Paul Dvorak
Senior Editor

A tensile load on the plate produces displacements and von Mises stresses.

A modal analysis produced the first four natural frequencies of vibration and their associated modes in an unsupported valve plate.

Deforming a beam supported by immovable hinges (top) may significantly change its stiffness due to membrane stresses, thereby requiring nonlinear analysis even when displacements are small. If one hinge is movable (bottom), membrane stresses won’t develop and linear analysis may suffice if displacements are small.

That's OK, but users work more confidently when they have a firm grip on the science behind the software. Hence, this second part of A designers intro to FEA. (The first appeared in the Aug. 10, 2006 issue.) It's useful to review correct terminology, examine how developers have organized the software, and learn what it can and cannot do.

"Structural analysis is a good place to start this review because it divides into different simulations based on whether loads and supports are time dependent or not," says Paul Kurowski, president of Design Generator Inc., London, Ontario, Canada (designgenerator.com). "If loads and supports do not change with time, it is static analysis. And it's dynamic or, more properly called vibration analysis, when loads, or supports, or both are time dependent. Vibration analysis is further divided into several types based on the nature of the time dependency. The most frequently encountered types include time response, frequency response, and random-vibration analyses."

Thermal analysis is also classified as to whether or not thermal loads are time dependent. "Unchanging thermal loads and boundary conditions point to a steady-state thermal analysis. Changing thermal loads and boundary conditions require a transient thermal analysis," he adds.

Structural and thermal analyses can also be linear or nonlinear. "Linear structural analysis assumes model stiffness, as described by a stiffness matrix [K] in FEA math, does not change with displacements caused by applied structural loads or supports. The stiffness matrix is calculated from structural shape, material properties, and applied restraints or supports. The matrix characterizes a structure's response to applied structural loads," says Kurowski.

Linear thermal analysis assumes temperature changes caused by applied thermal loads or boundary conditions do not change the model's conductivity, described by the conductivity matrix [C] in FEA math.

"Nonlinear analyses are quite another matter. Nonlinear structural analysis requires gradually applying loads while the software repeatedly recalculates the stiffness matrix during deformation," says Kurowski.

In nonlinear thermal analysis, changing temperatures can change the conductivity matrix. This requires dividing a solution into many steps. The conductivity matrix is thermally analogous to the stiffness matrix. It is calculated based on a structure's shape, material properties, and applied thermal-boundary conditions.

"The four most commonly preformed types of analyses are linear static, modal, linear buckling, and steady-state thermal," says Kurowski. (Linear buckling and steady-state thermal will be covered in the next installment.)

Linear-static analysis is often called static analysis because loads are not time dependent. "In linearstatic analysis, the structural-stiffness matrix is calculated just once for the original undeformed shape and is not updated during displacements under load when the model is deformed. Therefore, to solve a linear-static-analysis problem, analysts need only solve a set of linear algebraic equations once, even though there may be many of them," he adds.

Typical outputs for static analyses include displacements and stresses. "Displacements are vectors with three components in a 3D model. Users often plot resultant displacements rather than their X, Y, and Z components," he says.

Stresses in 3D models are tensors and so have six components: three normal and three shearstress components. "A scalar-stress measure called von Mises is often used to generate stress plots. Von Mises stress takes the magnitudes of all six stress components and calculates one value. The equation for von Mises stress is:

"In addition, von Mises stress is frequently used to calculate a safety factor for materials with properties close to elastoplastic," he says. Elastoplastic materials behave elastically up to a certain stress level. Von Mises stress are often used to express this stress level. After that point, elastoplastic materials become perfectly plastic, with strain increasing but stress staying constant.

Easy setups and quick executions may tempt engineers to use linear-static analysis even though the problem may involve large structural displacements. "So before executing a linear-static analysis, it's a good idea to verify that stiffness does not change significantly with displacements, and that loads can be assumed constant or changing so slowly that inertial effects can be ignored," advises Kurowski.

While checking for small displacements seems simple, correct answers require in-depth understanding of the problem. "True, small displacements cause negligible changes in the stiffness matrix. But it's not enough for displacements to be physically small to classify them as 'small.' The classic example is membrane effects in which displacements may be small, yet they significantly change the structural stiffness and so must not be treated as 'small.' It takes nonlinear analysis to solve these problems," says Kurowski.

Consider a beam bending under load and supported by immovable supports. While still straight, the beam can resist pressure only with a bending stiffness develops due to bending stresses. But once the beam deflects a bit, it acquires membrane stiffness due to tensile stresses in addition to the original bending stiffness, which now act in the deflected beam. Membrane stiffness is a mechanism for resisting pressure. It did not exist before deformation. To capture this new effect, the stiffness matrix of the beam must be updated during deformation. This calls for nonlinear analysis. Linear analysis cannot update the stiffness matrix.

"During deformation, the beam changes stiffness due to membrane effects. Hence, analysts should use a nonlinear analysis, even with small displacements. If one support is movable and can be represented as if on rollers, then there are no membrane effects and linear analysis may be sufficient if displacements are small.

Sources of nonlinear behavior in structural analysis include:

• Large deformations
• Membrane effect. Displacements that generate a new loadcarrying mechanism.
• Plasticity. Deformations that cause plastic strains, those above the elasticity limit
• Buckling. Displacements that cause the loss of elastic stability.

Modal analysis, also called frequency analysis, is often listed with different types of vibration analysis. "Modal analysis, however, is not a dynamic or static analysis. Mathematically, modal analysis finds eigenvalues from the sum of the stiffness and inertia matrices. Eigenvalues correspond to the natural frequencies of the structure," he says. Modal analysis looks for combinations of frequency and vibration shapes, which let elastic and inertial forces cancel each other. That cancellation produces resonance.

The only meaningful results of modal analysis are natural frequencies (also called resonance frequencies) and associated vibration shapes, also called modes of vibration.

Modal analysis does not calculate a vibration's amplitude even though many FEA programs display displacement results of modal analysis. In some programs, the maximum displacement from modal analyses are normalized to 1, while in other programs, displacements can be meaningless values (usually large). To find the amplitude of vibration, you must run vibration analysis that accounts for dynamic loading and damping.