Foundations — FEM software — NASTRAN, ABAQUS (concepts and use)

Everything in the parent note (FEM software — NASTRAN, ABAQUS (concepts and use)) is built on a handful of ideas. This page defines every symbol and word from absolute zero, in an order where each one leans only on the ones before it.

1. Displacement — "how far did a point move?"

Imagine drawing a dot on a metal bracket, then pushing on the bracket. The dot slides to a new spot. The little arrow from where the dot was to where it is now is called the displacement.

• Picture: the red arrow in the figure — tail at the old position, head at the new one.
• Why the topic needs it: displacement is the thing we solve for. Everything else (stress, whether the bracket cracks) is computed from how points moved. When the parent writes $\mathbf{u}$ in $\mathbf{Ku}=\mathbf{F}$, this is it.

The bold letter $\mathbf{u}$ means it is not a single number but a list of numbers — one for each direction a point can move. That leads straight to the next idea.

2. Degrees of Freedom (DOF) — "how many ways can it move?"

A single point in 3D space can slide three ways: left–right ($x$), forward–back ($y$), up–down ($z$). Each independent way of moving is one degree of freedom.

3. Nodes and Elements — "chop the smooth thing into pieces"

You cannot store a smooth continuous shape in a computer. So FEM covers the structure with a grid.

• Picture: the smooth L-bracket on the left, the same bracket covered in red quad elements on the right. The mesh approximates the smooth shape.
• Why the topic needs it: we only know how to write physics equations for a simple shape. So we solve simple shapes and glue the answers together. This gluing is called assembly: the element stiffness tables ${\mathbf{k}}_e$ are stacked into one global $\mathbf{K}$ by adding contributions at shared nodes (Section 7 shows where ${\mathbf{k}}_e$ comes from). This is literally the "finite" in Finite Element — finitely many pieces instead of infinitely smooth material.

4. Force $\mathbf{F}$ — "the push"

• Picture: an arrow pushing on a node; its length is the force size.
• Why bold? Same reason as $\mathbf{u}$ — one force number per DOF, stacked into a column.

5. Stiffness $\mathbf{K}$ — "how hard is it to push?"

Push a soft spring and it moves a lot; push a stiff spring and it barely budges. The ratio "force needed per unit of movement" is stiffness.

• Picture: the straight red line $F=ku$. Its slope is $k$. Steeper line = stiffer.
• Why a matrix, not a number? A real structure has many DOF, and pushing node 1 also moves node 2 (they are connected). So we need a table $\mathbf{K}$ where entry $K_{ij}$ says "force at DOF $i$ caused by a unit move of DOF $j$." That table is the stiffness matrix, and the many-spring version of $F=ku$ is exactly $\mathbf{Ku}=\mathbf{F}$.

6. Boundary conditions — "pin it down or it floats away"

A structure floating free in space has nothing to push against: give it any force and the whole thing drifts, so no unique displacement exists. Mathematically, the raw $\mathbf{K}$ is singular (its determinant is zero — you cannot invert it, and $\mathbf{Ku}=\mathbf{F}$ has infinitely many solutions).

7. Stress $\mathbf{\sigma }$, Strain $\mathbf{ϵ}$, and the matrices $\mathbf{N},\mathbf{B},\mathbf{D}$

Displacement tells you how points move. But to know if the metal cracks, you need what is happening inside it.

Now we need to connect the inside quantities (strain, stress) back to the nodal displacements we actually solve for. Two matrices do this, and both must be defined before use.

• Why the colon in $\mathbf{\sigma }:\delta \mathbf{ϵ}$ (parent's virtual-work integral)? Stress and strain are 2D tables (tensors), and "$:$" just means "multiply matching entries and add them up" — it produces energy per volume.

The chain the parent uses is: move the nodes → get strain → get stress → check margin. ${\mathbf{u}}_e\stackrel{\mathbf{B}}{\to }\mathbf{ϵ}\stackrel{\mathbf{D}}{\to }\mathbf{\sigma }$

8. Young's Modulus $E$ and Density $\rho$ — "the material facts"

• Why both $E$ and $\rho$? $E$ (via $\mathbf{D}$) builds the stiffness $\mathbf{K}$; $\rho$ builds the mass $\mathbf{M}$ (next). Stiffness resists bending; mass resists acceleration. Vibration is a fight between the two.

9. Mass $\mathbf{M}$, Frequency $\omega$, and Mode Shape $\mathbf{\varphi }$ — "how it vibrates"

Pluck a structure and it wobbles at special rates. FEM finds those.

• Why it matters for the parent's $f_1\approx 850$ Hz: if launch shaking contains energy at 850 Hz, the bracket resonates and can break. This is the bridge to Vibration and Modal Analysis and Launch Vehicle Loads.

10. Linear vs Nonlinear — "is $\mathbf{K}$ fixed or moving?"

There are three flavours of nonlinearity the parent relies on, and each one is just "$\mathbf{K}$ is no longer constant" for a different physical reason:

• Material nonlinearity: past a certain stress the material stops springing back proportionally (yielding/plasticity, composite damage), so its stiffness drops as load grows.
• Geometric nonlinearity: when parts bend or rotate a lot (deployable booms), the geometry itself changes enough that the stiffness depends on the current shape.
• Contact nonlinearity: two surfaces touch or separate (bolted joints), and stiffness jumps the instant they come into contact.

11. Factor of Safety & Margin — "how much did we over-build?"

Connects to Factors of Safety and Stress Analysis and Margins — the decision stage after $\mathbf{u}$, $\mathbf{\sigma }$ are known.

How these foundations feed the topic

Equipment checklist

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