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 The first buckling mode of a T beam under a bending load. Temperature distribution and resultant heat flux are typical output data requested in the steady-state thermal analysis. Modes of vibration calculated in modal analysis are prerequisites to vibration analysis using the modal superposition method. Combined transient thermal and linear static analyses allow calculating thermal stresses.

In the previous FE Update (Sept. 12), Design Generator President Paul Kurowski, remarked that the four most common types of analyses are linear static, modal, linear buckling, and steady-state thermal. Linear static and modal analysis were covered in the previous installment. This article continues with linear bucking and thermal simulations.

LINEAR-BUCKLING ANALYSIS
Like modal analysis, buckling analysis is neither dynamic or static. "Buckling analysis calculates a buckling load and associated shapes," says Kurowski. "The analysis is similar to modal analysis. It finds eigenvalues of the sum of stiffness and stress matrixes."

Beams buckle as a result of losing stress stiffness that is induced by compressive loads. "The resultant structure stiffness drops to zero. Even though software can calculate any number of buckling modes, engineers need only find the first mode and its associated load magnitude. Buckling in the first mode often causes catastrophic failure or renders the structure unusable even if it may still be able to hold the load in its buckled shape. Higher buckling modes are usually of no practical importance," says Kurowski.

Buckling analysis, more precisely called linearbuckling analysis, calculates a buckling load and shape but does not offer quantitative information on the deformed, postbuckling shape. "The buckling mode presents the shape of structure after buckling but says nothing about the actual magnitude of deformation. Nonlinear buckling analysis must be used to study postbuckling effects," he says.

To obtain the buckling-load magnitude, multiply or divide the applied load (this depends on the particular FEA program) by the bucking-load factor generated by the FEA program. This is closely analogous to modal analysis that provides information on modes of vibration (frequency and shape) but not on the actual displacement magnitude.

There are several reasons why buckling factors are nonconservative and must be interpreted with caution. "For instance, finiteelement models are stiffer than corresponding real-life structures because they are discretized or meshed. Also, models represent geometry without imperfections. In addition, loads and supports are applied with perfect accuracy, without offsets. Support elasticity is often ignored. In reality, a load is always applied with an offset, geometry always has imperfections, and supports are never perfectly rigid," he says.

Thermal analysis, more properly called steady-state thermal analysis, calculates a steady-state temperature field that establishes itself under applied thermal loads and boundary conditions. "'Steady state' implies that sufficient time has lapsed since the loads were applied and that the heat flow has stabilized. Analysis of a steady-state heat transfer knows nothing of the initial temperatures or of how long it took for the heat flow to reach stable conditions. It can take seconds, hours, or days to reach the steady state," says Kurowski.

As for thermal-analysis results, they are most often presented in the form of temperature distribution and resultant heat-flux plots. "Temperatures are scalars making them easy to show in a fringe plot. Resultant heat fluxes are vectors with three components, conveniently shown either as a fringe plot or a vector plot," he adds.

FEA, of course, describes heat flow by conduction inside a solid body. "The heat entering and exiting the solid body are defined as boundary conditions such as prescribed temperatures, heat flux, convection coefficients, or radiation emissivity. FEA cannot differentiate between natural and forced convection. The distinction is only possible by using different values of convection coefficients. Flow of fluid surrounding the solid body on which thermal analysis is performed cannot be modeled," says Kurowski.

COMBINING ANALYSES
Projects often require using more than one type of analysis. For example, a steady-state thermal analysis can be combined with a linear-static structural analysis to calculate thermal stress. "Analyses are performed sequentially, so that thermal analysis provides results for static analysis," he says.

Other frequently encountered examples of sequential analyses combine linear vibration analysis, which uses data provided by modal analysis. "Results from modal analysis, such as natural frequencies and modal shapes, allow for discretization of the dynamic response of structure so the response can be analyzed as a superposition of responses of several single-degree-of-freedom systems. This approach is called the modal superposition method and is effective for solving linear vibration problems," he adds.