Formula Sheet: Load Analysis Dead, Live, Wind, Seismic

Dead Loads

Basic Definitions

• Dead Load (D): The weight of all permanent structural and non-structural components of a building, including walls, floors, roofs, ceilings, stairways, built-in partitions, finishes, cladding, and fixed service equipment.
• Self-weight: The weight of the structural member itself.

Dead Load Calculation

Weight Calculation:

\[W = \gamma \times V\]

• W = weight (lb or kN)
• γ = unit weight or specific weight (lb/ft³ or kN/m³)
• V = volume (ft³ or m³)

For Area-Based Loads:

\[D = w \times A\]

• D = dead load (lb or kN)
• w = load per unit area (psf or kN/m²)
• A = area (ft² or m²)

For Linear Loads:

\[w = \gamma \times A_{cross-section}\]

• w = load per unit length (lb/ft or kN/m)
• A_{cross-section} = cross-sectional area (ft² or m²)

Common Material Unit Weights

• Concrete (normal weight): 145-150 pcf (23-24 kN/m³)
• Concrete (lightweight): 90-120 pcf (14-19 kN/m³)
• Steel: 490 pcf (77 kN/m³)
• Wood (softwood): 30-35 pcf (4.7-5.5 kN/m³)
• Masonry (brick): 120-130 pcf (19-20 kN/m³)
• Soil (dry): 100-130 pcf (16-20 kN/m³)
• Water: 62.4 pcf (9.81 kN/m³)

Live Loads

Basic Definitions

• Live Load (L): The load produced by the use and occupancy of the building or other structure, not including construction loads, environmental loads (wind, snow, rain, earthquake), or dead loads.
• Floor Live Load: Variable loads on floor systems due to occupancy and moveable equipment.
• Roof Live Load (L_r): Live load on roof surfaces for maintenance, construction, and repair activities.

Live Load Reduction for Floor Members

Reduced Live Load (ASCE 7):

\[L = L_0 \left(0.25 + \frac{15}{\sqrt{K_{LL} A_T}}\right)\]

• L = reduced design live load per unit area (psf or kN/m²)
• L₀ = unreduced design live load per unit area (psf or kN/m²)
• K_LL = live load element factor
• A_T = tributary area (ft² or m²)

Limitations:

• L shall not be less than \(0.50 L_0\) for members supporting one floor
• L shall not be less than \(0.40 L_0\) for members supporting two or more floors
• Live load reduction shall not exceed 40% for members receiving load from one level only (except for basic combinations)
• No reduction permitted where \(L_0 > 100\) psf (4.79 kN/m²), except for members supporting two or more floors
• No reduction for occupancies with live loads > 100 psf unless supporting two or more floors

Live Load Element Factor (K_LL)

• Interior columns: K_LL = 4
• Exterior columns without cantilever slabs: K_LL = 4
• Edge columns with cantilever slabs: K_LL = 3
• Corner columns with cantilever slabs: K_LL = 2
• Edge beams without cantilever slabs: K_LL = 2
• Interior beams: K_LL = 2
• All other members: K_LL = 1

Tributary Area

For beams:

\[A_T = L_{beam} \times s\]

• L_beam = span length of beam (ft or m)
• s = spacing of beams (ft or m)

For columns:

\[A_T = \text{sum of tributary areas from all floors supported}\]

Roof Live Load Reduction

Ordinary Flat, Pitched, and Curved Roofs:

\[L_r = L_0 R_1 R_2\]

• L_r = reduced roof live load (psf or kN/m²)
• L₀ = unreduced roof live load (typically 20 psf or 0.96 kN/m²)
• R₁ = reduction factor for tributary area
• R₂ = reduction factor for roof slope
• Minimum: 12 psf (0.58 kN/m²)

Tributary Area Reduction Factor:

\[R_1 = 1.2 - 0.001 A_T \text{ for } A_T \leq 200 \text{ ft}^2\] \[R_1 = 0.6 \text{ for } A_T > 600 \text{ ft}^2\]

