When it comes to tackling complex engineering challenges, the **Finite Element Method (FEM)** stands as a powerful computational tool. Often interchangeably referred to as **Finite Element Analysis (FEA)**, this method opens a gateway to unraveling the intricacies of physical problems in engineering analysis and design.

## The Essence of Finite Element Method

At its core, FEM involves transforming a real-world engineering problem into a manageable **mathematical model**. Picture this: an actual structure or structural component grappling with various loads becomes the focal point. Now, the magic of FEM lies in the conversion of this tangible challenge into a set of algebraic equations.

##### The Dance of Assumptions and Principles

As we delve into the mathematical realm, certain **assumptions** and **physical principles** come into play. These elements collaborate to mold the differential equations that govern the domain of the mathematical model. Imagine it as a choreography, where each step—each assumption—adds precision to the dance of equations.

##### The Boundary Value Problem Unveiled

Hold onto your hats; we’re about to venture into the heart of FEM—the **Boundary Value Problem (BVP)**. In simple terms, a BVP is a mathematical puzzle demanding that one or more dependent variables adhere to a differential equation throughout a defined domain. The twist? They must also meet specific conditions on the boundary of this domain—cue the term “boundary conditions.”

##### Where Fields and Problems Collide

Field problems, another alias for boundary value problems, bring us to the focal point of interest—the domain. Imagine this domain as the stage, where physical structures take center stage. Here, variables like **displacement**, temperature, heat flux, and fluid velocity emerge as the stars of the show.

##### Bridging the Physical and the Mathematical

As we navigate the terrain of FEM, it’s crucial to grasp the intricate connection between the physical and the mathematical. The domain of interest, the field, dictates the behavior of the dependent variables. The boundary conditions, on the other hand, set the rules these variables must abide by on the edges of the stage.

## Decoding the Jargon: Linear Static Structural Analysis

Embarking on the journey of Finite Element Analysis (FEA), let’s demystify the terminology, starting with **linear static structural analysis**. It sounds fancy, but fear not, we’ll break it down.

##### Unveiling Structural Analysis

Imagine a mechanical component shouldering external loads. Structural analysis dives into the realm of calculating **stress, strain, and displacements**—essentially understanding how a structure responds to outside forces.

##### The Static Symphony

In the realm of static structural analysis, the term “static” takes center stage. Picture external loads being applied gradually, sans the hustle of inertial and damping effects. The part sits in static equilibrium, meaning no wobbles or spins—everything is perfectly balanced.

##### Linear Harmony

Now, let’s talk “**linear**.” In this context, it denotes a straightforward relationship between force and displacement. Picture a graph plotting force against displacement, forming a neat, predictable straight line. To keep things linear, a few conditions must be met. Displacements should be small, steering clear of plastic deformations in the material. Friction-laden contacts are a no-go zone, and other types of contacts too. It’s a delicate dance of constraints.

##### The Field Variable Unveiled

In this analysis, the star of the show is the **displacement**. As external loads work their magic, the focus zeroes in on how the structure gracefully shifts and moves.

## Unveiling Real-world Challenges: A Simple Example

Now that we’ve unraveled the intricacies of linear static structural analysis, let’s dive into a tangible example. Our spotlight turns to a bracket tasked with supporting a vertical load.

This visual encapsulates our physical problem—a bracket under the influence of external forces. The next step? Translating this real-world scenario into a mathematical problem, the elusive **Boundary Value Problem.**

##### Crafting the Mathematical Landscape: Defining the Domain

To embark on this mathematical journey, we first define our **domain**. Picture the geometry of our bracket—simple, yet crucial. In this instance, envision a bracket with a constant, small thickness, bearing the weight of loads solely on the plane normal to the thickness direction. With the previous conditions attended, we opt for the **plane stress condition**. Imagine transforming our 3D solid domain into a 2D planar domain (surface).

