Statically determinate

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. 

An object is generally a three dimensional constmct whose position is dehned by its location (3 degrees of freedom- x, y, z) and by its orientation (3 rotations). Thus an object is constrained if six degrees of freedom of the object are constrained. If less than six degrees of freedom are constrained, the object is under constrained and can be viewed as a mechanism. It is also called under-determined. If the object is only considered in two dimensions, then three constraints are needed to dehne the object (x, y, rotation). When an object is just constrained it is called determinate or statically-determinate. 

The MOLWT-II program calculates the molecular weight of species in retention volume v(M(v)), where v is one of 256 equivalent volumes defined by a convenient data acquisition time which spans elution of the sample. I oment of the molecular weight distribution (e.g., Mz. Mw. Mn ) are calculated from summation across the chromatogram. Along with injected mass and chromatographic data, such as the flow rate and LALLS instruments constants, one needs to supply a value for the optical constant K (Equation la), and second virial coefficient Ag (Equation 1). The value of K was calculated for each of the samples after determination of the specific refractive index increment (dn/dc) for the sample in the appropriate solvent. Values of Ag were derived from off-line (static) determinations of Mw. 

The nanostructure of a solid, also referred to in terms such as "defects," "real structure," and "mosaic structure," depends strongly on its environment. As all aspects of the nanostructure may be relevant to catalytic functions, it is common to infer such properties from static determinations of the nanostructure, often carried out at about 300 K and in laboratory or autogeneous atmospheres. This approach neglects the dynamics and assumes incorrectly that the surface catalysis process should not modify the rigid crystalline bulk of a solid. 

Noting that the specimen is statically determinate, then the failure load P3 is related to P from Figure 4 via P = P3/ (1 + L1/L2). Figure 11 shows a comparison between the predicted (P3) and experimental failure loads as a function of the effective crack length. In general, failure loads are underestimated using this procedure, though there is some scatter in results. 

As demonstrated by the foregoing two formulations, some problems taken from mechanics can be formulated by using only Newton s laws of motion these are called mechanically determined problems. The dynamics of rigid bodies in the absence of friction, statically determined problems of rigid bodies, and mechanics of ideal fluids provide examples of this class. Some other mechanics problems, however, require knowledge beyond Newton s laws of motion. These are called mechanically undetermined problems. The dynamics of rigid bodies with friction and the mechanics of deformable bodies provide examples of this class. 

Detection of the exact temperature at which the mc.. s loss process starts is experimentally far more difficult than determination of a Td,l/2-value. Besides, the determination of the Td,o-value might be hampered by mass losses due to residual solvent/ monomer and/or oligomers. We expected that a kind of compromise would be possible by using semi-statically determined Td,o-values as shown already for polypropylene in chapter 2. 

For some problems, the stresses can sometimes be found simply by using statics and the loading is then termed statically determinate. For example, Fig. 4.2(a) shows a pin-jointed frame in which two rods are loaded by a force F. This frame is sometimes termed a perfect frame, as it has just sufficient linkages to prevent any rotations about the joints within the plane of the diagram. For three-dimensional constraint, an additional rod would be needed, e.g., using a tetrahedral configuration of rods. A frame that could not prevent a rotation is termed imperfect. In pin-jointed frames, the flexible joints ensure that the rods only transmit tensile and compressive forces. For Fig. 4.2(a), one can use statics to obtain. 

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

A simple example of a statically determinate problem is a thin-walled cylindrical pressure vessel with spherical end-caps, containing a gas or fluid exerting a pressure P. The geometry is shown in Figs. 4.3 and 4.4 the cylindrical section has a radius r, length L and thickness t, such that L r t. The cylinder is being expanded by the internal pressure and the principal axes are easily... 

Which of the following stress fields is (are) statically determinate i) Uniaxial tension, ii) Three-point bending, iii) Thin-walled pressure... 

There are many stress-analysis problems involving viscoelastic materials that are of a statically determinate class, i.e., the stresses in the body depend only on the applied forces and moments and not specifically on the elastic properties of the body. Such problems can be solved by invoking the correspondence principle. Then, the time and temperature dependences of the strains and flexures in the body can be obtained through the time temperature-shift properties of the viscoelastic polymer. 

The correspondence principle states that for problems of a statically determinate nature involving bodies of viscoelastic materials subjected to boundary forces and moments, which are applied initially and then held constant, the distribution of stresses in the body can be obtained from corresponding linear elastic solutions for the same body subjected to the same sets of boundary forces and moments. This is because the equations of equilibrium and compatibility that are satisfied by the linear elastic solution subject to the same force and moment boundary conditions of the viscoelastic body will also be satisfied by the linear viscoelastic body. Then the displacement field and the strains derivable from the stresses in the linear elastic body would correspond to the velocity field and strain rates in the linear viscoelastic body derivable from the same stresses. The actual displacements and strains in the linear viscoelastic body at any given time after the application of the forces and moments can then be obtained through the use of the shift properties of the relaxation moduli of the viscoelastic body. Below we furnish a simple example. 

