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Created: 2026/10/13 08:33
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1. [1. Static Indeterminacy & Stability](#1.%20Static%20Indeterminacy%20&%20Stability)
	1. [1.1 Total, External, and Internal DOSI](#1.1%20Total,%20External,%20and%20Internal%20DOSI)
	2. [1.2 Released Reactions ($R_r$ / Equations of Condition)](#1.2%20Released%20Reactions%20($R_r$%20/%20Equations%20of%20Condition))
2. [2. Governing Beam Relations & Kinematics](#2.%20Governing%20Beam%20Relations%20&%20Kinematics)
	1. [2.1 Euler-Bernoulli Hierarchy](#2.1%20Euler-Bernoulli%20Hierarchy)
3. [3. $4 \times 4$ Beam Element Stiffness Matrix (10-Second Recall)](#3.%20$4%20%5Ctimes%204$%20Beam%20Element%20Stiffness%20Matrix%20(10-Second%20Recall))
	1. [3.1 Matrix Formulation](#3.1%20Matrix%20Formulation)
	2. [3.2 Instant Memorization Rules](#3.2%20Instant%20Memorization%20Rules)
4. [4. Sign Conventions: Coordinate vs. BMD](#4.%20Sign%20Conventions:%20Coordinate%20vs.%20BMD)
5. [5. Direct Stiffness Formulation & Matrix Partitioning](#5.%20Direct%20Stiffness%20Formulation%20&%20Matrix%20Partitioning)
	1. [5.1 Superposition Equivalence](#5.1%20Superposition%20Equivalence)
	2. [5.2 Row & Column Interchange Rules](#5.2%20Row%20&%20Column%20Interchange%20Rules)
	3. [5.3 Exam Assembly Shortcut (Avoid Manual Swapping)](#5.3%20Exam%20Assembly%20Shortcut%20(Avoid%20Manual%20Swapping))
6. [6. Calculator Matrix Manipulation Tricks](#6.%20Calculator%20Matrix%20Manipulation%20Tricks)
	1. [6.1 Multiplying $(6 \times 3) \times (3 \times 1)$](#6.1%20Multiplying%20$(6%20%5Ctimes%203)%20%5Ctimes%20(3%20%5Ctimes%201)$)
	2. [6.2 Multiplying $(6 \times 6) \times (6 \times 1)$](#6.2%20Multiplying%20$(6%20%5Ctimes%206)%20%5Ctimes%20(6%20%5Ctimes%201)$)
7. [7. Plane Grids](#7.%20Plane%20Grids)
8. [8. 2D Truss Analysis via Direct Stiffness](#8.%202D%20Truss%20Analysis%20via%20Direct%20Stiffness)
9. [9. Inextensible Frames & Benchmark Portal Frame](#9.%20Inextensible%20Frames%20&%20Benchmark%20Portal%20Frame)
	1. [9.1 Inextensibility ($EA \to \infty$)](#9.1%20Inextensibility%20($EA%20%5Cto%20%5Cinfty$))
	2. [9.2 Benchmark Problem & Equilibrium Formulation](#9.2%20Benchmark%20Problem%20&%20Equilibrium%20Formulation)
10. [10. Software Modeling Nuances (SA6 Kassimali & Ftool)](#10.%20Software%20Modeling%20Nuances%20(SA6%20Kassimali%20&%20Ftool))
11. [11. Exam Time Optimization Protocol (Compress 50 min $\to$ 20 min)](#11.%20Exam%20Time%20Optimization%20Protocol%20(Compress%2050%20min%20$%5Cto$%2020%20min))

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# EWCE 411 (Section A) — Final Exam Quick Recall Summary

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## 1. Static Indeterminacy & Stability

### 1.1 Total, External, and Internal DOSI
* **Total Indeterminacy:** $I_s = I_{se} + I_{si}$
* **External ($I_{se}$):** Excess support reactions over rigid-body equilibrium equations.
  $$\text{2D: } I_{se} = R - (3 + r_c) \qquad (\text{Minimum } R = 3)$$
  * $R < 3 \implies$ Externally unstable.
  * $R \ge 3 \implies$ Stable **iff** reactions are not parallel, concurrent, or collinear.
  * Analytical Rank condition: $\text{rank}([A_{eq}]) = 3$ (2D) or $6$ (3D).
* **Internal ($I_{si}$):** Excess internal member forces after reactions are determined.
  $$I_{si} = 3m - 3j + 3 - R_r$$
* **Closed-Loop (Ring) Method:**
  $$C = m - j + 1 \implies I_{si} = 3C - R_r$$
  * $m$: Number of members, $j$: Number of joints, $C$: Number of closed cells.

