Formula Sheet: Temporary Structures

Table of Contents
1. Scaffolding and Shoring Systems
2. Formwork Design
3. Excavation Support Systems
4. Falsework and Centering
5. Trench Shoring and Shielding
6. Safety Factors and Load Factors
7. Special Considerations
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Scaffolding and Shoring Systems

Scaffold Design

Load Requirements

• Light-duty scaffold: Designed for loads up to 25 psf
• Medium-duty scaffold: Designed for loads up to 50 psf
• Heavy-duty scaffold: Designed for loads up to 75 psf
• Total design load: \(w_{total} = w_{dead} + w_{live}\)
• Minimum design load: Working load should not exceed 1/4 of the rated load capacity

Scaffold Platform Design

• Maximum permissible span: \(L_{max} = \sqrt{\frac{8 \times F_b \times S}{w}}\)
  • \(F_b\) = allowable bending stress (psi)
  • \(S\) = section modulus (in³)
  • \(w\) = uniform load (lb/ft)
  • \(L_{max}\) = maximum span (ft)
• Platform deflection: \(\Delta = \frac{5wL^4}{384EI}\)
  • \(\Delta\) = deflection (in)
  • \(w\) = uniform load (lb/in)
  • \(L\) = span length (in)
  • \(E\) = modulus of elasticity (psi)
  • \(I\) = moment of inertia (in⁴)
  • Maximum allowable deflection: \(L/60\) or \(L/120\) depending on application

Scaffold Post and Frame Design

• Axial load capacity: \(P_{allow} = \frac{P_{critical}}{FS}\)
  • \(P_{allow}\) = allowable axial load (lb)
  • \(P_{critical}\) = critical buckling load (lb)
  • \(FS\) = factor of safety (typically 3.0 to 4.0)
• Euler buckling load (long columns): \(P_{cr} = \frac{\pi^2 EI}{(KL)^2}\)
  • \(P_{cr}\) = critical buckling load (lb)
  • \(E\) = modulus of elasticity (psi)
  • \(I\) = moment of inertia (in⁴)
  • \(K\) = effective length factor (dimensionless)
  • \(L\) = unbraced length (in)
• Slenderness ratio: \(\lambda = \frac{KL}{r}\)
  • \(\lambda\) = slenderness ratio (dimensionless)
  • \(K\) = effective length factor
  • \(L\) = unbraced length (in)
  • \(r\) = radius of gyration (in)
  • \(r = \sqrt{\frac{I}{A}}\)

Shoring Design

Vertical Shores (Post Shores)

• Required number of shores: \(N = \frac{P_{total}}{P_{allow}}\)
  • \(N\) = number of shores required
  • \(P_{total}\) = total load to be supported (lb)
  • \(P_{allow}\) = allowable load per shore (lb)
• Shore spacing: \(s = \sqrt{\frac{P_{allow}}{w}}\)
  • \(s\) = shore spacing (ft)
  • \(P_{allow}\) = allowable load per shore (lb)
  • \(w\) = uniform load (lb/ft²)
• Wood shore capacity: \(P_{allow} = F_c \times A \times C_p\)
  • \(P_{allow}\) = allowable axial load (lb)
  • \(F_c\) = allowable compressive stress parallel to grain (psi)
  • \(A\) = cross-sectional area (in²)
  • \(C_p\) = column stability factor (dimensionless)
• Column stability factor: \(C_p = \frac{1 + \alpha}{2c} - \sqrt{\left(\frac{1 + \alpha}{2c}\right)^2 - \frac{\alpha}{c}}\)
  • \(\alpha = \frac{F_{cE}}{F_c^*}\)
  • \(c\) = 0.8 for sawn lumber, 0.85 for round timber, 0.9 for glulam
  • \(F_{cE} = \frac{0.822E'_{min}}{(L_e/d)^2}\)
  • \(L_e\) = effective length (in)
  • \(d\) = dimension in direction of buckling (in)

Horizontal Shoring (Raker Shores)

• Axial force in raker: \(P = \frac{H}{\cos\theta}\)
  • \(P\) = axial force in raker (lb)
  • \(H\) = horizontal force to be resisted (lb)
  • \(\theta\) = angle of raker from horizontal (degrees)
• Horizontal component: \(H = P \times \cos\theta\)
• Vertical component: \(V = P \times \sin\theta\)
• Optimal raker angle: 45° to 60° from horizontal for maximum efficiency

