Formula Sheet: Earth Retaining Structures

Lateral Earth Pressure Theory

At-Rest Earth Pressure

At-Rest Lateral Earth Pressure Coefficient:

\[K_0 = 1 - \sin \phi'\]

• \(K_0\) = at-rest lateral earth pressure coefficient (dimensionless)
• \(\phi'\) = effective angle of internal friction (degrees)
• Applies to normally consolidated soils with no lateral strain

For Overconsolidated Soils:

\[K_0(OC) = K_0(NC) \times \sqrt{OCR}\]

• \(K_0(OC)\) = at-rest coefficient for overconsolidated soil
• \(K_0(NC)\) = at-rest coefficient for normally consolidated soil
• OCR = overconsolidation ratio (dimensionless)

At-Rest Lateral Earth Pressure:

\[\sigma_h = K_0 \sigma_v'\]

• \(\sigma_h\) = horizontal (lateral) earth pressure (psf or kPa)
• \(\sigma_v'\) = effective vertical stress (psf or kPa)

Active Earth Pressure (Rankine Theory)

Active Earth Pressure Coefficient (Horizontal Backfill):

\[K_a = \frac{1 - \sin \phi'}{1 + \sin \phi'} = \tan^2\left(45° - \frac{\phi'}{2}\right)\]

• \(K_a\) = active earth pressure coefficient (dimensionless)
• \(\phi'\) = effective angle of internal friction (degrees)
• Applies when wall moves away from soil

Active Earth Pressure Coefficient (Sloping Backfill):

\[K_a = \cos \beta \frac{\cos \beta - \sqrt{\cos^2 \beta - \cos^2 \phi'}}{\cos \beta + \sqrt{\cos^2 \beta - \cos^2 \phi'}}\]

• \(\beta\) = angle of backfill slope to horizontal (degrees)
• Valid for \(\beta \leq \phi'\)

Active Lateral Earth Pressure (Cohesionless Soil):

\[\sigma_a = K_a \sigma_v' = K_a \gamma z\]

• \(\sigma_a\) = active lateral earth pressure (psf or kPa)
• \(\gamma\) = unit weight of soil (pcf or kN/m³)
• z = depth below ground surface (ft or m)

Active Lateral Earth Pressure (Cohesive Soil):

\[\sigma_a = K_a \gamma z - 2c\sqrt{K_a}\]

• c = cohesion (psf or kPa)
• Tensile stresses may develop near surface

Depth of Tension Crack:

\[z_c = \frac{2c}{\gamma \sqrt{K_a}}\]

• \(z_c\) = depth of tension crack (ft or m)
• No active pressure is assumed above this depth

Resultant Active Force per Unit Length (Cohesionless Soil):

\[P_a = \frac{1}{2} K_a \gamma H^2\]

• \(P_a\) = total active force per unit length of wall (lb/ft or kN/m)
• H = height of wall (ft or m)
• Acts at H/3 above base of wall

Resultant Active Force (Cohesive Soil):

\[P_a = \frac{1}{2} K_a \gamma H^2 - 2cH\sqrt{K_a}\]

Passive Earth Pressure (Rankine Theory)

Passive Earth Pressure Coefficient (Horizontal Backfill):

\[K_p = \frac{1 + \sin \phi'}{1 - \sin \phi'} = \tan^2\left(45° + \frac{\phi'}{2}\right)\]

• \(K_p\) = passive earth pressure coefficient (dimensionless)
• Applies when wall moves toward soil

Passive Earth Pressure Coefficient (Sloping Backfill):

\[K_p = \cos \beta \frac{\cos \beta + \sqrt{\cos^2 \beta - \cos^2 \phi'}}{\cos \beta - \sqrt{\cos^2 \beta - \cos^2 \phi'}}\]

• Valid for \(\beta \leq \phi'\)

Passive Lateral Earth Pressure (Cohesionless Soil):

\[\sigma_p = K_p \gamma z\]

• \(\sigma_p\) = passive lateral earth pressure (psf or kPa)

Passive Lateral Earth Pressure (Cohesive Soil):

\[\sigma_p = K_p \gamma z + 2c\sqrt{K_p}\]

Resultant Passive Force per Unit Length (Cohesionless Soil):

\[P_p = \frac{1}{2} K_p \gamma H^2\]

• \(P_p\) = total passive force per unit length of wall (lb/ft or kN/m)
• Acts at H/3 above base of wall

Resultant Passive Force (Cohesive Soil):

\[P_p = \frac{1}{2} K_p \gamma H^2 + 2cH\sqrt{K_p}\]

Coulomb Earth Pressure Theory

Active Earth Pressure Coefficient (Coulomb):

\[K_a = \frac{\sin^2(\alpha + \phi')}{\sin^2 \alpha \sin(\alpha - \delta) \left[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \beta)}{\sin(\alpha - \delta)\sin(\alpha + \beta)}}\right]^2}\]

• \(\alpha\) = angle of back face of wall from horizontal (degrees)
• \(\delta\) = angle of wall friction (degrees)
• \(\beta\) = angle of backfill slope to horizontal (degrees)
• \(\phi'\) = effective angle of internal friction (degrees)
• Accounts for wall friction and wall inclination

