Formulas

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(a) Development Length

L_{dc} = (\frac{f_{y} \cdot ψ_{r}}{50 \cdot λ \cdot \sqrt{{f^{'}}_{c}}}) \cdot d_{b} = (\frac{(60000 psi) \cdot (1.0)}{50 \cdot (1.0) \cdot \sqrt{4000 psi}}) \cdot (0.75 in) = 14.23 in

(b) Development Length

L_{dc} = 0.0003 \cdot f_{y} \cdot ψ_{r} \cdot d_{b} = 0.0003 \cdot (60000 psi) \cdot (1.0) \cdot (0.75 in) = 13.5 in

1. Iteration

{}^{t + Δ}R -^{t + Δt} F^{0}

100/30 Rule

E_{Ed} = 1, 0 \cdot E_{EdX} \oplus 0, 3 \cdot E_{EdY} \oplus 0, 3 \cdot E_{EdZ}

E_{Ed} = 0, 3 \cdot E_{EdX} \oplus 1, 0 \cdot E_{EdY} \oplus 0, 3 \cdot E_{EdZ}

E_{Ed} = 0, 3 \cdot E_{EdX} \oplus 0, 3 \cdot E_{EdY} \oplus 1, 0 \cdot E_{EdZ}

1st Core Width (Rhombus) of Biaxial Eccentricity

|e_{y}| \leq - \frac{\frac{w_{y}}{6}}{\frac{w_{x}}{6}} \cdot |e_{x}| + \frac{w_{y}}{6}

2nd Core Width (Ellipse) of Biaxial Eccentricity

e_{y} \leq \frac{w_{y}}{3} \cdot \sqrt{1 - {(\frac{e_{x}}{\frac{w_{x}}{3}})}^{2}}

Absolute Sum (Combination of Directions)

E_{Ed} = 1, 0 \cdot E_{EdX} \oplus 1, 0 \cdot E_{EdY} \oplus 1, 0 \cdot E_{EdZ}

Absolute Sum (Modal Superposition)

E_{AbsSum} = \sum_{i = 1}^{p} |E_{i}|

Acceleration: (Central Difference)

{\ddot{x}}_{n} = \frac{{\dot{x}}_{n + 0.5} - {\dot{x}}_{n - 0.5}}{{Δt}_{n}}

Accidental Eccentricity

e_{a} = e_{tot} - e_{int}

Accuracy

Δt < \frac{T}{20} = \frac{1}{20 f} = \frac{π}{10 ω}

Action Combination as per Eq.6.69

(\sum_{j \geq 1} G_{k, j} " + " P " + " ψ_{1, 1} Q_{k, 1} " + " \sum_{i > 1} ψ_{2, i} Q_{k, i}) " + " Q_{f a t}

Action Combination for Permanent and Transient Loading

E_{d} = \sum_{j \geq 1} γ_{G, j} \cdot E_{Gk, j} + γ_{Q, 1} \cdot E_{Qk, 1} + \sum_{i > 1} γ_{Q, i} \cdot ψ_{0, i} \cdot E_{Qk, i}

Action Combination SLS Frequent

\sum_{j \geq 1} G_{k, j} " + " P " + " ψ_{1, 1} Q_{k, 1} " + " \sum_{i > 1} ψ_{2, i} Q_{k, i}

Additional axial force and moment from shrinkage in reinforcement

n_{sh, ϕ} = - ε_{sh} \cdot E_{s} (a_{s 1} + a_{s 2})

m_{sh, ϕ} = n_{sh} \cdot e_{sh}

Additional Moment from Shift of Centroid

∆ M_{y, Ed} = N_{Ed} \cdot e_{Ny} = 130 \cdot 8.78 \cdot 10^{- 3} = 1.14 kNm

Adjusted Bending Design Value

\begin{array}{l} F'_{b} = F_{b} \cdot C_{D} \cdot C_{M} \cdot C_{t} \cdot C_{L} \cdot C_{F} \cdot C_{fu} \cdot C_{i} \cdot C_{r} \\ F'_{b} = 1192.32 psi \end{array}

Adjusted Bending Design Value

F_{b} = f_{b} \cdot (K_{D} \cdot K_{H} \cdot K_{s} \cdot K_{T})