Formwork Design

Concrete Pressure on Formwork

Lateral Pressure - ACI 347

• Maximum lateral pressure (columns): \(P_{max} = C_w \times C_c \times 150 + 9000 \times \frac{R}{T}\)
  • \(P_{max}\) = maximum lateral pressure (psf)
  • \(C_w\) = unit weight correction factor = \(\frac{w_c}{150}\)
  • \(w_c\) = concrete unit weight (pcf)
  • \(C_c\) = chemistry correction factor (1.0 for Type I cement, 1.2 for Type III)
  • \(R\) = rate of placement (ft/hr)
  • \(T\) = concrete temperature (°F)
  • Maximum value: \(P_{max} \leq 150 \times C_w \times h\) or \(3000\) psf, whichever is less
  • \(h\) = height of fresh concrete (ft)
• Alternative formula for columns/walls (rate ≤ 7 ft/hr): \(P = 150 + 9000 \times \frac{R}{T}\)
  • Maximum pressure: lesser of calculated value or \(150h\) or \(3000\) psf
• Alternative formula (rate > 7 ft/hr): \(P = 150 + 43,400/T + 2800 \times R/T\)
  • Maximum pressure: lesser of calculated value or \(150h\) or \(3000\) psf
• Hydrostatic pressure: \(P = \gamma_c \times h\)
  • \(P\) = lateral pressure (psf)
  • \(\gamma_c\) = unit weight of concrete (typically 150 pcf)
  • \(h\) = height of fresh concrete (ft)

Wall Formwork Design

• Sheathing bending moment: \(M = \frac{wl^2}{k}\)
  • \(M\) = bending moment (lb-ft or lb-in)
  • \(w\) = lateral pressure (psf or psi)
  • \(l\) = span between supports (ft or in)
  • \(k\) = coefficient based on support conditions
  • \(k = 8\) for simple spans
  • \(k = 10\) for continuous spans
  • \(k = 12\) for fixed ends
• Required section modulus: \(S_{req} = \frac{M}{F_b}\)
  • \(S_{req}\) = required section modulus (in³)
  • \(M\) = maximum bending moment (lb-in)
  • \(F_b\) = allowable bending stress (psi)
• Maximum deflection: \(\Delta = \frac{Kwl^4}{EI}\)
  • \(\Delta\) = deflection (in)
  • \(K\) = deflection coefficient
  • \(K = 5/384\) for simple span, uniform load
  • \(K = 1/384\) for continuous span, uniform load
  • \(w\) = uniform load (lb/in)
  • \(l\) = span (in)
  • \(E\) = modulus of elasticity (psi)
  • \(I\) = moment of inertia (in⁴)
  • Typical limit: \(\Delta \leq l/360\) or \(l/240\)

Wales and Tie Design

• Load on wale: \(w = P \times s_v\)
  • \(w\) = line load on wale (lb/ft)
  • \(P\) = lateral pressure (psf)
  • \(s_v\) = vertical spacing of wales (ft)
• Tie load: \(T = P \times s_v \times s_h\)
  • \(T\) = tension in tie (lb)
  • \(P\) = lateral pressure (psf)
  • \(s_v\) = vertical spacing of ties (ft)
  • \(s_h\) = horizontal spacing of ties (ft)
• Wale bending moment (two-span continuous): \(M = \frac{wl^2}{8}\)
  • \(M\) = bending moment (lb-ft)
  • \(w\) = uniform load (lb/ft)
  • \(l\) = span between ties (ft)
• Combined stress in wale: \(\frac{f_b}{F_b} + \frac{f_c}{F_c} \leq 1.0\)
  • \(f_b\) = actual bending stress (psi)
  • \(F_b\) = allowable bending stress (psi)
  • \(f_c\) = actual compressive stress (psi)
  • \(F_c\) = allowable compressive stress (psi)

Formwork for Slabs and Beams

Dead and Live Loads

• Minimum design live load: 50 psf or weight of workers, equipment, and impact loads, whichever is greater
• Dead load: \(w_d = t \times \gamma_c\)
  • \(w_d\) = dead load (psf)
  • \(t\) = slab thickness (ft)
  • \(\gamma_c\) = unit weight of concrete (pcf, typically 150 pcf)
• Total design load: \(w_{total} = w_d + w_L + w_{form}\)
  • \(w_L\) = live load (psf, minimum 50 psf)
  • \(w_{form}\) = weight of formwork (psf, typically 3-10 psf)

Joist and Beam Spacing

• Maximum joist spacing: \(s = \sqrt{\frac{8F_b S}{wL^2}} \times L\)
  • \(s\) = joist spacing (in)
  • \(F_b\) = allowable bending stress (psi)
  • \(S\) = section modulus of sheathing per foot width (in³/ft)
  • \(w\) = total load (psf)
  • \(L\) = span of sheathing (in)
• Joist reaction: \(R = w \times s \times L\)
  • \(R\) = reaction at support (lb)
  • \(w\) = total load (psf)
  • \(s\) = joist spacing (ft)
  • \(L\) = joist span (ft)

