Formula Sheet: Load Calculations

Table of Contents
1. Beam Loading and Reactions
2. Cantilever Beam Loading
3. Shear and Moment Diagrams
4. Combined Loading
5. Moving Loads and Influence Lines
6. Load Distribution and Transfer
7. Live Load Reduction
8. Dead Load Calculations
9. Load Combinations
10. Eccentric Loading
11. Impact and Dynamic Loads
12. Thermal Loading
13. Pressure Loads
14. Soil and Lateral Earth Pressure
15. Snow Loads
16. Seismic Loads
17. Crane Loads
18. Load Transfer and Analysis
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Beam Loading and Reactions

Concentrated Load Reactions

• Left Reaction: \[R_A = \frac{P \cdot b}{L}\]
• Right Reaction: \[R_B = \frac{P \cdot a}{L}\]
• Where:
  • P = concentrated load (lb or N)
  • a = distance from left support to load (ft or m)
  • b = distance from load to right support (ft or m)
  • L = total span length, \(L = a + b\) (ft or m)
• Maximum Shear: \(V_{max} = \max(R_A, R_B)\)
• Maximum Moment (at load location): \[M_{max} = \frac{P \cdot a \cdot b}{L}\]

• Left Reaction: \[R_A = \frac{\sum(P_i \cdot x_i)}{L}\]
• Right Reaction: \[R_B = \sum P_i - R_A\]
• Where:
  • P_i = individual concentrated load (lb or N)
  • x_i = distance from right support to load P_i (ft or m)
• Verification: \(\sum R = \sum P\) (equilibrium check)

Uniformly Distributed Load Reactions

• Total Load: \[W = w \cdot L\]
• Reactions (equal): \[R_A = R_B = \frac{w \cdot L}{2}\]
• Where:
  • w = uniformly distributed load intensity (lb/ft or N/m)
  • L = span length (ft or m)
  • W = total load (lb or N)
• Maximum Shear: \[V_{max} = \frac{w \cdot L}{2}\]
• Maximum Moment (at midspan): \[M_{max} = \frac{w \cdot L^2}{8}\]

• Equivalent Concentrated Load: \[P_{equiv} = w \cdot L_{loaded}\]
• Acts at centroid of loaded length
• Where:
  • L_loaded = length over which uniform load acts (ft or m)
• Then apply concentrated load reaction formulas with \(P = P_{equiv}\)

Triangular and Trapezoidal Load Reactions

• Total Load: \[W = \frac{1}{2} \cdot w_{max} \cdot L\]
• Centroid Location: Located at \(L/3\) from the maximum intensity end or \(2L/3\) from zero end
• Reaction at Maximum End: \[R_{max} = \frac{w_{max} \cdot L}{3}\]
• Reaction at Zero End: \[R_{zero} = \frac{w_{max} \cdot L}{6}\]
• Where:
  • w_max = maximum load intensity (lb/ft or N/m)

• Total Load: \[W = \frac{w_1 + w_2}{2} \cdot L\]
• Where:
  • w₁ = load intensity at left end (lb/ft or N/m)
  • w₂ = load intensity at right end (lb/ft or N/m)
• Centroid Distance from Left End: \[\bar{x} = \frac{L}{3} \cdot \frac{w_1 + 2w_2}{w_1 + w_2}\]
• Apply moment equilibrium to find reactions

Cantilever Beam Loading

Concentrated Load at Free End

• Reaction at Fixed End: \[R = P\]
• Moment at Fixed End: \[M_{fixed} = P \cdot L\]
• Where:
  • P = concentrated load at free end (lb or N)
  • L = cantilever length (ft or m)
• Maximum Shear: \(V_{max} = P\) (constant throughout)

Uniformly Distributed Load on Cantilever

• Total Load: \[W = w \cdot L\]
• Reaction at Fixed End: \[R = w \cdot L\]
• Moment at Fixed End: \[M_{fixed} = \frac{w \cdot L^2}{2}\]
• Where:
  • w = uniformly distributed load intensity (lb/ft or N/m)
• Maximum Shear: \(V_{max} = w \cdot L\) (at fixed end)