Passive Earth Pressure Coefficient (Coulomb):

\[K_p = \frac{\sin^2(\alpha - \phi')}{\sin^2 \alpha \sin(\alpha + \delta) \left[1 - \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' + \beta)}{\sin(\alpha + \delta)\sin(\alpha + \beta)}}\right]^2}\]

Wall Friction Angle:

\[\delta = k \phi'\]

• k = friction coefficient, typically 0.5 to 0.67 for active case, 0.33 to 0.50 for passive case
• Common assumption: \(\delta = \frac{2}{3}\phi'\) for active; \(\delta = \frac{1}{2}\phi'\) for passive

Retaining Wall Design

Gravity and Cantilever Retaining Walls

Overturning Moment:

\[M_O = P_a \times y_a\]

• \(M_O\) = overturning moment about toe (lb-ft/ft or kN-m/m)
• \(y_a\) = vertical distance from toe to point of application of \(P_a\) (ft or m)

Resisting Moment:

\[M_R = \sum (W_i \times x_i)\]

• \(M_R\) = resisting moment about toe (lb-ft/ft or kN-m/m)
• \(W_i\) = weight of individual wall component or soil wedge (lb/ft or kN/m)
• \(x_i\) = horizontal distance from toe to centroid of \(W_i\) (ft or m)

Factor of Safety Against Overturning:

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

• \(FS_{OT}\) = factor of safety against overturning (dimensionless)
• Minimum typically 1.5 to 2.0

Factor of Safety Against Sliding:

\[FS_{SL} = \frac{\sum V \tan \delta_b + c_b B + P_p}{\sum H}\]

• \(FS_{SL}\) = factor of safety against sliding (dimensionless)
• \(\sum V\) = sum of vertical forces (lb/ft or kN/m)
• \(\delta_b\) = angle of friction at base (degrees)
• \(c_b\) = cohesion at base (psf or kPa)
• B = base width (ft or m)
• \(P_p\) = passive resistance at toe (lb/ft or kN/m)
• \(\sum H\) = sum of horizontal forces (lb/ft or kN/m)
• Minimum typically 1.5 to 2.0
• Passive resistance often neglected or heavily factored

Simplified Sliding (Without Passive):

\[FS_{SL} = \frac{\sum V \tan \delta_b}{\sum H}\]

Location of Resultant Force:

\[x = \frac{M_R - M_O}{\sum V}\]

• x = horizontal distance from toe to resultant force (ft or m)
• For middle third: \(\frac{B}{6} \leq x \leq \frac{5B}{6}\)

Eccentricity:

\[e = \frac{B}{2} - x\]

• e = eccentricity of resultant from center of base (ft or m)
• For no tension: \(e \leq \frac{B}{6}\)

Base Pressure Distribution

Maximum and Minimum Base Pressure (e ≤ B/6):

\[q_{max,min} = \frac{\sum V}{B}\left(1 \pm \frac{6e}{B}\right)\]

• \(q_{max}\) = maximum base pressure at toe (psf or kPa)
• \(q_{min}\) = minimum base pressure at heel (psf or kPa)
• Both values should be positive (no tension)

Maximum Base Pressure (e > B/6):

\[q_{max} = \frac{2\sum V}{3x}\]

• Applies when resultant falls outside middle third
• Linear distribution with zero pressure at heel

Factor of Safety Against Bearing Capacity Failure:

\[FS_B = \frac{q_{ult}}{q_{max}}\]

• \(FS_B\) = factor of safety against bearing capacity failure (dimensionless)
• \(q_{ult}\) = ultimate bearing capacity (psf or kPa)
• Minimum typically 2.5 to 3.0

Structural Design of Wall Components

Maximum Moment in Stem:

\[M_{max} = \frac{1}{6}K_a \gamma H^3\]

• For cantilever stem with triangular pressure distribution
• Occurs at base of stem

Maximum Shear in Stem:

\[V_{max} = \frac{1}{2}K_a \gamma H^2\]

Sheet Pile Walls

Cantilever Sheet Pile Walls

Design Depth (Free Earth Support Method):

\[D = \frac{H}{\sqrt{\frac{K_p}{K_a} - 1}}\]

• D = depth of embedment (ft or m)
• H = height of retained soil (ft or m)
• Simplified relationship for preliminary design

Theoretical Depth of Embedment:

• Obtained by equating moments about point of zero shear
• Actual design depth = theoretical depth × factor (typically 1.2 to 1.4)

Maximum Bending Moment:

• Occurs at point of zero shear
• Calculated by taking moment of all forces above point of zero shear

Anchored Sheet Pile Walls

Anchor Force (Free Earth Support Method):

\[T = P_a - P_p\]

• T = anchor force per unit length (lb/ft or kN/m)
• \(P_a\) = active force above dredge line
• \(P_p\) = passive force below dredge line

Embedment Depth (Free Earth Support):

• Determined by moment equilibrium about anchor point
• Design depth = theoretical depth × factor (typically 1.2)

Embedment Depth (Fixed Earth Support):