• For \(200 < a_t="" \leq="" 600\)="" ft²,="" interpolate="">
• \(0.6 \leq R_1 \leq 1.2\)

Roof Slope Reduction Factor:

\[R_2 = 1.2 - 0.05F \text{ for } F < 4\]="" \[r_2="0.6" \text{="" for="" }="" f="" \geq="" 12\]="">

• F = number of inches of rise per foot for pitched roof
• For arch or dome: F = rise-to-span ratio × 32
• For \(4 \leq F < 12\),="" interpolate="">
• \(0.6 \leq R_2 \leq 1.2\)

Wind Loads

Basic Wind Speed and Exposure

• V = basic wind speed (mph or m/s), 3-second gust at 33 ft (10 m) above ground in Exposure C
• Exposure Categories:
  • B: Urban and suburban areas, wooded areas, or terrain with numerous closely spaced obstructions
  • C: Open terrain with scattered obstructions; grasslands
  • D: Flat, unobstructed areas exposed to wind flowing over open water for at least 1 mile

Directional Procedure - Main Wind Force Resisting System (MWFRS)

Design Wind Pressure:

\[p = q G C_p - q_i (GC_{pi})\]

• p = design wind pressure (psf or Pa)
• q = velocity pressure at height z (psf or Pa)
• q_i = velocity pressure for internal pressure (psf or Pa)
• G = gust effect factor (0.85 for rigid structures)
• C_p = external pressure coefficient
• GC_pi = internal pressure coefficient

Velocity Pressure

Velocity Pressure at Height z:

\[q_z = 0.00256 K_z K_{zt} K_d V^2 \text{ (US Customary, psf)}\] \[q_z = 0.613 K_z K_{zt} K_d V^2 \text{ (SI, Pa when V in m/s)}\]

• q_z = velocity pressure at height z (psf or Pa)
• K_z = velocity pressure exposure coefficient
• K_zt = topographic factor
• K_d = wind directionality factor
• V = basic wind speed (mph or m/s)

Velocity Pressure Exposure Coefficient (K_z)

For Exposures B, C, and D:

\[K_z = 2.01 \left(\frac{z}{z_g}\right)^{2/\alpha} \text{ for } z \geq 15 \text{ ft (4.6 m)}\]

• z = height above ground level (ft or m)
• z_g = gradient height
• α = power law exponent

Exposure Parameters:

• Exposure B: α = 7.0, z_g = 1200 ft (365.8 m)
• Exposure C: α = 9.5, z_g = 900 ft (274.3 m)
• Exposure D: α = 11.5, z_g = 700 ft (213.4 m)

For heights below 15 ft (4.6 m), use K_z evaluated at 15 ft.

Topographic Factor (K_zt)

For Hills, Ridges, and Escarpments:

\[K_{zt} = (1 + K_1 K_2 K_3)^2\]

• K₁ = factor for shape of topographic feature
• K₂ = factor for horizontal distance from crest
• K₃ = factor for height above local grade
• For flat terrain or where topographic effects are minimal: K_zt = 1.0

Wind Directionality Factor (K_d)

• Buildings - MWFRS: K_d = 0.85
• Buildings - Components and Cladding: K_d = 0.85
• Chimneys, tanks, and similar structures: K_d = 0.95
• Solid signs: K_d = 0.85
• Open signs and lattice frameworks: K_d = 0.85

Gust Effect Factor (G or G_f)

Rigid Structures:

\[G = 0.85\]

Flexible or Dynamically Sensitive Structures:

\[G_f = 0.925 \left(\frac{1 + 1.7 g_Q I_z \sqrt{Q^2 + R^2}}{1 + 1.7 g_v I_z}\right)\]

• g_Q = peak factor for background response (typically 3.4)
• g_v = peak factor for wind response (typically 3.4)
• I_z = intensity of turbulence
• Q = background response factor
• R = resonant response factor

