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In the broad field of solid mechanics, structural analysis refers to the response of a structure and the analysis of its behavior under certain loading conditions. A structure or an assembly may consist of several sub-components. The goal of structural engineers is to analyze the behavior of all the sub-components but also the structure as a whole by typically trying to answer the following question: “Is it going to withstand the loading or is it going to fail?”

Stress and strain are two really important quantities when it comes to structural analysis.

In a solid, stress (σ) is defined as the internal force (F) within the material that opposes any external loading, and runs through any given cross-section (A):

$$\sigma = \frac{F}{A}$$

Any solid under stress will experience deformation. Strain (ε) is a dimensionless quantity that quantifies the relative deformation of materials under stress and is expressed as:

$$\epsilon = \frac{dL}{L}$$

where L is the undeformed length of the solid element and dL is its relative deformation.

Deriving the stresses and strains of a given component is the major goal of structural analysis. However, there are other, equally important quantities and analysis types such as natural frequencies and mode shapes, vibration responses, buckling, fracture/crack propagation, and material fatigue.

The relationship between stress and strain is often expressed as:

$$\epsilon = \frac{\sigma}{E}$$

where E is a characteristic material constant known as modulus of elasticity, elastic modulus, or simply Young’s modulus, named after the British scientist Thomas Young.

It is a mechanical property of linear elastic solid materials which quantifies how rigid/stiff a material is, and is often expressed in pressure/stress units (Pa or psi). This value is typically quite large so we may encounter it being expressed in MPa or even GPa units as well.

Talking about materials, it’s worth mentioning the stress-strain curve. In engineering structural analysis, the most commonly encountered materials are ductile. A material is ductile if it is able to withstand excessive strains before failure. Figure 1 demonstrates a typical stress-strain curve of a ductile material.

The stress-strain curve is unique for any material and it is typically defined through physical testing. There are several types of tests, for example, tensile, compression, and bending tests which are used for material characterization. The stress-strain curve provides engineers and designers with important information about material limits and the risk of failure.

If you want to learn more about stresses and strains, and the various stress-strain relationships based on different material types, make sure to read our article on the Stress-Strain curve**.**

During the design process of a component or a structure, the structural analyst has to ensure that the designed model will withstand all sorts of loads that it is expected to experience throughout its designated lifetime. This very much depends on the material properties and the stress-strain characteristic curve that comes with it. There are two main concerns when it comes to material strength:

- What is the limit stress (better known as Yield Strength) after which a material will experience large and remaining deformation (plastic deformations)?
- What is the material strength (or Ultimate Strength) which describes the maximum stress that a material can take before it fails?

Based on the application, the design must ensure that the developed stresses will never surpass the material’s Ultimate Strength and/or Yield Strength. Of course, the maximum allowable stress should be somewhat lower than these two limits in order to account for uncertainties or unpredicted extreme loading conditions. This defines the maximum allowable stress and in turn the factor of safety (FoS) of the design. This method is commonly used to ensure that safety-critical lifting equipment has a high safety factor to account for unexpected loading regimes.

$$FoS=\frac{\text{max. allowable stress}}{\text{max. working stress}}$$

In the case of a simple, statically determinate problem, the reaction and internal forces can be found employing simple statics. This case is simple since the number of unknowns is typically equal to the number of equations to maintain equilibrium. However, there are cases where the number of unknowns exceeds the number of equilibrium equations as in the case of a propped cantilever beam for instance. This is now a statically indeterminate problem and it can only be solved with alternative methods.

The principle of virtual work or the virtual force method is one of the very first and most powerful methods available for the analysis of statically in-determinate problems and can extend beyond the linear elastic material range. Then, there come the energy methods (based on strain energy) which are more general and can provide estimated solutions to complex problems when exact solutions do not exist or are difficult to obtain. Finally, with the advancement of computers and numerical computations, computer-based techniques were developed, for example, the flexibility and stiffness methods. The former (work and energy methods) oftentimes can be solved by hand and are generally good for smaller and simpler (in shape) components while the latter (matrix methods) are quite useful in modern structural engineering where the need for designing and analyzing larger assemblies (i.e. aircraft) is desired. A subcategory or in other words, the evolution of the matrix methods, is the finite element method (FEM) which is particularly good for continuum structures representing real-life models. Make sure to read our article about Finite Element Analysis and how FEM is applied in practical engineering problems to gain more understanding about this topic.