Shear and Moment Diagrams

Fundamental Relationships

• \[\frac{dV}{dx} = -w(x)\]
• Where:
  • V = shear force (lb or N)
  • w(x) = distributed load intensity at position x (lb/ft or N/m)
  • x = distance along beam (ft or m)
• The slope of the shear diagram equals the negative of the applied load intensity

• \[\frac{dM}{dx} = V(x)\]
• Where:
  • M = bending moment (lb-ft or N⋅m)
  • V(x) = shear force at position x (lb or N)
• The slope of the moment diagram equals the shear force
• Maximum moment occurs where \(V = 0\)

• \[\Delta V = -\int w(x) \, dx\]
• \[\Delta M = \int V(x) \, dx\]
• Change in shear between two points equals negative of area under load diagram
• Change in moment between two points equals area under shear diagram

Sign Conventions

• Shear Force: Positive when it causes clockwise rotation of beam segment
• Bending Moment: Positive when it causes compression in top fiber (sagging/concave up)
• Distributed Load: Positive when acting downward (most common convention)

Combined Loading

Superposition Principle

• For linear elastic systems, effects of multiple loads can be superimposed
• Total Reaction: \[R_{total} = R_1 + R_2 + R_3 + \cdots\]
• Total Shear: \[V_{total}(x) = V_1(x) + V_2(x) + V_3(x) + \cdots\]
• Total Moment: \[M_{total}(x) = M_1(x) + M_2(x) + M_3(x) + \cdots\]
• Where subscripts 1, 2, 3, etc. represent individual loading cases
• Note: Only valid for linear elastic behavior

Moving Loads and Influence Lines

Moving Concentrated Load

• Maximum reaction occurs when load is directly over the support
• \[R_{max} = P\]
• Where P = magnitude of moving load (lb or N)

• Maximum moment under load occurs when load is at midspan
• \[M_{max} = \frac{P \cdot L}{4}\]
• For a single concentrated load on simply supported beam

Influence Line Concepts

• Influence Line: Diagram showing variation of a response (reaction, shear, or moment) at a specific location as a unit load moves across the structure
• Response due to Load P at position x: \[Response = P \times I.L.(x)\]
• Where I.L.(x) = ordinate of influence line at position x
• Response due to Distributed Load w: \[Response = w \times Area_{I.L.}\]
• Where Area_I.L. = area under influence line over loaded length

Load Distribution and Transfer

Tributary Area Method

• Tributary Width: Half the distance to adjacent parallel supports on each side
• \[w_{beam} = w_{slab} \times T_w\]
• Where:
  • w_beam = line load on beam (lb/ft or N/m)
  • w_slab = slab load intensity (lb/ft² or N/m²)
  • T_w = tributary width (ft or m)

• Tributary Area: Area bounded by lines at 45° from corners or midpoints between supports
• \[P_{column} = w_{slab} \times A_T\]
• Where:
  • P_column = column load (lb or N)
  • A_T = tributary area (ft² or m²)

Load Distribution Criteria

• One-way action: If \(\frac{L_{long}}{L_{short}} > 2\)
• Two-way action: If \(\frac{L_{long}}{L_{short}} \leq 2\)
• Where:
  • L_long = longer span dimension
  • L_short = shorter span dimension

Live Load Reduction

Floor Live Load Reduction

• \[L = L_0 \left(0.25 + \frac{15}{\sqrt{K_{LL} \cdot A_T}}\right)\]
• Where:
  • L = reduced design live load per ft² (or m²)
  • L₀ = unreduced design live load per ft² (or m²)
  • K_LL = live load element factor
  • A_T = tributary area (ft² or m²)
• Limitations: \(L \geq 0.50 \cdot L_0\) for members supporting one floor
• Limitations: \(L \geq 0.40 \cdot L_0\) for members supporting two or more floors
• Exception: \(L\) shall not be less than \(0.40 \cdot L_0\) for live loads greater than 100 psf