Internal Pressure Coefficient (GC_pi)

• Enclosed buildings: GC_pi = ±0.18
• Partially enclosed buildings: GC_pi = +0.55 or -0.55
• Open buildings: GC_pi = 0.00

Enclosure Classification Criteria:

• Enclosed: Does not comply with requirements for open or partially enclosed
• Partially Enclosed: Total area of openings in one wall exceeds the sum of areas of openings in balance of building envelope by more than 10%, and total area of openings exceeds 4 ft² or 1% of wall area
• Open: Each wall is at least 80% open

External Pressure Coefficient (C_p) - Low-Rise Buildings

Windward Wall:

\[C_p = 0.8\]

Leeward Wall:

• For L/B ≤ 1: C_p = -0.5
• For L/B = 2: C_p = -0.3
• For L/B ≥ 4: C_p = -0.2
• Interpolate for intermediate values
• L = length of building parallel to wind direction
• B = width of building perpendicular to wind direction

Side Walls:

\[C_p = -0.7\]

Roof (depends on slope and wind direction):

• Values range from -0.9 to +0.3 depending on roof angle and area

Simplified Wind Pressure (Envelope Procedure for Low-Rise Buildings)

Design Wind Pressure:

\[p_s = \lambda K_{zt} p_{s30}\]

• p_s = simplified design wind pressure (psf or Pa)
• λ = adjustment factor for building height and exposure
• K_zt = topographic factor
• p_s30 = simplified design wind pressure for Exposure B, h = 30 ft, K_zt = 1.0

Design Wind Force

Total Horizontal Force:

\[F = q_z G C_f A_f\]

• F = design wind force (lb or N)
• q_z = velocity pressure at height z (psf or Pa)
• G = gust effect factor
• C_f = force coefficient
• A_f = projected area normal to wind (ft² or m²)

Seismic Loads

Seismic Design Parameters

• S_S = mapped maximum considered earthquake (MCE) spectral response acceleration at short periods
• S₁ = mapped MCE spectral response acceleration at 1-second period
• S_MS = MCE spectral response acceleration at short periods adjusted for site class
• S_M1 = MCE spectral response acceleration at 1-second period adjusted for site class
• S_DS = design spectral response acceleration at short periods
• S_D1 = design spectral response acceleration at 1-second period

Site-Specific Ground Motion

Site Coefficients:

\[S_{MS} = F_a S_S\] \[S_{M1} = F_v S_1\]

• F_a = site coefficient for short period (function of Site Class and S_S)
• F_v = site coefficient for long period (function of Site Class and S₁)

Design Spectral Accelerations:

\[S_{DS} = \frac{2}{3} S_{MS}\] \[S_{D1} = \frac{2}{3} S_{M1}\]

Seismic Design Category (SDC)

Based on Risk Category and design spectral accelerations (S_DS and S_D1):

• SDC A: Minimal seismic risk
• SDC B, C: Moderate seismic risk
• SDC D, E, F: High seismic risk

Equivalent Lateral Force Procedure

Seismic Base Shear:

\[V = C_s W\]

• V = seismic base shear (lb or kN)
• C_s = seismic response coefficient
• W = effective seismic weight (lb or kN)

Seismic Response Coefficient

General Formula:

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

• S_DS = design spectral response acceleration at short periods
• R = response modification coefficient
• I_e = importance factor

Need not exceed (for T ≤ T_L):

\[C_s = \frac{S_{D1}}{T(R/I_e)}\]

• T = fundamental period of the structure (seconds)
• T_L = long-period transition period (seconds)

For T > T_L:

\[C_s = \frac{S_{D1} T_L}{T^2 (R/I_e)}\]

Shall not be less than:

\[C_s = 0.044 S_{DS} I_e \geq 0.01\]

For structures where S₁ ≥ 0.6g:

\[C_s \geq \frac{0.5 S_1}{R/I_e}\]