##### The Magic of Rigid Column Boundary Conditions

In our journey through linear static structural analysis, we encounter a fascinating twist—the bracket’s connection to a “very thick” steel column. This relative term guides us to assume the column’s rigidity, allowing us to simplify our model using a “rigid column boundary condition.” Imagine it as a magician’s trick, where the column becomes a supporting character, and the bracket takes center stage. By **fixing displacements in the hole areas**, we streamline our analysis, focusing on the bracket’s response to external forces without getting entangled in unnecessary details.

##### Unveiling Linear Static Structural Analysis under Controlled Loads

We extend our assumptions to the application of a **slow load**, denoted as W. This deliberate deceleration prompts a static analysis, steering **clear of dynamic considerations**. The calculated load must remain within limits to prevent material yielding or significant displacements, ensuring a linear analysis. In this delicate balance, the pace of the load dictates the nature of our exploration—methodically unraveling the bracket’s response while maintaining the linear framework of our analysis. This meticulous approach guarantees a comprehensive understanding of structural behavior without succumbing to nonlinear complexities.

As we dissect the intricacies of controlled loads in structural analysis, the spotlight shifts to the symbolic force, load W. Its significance lies not only in its nature as a force but in its representation of a **pin traversing that hole **on the right side of the part. A strategic decision unfolds – we choose not to explicitly model the pin and the contact points. Why? To safeguard the sanctity of our linear assumption. Contacts, when detailed, can usher in non-linear complexities, a detour we purposefully sidestep. In this ballet of precision, load W emerges not just as an influential force but also dons the hat of a **boundary condition**, shaping our exploration into a controlled, linear venture through the realm of structural analysis.

**Unraveling the Math Behind Static Analysis**

Delving into the realm of static analysis unveils a world governed by equations shaped by the physical principles of structural equilibrium. At its core lies Newton’s second law, a guiding force stating F = ma, where acceleration (a) is set to zero. In simpler terms, the summation of all forces in any direction equals zero. While I won’t inundate you with these equations here, it’s crucial to note that they aren’t complex for those well-versed in structural analysis. However, for those unfamiliar, they might seem a tad intimidating. In a nutshell, the defition of our mathemetical problem is the following:

**Navigating the Numerical Landscape: Introducing FEM**

Now that your mathematical model is in place, it’s time to solve the puzzle. Enter the finite element method (FEM), a **numerical** superhero that swoops in when analytical solutions wave the white flag. Yes, it doesn’t provide a perfect solution – there are some introduced errors.

**The Dance of Elements: Demystifying Meshing in FEM**

Wondering how FEM orchestrates its problem-solving symphony? Step one: break down your 2D planar surface into elements. These aren’t your fancy differentials; think **finite-sized** subdomains enclosed by triangles or quadrilaterals. We call this **meshing**, where the mesh becomes the grid guiding FEM’s magic. The figure below shows our domain divided into elements, that is, meshed.

**Triangular Marvels: Understanding Element Configuration**

In our FEM escapade, the choice of elements plays a key role. In this case, say hello to triangular elements, specifically the 3-node linear triangular elements. Picture nodes stationed at the vertices of these triangles, giving us a configuration known as **constant strain triangle**. Now, you might wonder why constant strain? Well, that’s a story for another day. The crucial bit is understanding how these nodes dance across the elements. Take a peek at the highlighted nodes in the image below, and you’ll start to see the symphony of FEM unfold.

Let’s zoom into the world of nodes – the unsung heroes that weave **connections between our elements**. Now, in the realm of structural analysis, **our field variable is displacement**. Here’s the twist: these displacements at each node are unknown, and that’s precisely what we’re on a mission to calculate. In our 2D plane stress scenario, each node can potentially undergo two displacements – one in the vertical (typically y direction) and one in the horizontal (usually x direction).

