressing anchorages too to avoid cracks under stressing loads. 

The stress trajectories at normal operation and at 2.5 times design pressures can be obtained in the same manner as for vessels discussed previously. The vessel design pressure GD here is 5.68 MN/m. At a pressure of 7.48 MN/m (1.315Pgd). this vessel s behaviour (Fig. 5.36) is identical to that of the Hartlepool vessel. The only change is the plastic zone developed around the gas inlet duct under mark 6. 

Thermal analysis has been carried out for all vessels noted in this chapter. Figures 5.50, 5.51, 5.52 and 5.53 give the result of temperature distributions for the vessels. Where thermal loads are to be included in the overall analysis they are computed from these figures and are put at the respective nodes of the elements as concentrated loads or patch loads. Alternatively strains are evaluated from these temperatures and they are included in the finite element formulations as initial strains. Stress trajectories can be drawn for the temperature-only case on the lines given for vessels in normal operation - cases which in fact include temperature effects. 

The same analytical procedures are adopted as stated in the text and the nonlinear finite-element analysis given in the text under various loading conditions the boiler cavities are examined. Plates AIB.4 and AIB.5 summarized the results in the forms of stress-trajectories. These boiler holes or cavities have been discussed already in the text. 

Stress trajectories are the paths stresses tend to follow, and are akin to the more familiar concept of streamlines in fluid flow. 

It seems that in concrete composites with brittle matrix, in the crack tip area, the most probable directions of microcracks formation are those which are perpendicular to the isostatic lines. Principal stress trajectories in the vicinity of a notch are connected with the polar coordinate system -microcracks initiate radially and circumferentially, so the body may be considered as cy1indrically-anisotropic. 

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