Adjusted Compression Design Value Perpendicular to Grain

F'_{c ⟂} = F_{c ⟂} \cdot C_{M} \cdot C_{t} \cdot C_{b} \cdot K_{f} \cdot φ

Adjusted Compressive Design Value

F_{c} = f_{c} \cdot (K_{D} \cdot K_{H} \cdot K_{SC} \cdot K_{T})

Adjusted Compressive Design Value Parallel to Grain, Fc'

F_{c}' = F_{cy}' = F_{c} \cdot C_{D} \cdot C_{M} \cdot C_{t} \cdot C_{F} \cdot C_{i} \cdot C_{P}

F_{c}' = F_{cy}' = 1, 450 psi \cdot 1.6 \cdot 1.0 \cdot 1.0 \cdot 1.0 \cdot 1.0 \cdot 0.29

F_{c}' = F_{cy}' = 673 psi

Adjusted Major Axis Bending Design Value, Fbx'

F_{bx}' = F_{b 1} = F_{b} \cdot C_{D} \cdot C_{M} \cdot C_{L} \cdot C_{t} \cdot C_{F} \cdot C_{i} \cdot C_{r}

F_{bx}' = F_{b 1} = 1, 100 psi \cdot 1.6 \cdot 1.0 \cdot 0.982 \cdot 1.0 \cdot 1.0 \cdot 1.0 \cdot 1.0

F_{bx}' = F_{b 1} = 1, 729 psi

Adjusted Minor Axis Bending Design Value, Fby'

F_{by}' = F_{b 2} = F_{b} \cdot C_{D} \cdot C_{M} \cdot C_{L} \cdot C_{t} \cdot C_{fu} \cdot C_{F} \cdot C_{i} \cdot C_{r}

F_{by}' = F_{b 2} = 1, 100 psi \cdot 1.6 \cdot 1.0 \cdot 1.0 \cdot 1.0 \cdot 1.1 \cdot 1.0 \cdot 1.0 \cdot 1.0

F_{by}' = F_{b 2} = 1, 936 psi

Adjusted Modulus of Elasticity

\begin{array}{l} E' = E \cdot C_{M} \cdot C_{t} \cdot C_{i} \\ E' = 160, 550 psi \\ E'_{\min} = E_{\min} \cdot C_{M} \cdot C_{t} \cdot C_{i} \cdot C_{T} \\ E'_{\min} = 690, 000 psi \end{array}

Adjusted Reduction Factor According to Annex E for Cross-Section Parts Supported on Two Sides

(a) for cross - section parts supported on two sides :

ρ = \frac{1 - 0.055 \cdot \frac{(3 + ψ)}{{\bar{λ}}_{p, red}}}{{\bar{λ}}_{p, red}} + 0, 18 \cdot \frac{({\bar{λ}}_{p} - {\bar{λ}}_{p, red})}{({\bar{λ}}_{p} - 0.6)} \leq 1.0

(b) for cross - section parts under compression supported on one side :

ρ = \frac{1 - \frac{0.188}{{\bar{λ}}_{p, red}}}{{\bar{λ}}_{p, red}} + 0, 18 \cdot \frac{({\bar{λ}}_{p} - {\bar{λ}}_{p, red})}{({\bar{λ}}_{p} - 0.6)} \leq 1.0

Adjustment Factor α₁

α_{1} = {(\frac{35}{f_{cm}})}^{0.7}

Adjustment Factor α₂

α_{2} = {(\frac{35}{f_{cm}})}^{0.2}

Adjustment Factor α_3rd

α_{3} = {(\frac{35}{f_{cm}})}^{0.5}

AISC Weld Limit Stress

\emptyset R_{n} = \emptyset \cdot β \cdot F_{nw} \cdot k_{ds}

\emptyset R_{n} = 0.75 \cdot 0.912 \cdot (0.60 \cdot 70 ksi) \cdot 1.0

\emptyset R_{n} = 28.7 ksi

AISC Weld Limit Stress

\emptyset R_{n} = \emptyset \cdot F_{nw} \cdot k_{ds}

\emptyset R_{n} = 0.75 \cdot (0.60 \cdot 70 ksi) \cdot 1.40

\emptyset R_{n} = 44.1 ksi