Approximate Fundamental Period

Method A (Approximate):

\[T_a = C_t h_n^x\]

• T_a = approximate fundamental period (seconds)
• C_t = building period coefficient
• h_n = height above base to highest level of structure (ft or m)
• x = exponent

Building Period Coefficients (C_t and x):

• Moment-resisting steel frames: C_t = 0.028 (0.0724), x = 0.8
• Moment-resisting concrete frames: C_t = 0.016 (0.0466), x = 0.9
• Eccentrically braced steel frames: C_t = 0.03 (0.0731), x = 0.75
• All other structural systems: C_t = 0.02 (0.0488), x = 0.75
• Values in parentheses are for SI units (h_n in meters)

Upper Limit on Calculated Period:

\[T \leq C_u T_a\]

• C_u = coefficient for upper limit (function of S_D1)
• For S_D1 ≥ 0.4: C_u = 1.4
• For S_D1 = 0.3: C_u = 1.4
• For S_D1 = 0.2: C_u = 1.5
• For S_D1 = 0.15: C_u = 1.6
• For S_D1 ≤ 0.1: C_u = 1.7

Vertical Distribution of Seismic Forces

Lateral Force at Level x:

\[F_x = C_{vx} V\]

• F_x = lateral seismic force at level x (lb or kN)
• C_vx = vertical distribution factor
• V = total design lateral force or shear at base (lb or kN)

Vertical Distribution Factor:

\[C_{vx} = \frac{w_x h_x^k}{\sum_{i=1}^{n} w_i h_i^k}\]

• w_x, w_i = portion of effective seismic weight at level x or i (lb or kN)
• h_x, h_i = height from base to level x or i (ft or m)
• k = distribution exponent related to structure period
• n = number of levels

Distribution Exponent k:

• For T ≤ 0.5 seconds: k = 1
• For T ≥ 2.5 seconds: k = 2
• For 0.5 < t="">< 2.5="" seconds:="" interpolate="" linearly="" or="" use="" \(k="1" +="" \frac{t="" -="">

Horizontal Distribution of Seismic Forces

Story Shear:

\[V_x = \sum_{i=x}^{n} F_i\]

• V_x = story shear at level x (lb or kN)
• F_i = lateral force at level i (lb or kN)
• Sum from level x to top level n

Torsional Moment:

\[M_t = V_x e\]

• M_t = torsional moment at level x (lb-ft or kN-m)
• V_x = story shear at level x (lb or kN)
• e = eccentricity between center of mass and center of rigidity (ft or m)

Accidental Torsion:

\[e_{acc} = \pm 0.05 D\]

• e_acc = accidental eccentricity (ft or m)
• D = building plan dimension perpendicular to direction of force (ft or m)

Redundancy Factor (ρ)

For SDC D, E, or F:

\[\rho = 1.0 \text{ to } 1.3\]

• ρ = 1.0: For structures with adequate redundancy
• ρ = 1.3: For structures lacking redundancy (each story resisting > 35% of base shear in single element)
• For SDC A, B, C: ρ = 1.0

Effective Seismic Weight (W)

Total Effective Seismic Weight:

\[W = \sum W_i\]

• W = effective seismic weight (lb or kN)
• Includes total dead load and applicable portions of other loads

Components of Effective Seismic Weight:

• Total dead load of structure
• In areas used for storage: minimum 25% of floor live load (up to 100% if required by authority)
• Partition loads if provided in design or ≥ 10 psf
• Total weight of permanent equipment
• 20% of flat roof snow load where flat roof snow load > 30 psf

Diaphragm Design Force

Diaphragm Force:

\[F_{px} = \frac{\sum_{i=x}^{n} F_i}{\sum_{i=x}^{n} w_i} w_{px}\]

• F_px = diaphragm design force at level x (lb or kN)
• F_i = lateral force at level i (lb or kN)
• w_i = weight tributary to level i (lb or kN)
• w_px = weight tributary to diaphragm at level x (lb or kN)