Excavation Support Systems

Lateral Earth Pressure

Active Earth Pressure

• Rankine active earth pressure coefficient: \(K_a = \frac{1 - \sin\phi}{1 + \sin\phi} = \tan^2\left(45° - \frac{\phi}{2}\right)\)
  • \(K_a\) = active earth pressure coefficient (dimensionless)
  • \(\phi\) = angle of internal friction (degrees)
• Active lateral earth pressure: \(p_a = K_a \times \gamma \times h\)
  • \(p_a\) = active lateral pressure (psf)
  • \(K_a\) = active earth pressure coefficient
  • \(\gamma\) = unit weight of soil (pcf)
  • \(h\) = depth below surface (ft)
• Total active force: \(P_a = \frac{1}{2} K_a \gamma H^2\)
  • \(P_a\) = total active force per unit length (lb/ft)
  • \(H\) = total height of wall (ft)
  • Acts at \(H/3\) from bottom of wall
• Coulomb active earth pressure coefficient: \(K_a = \frac{\sin^2(\alpha + \phi)}{\sin^2\alpha \sin(\alpha - \delta) \left[1 + \sqrt{\frac{\sin(\phi + \delta)\sin(\phi - \beta)}{\sin(\alpha - \delta)\sin(\alpha + \beta)}}\right]^2}\)
  • \(\alpha\) = angle of back face of wall from horizontal (degrees)
  • \(\phi\) = angle of internal friction (degrees)
  • \(\delta\) = angle of wall friction (degrees)
  • \(\beta\) = angle of backfill slope from horizontal (degrees)

Passive Earth Pressure

• Rankine passive earth pressure coefficient: \(K_p = \frac{1 + \sin\phi}{1 - \sin\phi} = \tan^2\left(45° + \frac{\phi}{2}\right)\)
  • \(K_p\) = passive earth pressure coefficient (dimensionless)
  • \(\phi\) = angle of internal friction (degrees)
• Passive lateral earth pressure: \(p_p = K_p \times \gamma \times h\)
  • \(p_p\) = passive lateral pressure (psf)
  • \(K_p\) = passive earth pressure coefficient
  • \(\gamma\) = unit weight of soil (pcf)
  • \(h\) = depth below surface (ft)
• Total passive force: \(P_p = \frac{1}{2} K_p \gamma H^2\)
  • \(P_p\) = total passive force per unit length (lb/ft)
  • \(H\) = total height of wall (ft)
  • Acts at \(H/3\) from bottom of wall

At-Rest Earth Pressure

• At-rest earth pressure coefficient: \(K_0 = 1 - \sin\phi\)
  • \(K_0\) = at-rest earth pressure coefficient (dimensionless)
  • \(\phi\) = angle of internal friction (degrees)
  • Applies when wall does not move
• At-rest lateral pressure: \(p_0 = K_0 \times \gamma \times h\)
  • \(p_0\) = at-rest lateral pressure (psf)
• Total at-rest force: \(P_0 = \frac{1}{2} K_0 \gamma H^2\)

Water Pressure Effects

• Hydrostatic pressure: \(p_w = \gamma_w \times h_w\)
  • \(p_w\) = water pressure (psf)
  • \(\gamma_w\) = unit weight of water (62.4 pcf)
  • \(h_w\) = height of water above point (ft)
• Total lateral pressure with water: \(p_{total} = K_a \times \gamma' \times h + \gamma_w \times h_w\)
  • \(\gamma'\) = effective (submerged) unit weight of soil (pcf)
  • \(\gamma' = \gamma_{sat} - \gamma_w\)

Sheet Pile Walls

Cantilever Sheet Pile Walls

• Depth of embedment (cantilever): Solve by moment equilibrium about bottom of wall
• Moment at dredge line: \(M_0 = P_a \times y_a - P_p \times y_p\)
  • \(M_0\) = maximum moment (lb-ft/ft)
  • \(P_a\) = active force (lb/ft)
  • \(P_p\) = passive force (lb/ft)
  • \(y_a\), \(y_p\) = moment arms from point of interest (ft)
• Required section modulus: \(S_{req} = \frac{M_{max}}{F_y/FS}\)
  • \(S_{req}\) = required section modulus (in³/ft)
  • \(M_{max}\) = maximum moment (lb-in/ft)
  • \(F_y\) = yield strength (psi)
  • \(FS\) = factor of safety (typically 1.5 to 2.0)