### 1.2 Released Reactions ($R_r$ / Equations of Condition)
* **Internal Hinge (pin) connecting $n$ members:** $R_r = n - 1$ (Moment $M = 0$).
* **Internal Shear Release (guided slider):** $R_r = 1$ (Shear $V = 0$).
* **Internal Axial Release:** $R_r = 1$ (Axial $P = 0$).

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## 2. Governing Beam Relations & Kinematics

### 2.1 Euler-Bernoulli Hierarchy
$$\text{Deflection: } v(x) \xrightarrow{\frac{d}{dx}} \text{Slope: } \theta(x) = v' \xrightarrow{\times EI\frac{d}{dx}} \text{Moment: } M(x) = EIv'' \xrightarrow{\frac{d}{dx}} \text{Shear: } V(x) = EIv''' \xrightarrow{\frac{d}{dx}} \text{Load: } p(x) = EIv''''$$
* **Governing ODE:** $EI \frac{d^4 v(x)}{dx^4} = p(x)$.
* **Homogeneous solution ($p=0$):** $v(x) = C_1 x^3 + C_2 x^2 + C_3 x + C_4$ (Cubic Hermite polynomial).
* **End Moment Derivation:** Applying $\theta_1 = 1$, all other end displacements zero $\implies M_1 = \frac{4EI}{L}, \; M_2 = \frac{2EI}{L}, \; V = \pm\frac{6EI}{L^2}$.
* **Prismatic Member:** $A, I, E$ remain constant along the entire length.
* **Compatibility Conditions:** Geometric continuity constraints ensuring no gaps, voids, or boundary constraint violations.

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## 3. $4 \times 4$ Beam Element Stiffness Matrix (10-Second Recall)

### 3.1 Matrix Formulation
Degrees of Freedom: $[v_1, \; \theta_1, \; v_2, \; \theta_2]^T$ (Shear, Moment, Shear, Moment):

$$k_e = EI \begin{bmatrix}
\frac{12}{L^3} & \frac{6}{L^2} & -\frac{12}{L^3} & \frac{6}{L^2} \\[6pt]
\frac{6}{L^2} & \frac{4}{L} & -\frac{6}{L^2} & \frac{2}{L} \\[6pt]
-\frac{12}{L^3} & -\frac{6}{L^2} & \frac{12}{L^3} & -\frac{6}{L^2} \\[6pt]
\frac{6}{L^2} & \frac{2}{L} & -\frac{6}{L^2} & \frac{4}{L}
\end{bmatrix}$$

### 3.2 Instant Memorization Rules
1. **The 4 Base Numbers & $L$-Powers:**
   * Shear $\leftrightarrow$ Shear (Odd $\times$ Odd): $\mathbf{12 / L^3}$
   * Shear $\leftrightarrow$ Moment (Mixed): $\mathbf{6 / L^2}$
   * Moment $\leftrightarrow$ Same End (Main Diag): $\mathbf{4 / L}$
   * Moment $\leftrightarrow$ Far End (Carryover): $\mathbf{2 / L}$
2. **"Index 3" Sign Rule:**
   * Every single entry is **POSITIVE (+)** by default.
   * Only **Row 3** and **Column 3** are **NEGATIVE ($-$)**.
   * Diagonal entry $(3,3)$ remains **POSITIVE (+)**.

