Statically indeterminate

The foregoing designs were discussed as ring expansion joints by Kopp and Sayre, Expansion Joints for Heat Exchangers (ASME Misc. Pap., vol. 6, no. 211). All are statically indeterminate but are subjected to analysis by introducing various simplifying assumptions. Some joints in current industrial use are of lighter wall constniction than is indicated by the method of this paper. 

Not all structures can be fully analyzed by the methods of statics. If the number of discrete equilibrium equations is equal to the number of unknown loads, then the structure is said to be statically determinate and rigid. If there are more unknowns than equations, then the structure is statically indeterminate. If there are more equations than unknowns, then the structure is said to be statically indeterminate and nonrigid. 

Rigid Systems. Literature pertaining to the theoretical analysis of the three-plane rigid piping system is voluminous (30). This literature is expanding steadily and, as it is becoming more abstract, tends to obscure the basic problem which is the analysis of a three-dimensional statically indeterminate structure. 

The pipe wall in Fig. 14.7a is a statically indeterminate structure the bending moment depends both on the external loads and the wall bending stiffness. A distributed load q N m acts at the top and bottom of the pipe of diameter D. If the pipe were cut in half horizontally and the cut ends supported on a frictionless surface (Fig. 14.7c), the bending moment me would be related to the angular distance 6 by... 

Figure 4.2 Pin-jointed frames in which the stresses are a) statically determinate and b) statically indeterminate.

The simultaneous solution of Eqs. (4.2) and (4.3) allows the forces and stresses in the various rods to be determined. It should be noted that even though linear elasticity is assumed, the terms for stresses do not involve the elastic constants. This is not true, however, for the strains. In the last chapter, the geometry shown in Fig. 3.25 was statically indeterminate, and to solve the problem the rods were assumed to be linear elastic. 

Experimental creep data for ceramics have been obtained using mainly flexural or uniaxial compression loading modes. Both approaches can present some important difficulties in the interpretation of the data. For example, in uniaxial compression it is very difficult to perform a test without the presence of friction between the sample and the loading rams. This effect causes specimens to barrel and leads to the presence of a non-uniform stress field. As mentioned in Section 4.3, the bend test is statically indeterminate. Thus, the actual stress distribution depends on the (unknown) deformation behavior of the material. Some experimental approaches have been suggested for dealing with this problem. Unfortunately, the situation can become even more intractable if asymmetric creep occurs. This effect will lead to a shift in the neutral axis during deformation. It is now recommended that creep data be obtained in uniaxial tension and more workers are taking this approach. 

Professor of Applied Mechanics. He was occupied not only with theoretical work and editing many books, but also with practical work, particularly bridges. In fact Navier brought together many of the isolated discoveries of his predecessors in the fields of applied mechanics and related subjects into one subject, structural analysis. He also added many new ideas such as the solution of simple statically indeterminate structures by considering the elastic deformations of individual members, and he calculated results for beams with fixed ends and for beams continuous over three supports. He was the first to develop the formula... 

Determination of the load distribution in the joint is a statically indeterminate problem which depends on a number of factors ... 

The beam shown in Fig. P8.21 is statically indeterminate. We will first compute the defiection of the simply supported beam by releasing the two bents. Then, weTl solve the unknown reactions in the two bents and the total defiection. This is the flexibility method. 

In the analysis of statically Indeterminate structures, static Indeterminate forces Bj, (1=1,...,n) are Introduced In order to satisfy n boundary conditions which cannot be satisfied from equilibrium only. Given the bending moment Mo(x) of the associated statically determinate beam, the bending moment M(x) is... 

In which M (x) are the bending moments within the associted statically determinate beam due to statically Indeterminate forces B < equal to unity. Since the Bj are determined from deflection boundary conditions and deflections depend on the elastic properties of the structure, the constants B will be random. Let the prescribed boundary conditions be... 

In which V() denotes deflection or derivatives of deflection. Then the statically indeterminate forces follow from a set of linear equations... 

The Indices k,l refer to the statically Indeterminate forces and the boundary conditions, respectively,... 

In which Wq(x) is the deflection of the statically determinate system and W (x) denote the deflection due to unit statically Indeterminate forces, it is obvious that w(x) Is a function of Gaussian variables. Thus, the statistics of w(x) can be obtained analytically or through Monte Carlo simulation. The analytical solution, however, may require approximation. 

Fig. 2 Coefficient of Variation of Midspan Deflection for Statically Indeterminate Beam as Function of Correlation Parameter b...

The present study shows that It is possible to evaluate the variability of statically determinate and statically indeterminate structures due to spatial variation of elastic properties without resort to finite element analysis. If a Green s function formulation is used, the mean square statistics of the indeterminate forces are obtained in a simple Integral form which is evaluated by numerical methods in negligible computer time. It was shown that the response variability problem becomes a problem Involving only few random variables, even if the material property is considered to constitute stochastic fields. The response variability was estimated using two methods, the First-Order Second Moment method, and the Monte Carlo simulation technique. 

The connection of closed cells through the sharing of a common branch represents a statically indeterminate system. Due to the excess of branches with regard to cells and junctions respectively, the unknown warping resultants may only be determined by both continuity requirements of the cells and axial equilibrium conditions at the junctions. 

To illustrate the rather abstract formulation of the general cross-section outlined above, the essential relations for two examples will be given. The first is a closed cross-section with two cells and thus represents the elementary case of a statically indeterminate system. The second examines the differences induced by a slit in one of these cells and therefore is concerned with the combination of a closed cell and two open branches. 

Solutions for stresses and strains within even simple structures discussed above often involve statically indeterminate solutions requiring information beyond that available through the equations of equilibrium alone. In addition to equilibrium equations, kinematic relations linking displacements, and constitutive relations relating stresses and strains are normally required for these solutions. Such solutions may be obtained at the mechanics of materials level, involving simple solutions for basic structural elements as given above, or they may require more sophisticated analytical or numerical methods. 

The upper and lower adherends are denoted by 1 and 2, respectively. Each adherend has a Young s modulus , and a thickness r,. The adhesive has a shear modulus of G and a thickness of h. The joint length is I, as shown in Fig. 12a. Solving this statically indeterminate problem involves the equilibrium equations based on a differential element as shown in Fig. 12a ... 

DYNAMITE will be extended to perform parameter-sensitivity analysis (with analytically generated sensitivity matrices) and to handle small and large elastic deformations. Other possible extensions are numerical methods for statically indetermined systems, adequate methods for stiff systems. 

Suppose you have a four-leg bridle. The loads in the slings vary because the shngs are almost always unequal in length. The loads are statically indeterminate, meaning that the true load in each shng cannot be mathematically solved. In reality, the load is carried by two slings, while the other two act to balance the load. To solve this problem, you must size the bridle such that just two legs carry the fuU load, or you must use a spreader. 

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