Anchored Sheet Pile Walls

• Anchor force (Free Earth Support Method): \(T = P_a - P_p\)
  • \(T\) = anchor force per unit length (lb/ft)
  • \(P_a\) = total active force above dredge line (lb/ft)
  • \(P_p\) = total passive force above dredge line (lb/ft)
• Depth of embedment (anchored): Determined from moment equilibrium about anchor point
• Maximum moment location: At point of zero shear between anchor and dredge line
• Factor of safety on embedment: Actual embedment ≥ 1.2 to 1.5 × calculated theoretical embedment

Braced Excavations

Apparent Earth Pressure Diagrams (Peck)

• Sand (Peck envelope): \(p_a = 0.65 \times K_a \times \gamma \times H\)
  • \(p_a\) = uniform apparent pressure (psf)
  • \(K_a\) = active earth pressure coefficient
  • \(\gamma\) = unit weight of soil (pcf)
  • \(H\) = excavation depth (ft)
  • Applied as rectangular distribution
• Soft to medium clay: Trapezoidal distribution
  • Top: \(p_1 = 0.25\gamma H\) to depth \(0.25H\)
  • Middle: \(p_2 = 0.5\gamma H\) to depth \(0.75H\)
  • Bottom: \(p_3 = 0.75\gamma H\) to depth \(H\)
  • For \(\gamma H/c > 4\)
• Stiff fissured clay: Trapezoidal distribution
  • Top: \(p_1 = 0.2\gamma H\) to \(0.5\gamma H\)
  • Bottom: increases linearly

Strut Loads

• Strut load: \(P_{strut} = p_a \times A_{tributary}\)
  • \(P_{strut}\) = axial load in strut (lb)
  • \(p_a\) = apparent earth pressure at strut level (psf)
  • \(A_{tributary}\) = tributary area (ft²)
• Tributary area: \(A_{trib} = s_h \times s_v\)
  • \(s_h\) = horizontal spacing of struts (ft)
  • \(s_v\) = vertical spacing of struts (ft)
• Strut capacity (buckling): \(P_{cr} = \frac{\pi^2 EI}{(KL)^2}\)
  • Use reduced modulus for inelastic buckling if applicable
  • \(K\) typically 1.0 for pinned-pinned struts

Wales Design in Braced Excavations

• Load on wale: \(w = p_a \times s_v\)
  • \(w\) = line load on wale (lb/ft)
  • \(p_a\) = apparent earth pressure (psf)
  • \(s_v\) = vertical spacing of wales (ft)
• Maximum moment in wale: \(M = \frac{wL^2}{8}\) (simple span) or \(M = \frac{wL^2}{10}\) (continuous)

Falsework and Centering

Bridge Falsework

Load Analysis

• Dead load: Weight of concrete, reinforcing steel, formwork, and falsework
• Construction live load: Minimum 50 psf or actual equipment loads
• Impact factor: 1.0 to 1.5 times static load for equipment and material placement
• Wind load on falsework: \(F = q_z \times G \times C_f \times A_f\)
  • \(F\) = wind force (lb)
  • \(q_z\) = velocity pressure (psf)
  • \(G\) = gust effect factor (typically 0.85)
  • \(C_f\) = force coefficient (1.4 to 2.0 for structural frames)
  • \(A_f\) = projected area (ft²)
• Load combinations:
  • Dead + Live + Impact
  • Dead + Live + Wind
  • Dead + Wind

Tower and Post Design

• Axial load from tributary area: \(P = w \times A_{trib}\)
  • \(P\) = axial load (lb)
  • \(w\) = total load (psf)
  • \(A_{trib}\) = tributary area (ft²)
• Allowable axial load (wood post): \(P_{allow} = F_c' \times A \times C_p\)
  • \(F_c'\) = adjusted compressive stress (psi)
  • \(A\) = cross-sectional area (in²)
  • \(C_p\) = column stability factor
• Lateral stability bracing requirement: Brace points required when \(L/d > 50\)
  • \(L\) = unbraced length (in)
  • \(d\) = least dimension (in)

Mudsills and Foundation

• Bearing pressure: \(q = \frac{P}{A}\)
  • \(q\) = bearing pressure (psf)
  • \(P\) = vertical load (lb)
  • \(A\) = contact area (ft²)
  • Must be ≤ allowable soil bearing capacity
• Required mudsill area: \(A_{req} = \frac{P}{q_{allow}}\)
  • \(A_{req}\) = required contact area (ft²)
  • \(q_{allow}\) = allowable bearing pressure (psf)
• Settlement check: Verify settlement is within acceptable limits using elastic settlement equations

