Linear elastic analysis

When required, combined with the use of computers, the finite element analysis (FEA) method can greatly enhanced the capability of the structural analyst to calculate displacement and stress-strain values in complicated structures subjected to arbitrary loading conditions. In its fundamental form, the FEA technique is limited to static, linear elastic analysis. However, there are advanced FEA computer programs that can treat highly nonlinear dynamic problems efficiently. 

The crack-driving force G may be estimated from energy considerations. Consider an arbitrarily shaped body containing a crack, with area A, loaded in tension by a force P applied in a direction perpendicular to the crack plane as illustrated in Fig. 2.6. For simplicity, the body is assumed to be pinned at the opposite end. Under load, the stresses in the body will be elastic, except in a small zone near the crack tip i.e., in the crack-tip plastic zone). If the zone of plastic deformation is small relative to the size of the crack and the dimensions of the body, a linear elastic analysis may be justihed as being a good approximation. The stressed body, then, may be characterized by an elastic strain energy function U that depends on the load P and the crack area A i.e., U = U(P, A)), and the elastic constants of the material. 

In principle then, a fracture toughness parameter has been defined in terms of linear elastic analysis of a cracked body involving the strain energy release rate G, or the stress intensity factor K. For thick sections, the fracture toughness is defined as Gic, and for thinner sections, as Gc or R (referred only to mode 1 loading here). This value is to be measured in the laboratory and applied to design. The validity of... 

Most of the previous works used a static debonding mechanism. Thus it is mostly based on linear elastic analysis. The importance and usefulness of linear elastic fracture mechanics is well recognized, especially for the case of interfacial fracture. However, the analysis requires serious theoretical approximations (I) It... 

FWD provides the ability to estimate the stiffness moduli of each individual layer after back-calculation analysis of the deflection data. This is carried out using specially developed computer programs, almost all using the principles and assumptions of the multi-layer system linear elastic analysis. Some guidance for calculating in situ equivalent elastic moduli of pavement materials using layered elastic theory is provided by ASTM D 5858 (2008). 

The analysis methods used in the design process make use of hand computations, three dimensional structural analysis software, STAAD Pro (2009) and spreadsheets. The structural analysis software is required to be capable of conducting static linear elastic analysis, non-linear second order analysis, buckling/stability, and modal (frequency) analysis. The steps and procedures used for preliminary design, detailed analysis, and member optimization, are described in the following sections. 

P(2) Linear-elastic analysis shall generally be used in the design of the connection. Non-linear analysis of the connection may be employed provided that it takes into account the load-deformation characteristics of all components of the connection. 

Albat A M and Romilly D P (1999), A direct linear-elastic analysis of double symmetric bonded joints and reinforcements . Compos Sci Technol, 59(7), 1127-1137. 

Where Young s modulus is needed in the above stress analysis, the tensile creep modulus is inserted, using the value for the appropriate temperature and duration of loading, and for the maximum tensile strain in the component (as determined by the linear elastic analysis). 

Program name Material properties Linear elastic Analysis linear visco- elastic Nonlinear visco- elastic Geometric nonlinearity Loading Time function Nonlinear diffusion... 

In this division, there is the possibility to estimate the results of an FE analysis based on the KD-230 Elastic-plastic analysis and KD-240 Linear elastic analysis. KD-3 Fatigue evaluation and KD-4 Fracture mechanics evaluation also give a guideline to assess the protection against failure from cyclic loading. 

The minimum required thickness can be determined by using closed-form equations for cylindrical and spherical shells, blind ends, threaded closures, and clamp connections. The elastic-plastic analysis can be used in most cases and for all D/d ratios. The linear elastic analysis is permitted only if D/d< 1.25 [17]. 

The behavior factor, q, accounts for the capacity of the structural system to dissipate seismic energy. This results in a reduction of the forces obtained from a linear elastic analysis, in order to account for the nonlinear response of the structure, associated with the material, the structural system, and the design procedure. However, the structural system should be able to sustain such nonelastic behavior and dissipate seismic energy. 

The prevailing approach proposed in modem codes for carrying out ultimate limit state checks consists of obtaining action effects by means of linear elastic analysis of the stracture subjected to design loads and comparing them to resistances that account for both types of eventual nonlinearity, material, and geometric. Thus, prediction of coUapse, a highly nonlinear phenomenon, is accomplished by means of linear elastic analysis. 

The most common approach to perform seismic design of structures is the traditional one based on a linear elastic analysis assumption. On the other... 

Eurocode 8, linear elastic analysis can be performed either with (a) the lateral force method of analysis for buildings meeting specific conditions or (b) with the modal response spectrum analysis, applicable to all types of builduigs and being the reference method for obtaining seismic response estimates. 

According to the lateral force method of analysis, a lateral load pattern that follows the first mode is applied (e.g., Eq. 35). In Eurocode 8, the method is applicable only if the fundamental period of vibration is less than 2s or 4Tc, where Tc is the comer period of the design spectrum. A second condition that also has to be satisfied is that the stiucture should meet the criteria for regularity in plan. Different criteria may be found in other codes. The modal response spectrum analysis is reconunended for all other cases, with the only exception of buildings with seismic isolation provided by highly nmilinear devices. Practically every design code in the world uses these two methods for linear elastic analysis and recommends criteria to determine if their applicable. 

Linear elastic analysis methods can also be adopted for ultimate limit state verifications of URM buildings, provided the stmcture does not have excessive irregularities (EC6 2005) and lacks significant torsional effects, thereby... 

Linear elastic analysis includes both equivalent static and modal analysis ... 

The joint efficiencies calculated from each of these separate failure modes are given in Table 5 for a double-lap joint. On the basis of a linear elastic analysis, it appears that adhesive failure is most likely to occur before failure in the adherends. 

If we now allow for non-linear adhesive behaviour, the high adhesive stress concentrations predicted by the linear elastic analysis will be relieved to some extent. Figure 54 shows the predicted spread of the yield zone of adhesive at the tension end of a double-lap joint as the load is increased. As would be expected, plastic flow begins near the adherend corner and the load corresponds to a joint efficiency of 21%. Each subsequent load increment represents an increase in joint efficiency of 4 4%. When elastic perfectly-plastic behaviour is assumed for the adhesive, a maximum strain criterion for failure seems appropriate. In Fig. 55 the joint efficiency is plotted against the maximum principal strain in the adhesive at each end of a double-lap joint. Assuming a failure strain for the adhesive of 5%, the analysis predicts a joint efficiency of 31% for a double-lap joint compared with 16% predicted by the linear elastic analysis. Similarly, the non-linear analysis predicts an efficiency of 39% for the double-scarf joint compared with 20% predicted by the linear elastic analysis. Although the predicted efficiencies are almost doubled by allowing for non-linear behaviour in the adhesive, failure in the adhesive is still predicted to be more probable than failure in the adherends (Table 5). 

The linear elastic analysis of section II predicts infinite stresses at the crack tip (eqn (2.5)). In reality, this divergence is relaxed by the dissipative pull-out process which takes place at the crack tip. The pull-out process is expected to occur in an approximately planar cohesive zone directly ahead of the crack tip. A thorough investigation of cohesive zone models can be found in Eager et al. (1991). See also Xu et al. (1991). 

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