Minimum and Maximum Limits:

\[F_{px(min)} = 0.2 S_{DS} I_e w_{px}\] \[F_{px(max)} = 0.4 S_{DS} I_e w_{px}\]

Importance Factor (I_e)

• Risk Category I: I_e = 1.0
• Risk Category II: I_e = 1.0
• Risk Category III: I_e = 1.25
• Risk Category IV: I_e = 1.5

Response Modification Coefficient (R)

• Special moment-resisting frame (steel or concrete): R = 8
• Intermediate moment-resisting frame (concrete): R = 5
• Ordinary moment-resisting frame (steel): R = 3.5
• Special concentrically braced frame (steel): R = 6
• Eccentrically braced frame (steel): R = 8
• Shear walls (reinforced concrete): R = 5 to 6
• Light-frame wood shear walls: R = 6.5
• Bearing wall systems: R = 2 to 5 (varies by material)

Story Drift and P-Delta Effects

Design Story Drift:

\[\Delta = \frac{C_d \delta_{xe}}{I_e}\]

• Δ = design story drift (in or mm)
• C_d = deflection amplification factor
• δ_xe = deflection at level x determined by elastic analysis (in or mm)
• I_e = importance factor

Allowable Story Drift (Δ_a):

• Structures 4 stories or less with interior walls, partitions, etc.: 0.025h_sx
• Structures 4 stories or less, other: 0.020h_sx
• All other structures: 0.020h_sx for Risk Category I or II; 0.015h_sx for Risk Category III or IV
• h_sx = story height below level x (in or mm)

Stability Coefficient (P-Delta):

\[\theta = \frac{P_x \Delta}{V_x h_{sx} C_d}\]

• θ = stability coefficient
• P_x = total vertical design load at and above level x (lb or kN)
• Δ = design story drift at level x (in or mm)
• V_x = seismic shear force at level x (lb or kN)
• h_sx = story height below level x (in or mm)
• C_d = deflection amplification factor

P-Delta Criteria:

• If θ ≤ 0.10: P-Delta effects may be ignored
• If 0.10 < θ="" ≤="">_max: P-Delta effects must be considered
• If θ > θ_max: Structure is potentially unstable
• \(\theta_{max} = \frac{0.5}{\beta C_d} \leq 0.25\)
• β = ratio of shear demand to shear capacity (conservatively = 1.0)

Load Combinations

Strength Design (LRFD) Load Combinations

Basic Combinations (ASCE 7):

1. \(1.4D\)
2. \(1.2D + 1.6L + 0.5(L_r \text{ or } S \text{ or } R)\)
3. \(1.2D + 1.6(L_r \text{ or } S \text{ or } R) + (1.0L \text{ or } 0.5W)\)
4. \(1.2D + 1.0W + 1.0L + 0.5(L_r \text{ or } S \text{ or } R)\)
5. \(1.2D + 1.0E + 1.0L + 0.2S\)
6. \(0.9D + 1.0W\)
7. \(0.9D + 1.0E\)

• D = dead load
• L = live load
• L_r = roof live load
• S = snow load
• R = rain load
• W = wind load
• E = seismic load effect

Allowable Stress Design (ASD) Load Combinations

Basic Combinations (ASCE 7):

1. \(D\)
2. \(D + L\)
3. \(D + (L_r \text{ or } S \text{ or } R)\)
4. \(D + 0.75L + 0.75(L_r \text{ or } S \text{ or } R)\)
5. \(D + (0.6W \text{ or } 0.7E)\)
6. \(D + 0.75L + 0.75(0.6W) + 0.75(L_r \text{ or } S \text{ or } R)\)
7. \(D + 0.75L + 0.75(0.7E) + 0.75S\)
8. \(0.6D + 0.6W\)
9. \(0.6D + 0.7E\)

Seismic Load Effect (E)

Horizontal and Vertical Seismic Load Effect:

\[E = E_h + E_v\]