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## 4. Sign Conventions: Coordinate vs. BMD

| System | Basis | Positive Rule | Conversion to BMD |
| :--- | :--- | :--- | :--- |
| **Matrix / Coordinate** | Directional Rotation | Counterclockwise (CCW) $= (+)$ everywhere | — |
| **BMD (Mechanics)** | Fiber Stress / Curvature | Sagging (Tension bottom) $= (+)$<br>Hogging (Tension top) $= (-)$ | **Left End ($A$):** $M_{BMD} = -M_{\text{matrix}}$<br>**Right End ($C$):** $M_{BMD} = +M_{\text{matrix}}$ |

* **Cut Statics Proof:**
  * Left cut: $\sum M_{\text{cut}} = 0 \implies M(0) + M_A = 0 \implies M_{BMD} = -M_A$
  * Right cut: $\sum M_{\text{cut}} = 0 \implies -M(L) + M_C = 0 \implies M_{BMD} = +M_C$

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## 5. Direct Stiffness Formulation & Matrix Partitioning

### 5.1 Superposition Equivalence
$$P_j = P_m + K U \iff Q = Q_f + k u$$
* $P_j, Q$: Total joint / node forces.
* $P_m, Q_f$: Restraint forces due to member span loads (Fixed-End Actions).
* $K, k$: Structural / element stiffness.
* $U, u$: Nodal displacements / rotations.

### 5.2 Row & Column Interchange Rules
* **Row Swap (Equation Swap):** Swaps equations $\implies$ Swap rows of $P_j$, $P_m$, and $K$. **Vector $U$ order remains unchanged.**
* **Column Swap (Variable Swap):** Swaps variables $\implies$ Swap columns of $K$ **AND** swap corresponding rows of $U$. $P_j$ and $P_m$ remain unchanged.

### 5.3 Exam Assembly Shortcut (Avoid Manual Swapping)
* **Rule:** Number **unconstrained DOFs first** ($1 \dots n_u$), then **restrained DOFs** ($n_u+1 \dots n$).
* Direct assembly automatically generates the partitioned blocks:
  $$\begin{bmatrix} P_{j,u} \\ P_{j,r} \end{bmatrix} = \begin{bmatrix} P_{m,u} \\ P_{m,r} \end{bmatrix} + \begin{bmatrix} K_{uu} & K_{ur} \\ K_{ru} & K_{rr} \end{bmatrix} \begin{bmatrix} U_u \\ U_r \end{bmatrix}$$
* **Solving Displacements:**
  $$U_u = K_{uu}^{-1} (P_{j,u} - P_{m,u} - K_{ur} U_r)$$
  *(If unyielding supports: $U_r = 0$, so $U_u = K_{uu}^{-1}(P_{j,u} - P_{m,u})$)*
* **Solving Support Reactions:**
  $$P_{j,r} = P_{m,r} + K_{ru} U_u + K_{rr} U_r$$
* **Support Settlement ($\Delta, \theta_r$):** Enter settlement values directly into $U_r$.

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## 6. Calculator Matrix Manipulation Tricks

### 6.1 Multiplying $(6 \times 3) \times (3 \times 1)$
Partition $M$ horizontally into two $3 \times 3$ matrices ($M_1, M_2$):
$$\begin{bmatrix} M_1 \\ M_2 \end{bmatrix} v = \begin{bmatrix} M_1 v \\ M_2 v \end{bmatrix}_{6 \times 1}$$
* Compute $M_1 v \to$ top 3 values. Compute $M_2 v \to$ bottom 3 values.

### 6.2 Multiplying $(6 \times 6) \times (6 \times 1)$
Partition into $3 \times 3$ and $3 \times 1$ blocks:
$$\begin{bmatrix} A_{11} & A_{12} \\ A_{21} & A_{22} \end{bmatrix} \begin{bmatrix} B_1 \\ B_2 \end{bmatrix} = \begin{bmatrix} A_{11} B_1 + A_{12} B_2 \\ A_{21} B_1 + A_{22} B_2 \end{bmatrix}$$
* **Key requirement:** Compute each dot product pair and **add them together**; do not just stack.