• 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: \(K_{LL} = 1\)

• Live load reduction is not permitted when:
  • Live load exceeds 100 psf (4.79 kN/m²), except for members supporting two or more floors
  • Occupancies used for public assembly (e.g., places of assembly)
  • Garage live loads
  • Roof live loads (separate reduction method applies)

Roof Live Load Reduction

• \[L_r = L_0 \cdot R_1 \cdot R_2\]
• Where:
  • L_r = reduced roof live load per ft² (or m²)
  • L₀ = unreduced roof live load (typically 20 psf minimum)
  • R₁ = reduction factor for tributary area
  • R₂ = reduction factor for roof slope
• Minimum: \(L_r \geq 12\) psf for ordinary roofs

• For \(A_T \leq 200\) ft²: \[R_1 = 1.0\]
• For \(200 < a_t="">< 600\)="" ft²:="" \[r_1="1.2" -="" 0.001="" \cdot="">
• For \(A_T \geq 600\) ft²: \[R_1 = 0.6\]
• Where A_T = tributary area in ft²

• For slope \(F < 4\)="" on="" 12:="" \[r_2="">
• For slope \(4 \leq F < 12\)="" on="" 12:="" \[r_2="1.2" -="" 0.05="" \cdot="">
• For slope \(F \geq 12\) on 12: \[R_2 = 0.6\]
• Where F = rise-to-run ratio multiplied by 12 (slope in inches per foot)

Dead Load Calculations

Self-Weight of Structural Elements

• \[w_{self} = \gamma \times A\]
• Where:
  • w_self = self-weight per unit length (lb/ft or N/m)
  • γ = unit weight of material (lb/ft³ or N/m³)
  • A = cross-sectional area (ft² or m²)

• \[w_{slab} = \gamma \times t\]
• Where:
  • w_slab = self-weight per unit area (lb/ft² or N/m²)
  • t = slab thickness (ft or m)

Typical Unit Weights

• Structural Steel: 490 lb/ft³ (77 kN/m³)
• Reinforced Concrete: 150 lb/ft³ (23.6 kN/m³)
• Plain Concrete: 145 lb/ft³ (22.8 kN/m³)
• Lightweight Concrete: 90-120 lb/ft³ (14-19 kN/m³)
• Wood (average): 35-50 lb/ft³ (5.5-7.9 kN/m³)
• Masonry: 120-140 lb/ft³ (19-22 kN/m³)
• Aluminum: 170 lb/ft³ (26.7 kN/m³)

Load Combinations

LRFD Load Combinations (ASCE 7)

• 1. \(1.4D\)
• 2. \(1.2D + 1.6L + 0.5(L_r \text{ or } S \text{ or } R)\)
• 3. \(1.2D + 1.6(L_r \text{ or } S \text{ or } R) + (L \text{ or } 0.5W)\)
• 4. \(1.2D + 1.0W + L + 0.5(L_r \text{ or } S \text{ or } R)\)
• 5. \(1.2D + 1.0E + L + 0.2S\)
• 6. \(0.9D + 1.0W\)
• 7. \(0.9D + 1.0E\)
• Where:
  • D = dead load
  • L = live load
  • L_r = roof live load
  • S = snow load
  • R = rain load
  • W = wind load
  • E = seismic load

ASD Load 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. \(0.6D + 0.7E\)

Eccentric Loading

Combined Axial and Bending

• An eccentric load P at eccentricity e creates both axial force and moment
• Axial Force: \[N = P\]
• Bending Moment: \[M = P \times e\]
• Where:
  • P = magnitude of load (lb or N)
  • e = eccentricity (distance from centroid) (ft or m)

• \[\sigma = \frac{P}{A} \pm \frac{M \cdot c}{I}\]
• Or equivalently: \[\sigma = \frac{P}{A} \pm \frac{P \cdot e \cdot c}{I}\]
• Where:
  • σ = normal stress (psi or Pa)
  • A = cross-sectional area (in² or m²)
  • M = bending moment (lb-in or N⋅m)
  • c = distance from neutral axis to extreme fiber (in or m)
  • I = moment of inertia (in⁴ or m⁴)
• ± sign indicates tension on one side, compression on the other