• E = combined seismic load effect
• E_h = effect of horizontal seismic forces
• E_v = effect of vertical seismic forces

Horizontal Seismic Load Effect:

\[E_h = \rho Q_E\]

• ρ = redundancy factor
• Q_E = effect of horizontal seismic forces

Vertical Seismic Load Effect:

\[E_v = 0.2 S_{DS} D\]

• S_DS = design spectral response acceleration at short periods
• D = dead load

Combined Effect in LRFD:

\[E = \rho Q_E + 0.2 S_{DS} D\] \[E = \rho Q_E - 0.2 S_{DS} D\]

Use the combination that produces the more severe load effect.

Wind Load (W)

For LRFD:

• Use full calculated wind pressure or force

For ASD:

• Wind load is typically reduced by factor of 0.6 in load combinations

Other Load Types

Snow Load

Flat Roof Snow Load:

\[p_f = 0.7 C_e C_t I_s p_g\]

• p_f = flat roof snow load (psf or kN/m²)
• C_e = exposure factor
• C_t = thermal factor
• I_s = importance factor for snow loads
• p_g = ground snow load (psf or kN/m²)

Minimum Snow Load:

\[p_f(min) = I_s p_g \text{ for } p_g \leq 20 \text{ psf}\] \[p_f(min) = 20 I_s \text{ for } p_g > 20 \text{ psf}\]

Sloped Roof Snow Load:

\[p_s = C_s p_f\]

• p_s = sloped roof snow load (psf or kN/m²)
• C_s = slope factor (function of roof slope, surface, and thermal condition)
• p_f = flat roof snow load (psf or kN/m²)

Rain Load

Rain Load on Undeflected Roof:

\[R = 5.2(d_s + d_h)\]

• R = rain load (psf), US Customary only
• d_s = depth of water on undeflected roof up to inlet of secondary drainage system (in)
• d_h = additional depth of water on undeflected roof above inlet of secondary drainage (in)

Flood Load

Hydrostatic Load:

\[F_{hydro} = \gamma_w h A\]

• F_hydro = hydrostatic load (lb or kN)
• γ_w = unit weight of water (62.4 pcf or 9.81 kN/m³)
• h = depth of water above point of interest (ft or m)
• A = area over which load acts (ft² or m²)

Hydrodynamic Load (simplified):

\[F_{hydyn} = C_d \frac{1}{2} \gamma_w V^2 A\]

• F_hydyn = hydrodynamic load (lb or kN)
• C_d = drag coefficient
• γ_w = unit weight of water (divide by g for mass density)
• V = velocity of water (ft/s or m/s)
• A = projected area perpendicular to flow (ft² or m²)

Lateral Earth Pressure

Active Earth Pressure (Rankine):

\[p_a = K_a \gamma h\] \[K_a = \tan^2\left(45° - \frac{\phi}{2}\right)\]

• p_a = active lateral earth pressure (psf or kPa)
• K_a = coefficient of active earth pressure
• γ = unit weight of soil (pcf or kN/m³)
• h = depth below surface (ft or m)
• φ = angle of internal friction of soil (degrees)

Passive Earth Pressure (Rankine):

\[p_p = K_p \gamma h\] \[K_p = \tan^2\left(45° + \frac{\phi}{2}\right)\]

• p_p = passive lateral earth pressure (psf or kPa)
• K_p = coefficient of passive earth pressure

At-Rest Earth Pressure:

\[p_0 = K_0 \gamma h\] \[K_0 = 1 - \sin\phi\]

• p₀ = at-rest lateral earth pressure (psf or kPa)
• K₀ = coefficient of at-rest earth pressure

Total Lateral Force:

\[F = \frac{1}{2} K \gamma H^2\]

• F = total lateral force per unit length of wall (lb/ft or kN/m)
• K = earth pressure coefficient (K_a, K_p, or K₀)
• H = total height of wall (ft or m)
• Resultant acts at H/3 from base