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## 7. Plane Grids

* **Definition:** All members lie in $x-y$ plane; all loads act **normal to plane** (along $z$-axis) or moments about in-plane axes.
* **In-Plane DOFs ($u_x, u_y, \theta_z$):** Decoupled and identically zero.
* **Active DOFs (3 per joint):**
  1. $w$ (or $\Delta_z$): Out-of-plane translation.
  2. $\theta_x$: Rotation about $x$-axis (twists member along $x$, bends member along $y$).
  3. $\theta_y$: Rotation about $y$-axis (bends member along $x$, twists member along $y$).
* **Internal Actions:** Bending moment ($M$), Torsional moment ($T$), Out-of-plane shear ($V_z$). Axial force $P = 0$.
* **Rigidity Ratio ($GJ = EI/3$):**
  * $G = \frac{E}{2(1+\nu)} \approx 0.38E - 0.4E$.
  * Torsion constant $J$ for rectangular sections is significantly smaller than flexural inertia $I$.
  * Typical standard design/course assumption: $GJ \approx \frac{1}{3} EI$.

---

## 8. 2D Truss Analysis via Direct Stiffness

* **Assumptions:** Pin-connected, axial force only ($P = \frac{AE}{L}\Delta L$), loads at joints only ($P_m = 0$).
* **DOFs per joint:** 2 translational ($u_x, u_y$).
* **Transformation & Derivation of $c^2, cs, s^2$:**
  1. **Kinematics:** Unit displacement $u_x = 1 \implies$ Axial elongation $\Delta L = \cos\gamma$.
  2. **Constitutive:** Axial force $F = \frac{AE}{L} \Delta L = \frac{AE}{L} \cos\gamma$.
  3. **Statics:** Resolving back to global axes:
     * $F_x = F \cos\gamma = \frac{AE}{L}\cos^2\gamma$
     * $F_y = F \sin\gamma = \frac{AE}{L}\cos\gamma\sin\gamma$
* **Global Member Stiffness Matrix:**
  $$k_e = \frac{AE}{L} \begin{bmatrix}
  c^2 & cs & -c^2 & -cs \\
  cs & s^2 & -cs & -s^2 \\
  -c^2 & -cs & c^2 & cs \\
  -cs & -s^2 & cs & s^2
  \end{bmatrix} \qquad (c = \cos\gamma, \; s = \sin\gamma)$$

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## 9. Inextensible Frames & Benchmark Portal Frame

### 9.1 Inextensibility ($EA \to \infty$)
* Member axial deformation $\Delta L = \frac{PL}{EA} = 0$.
* **Area $A$ is extraneous data / distractor:** Drop all $\frac{EA}{L}$ terms.
* Columns do not shorten vertically $\implies$ joints at beam level have $\Delta_y = 0$.
* Beams do not stretch $\implies$ horizontal sidesway is identical across all joints at that floor level ($\Delta_x = \Delta$).

### 9.2 Benchmark Problem & Equilibrium Formulation
* **Frame:** Fixed bases at 1 $(0,0)$ and 4 $(8,0)$. Joint 2 at $(0,6)$, Joint 3 at $(8,6)$.
  * Member 1 (Col 1–2): $L=6\text{ m}, I$.
  * Member 2 (Beam 2–3): $L=8\text{ m}, 2I$.
  * Member 3 (Col 3–4): $L=6\text{ m}, I$.
  * Load: $100\text{ kN} (\to)$ at joint 2; UDL $37.5\text{ kN/m} (\downarrow)$ on Member 2.
  * $E = 200\text{ GPa}, I = 375 \times 10^6\text{ mm}^4 \implies EI = 75,000\text{ kN}\cdot\text{m}^2$.
* **Unknowns (DOKI = 3):** $[\theta_2, \; \theta_3, \; \Delta]^T$.
* **Fixed-End Moments (Beam 2–3):**
  $$FEM_{23} = -\frac{wL^2}{12} = -200\text{ kN}\cdot\text{m}, \qquad FEM_{32} = +200\text{ kN}\cdot\text{m}$$
* **Slope-Deflection Equations:**
  $$M_{21} = EI\left(\frac{2}{3}\theta_2 - \frac{1}{6}\Delta\right), \qquad M_{12} = EI\left(\frac{1}{3}\theta_2 - \frac{1}{6}\Delta\right)$$
  $$M_{34} = EI\left(\frac{2}{3}\theta_3 - \frac{1}{6}\Delta\right), \qquad M_{43} = EI\left(\frac{1}{3}\theta_3 - \frac{1}{6}\Delta\right)$$
  $$M_{23} = -200 + EI(\theta_2 + 0.5\theta_3), \qquad M_{32} = +200 + EI(0.5\theta_2 + \theta_3)$$
* **Equilibrium System ($K_{uu} U = P_{\text{net}}$):**
  $$EI \begin{bmatrix}
  \frac{5}{3} & \frac{1}{2} & -\frac{1}{6} \\[4pt]
  \frac{1}{2} & \frac{5}{3} & -\frac{1}{6} \\[4pt]
  -\frac{1}{6} & -\frac{1}{6} & \frac{1}{9}
  \end{bmatrix}
  \begin{bmatrix} \theta_2 \\ \theta_3 \\ \Delta \end{bmatrix}
  = \begin{bmatrix} 200 \\ -200 \\ 100 \end{bmatrix}$$
  *(Row 3 is Story Shear: $H_1 + H_4 = \frac{M_{12}+M_{21}}{6} + \frac{M_{43}+M_{34}}{6} = 100$)*