Of course, talking about structural analysis solution methods could be endless and cannot be covered in the present article. One academic textbook I possess and I like to retrieve when looking for more information about structural problems is Aircraft Structures for Engineering Students by T.H.G. Megson. [2]

With the development of modern computing, engineers and designers implement structural analysis methods more regularly in their daily jobs. They can use dedicated FEA or structural analysis software to reduce the number of physical prototypes required for testing at the component or assembly level. The set of mathematical equations and unknowns are now handled by a computer allowing engineers to perform speedy computations and reach design decisions faster.

New technology has flattened the barriers; now structural analysis simulation is truly accessible through any web browser leveraging the power of cloud computing. Engineers can now run multiple designs in parallel and post-process simulation results online. SimScale has made engineering simulation easily accessible in every industry and truly scalable using premium cloud computing infrastructure.

As discussed, there are several structural analysis types. Based on the application type, structural analysis can be employed to solve different problems. It can either be a simple linear static analysis or a more complex non-linear analysis with elastoplastic material properties and large deformations.

The static analysis type allows time-independent calculation of displacements, stresses, and strains in one or multiple solid components. The results are a consequence of the applied constraints and loads, for example, bearings, gravity, forces, etc. The results enable engineers to evaluate whether the component of interest is deformed in an undesired manner or if a critical stress state occurs that will pose threat and risk failure. Feel free to check the following bolted flange example for an idea of static structural analysis.

The dynamic analysis type allows the time-dependent calculation of displacements as well as stresses and strains in one or multiple solid components. When compared to a simple static analysis it now differs due to the fact that inertia effects are also taken into consideration through a variation in time. The results allow engineers to analyze single timesteps as well as the dynamic performance as a function of time. Similar to static analysis, the model adequacy with respect to maximum allowable stresses and deformations can be assessed through time. Drop-test and impact assessments are characteristic examples of dynamic analysis.

The thermomechanical analysis is beneficial when trying to compute the thermal and structural behavior of a component or assembly at once. It is actually helpful in calculating the stresses of a solid body caused by combined thermal and structural loads. The thermal and structural results are calculated sequentially. The process typically starts with a thermal step which then serves as input to a consecutive structural step. It is essentially a thermal and a structural analysis combined. Studying the thermal shock of components, like a spark plug or a globe valve, for example, can be typical applications of interest in this case.

The frequency Analysis simulation type allows the computation of natural (under no external load excitation) frequencies (eigenfrequencies) of oscillation of a structure and the corresponding oscillation mode shapes (eigenmodes). The resulting frequencies and deformation modes are dependent on the geometry and material distribution of the structure, with or without displacement constraints. Frequency analysis results help evaluate the overall rigidity of the structure and understand what are the “dangerous” frequencies, at which if the structure is excited, might cause resonance. The results of a frequency analysis can be used as valuable input to seismic studies, wind loading studies, or vibration studies of electronic equipment like this Electric Vehicle (EV) battery module for example.

The frequency analysis is often used as the first step to a later Harmonic Analysis. The harmonic analysis enables the engineers to simulate the steady-state structural response of solids excited by periodical (sinusoidal) loads. It is quite similar to a dynamic analysis where inertia effects are taken into account, but it is different in the sense that now the results are not time-dependent but frequency-dependent. In other words, the harmonic analysis makes it possible to compute the response of a structure under vibrating forces or displacements over a defined frequency spectrum. Damping effects can also be taken into account. The EV battery module example mentioned above is a good example of a combined frequency and harmonic analysis simulation for electronic equipment.

Eigenmode | Eigenfrequency |

1.0 | 233.433 |

2.0 | 236.206 |

3.0 | 251.691 |

4.0 | 362.203 |

5.0 | 364.132 |

You can start exploring all the different structural analysis types with SimScale. You can leverage cloud computing and simulate hundreds of designs in parallel on the cloud without the need for any sophisticated hardware — right from your web browser.

If you liked this article and you are new to FEA, then do not miss our FEA Guide which will give you enough material to get you started. Try our structural and thermomechanical tutorials now and explore the different plans that can bring you onboard.

References

- “Wikimedia commons: Stress Strain Ductile Material.”
- T.H.G. Megson, Aircraft Structures for Engineering Students

Last updated: December 27th, 2022

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