Impact and Dynamic Load Factors

Impact Loads for Bridges and Industrial Structures

Impact Factor (AASHTO - historical):

\[I = \frac{50}{L + 125} \leq 0.3\]

• I = impact fraction
• L = loaded length of span (ft)
• Modern AASHTO uses dynamic load allowance instead

Dynamic Load Allowance (AASHTO LRFD):

• Deck joints: 75%
• All other components: 33%
• Fatigue and fracture limit state: 15%

Crane Loads

Vertical Wheel Load with Impact:

\[P_{impact} = P_{static} \times (1 + I)\]

• I = impact factor, typically 0.25 for pendant-operated cranes, 0.10 for cab-operated cranes

Lateral Load (perpendicular to rail):

• Typically 20% of sum of lifted load and crane weight

Longitudinal Load (parallel to rail):

• Typically 10% of maximum wheel loads

Tributary Area and Influence Coefficients

Tributary Area Concepts

One-Way Slab Tributary Width:

\[w_{trib} = \frac{s}{2} + \frac{s}{2} = s\]

• w_trib = tributary width (ft or m)
• s = spacing between supports (ft or m)
• For interior members only

Two-Way Slab Tributary Area (rectangular bay):

\[A_{trib} = \frac{L_1 \times L_2}{n}\]

• A_trib = tributary area (ft² or m²)
• L₁, L₂ = bay dimensions (ft or m)
• n = number of columns/supports in bay (typically 4)

Influence Line Concept:

• Value of influence line at point × load = effect at point of interest
• For uniform loads: integrate influence line over loaded length

Load Path and Transfer

Load Path Hierarchy

1. Applied loads (roof, floor live loads)
2. Slab/deck (distributes to supporting beams)
3. Beams/joists (transfer to girders or columns)
4. Girders (transfer to columns)
5. Columns (transfer to foundations)
6. Foundations (transfer to soil)

Diaphragm Action

Diaphragm Unit Shear:

\[v = \frac{V}{b}\]

• v = unit shear in diaphragm (lb/ft or kN/m)
• V = total shear force in diaphragm (lb or kN)
• b = depth/width of diaphragm (ft or m)

Chord Force in Diaphragm:

\[T = C = \frac{M}{d}\]

• T = tension in chord (lb or kN)
• C = compression in chord (lb or kN)
• M = bending moment in diaphragm (lb-ft or kN-m)
• d = depth of diaphragm (distance between chords) (ft or m)

Special Load Cases

Overturning and Stability

Overturning Moment:

\[M_{OT} = F \times h\]

• M_OT = overturning moment (lb-ft or kN-m)
• F = lateral force (wind or seismic) (lb or kN)
• h = height above base to point of force application (ft or m)

Resisting Moment:

\[M_R = W \times d\]

• M_R = resisting moment (lb-ft or kN-m)
• W = stabilizing weight (lb or kN)
• d = horizontal distance from pivot point to center of weight (ft or m)

Factor of Safety Against Overturning:

\[FS_{OT} = \frac{M_R}{M_{OT}}\]

• Typical minimum FS_OT = 1.5 for service loads
• For LRFD, use factored loads and check equilibrium

Sliding Resistance

Sliding Resistance:

\[F_R = \mu N\]

• F_R = friction resistance to sliding (lb or kN)
• μ = coefficient of friction
• N = normal force (lb or kN)

Factor of Safety Against Sliding:

\[FS_{slide} = \frac{F_R}{F_H}\]

• F_H = horizontal applied force (lb or kN)
• Typical minimum FS_slide = 1.5 for service loads

Notional Loads (for frame stability)

Notional Load:

\[N_i = 0.002 Y_i\]

• N_i = notional load applied at level i (lb or kN)
• Y_i = gravity load applied at level i from LRFD load combination (lb or kN)
• Applied as lateral load at each level to account for geometric imperfections
• Used when direct analysis method is employed