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{(60, 000 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 .

{}^{t + ∆ t}R - {}^{t + ∆ t}F^{0}

E_{E d} = 1, 0 \cdot E_{E d X} \oplus 0, 3 \cdot E_{E d Y} \oplus 0, 3 \cdot E_{E d Z}

E_{E d} = 0, 3 \cdot E_{E d X} \oplus 1, 0 \cdot E_{E d Y} \oplus 0, 3 \cdot E_{E d Z}

E_{E d} = 0, 3 \cdot E_{E d X} \oplus 0, 3 \cdot E_{E d Y} \oplus 1, 0 \cdot E_{E d Z}

Absolute Sum (Combination of Directions)

E_{E d} = 1.0 \cdot E_{E d X} \oplus 1.0 \cdot E_{E d Y} \oplus 1.0 \cdot E_{E d Z}

Absolute Sum (Modal Superposition)

E_{A b s S u m} = \sum_{i = 1}^{p} |E_{i}|

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 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}

Adjusted Compression Design Value Perpendicular to Grain

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

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 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

Allowable Deformation

\frac{L}{500} = \frac{18 m}{500} = 36.00 mm \geq 4.74 mm

Angle α_- Between Local Axis x (or y) and Principal Axis 1 (or 2) for Strains on Negative Surface Side

α_{-} = \frac{1}{2} [\arctan (\frac{γ_{x y, -}}{ε_{x, -} - ε_{y, -}})]

Angle α_- Between Local Axis x (or y) and Principal Axis 1 (or 2) for Stresses on Negative Surface Side

α_{-} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, -}}{σ_{x, -} - σ_{y, -}}) \in (- 90 °, 90 °]

Angle α₊ Between Local Axis x (or y) and Principal Axis 1 (or 2) for Strains on Positive Surface Side

α_{+} = \frac{1}{2} [\arctan (\frac{γ_{x y, +}}{ε_{x, +} - ε_{y, +}})]

Angle α₊ Between Local Axis x (or y) and Principal Axis 1 (or 2) for Stresses on Positive Surface Side

α_{+} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, +}}{σ_{x, +} - σ_{y, +}}) \in (- 90 °, 90 °]

Angle α_b Between Local Axis x (or y) and Principal Axis 1 (or 2)

α_{b} = \frac{1}{2} [\arctan (\frac{2 m_{x y}}{m_{x} - m_{y}})]

Angle α_m Between Axis x and Principal Axis 1 (for Axial Force n₁)

α_{m} = \frac{1}{2} [\arctan (\frac{2 n_{x y}}{n_{x} - n_{y}})]

Angle α_m Between Local Axis x and Principal Axis 1 for Membrane Stresses

α_{m} = \frac{1}{2} \arctan 2 (\frac{2 τ_{x y, m}}{σ_{x, m} - σ_{y, m}}) \in (- 90 °, 90 °]

Angle β_b Between Principal Shear Force v-max,b and Local Axis x

β_{b} = \arctan \frac{v_{y}}{v_{x}}

Applicable Shear Force

η = 0.8

F_{s, k} = η \cdot E_{a h, k} \cdot \tan φ_{α}

F_{s, k} = 0.8 \cdot (0.5 \cdot 7.1 k N / m^{2} \cdot 1.5 m \cdot \tan \frac{2}{3} \cdot 30 ° + 0.5 \cdot (7.1 k N / m^{2} + 20.5 k N / m^{2}) \cdot 5.3 m \cdot \tan \frac{2}{3} \cdot 30 ° + 0.5 \cdot (18.3 k N / m^{2} + 27.9 k N / m^{2}) \cdot 3.2 m \cdot \tan \frac{2}{3} \cdot 32.5 °

F_{s, k} = 46.32 k N / m

Average Value of Pressure

P = \frac{\sum_{i} P_{i} \cdot A_{i}}{\sum_{i} A_{i}}

Axial force - calculated by standard SRSS rule

N_{SRSS} = \sqrt{{(1.361 kN)}^{2} + {(- 0.246 kN)}^{2} + {(0.815 kN)}^{2} + {(- 2.322 kN)}^{2}} = 2.823 kN

Axial Force n₁ in Direction of Principal Axis 1

n_{1} = \frac{1}{2} [n_{x} + n_{y} + \sqrt{{(n_{x} - n_{y})}^{2} + 4 {n_{x y}}^{2}}]

Axial Force n₂ in Direction of Principal Axis 2

n_{2} = \frac{1}{2} [n_{x} + n_{y} - \sqrt{{(n_{x} - n_{y})}^{2} + 4 {n_{x y}}^{2}}]

Axial Force n_x in Direction of Local x-Axis

n_{x} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{x} d z

Axial Force n_y in Direction of Local y-Axis

n_{y} = \int_{- \frac{d}{2}}^{+ \frac{d}{2}} σ_{y} d z

Axis Transformation

T = [\begin{matrix} 1 & 0 & 0 & 0 & (Z - Z_{0}) & - (Y - Y_{0}) \\ 0 & 1 & 0 & - (Z - Z_{0}) & 0 & (X - X_{0}) \\ 0 & 0 & 1 & (Y - Y_{0}) & - (X - X_{0}) & 0 \\ 0 & 0 & 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 1 \end{matrix}] \begin{matrix}  \end{matrix}