Kern Region

• Kern: Region within which an eccentric load must be applied to avoid tensile stresses
• Rectangular Section - Kern Dimension: \[e_{max} = \frac{h}{6}\]
• Where:
  • h = depth of rectangular section
  • Load must be within middle third to avoid tension
• Circular Section - Kern Radius: \[e_{max} = \frac{d}{8}\]
• Where d = diameter of circular section

Impact and Dynamic Loads

Impact Factor

• \[P_{equiv} = P \times (1 + IF)\]
• Where:
  • P_equiv = equivalent static load (lb or N)
  • P = nominal static load (lb or N)
  • IF = impact factor (dimensionless)

• Elevators: IF = 1.0 (100%)
• Reciprocating machinery: IF = 0.5 (50%)
• Light machinery, shaft-driven: IF = 0.2 (20%)
• Craneways - vertical loads: IF = 0.25 (25%)
• Craneways - lateral loads: 20% of lifted load
• Highway bridges: Varies by code (typically 15-30%)

Free-Falling Weight

• \[P_{max} = P_{static} \left(1 + \sqrt{1 + \frac{2h}{\delta_{static}}}\right)\]
• Where:
  • P_max = maximum dynamic force (lb or N)
  • P_static = weight of falling object (lb or N)
  • h = drop height (ft or m)
  • δ_static = static deflection under weight P_static (ft or m)
• Special Case - Suddenly Applied Load (h = 0): \[P_{max} = 2 \times P_{static}\]

Thermal Loading

Thermal Expansion and Stress

• \[\delta_T = \alpha \times L \times \Delta T\]
• Where:
  • δ_T = thermal expansion (ft or m)
  • α = coefficient of thermal expansion (per °F or per °C)
  • L = original length (ft or m)
  • ΔT = temperature change (°F or °C)

• \[\sigma_T = E \times \alpha \times \Delta T\]
• Where:
  • σ_T = thermal stress (psi or Pa)
  • E = modulus of elasticity (psi or Pa)
• Note: Compression if temperature increases, tension if decreases (for restrained member)

• \[F_T = E \times A \times \alpha \times \Delta T\]
• Where:
  • F_T = thermal force (lb or N)
  • A = cross-sectional area (in² or m²)

Typical Coefficients of Thermal Expansion

• Steel: α = 6.5 × 10^-6 /°F (11.7 × 10^-6 /°C)
• Aluminum: α = 13 × 10^-6 /°F (23.4 × 10^-6 /°C)
• Concrete: α = 5.5 × 10^-6 /°F (9.9 × 10^-6 /°C)
• Wood (parallel to grain): α = 2-3 × 10^-6 /°F (3.6-5.4 × 10^-6 /°C)
• Brass/Bronze: α = 10 × 10^-6 /°F (18 × 10^-6 /°C)

Pressure Loads

Hydrostatic Pressure

• \[p = \gamma \times h\]
• Where:
  • p = pressure (psf or Pa)
  • γ = specific weight of fluid (lb/ft³ or N/m³)
  • h = depth below surface (ft or m)
• For water: γ = 62.4 lb/ft³ (9.81 kN/m³)

• \[F = \frac{1}{2} \times \gamma \times H^2 \times b\]
• Where:
  • F = total resultant force (lb or N)
  • H = total depth of fluid (ft or m)
  • b = width of surface perpendicular to page (ft or m)
• Location of resultant: Acts at \(H/3\) from bottom (centroid of pressure triangle)

• \[F = \gamma \times h_c \times A\]
• Where:
  • h_c = depth to centroid of submerged area (ft or m)
  • A = area of submerged surface (ft² or m²)