**Decoding Freedom: The Language of Displacement**

Enter the realm of **degrees of freedom** – a term we use to describe possible displacements at each node. Picture this: at every node, we have two degrees of freedom – one in the vertical and one in the horizontal direction. Sum it up for an element, and you’ve got a total of six degrees of freedom. Now, initially, these displacements are like mystery novels waiting to be unraveled. But let’s say we’ve cracked the code and calculated them somehow. The next chapter poses a new question: how do we extend these calculated displacements beyond the nodes, into the heart of the elements? This is where interpolation waltzes in. Picture known displacements at nodes working hand-in-hand with **interpolation functions**, or **shape functions**. For our 3-node triangle, it’s expressed as:

Here, N_{1}, N_{2} and N_{3} are the interpolation functions, while d_{1}, d_{2} and d_{3} represent the displacements of nodes 1, 2, and 3 respectively. Each interpolation function acts as a weighting factor, determining the influence of each node in the dance of displacement calculation.

**Unveiling Nodal Forces**

The intricate dance of displacement calculation beckons the need for expressions connecting nodal displacements and nodal forces. Enter the concept of **nodal forces** – counterparts to displacements in this numerical waltz. Picture this: just as we can have two possible nodal displacements, we can also entertain two potential nodal forces, mirroring the direction of the displacements. Here’s a key insight in FEA – **forces can only be applied at the nodes of the elements**. But what if forces want to wander over areas or volumes? Fear not; they transform into equivalent nodal forces. There’s a technique behind this transformation, a mystery for another day. For now, let’s acknowledge the intricate relationship unfolding in our triangular element.

**Behind the Curtain: The Principle of Minimum Potential Energy**

As we venture into the labyrinth of force-displacement relations, the intricate details of finding expressions won’t unfold here, for it’s a journey not so straightforward. However, for those with a penchant for the intricate workings, enter the **Principle of Minimum Potential Energy**. This principle, a guiding light in the realm of FEA, guides us through the intricate process. Imagine applying this principle, and voila – expressions emerge, connecting forces and displacements in a dance of equilibrium.

Each nodal force unravels its mystery through a simple equation – the **nodal displacements multiplied by the coefficients a _{ij}**. However, to enhance organization and visual appeal, we transform the six algebraic equations into a more compact matrix notation. This transformation doesn’t just streamline the representation; it’s like donning a well-tailored suit, presenting the equations in a sleek, structured fashion. In matrix notation, our system takes on a refined form, offering a clearer view into the intricate relationships between nodal forces and displacements.

The above expression can also be represented by the matrix equation: **{f} = [k]{u}**. Here, {f} represents the vector matrix of element nodal forces, {u} embodies the vector matrix of element nodal displacements, and [k] stands as the **element stiffness matrix**. This strategy unfolds across the mesh, generating a matrix equation for each element. Yet, the real magic lies in the amalgamation of these equations into a grand expression that considers the connectivity between elements.

**Global Unification: The Emergence of {F}=[K]{U}**

Behold the grandeur of unification as we transition to a comprehensive equation: **{F} = [K]{U}**. The vectors {F} and {U} now encapsulate the nodal forces and displacements of all nodes in the entire part. Picture a part with 1000 nodes, each potentially holding two displacements. Here, {F} and {U} stretch to 2000 lines, each with a solitary column. The protagonist of this equation, [K], steps forward as the **global stiffness matrix – square, symmetric, and singular**, with a realm of order N (total degrees of freedom). In our case, N sways to 2000, rendering [K] a formidable matrix boasting 2000 lines and columns.

**Techniques for Global Matrix Assembly**

While we won’t delve into the technicalities here, it’s essential to acknowledge the existence of techniques that assemble the global stiffness matrix. These methodologies navigate the intricacies of connectivity, recognizing the shared nodes as vital intersections in the matrix’s grand design.

**Boundary Ballet: Applying Constraints and Forces**

As the stage is set, the next act involves applying boundary conditions. Picture all displacements in the left holes rendered as zero, while a force gracefully enters the scene on the right. This force, if distributed over the hole, transforms into equivalent nodal forces in the y direction.

