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Created: 2026/10/13 08:33

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

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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. $$D:\text{[PARSE ERROR: Undefined("RBrace")]}{I}_{se}=R-(3+r_c)(\text{Minimum}R=3)$$
  • $R<3⟹$ Externally unstable.
  • $R\text{[PARSE ERROR: Undefined("Command("ge")")]}3⟹$ Stable iff reactions are not parallel, concurrent, or collinear.
  • Analytical Rank condition: $\text{rank}\left(\right[A_{eq}\left]\right)=3$ (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: $$C=m-j+1⟹I_{si}=3C-R_r$$
  • $m$: 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

$$\text{Deflection}\text{[PARSE ERROR: Undefined("RBrace")]}v\left(x\right)\text{[PARSE ERROR: Undefined("Command("xrightarrow")")]}\frac{d}{dx}\text{Slope}\text{[PARSE ERROR: Undefined("RBrace")]}\theta \left(x\right)=v^{\prime }\text{[PARSE ERROR: Undefined("Command("xrightarrow")")]}\times EI\frac{d}{dx}\text{Moment}\text{[PARSE ERROR: Undefined("RBrace")]}M\left(x\right)=EI{v}^{\prime }\text{'}\text{[PARSE ERROR: Undefined("Command("xrightarrow")")]}\frac{d}{dx}\text{Shear}\text{[PARSE ERROR: Undefined("RBrace")]}V\left(x\right)=EI{v}^{\prime }{\text{'}}^{\prime }\text{[PARSE ERROR: Undefined("Command("xrightarrow")")]}\frac{d}{dx}\text{Load}\text{[PARSE ERROR: Undefined("RBrace")]}p\left(x\right)=EI{v}^{\prime }{\text{'}}^{\prime }\text{'}$$

• Governing ODE: $EI\frac{d^{4}v\left(x\right)}{d{x}^4}=p\left(x\right)$.
• Homogeneous solution ($p=0$): $v\left(x\right)=C_1{x}^3+C_2{x}^2+C_{3}x+C_4$ (Cubic Hermite polynomial).
• End Moment Derivation: Applying ${\theta }_1=1$, all other end displacements zero $⟹M_1=\frac{4EI}{L},M_2=\frac{2EI}{L},V=\pm \frac{6EI}{L^2}$.
• 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):

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

3.2 Instant Memorization Rules

1. The 4 Base Numbers & $L$-Powers:
   • Shear $\leftrightarrow$ Shear (Odd $\times$ Odd): $12/L^3$
   • Shear $\leftrightarrow$ Moment (Mixed): $6/L^2$
   • Moment $\leftrightarrow$ Same End (Main Diag): $4/\mathbf{L}$
   • Moment $\leftrightarrow$ Far End (Carryover): $2/\mathbf{L}$
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) $=(+)$ Hogging (Tension top) $=(-)$	Left End ($A$): $M_{BMD}=-M_{\text{matrix}}$ Right End ($C$): $M_{BMD}=+M_{\text{matrix}}$

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

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5. Direct Stiffness Formulation & Matrix Partitioning

5.1 Superposition Equivalence

$$P_j=P_m+KU⟺Q=Q_f+ku$$

• $P_j,Q$: 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 $⟹$ Swap rows of $P_j$, $P_m$, and $K$. Vector $U$ order remains unchanged.
• Column Swap (Variable Swap): Swaps variables $⟹$ 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 ($1\text{[PARSE ERROR: Undefined("Command("dots")")]}{n}_u$), then restrained DOFs ($n_u+1\text{[PARSE ERROR: Undefined("Command("dots")")]}n$).
• Direct assembly automatically generates the partitioned blocks: $$\left[\begin{array}{c}{P}_{j,u}\\ P_{j,r}\end{array}\right]=\left[\begin{array}{c}{P}_{m,u}\\ P_{m,r}\end{array}\right]+\left[\begin{array}{cc}{K}_{uu}& K_{ur}\\ K_{ru}& K_{rr}\end{array}\right]\left[\begin{array}{c}{U}_u\\ U_r\end{array}\right]$$
• 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 ($\mathrm{\Delta },{\theta }_r$): 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 ($M_1,M_2$): $$\left[\begin{array}{c}{M}_1\\ M_2\end{array}\right]v={\left[\begin{array}{c}{M}_{1}v\\ M_{2}v\end{array}\right]}_{6\times 1}$$

• 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: $$\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]=\left[\begin{array}{c}{A}_{11}{B}_1+A_{12}{B}_2\\ A_{21}{B}_1+A_{22}{B}_2\end{array}\right]$$

• 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 ${\mathrm{\Delta }}_z$): 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: $GJ\approx \frac{1}{3}EI$.