We note that neither nor is affected by the elastic modulus of the material because the problem is statically determinate. Thus, when additional creep strains develop in time, they will be linearly proportional to the stresses and the strain distributions through the bar, and will remain similar to the elastic distribution, while the stress distributions will remain unaltered from the linear elastic... 

We must note in passing that, if the beam of interest had been constrained at any point by a local reaction support of a different stiffness with a different time dependence, then the problem would not have been of a statically determinate character and the simple procedure would not apply. The solution of such problems would require energy methods as discussed, e.g., by Timoshenko (1930). 

Let V be aL finite set of processes indexed by an index set I, and let S and V be the sets of signals and variables in the program respectively. The cardinalities of the sets V, 5, and V are notated rip, ns, and ny, respectively. Variables, signals, boolean expressions, and statically determined time delays are denoted as v, s, e, and n respectively. Variables and signals in the elaborated VHDL program are renamed to vi and Si, where i is a positive index. Each line is assigned a label where i is the process index and j is the line index within the process (i, j are positive integers). 

V.16. To show the level of consideration required, even for simple statically determinate retention systems, the following two examples, with their simplifying assumptions, are presented. 

V. 17. Consider a rigid package restrained by four symmetrically disposed tension tiedowns. A requirement of the simplified method is to predict upper bound values of tiedown force and hence, by reaction, forces on the package attachment and the conveyance. This method is apphcable only to statically determinate systems, and simple iterative assumptions are made on the system behaviour to derive upper bound forces. 

We will make the structure statically determinate by releasing the two bents. The resulting primary structure is a simply supported beam. 

Because the structure is statically determinate, the quasistatic displacements do not cause any bending moment. 

A distinction can be made among the available methods between static and dynamic contact angle determination methods. In the case of a static determination the contact angle of a drop with an immobile solid/liquid/gas interface is determined microscopically (sessile drop). In the captive bubble method the contact angle of an air bubble, which is located under the solid surface in contact with the liquid, is determined. In contrast to the sessile drop method, in the captive bubble method the contact angle is measured at a completely wet surface. 

However, two major objections could be foreseen and they were very soon subject of controversy. Indeed, the two indicators allow for a preliminary design, achieving the required performances of strength and stiffness with a minimum volume of material (a fully stressed design of statically determinate structures, subject to classical load cases). .. but what about other phenomena like e.g. (in)stability and possible resonance Let us state here that we are convinced that conceptual design should take into account the totality of the performance criteria to be satisfied by any structure ... 

Vandenbergh, T., Verbeeck, B. De Wilde, WP. 2007. Dynamical analysis and optimisation of statically determinate trusses at conceptual design stage, Compdyn 2007, Rethymnon, June. 

Dynamic binding (also known as dynamic dispatch) the principle that when a variable or similar entity is polymorphic and is passed as a parameter in a call to a function or procedure, the exact function or procedure called may not be statically determined but may depend at run-time on the class of which the variable is an instance. Most object-oriented languages support single dynamic dispatch (i.e. the choice of function or procedure called depends on at most one parameter, which is typically distinguished syntactically from the other parameters) a few support multiple dynamic dispatch. 

A further simplification can often arise if the stress analysis problem required in step (a) is statically determinate. In particular, this requires that the externally applied constraints (or boimdary conditions) can all be expressed in the form of applied loads and not in terms of imposed relative displacements. The stress distribution depends on the applied loads and on the component geometry, but not on the material stiffness properties. Thus, it is identical for all materials, whether they be elastic, rigid, or any other form, provided only that the material is sufficiently stiff for satisfaction of the assumption that the applied loads can be considered to be applied to the imdeformed, rather than deformed, component geometry. 

Thus, for metallic materials in many idealized practical situations, the design process is simplified to a stress (but not strain or displacement) analysis followed by comparison and optimization with critical stress values. Wlien the problem is not statically determinate, the stress analysis requires specification of material stiffness values, but the associated strain and deformation values are usually not required. Since the material behavior is usually represented adequately by linear isotropic elasticity, the stress analysis can be limited to that form, and there are many standard formulae available to aid the designer. 

In which w Is the deflection, p(x) the applied distributed load and x denotes the coordinate along the beam axis. For a statically determinate beam, Integration of eg. (3) yields... 

Thus, the deflection w(x) of a statically determinate beam may be expressed In terms of a convolution integral, 1.e.,... 

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... 

The quantities Vi Cxj) are calculated from the associated statically determinate system, hence their statistics may easily be obtained. The solution to eq.(15) yields as functions of VgCxj) V (Xj), (k=l,...,n) which are Gaussian random variables. Since any desired deflection w(x) may be written as... 

---