* **Displacement Solutions:**
  $$EI\theta_2 = +\frac{1830}{7} \approx +261.43\text{ kN}\cdot\text{m} \implies \theta_2 = +0.003486\text{ rad}$$
  $$EI\theta_3 = -\frac{570}{7} \approx -81.43\text{ kN}\cdot\text{m} \implies \theta_3 = -0.001086\text{ rad}$$
  $$EI\Delta = 1170\text{ kN}\cdot\text{m}^2 \implies \Delta = 0.0156\text{ m} = 15.6\text{ mm}$$

* **Final Support Reactions:**
  * **Support 1:** $H_1 = 21.43\text{ kN} \;(\leftarrow), \quad V_1 = 116.25\text{ kN} \;(\uparrow), \quad M_1 = 107.86\text{ kN}\cdot\text{m} \;(\text{CW } \circlearrowright)$
  * **Support 4:** $H_4 = 78.57\text{ kN} \;(\leftarrow), \quad V_4 = 183.75\text{ kN} \;(\uparrow), \quad M_4 = 222.14\text{ kN}\cdot\text{m} \;(\text{CW } \circlearrowright)$

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## 10. Software Modeling Nuances (SA6 Kassimali & Ftool)

* **Unit Handling:** SA6 is **dimensionless**; consistency of input units is strictly required.
* **Symbolic / Relative Inertia ($I, 2I$):** Enter dummy numbers ($E=1, I=1, 2I=2$). Reactions and internal moments are scale-invariant and 100% exact.
* **Extensible Default:** Computer software always uses $6 \times 6$ extensible formulations.
* **"Large Area Trick" (Simulating Inextensibility in Software):**
  * Keep true $E$ and $I$.
  * Set cross-sectional area $A$ arbitrarily large ($A = 10^8\text{ mm}^2$ or $10^{10}$).
  * Axial deformation $\frac{PL}{EA} \to 0$, forcing software output to match inextensible hand calculations to the exact decimal.

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## 11. Exam Time Optimization Protocol (Compress 50 min $\to$ 20 min)

1. **Do not write symbolic element matrices:** Compute numbers or decimals directly into Destination Vectors.
2. **Number active/unknown DOFs first:** Eliminates matrix rewrites and row/column permutation steps.
3. **Assemble only necessary submatrices:**
   * Only write $K_{uu}$ ($2 \times 2$ or $3 \times 3$) and solve displacements on calculator.
   * Do not assemble $K_{rr}$ unless explicitly asked by the question.
4. **Member Force & Reaction Recovery:**
   * Fast track: Use slope-deflection member equations to find span end-moments ($M_{ij} = FEM_{ij} + \frac{4EI}{L}\theta_i + \frac{2EI}{L}\theta_j - \frac{6EI}{L^2}\Delta$).
   * Calculate shears and reactions using simple free-body statics ($\sum M = 0, \sum F_y = 0$) in under 3 minutes.