##### Stiffness **Matrix Partition: The Art of Isolation**

Solving the problem requires the delicate art of **matrix partitioning**. Though the details elude this narrative, the essence involves splitting the stiffness matrix into four parts. In our scenario, with fixed displacements as boundary conditions, we embrace one of these parts **[K _{aa}]**. The force vector

**{F**gets a makeover, containing only the known forces on the right hole. Simultaneously, the displacement vector sheds its knowns, leaving only the enigmatic

_{a}}**active displacements (unkown displacements)**, dubbed

**{U**.

_{a}}**{F _{a}} = [K_{aa}]{U_{a}}**

**Active Displacements Revealed**: **Isolating the Unknown with Matrix Inversion**

To unveil the mystery of active displacements (unknown displacements), we perform a symphony of matrix operations. Multiplying both sides of the equation by the inverse of the partitioned stiffness matrix, we isolate **{U _{a}}**. In the realm of matrices, the inverse of a matrix multiplied by itself equals the identity matrix. Thus,

**[K**. The identity matrix multiplied by any matrix is equal to the matrix itself. The result:

_{aa}]⁻¹{F_{a}} =**[K**[K_{aa}]⁻¹_{aa}]{U_{a}} =**[I]**{U_{a}}**[K**. In simpler terms, to find the active displacements, we need to inverse a specific part of the stiffness matrix and multiply it by the vector of known external forces.

_{aa}]⁻¹{F_{a}} = {U_{a}}**Closing Notes: The Essence of FEM**

In essence, this overview provides a glimpse into the workings of Finite Element Method (FEM). While the intricacies of specific procedures may elude this narrative, the fundamental idea stands – FEM, a **powerful numerical tool**, unveils structural insights by weaving together matrices, displacements, forces, and a touch of matrix magic.

**Nodal Displacements Across the Domain**

With nodal displacements intricately woven into the fabric of our analysis, a fascinating journey begins. Leveraging the interpolation technique and the graceful dance of shape functions, we extend our understanding beyond the nodes. Picture a symphony where nodal displacements take center stage, and interpolation elegantly reveals the intricate patterns within the elements. This ballet unfolds in a displacement contour, a visual masterpiece portraying the nuanced movements across the entire structure, offering a holistic view of the calculated displacements.

In our visual symphony, the image above becomes a canvas of insights. The plotted Y-direction displacements, showcased in hues of blue and red, narrate a story of structural response. The maximum displacement of -0.015 mm, a deep blue hue, elegantly marks the impact zone on the right—aligned with the applied downward force. It’s a dance of numbers visualized, where each color whispers the structural secrets it holds.

**Beyond Displacements: Unveiling the Stress Palette**

Now that every nook and cranny holds calculated displacements, our exploration delves deeper. These displacements serve as a gateway to post-process other vital results like **stress and strain**. Take, for instance, the image below unveiling the **von-Mises stress** distribution—an intricate tapestry of structural responses. This journey, initiated by displacements, unravels a narrative where colors represent not just movement but the nuanced stresses woven into the very fabric of our structure.

## Conclusion: **Revealing the Harmonies of Structural Analysis with Finite Element Method (FEM)**

In this journey through the Finite Element Method (FEM), we’ve unveiled the method’s prowess in dissecting complex engineering challenges. By transforming real-world problems into manageable mathematical models, FEM allows us to navigate the intricate dance of forces and displacements, bridging the physical and mathematical realms. The culmination of this numerical symphony lies in the comprehensive equation {F} = [K]{U}, where vectors {F} and {U} encapsulate nodal forces and displacements across the entire structure.

As our exploration extends beyond nodal displacements, we witness the ballet of shape functions and interpolation techniques, creating a visual masterpiece of displacement contours. This canvas not only reveals the calculated movements but also sets the stage for post-processing insights like stress and strain distributions, weaving a holistic narrative of structural behavior.

Would you like to learn more about FEA. EnterFEA’s blog has a very nice post about it. Click here to check this out.

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