Seismic Base Shear:

\\[ V = C_s \cdot W \\]

Seismic Response Coefficient:

\\[ C_s = \frac{S_{DS} \cdot I_e}{R} \\]

Constrained by:

\\[ C_s \leq \frac{S_{D1}}{T \cdot R / I_e}, \quad C_s \geq 0.01 \\]

Fundamental Period:

\\[ T = C_t \cdot h_n^x \\]

Vertical Distribution of Force:

\\[ F_x = C_{vx} \cdot V, \quad C_{vx} = \frac{w_x \cdot h_x^k}{\sum_{i=1}^n w_i \cdot h_i^k} \\]

Seismic Design Category (SDC):

• SDC A: \\( S_{DS} < 0.167 \\), \\( S_{D1} < 0.067 \\)
• SDC B: \\( S_{DS} < 0.33 \\), \\( S_{D1} < 0.133 \\)
• SDC C: \\( S_{DS} < 0.5 \\), \\( S_{D1} < 0.2 \\)
• SDC D: \\( S_{DS} < 0.75 \\), \\( S_{D1} < 0.3 \\)
• SDC E/F: Higher values or critical facilities

Where:

• \\( V \\): Base shear force (kN)
• \\( C_s \\): Seismic response coefficient
• \\( W \\): Effective seismic weight (kN)
• \\( S_{DS} \\): Design spectral acceleration, short period (g)
• \\( S_{D1} \\): Design spectral acceleration, 1-second period (g)
• \\( I_e \\): Importance factor
• \\( R \\): Response modification factor
• \\( T \\): Fundamental period (s)
• \\( C_t \\): Period coefficient (0.028 for steel, 0.016 for concrete, 0.02 for others)
• \\( h_n \\): Structure height (m)
• \\( x \\): Period exponent (0.8 for steel/concrete, 0.75 for others)
• \\( F_x \\): Force at level \\( x \\) (kN)
• \\( C_{vx} \\): Vertical distribution factor
• \\( w_x \\): Weight at level \\( x \\) (kN)
• \\( h_x \\): Height of level \\( x \\) (m)
• \\( k \\): Distribution exponent (1 to 2)

\\[ T = 0.028 \cdot 20^{0.8} \approx 0.49 \ \text{s} \\] \\[ C_s = \frac{0.5 \cdot 1.0}{8} = 0.0625 \\] \\[ C_s \leq \frac{0.2}{0.49 \cdot 8 / 1.0} \approx 0.051, \quad C_s = 0.051 \\] \\[ V = 0.051 \cdot 5000 \approx 255 \ \text{kN} \\] \\[ k = 1 \ (T \leq 0.5) \\] \\[ C_{v5} = \frac{1000 \cdot (4 \cdot 5)^1}{\sum (1000 \cdot (4i)^1)} \approx 0.333, \quad F_5 = 0.333 \cdot 255 \approx 85 \ \text{kN} \\]

SDC: C (\\( S_{DS} = 0.5 \\), \\( S_{D1} = 0.2 \\))

\\[ T = 0.016 \cdot 10^{0.8} \approx 0.28 \ \text{s} \\] \\[ C_s = \frac{0.3 \cdot 1.0}{5} = 0.06 \\] \\[ C_s \leq \frac{0.12}{0.28 \cdot 5 / 1.0} \approx 0.086, \quad C_s = 0.06 \\] \\[ V = 0.06 \cdot 3000 \approx 180 \ \text{kN} \\] \\[ k = 1 \ (T \leq 0.5) \\] \\[ C_{v3} = \frac{1000 \cdot (3.33 \cdot 3)^1}{\sum (1000 \cdot (3.33i)^1)} \approx 0.5, \quad F_3 = 0.5 \cdot 180 \approx 90 \ \text{kN} \\]

SDC: B (\\( S_{DS} = 0.3 \\), \\( S_{D1} = 0.12 \\))

\\[ T = 0.02 \cdot 40^{0.75} \approx 0.81 \ \text{s} \\] \\[ C_s = \frac{0.7 \cdot 1.25}{5} = 0.175 \\] \\[ C_s \leq \frac{0.28}{0.81 \cdot 5 / 1.25} \approx 0.086, \quad C_s = 0.086 \\] \\[ V = 0.086 \cdot 10000 \approx 860 \ \text{kN} \\] \\[ k = 1.12 \ (T = 0.81) \\] \\[ C_{v10} = \frac{1000 \cdot (4 \cdot 10)^{1.12}}{\sum (1000 \cdot (4i)^{1.12})} \approx 0.208, \quad F_{10} = 0.208 \cdot 860 \approx 179 \ \text{kN} \\]

SDC: D (\\( S_{DS} = 0.7 \\), \\( S_{D1} = 0.28 \\))