Wind Pressure

• \[q_z = 0.00256 \times K_z \times K_{zt} \times K_d \times V^2\] (US Customary)
• Where:
  • q_z = velocity pressure at height z (psf)
  • K_z = velocity pressure exposure coefficient
  • K_zt = topographic factor
  • K_d = wind directionality factor
  • V = basic wind speed (mph)

• \[p = q \times G \times C_p\] (simplified for main wind force resisting system)
• Where:
  • p = design wind pressure (psf or Pa)
  • q = velocity pressure (psf or Pa)
  • G = gust effect factor
  • C_p = external pressure coefficient

Soil and Lateral Earth Pressure

Active Earth Pressure

• \[K_a = \frac{1 - \sin \phi}{1 + \sin \phi} = \tan^2\left(45° - \frac{\phi}{2}\right)\]
• Where:
  • K_a = active earth pressure coefficient
  • φ = angle of internal friction of soil (degrees)

• \[p_a = K_a \times \gamma_{soil} \times h\]
• Where:
  • p_a = active lateral pressure at depth h (psf or Pa)
  • γ_soil = unit weight of soil (lb/ft³ or N/m³)
  • h = depth below surface (ft or m)

• \[P_a = \frac{1}{2} \times K_a \times \gamma_{soil} \times H^2\]
• Where:
  • P_a = total active force per unit width (lb/ft or N/m)
  • H = total height of wall (ft or m)
• Acts at H/3 from bottom

At-Rest Earth Pressure

• \[K_0 = 1 - \sin \phi\]
• Where:
  • K₀ = at-rest earth pressure coefficient
• Alternative for normally consolidated clay: \(K_0 \approx 0.4\) to \(0.5\)

• \[p_0 = K_0 \times \gamma_{soil} \times h\]
• Used for non-yielding walls (rigid basement walls, bridge abutments)

Passive Earth Pressure

• \[K_p = \frac{1 + \sin \phi}{1 - \sin \phi} = \tan^2\left(45° + \frac{\phi}{2}\right)\]
• Where:
  • K_p = passive earth pressure coefficient

• \[p_p = K_p \times \gamma_{soil} \times h\]
• Develops when soil is compressed (resists movement)

Snow Loads

Flat Roof Snow Load

• \[p_f = 0.7 \times C_e \times C_t \times I_s \times p_g\]
• Where:
  • p_f = flat roof snow load (psf or kN/m²)
  • C_e = exposure factor
  • C_t = thermal factor
  • I_s = importance factor for snow
  • p_g = ground snow load (psf or kN/m²)
• Minimum: \(p_f \geq 20 \times I_s\) psf for low-slope roofs where \(p_g > 20\) psf

Sloped Roof Snow Load

• \[p_s = C_s \times p_f\]
• Where:
  • p_s = sloped roof snow load (psf or kN/m²)
  • C_s = slope factor

• For slope ≤ 30°: \[C_s = 1.0\]
• For 30° < slope="">< 70°:="" \[c_s="\frac{70°" -="">
• For slope ≥ 70°: \[C_s = 0\]

Snow Drift Loads

• Occurs on lower roof downwind of higher roof or obstruction
• Drift height: \[h_d = 0.43 \times \left(\frac{l_u}{\gamma}\right)^{1/3} \times (p_g + 10)^{1/4} - 1.5\]
• Where:
  • h_d = drift height (ft)
  • l_u = length of upper roof (windward) (ft)
  • γ = snow density (pcf), typically 15-30 pcf
• Maximum drift height: \(h_d \leq h_c\) (clear height difference between roofs)

• \[p_d = h_d \times \gamma\]
• Where:
  • p_d = drift surcharge load (psf)
• Applied over width \(w = 4 h_d\) (triangular distribution)

Seismic Loads

Equivalent Lateral Force Method

• \[V = C_s \times W\]
• Where:
  • V = seismic base shear (lb or N)
  • C_s = seismic response coefficient
  • W = effective seismic weight (lb or N)