Trench Shoring and Shielding

OSHA Trench Requirements

Soil Classifications

• Type A soil: Cohesive, unconfined compressive strength ≥ 1.5 tsf (144 kPa)
• Type B soil: Cohesive, unconfined compressive strength 0.5 to 1.5 tsf (48 to 144 kPa), or granular soils (silt, sandy loam)
• Type C soil: Cohesive, unconfined compressive strength < 0.5="" tsf="" (48="" kpa),="" or="" submerged="" soil,="" or="" soil="" with="">

Maximum Allowable Slopes

• Type A soil: 3/4:1 (53° from horizontal) or 0.75H:1V
• Type B soil: 1:1 (45° from horizontal) or 1H:1V
• Type C soil: 1.5:1 (34° from horizontal) or 1.5H:1V
• Vertical rise for short-term: Maximum 3.5 ft for Type A, none for Type B or C without shoring

Timber Trench Shoring

Design Loads

• Lateral soil load: Based on equivalent fluid pressure
  • Type A: 25 pcf equivalent fluid weight
  • Type B: 45 pcf equivalent fluid weight
  • Type C: 80 pcf equivalent fluid weight
• Pressure at depth: \(p = \gamma_{eq} \times h\)
  • \(p\) = lateral pressure (psf)
  • \(\gamma_{eq}\) = equivalent fluid weight (pcf)
  • \(h\) = depth (ft)

Timber Shoring Components

• Wales: Horizontal members supporting sheeting
• Struts (cross braces): Horizontal members between wales on opposite sides
• Uprights (sheeting): Vertical members against trench walls
• Strut spacing: Based on soil type and trench depth per OSHA tables
• Wales spacing: Typically 4 ft vertically for depths up to 20 ft

Safety Factors and Load Factors

General Safety Factors

• Scaffold components: FS = 4.0 on ultimate strength
• Formwork structural members: FS = 1.5 to 2.0 depending on material and loading
• Shoring posts: FS = 3.0 to 4.0
• Excavation support: FS = 1.5 on passive resistance, 1.0 on active pressure
• Anchor systems: FS = 2.0 to 3.0

Load Duration Factors (Wood Design)

• Permanent loads: \(C_D = 0.9\)
• Normal (10 years): \(C_D = 1.0\)
• Construction (7 days): \(C_D = 1.25\)
• Impact: \(C_D = 2.0\)
• Adjusted allowable stress: \(F' = F \times C_D \times C_M \times C_t \times ...\)
  • \(F'\) = adjusted allowable stress (psi)
  • \(F\) = reference design value (psi)
  • \(C_D\) = load duration factor
  • \(C_M\) = wet service factor
  • \(C_t\) = temperature factor

Special Considerations

Formwork Removal Criteria

• Minimum concrete strength: Forms may be removed when concrete reaches required strength
• Walls and columns: Forms may be removed when concrete strength ≥ 2 × design load
• Slab soffits (props remaining): Minimum 50% of design strength
• Slab soffits (props removed): Minimum 100% of design strength
• Maturity method: \(M = \sum(T - T_0)\Delta t\)
  • \(M\) = maturity index (°C-hours or °F-hours)
  • \(T\) = average concrete temperature during time interval (°C or °F)
  • \(T_0\) = datum temperature (-10°C or 14°F for Type I cement)
  • \(\Delta t\) = time interval (hours)

Deflection Limits

• Formwork deflection limits:
  • Horizontal members: \(L/360\) or \(L/240\)
  • Vertical shores: \(L/240\)
  • Where appearance is critical: \(L/600\)
• Scaffold platform deflection: \(L/60\) under total load
• Wales and soldier beams: \(L/240\) to \(L/360\)

Connection Design

Bolt Connections in Formwork

• Bolt shear capacity: \(V_{allow} = A_b \times F_v / FS\)
  • \(V_{allow}\) = allowable shear force (lb)
  • \(A_b\) = bolt cross-sectional area (in²)
  • \(F_v\) = shear strength (psi)
  • \(FS\) = factor of safety (typically 2.0)
• Bolt bearing capacity: \(P_{allow} = d \times t \times F_p / FS\)
  • \(d\) = bolt diameter (in)
  • \(t\) = thickness of thinner connected part (in)
  • \(F_p\) = bearing strength (psi)

Form Ties

• Tie spacing: \(s = \sqrt{\frac{T_{allow}}{p}}\)
  • \(s\) = tie spacing (ft)
  • \(T_{allow}\) = allowable tie capacity (lb)
  • \(p\) = lateral pressure (psf)
• Snap tie capacity: Manufacturer rated capacity with FS ≥ 2.0
• She-bolt systems: Must account for thread stripping and pull-out