Moment, and Stress

The force and moment ia a constrained system can be estimated by the cantilever formula. Leg MB is a cantilever subject to a displacement of and leg CB subject to a displacement Av. Taking leg CB, for example, the task has become the problem of a cantilever beam with length E and displacement of Av. This problem caimot be readily solved, because the end condition at is an unknown quantity. However, it can be conservatively solved by assuming there is no rotation at poiat B. This is equivalent to putting a guide at poiat B, and results ia higher estimate ia force, moment, and stress. The approach is called guided-cantilever method. 

Acceptable comprehensive methods of analysis include analytical and chart methods which provide an evaluation of the forces, moments, and stresses caused by displacement strains. 

Acceptable comprehensive methods of analysis are analytical, model-test, and chart methods, which evaluate for the entire piping system under consideration the forces, moments, and stresses caused by bending and torsion from a simultaneous consideration of terminal and intermediate restraints to thermal expansion and include all external movements transmitted under thermal change to the piping by its terminal and intermediate attachments. Correction factors, as provided by the details of these rules, must be applied for the stress intensification of curved pipe and branch connections and may be applied for the increased flexibihty of such component parts. 

Figure 27.1 summarises the methodology for designing a component which must carry load. At the start there are two parallel streams materials selection and component design. A tentative material is chosen and data for it are assembled from data sheets like the ones given in this book or from data books (referred to at the end of this chapter). At the same time, a tentative component design is drawn up, able to fill the function (which must be carefully defined at the start) and an approximate stress analysis is carried out to assess the stresses, moments, and stress concentrations to which it will be subjected. 

Figure 7-13. Variation of bending moment and stress with beta and alpha.

If the calculation, based on either this method or an analytical method, indicates that the moments and stresses exceed the allowable limits, then Figure 7-13 can be used to predict the guide distance G which will allow the shape to become flexible enough and to yield moments and stresses equal to the allowable limits. 

Design a 17-type symmetrical expansion loop to yield the maximum moment and stress equal to the allowable limits. 

Solution, Step 1. Determine from Figure 7-13 when a = 3 and j3 = 5. Follow the arrows and read value of 7 = 0.17 which is the moment and stress factor on the left hand vertical scale. 

It may be assumed, in determining the stress caused by wind pressure, that the tower is a vertical beam, and that the wind produces a bending moment. The ordinary formulas for determining bending moment and stress may therefore be applied, as follows ... 

Step 4 After all of the preliminary dimensions and details are selected, proceed with the detailed analysis of the flange by calculating the balance of forces, moments, and stresses in the appropriate design form. 

Derivations for almost all analytical models for FRP strengthened flexural members are based on the typical schematic FBDs of Fig. 10.14. This particular case represents a differential segment of an FRP strengthened beam under uniformly distributed load, and the bending stiffness of the FRP laminate is assumed to be much smaller than that of the beam to be strengthened. Forces, moments and stresses acting on these basic FBDs reflect the individual assumptions preset for any analysis. The interfacial adhesive shear and normal stress are denoted by t x) and a(x), respectively. Equation [10.19] is the mathematical representation of the basic definition of shear stress t(x) in the adhesive layer, which is directly related to the difference in longitudinal deformation between the FRP laminate at its interface with the adhesive and the beam s soffit. 

Figure 26.18 The single lap shear joint has bending moment and stress concentrations at the ends of the overlap.

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