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8. 2D Truss Analysis via Direct Stiffness

• Assumptions: Pin-connected, axial force only ($P=\frac{AE}{L}\mathrm{\Delta }L$), 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 $u_x=1⟹$ Axial elongation $\mathrm{\Delta }L=\cos \gamma$.
  2. Constitutive: Axial force $F=\frac{AE}{L}\mathrm{\Delta }L=\frac{AE}{L}\cos \gamma$.
  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: $$k_e=\frac{AE}{L}\left[\begin{array}{cccc}{c}^2& cs& -c^2& -cs\\ cs& s^2& -cs& -s^2\\ -c^2& -cs& c^2& cs\\ -cs& -s^2& cs& s^2\end{array}\right](c=\cos \gamma ,s=\sin \gamma )$$

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9. Inextensible Frames & Benchmark Portal Frame

9.1 Inextensibility ($EA\to \mathrm{\infty }$)

• Member axial deformation $\mathrm{\Delta }L=\frac{PL}{EA}=0$.
• Area $A$ is extraneous data / distractor: Drop all $\frac{EA}{L}$ terms.
• Columns do not shorten vertically $⟹$ joints at beam level have ${\mathrm{\Delta }}_y=0$.
• Beams do not stretch $⟹$ horizontal sidesway is identical across all joints at that floor level (${\mathrm{\Delta }}_x=\mathrm{\Delta }$).

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 $37.5\text{kN}m\text{[PARSE ERROR: Undefined("RBrace")]}(\downarrow )$ on Member 2.
  • $E=200\text{GPa},I=375\times {10}^6{\text{mm}}^4⟹EI=75,000\text{kN}·{\text{m}}^2$.

• Unknowns (DOKI = 3): $[{\theta }_2,{\theta }_3,\mathrm{\Delta }{]}^T$.

• Fixed-End Moments (Beam 2–3): $$FE{M}_{23}=-\frac{w{L}^2}{12}=-200\text{kN}·\text{m},FE{M}_{32}=+200\text{kN}·\text{m}$$

• Slope-Deflection Equations: $$M_{21}=EI\left(\frac{2}{3}{\theta }_2-\frac{1}{6}\mathrm{\Delta }\right),M_{12}=EI\left(\frac{1}{3}{\theta }_2-\frac{1}{6}\mathrm{\Delta }\right)$$ $$M_{34}=EI\left(\frac{2}{3}{\theta }_3-\frac{1}{6}\mathrm{\Delta }\right),M_{43}=EI\left(\frac{1}{3}{\theta }_3-\frac{1}{6}\mathrm{\Delta }\right)$$ $$M_{23}=-200+EI({\theta }_2+0.5{\theta }_3),M_{32}=+200+EI(0.5{\theta }_2+{\theta }_3)$$

• Equilibrium System ($K_{uu}U=P_{\text{net}}$): $$EI\left[\begin{array}{ccc}\frac{5}{3}& \frac{1}{2}& -\frac{1}{6}\\ \left[4pt\right]\frac{1}{2}& \frac{5}{3}& -\frac{1}{6}\\ \left[4pt\right]-\frac{1}{6}& -\frac{1}{6}& \frac{1}{9}\end{array}\right]\left[\begin{array}{c}{\theta }_2\\ {\theta }_3\\ \mathrm{\Delta }\end{array}\right]=\left[\begin{array}{c}200\\ -200\\ 100\end{array}\right]$$ (Row 3 is Story Shear: $H_1+H_4=\frac{M_{12}+M_{21}}{6}+\frac{M_{43}+M_{34}}{6}=100$)

• Displacement Solutions: $$EI{\theta }_2=+\frac{1830}{7}\approx +261.43\text{kN}·\text{m}⟹{\theta }_2=+0.003486\text{rad}$$ $$EI{\theta }_3=-\frac{570}{7}\approx -81.43\text{kN}·\text{m}⟹{\theta }_3=-0.001086\text{rad}$$ $$EI\mathrm{\Delta }=1170\text{kN}·{\text{m}}^2⟹\mathrm{\Delta }=0.0156\text{m}=15.6\text{mm}$$

• Final Support Reactions:

  • Support 1: $H_1=21.43\text{kN}(\leftarrow ),V_1=116.25\text{kN}(\uparrow ),M_1=107.86\text{kN}·\text{m}(\text{CW}\circlearrowright )$
  • Support 4: $H_4=78.57\text{kN}(\leftarrow ),V_4=183.75\text{kN}(\uparrow ),M_4=222.14\text{kN}·\text{m}(\text{CW}\circlearrowright )$

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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 ($M_{ij}=FE{M}_{ij}+\frac{4EI}{L}{\theta }_i+\frac{2EI}{L}{\theta }_j-\frac{6EI}{L^2}\mathrm{\Delta }$).
   • Calculate shears and reactions using simple free-body statics ($\sum M=0,\sum F_y=0$) in under 3 minutes.