• \[C_s = \frac{S_{DS}}{R / I_e}\]
• Where:
  • S_DS = design spectral response acceleration parameter at short periods
  • R = response modification coefficient
  • I_e = seismic importance factor
• Need not exceed: \[C_s = \frac{S_{D1}}{T \times (R / I_e)}\] for \(T \leq T_L\)
• Minimum: \[C_s \geq 0.044 \times S_{DS} \times I_e\] (but not less than 0.01)
• Where:
  • S_D1 = design spectral response acceleration at 1-second period
  • T = fundamental period of structure (seconds)
  • T_L = long-period transition period (seconds)

• \[T_a = C_t \times h_n^x\]
• Where:
  • T_a = approximate fundamental period (seconds)
  • h_n = height from base to highest level (ft)
  • C_t = building period coefficient (varies by structure type)
  • x = exponent (varies by structure type)
• Steel moment frames: \(C_t = 0.028\), \(x = 0.8\)
• Concrete moment frames: \(C_t = 0.016\), \(x = 0.9\)
• Steel eccentrically braced frames: \(C_t = 0.03\), \(x = 0.75\)
• All other buildings: \(C_t = 0.02\), \(x = 0.75\)

Vertical Distribution of Seismic Force

• \[F_x = C_{vx} \times V\]
• Where:
  • F_x = lateral seismic force at level x (lb or N)
  • C_vx = vertical distribution factor at level x

• \[C_{vx} = \frac{w_x \times h_x^k}{\sum_{i=1}^{n} w_i \times h_i^k}\]
• Where:
  • w_x = portion of effective seismic weight at level x (lb or N)
  • h_x = height from base to level x (ft or m)
  • k = exponent related to structure period
• Exponent k:
  • For \(T \leq 0.5\) sec: \(k = 1.0\)
  • For \(T \geq 2.5\) sec: \(k = 2.0\)
  • For \(0.5 < t="">< 2.5\)="" sec:="" linear="">

Crane Loads

Vertical Wheel Loads

• \[P_{wheel,max} = \frac{(W_{bridge} + W_{trolley} + W_{lifted}) \times x_{max}}{span} + \frac{W_{bridge}}{n_{wheels}}\]
• Where:
  • P_wheel,max = maximum vertical wheel load (lb or N)
  • W_bridge = weight of bridge (lb or N)
  • W_trolley = weight of trolley (lb or N)
  • W_lifted = weight of lifted load (lb or N)
  • x_max = maximum lateral position of trolley from runway centerline (ft or m)
  • span = crane span (ft or m)
  • n_wheels = number of wheels on one side
• Simplified conservative approach: distribute total load to wheels

• \[P_{design} = P_{static} \times (1 + 0.25)\]
• 25% impact increase for powered cranes (vertical direction)

Lateral and Longitudinal Forces

• \[F_{lateral} = 0.20 \times (W_{trolley} + W_{lifted})\]
• 20% of combined trolley and lifted load weight
• Applied horizontally at top of rail, perpendicular to runway

• \[F_{long} = 0.10 \times W_{max,wheels}\]
• 10% of maximum wheel loads
• Applied parallel to direction of crane travel
• Where W_max,wheels = sum of all maximum vertical wheel loads on one rail

Load Transfer and Analysis

Load Path Concepts

• Load path: Route by which loads are transferred from point of application to foundation
• General sequence: Applied load → Slab/Deck → Beam → Girder → Column → Foundation → Soil
• Continuity requirement: Each element must have adequate capacity and connections

Point Load Distribution Through Slabs

• Load spreads at 45° through slab thickness
• Effective loaded area at slab bottom: \[A_{eff} = (a + t) \times (b + t)\]
• Where:
  • a, b = dimensions of loaded area at top (ft or m)
  • t = slab thickness (ft or m)
• Distributed pressure: \[p = \frac{P}{A_{eff}}\]

Axial Load Transfer in Columns

• \[P_{column,level\,i} = P_{column,level\,i+1} + P_{beams,level\,i} + P_{slab,level\,i} + P_{column,self,level\,i}\]
• Load accumulates from top to bottom
• Each floor adds tributary loads from beams, slabs, and column self-weight