• Assumes fixity at some point below dredge line
• Requires moment and shear equilibrium
• Design depth = theoretical depth × factor (typically 1.4)

Braced Excavations

Lateral Earth Pressure Distribution

Apparent Pressure Diagram (Peck's Envelopes)

For Sands:

\[p_a = 0.65 K_a \gamma H\]

• \(p_a\) = apparent pressure (psf or kPa)
• Rectangular distribution from 0 to H

For Soft to Medium Clay:

\[p_a = \gamma H \left(1 - \frac{4c}{\gamma H}\right) \geq 0.3\gamma H\]

• Trapezoidal distribution
• c = undrained shear strength (psf or kPa)

For Stiff Clay:

\[p_a = 0.2\gamma H \text{ to } 0.4\gamma H\]

• Trapezoidal distribution
• Values depend on stability number

Strut Loads

Tributary Area Method:

\[P_{strut} = p_a \times A_{tributary} \times s\]

• \(P_{strut}\) = load on individual strut (lb or kN)
• \(A_{tributary}\) = tributary area for strut (ft² or m²)
• s = horizontal spacing of struts (ft or m)

Basal Heave Stability

Factor of Safety Against Basal Heave (Terzaghi):

\[FS_{BH} = \frac{5.7c}{q}\]

• \(FS_{BH}\) = factor of safety against basal heave (dimensionless)
• c = undrained shear strength of clay (psf or kPa)
• q = net vertical stress at base of excavation (psf or kPa)
• Minimum typically 1.5

Net Vertical Stress:

\[q = \gamma H - \gamma' D\]

• \(\gamma'\) = effective unit weight of soil below excavation (pcf or kN/m³)
• D = depth of wall penetration below excavation (ft or m)

Factor of Safety Against Basal Heave (Bjerrum and Eide):

\[FS_{BH} = \frac{N_c c}{\gamma H}\]

• \(N_c\) = bearing capacity factor (function of H/B ratio)
• B = width of excavation (ft or m)

Mechanically Stabilized Earth (MSE) Walls

Internal Stability

Maximum Tension in Reinforcement:

\[T_{max} = K_a \gamma z S_v S_h\]

• \(T_{max}\) = maximum tension force in reinforcement (lb/ft or kN/m)
• z = depth below top of wall (ft or m)
• \(S_v\) = vertical spacing of reinforcement (ft or m)
• \(S_h\) = horizontal spacing of reinforcement (ft or m)

Factor of Safety Against Tensile Failure:

\[FS_T = \frac{T_{ult}}{T_{max}}\]

• \(FS_T\) = factor of safety against tensile failure (dimensionless)
• \(T_{ult}\) = ultimate tensile strength of reinforcement (lb/ft or kN/m)
• Minimum typically 1.5 for metallic, 1.5 to 2.0 for geosynthetic

Allowable Tensile Strength (Geosynthetic):

\[T_{allow} = \frac{T_{ult}}{RF_{ID} \times RF_{CR} \times RF_{CD}}\]

• \(T_{allow}\) = allowable tensile strength (lb/ft or kN/m)
• \(RF_{ID}\) = reduction factor for installation damage
• \(RF_{CR}\) = reduction factor for creep
• \(RF_{CD}\) = reduction factor for chemical/biological degradation

Pullout Resistance

Pullout Resistance (Extensible Reinforcement):

\[P_R = 2L_e b \sigma_v' \tan \phi' F^*\]

• \(P_R\) = pullout resistance (lb/ft or kN/m)
• \(L_e\) = embedment length in resistant zone (ft or m)
• b = width of reinforcement strip (ft or m)
• \(\sigma_v'\) = effective vertical stress at reinforcement level (psf or kPa)
• \(F^*\) = pullout friction factor (typically 0.67 to 1.0)
• Factor 2 accounts for top and bottom surfaces

Factor of Safety Against Pullout:

\[FS_{PO} = \frac{P_R}{T_{max}}\]

• \(FS_{PO}\) = factor of safety against pullout (dimensionless)
• Minimum typically 1.5

Required Embedment Length:

\[L_e = \frac{L_r}{2} \geq \frac{FS_{PO} \times T_{max}}{2b\sigma_v' \tan \phi' F^*}\]

• \(L_r\) = total length of reinforcement (ft or m)
• Active zone typically extends 0.3H to 0.5H from wall face

External Stability

Reinforcement Length for External Stability:

\[L_r \geq 0.7H \text{ to } 1.0H\]

• Common practice: \(L_r = 0.7H\) minimum
• May need longer for special conditions

Factor of Safety Against Sliding:

\[FS_{SL} = \frac{\sum V \tan \delta + cB}{\sum H}\]

• Analyzed same as conventional retaining wall
• Minimum typically 1.5

Factor of Safety Against Overturning:

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

• Minimum typically 2.0

Bearing Capacity:

\[FS_B = \frac{q_{ult}}{q_{max}}\]

• Minimum typically 2.5

Soil Nailing

Nail Tensile Capacity

Ultimate Tensile Capacity:

\[T_{ult} = f_y A_s\]

• \(f_y\) = yield strength of steel (psi or MPa)
• \(A_s\) = cross-sectional area of nail bar (in² or mm²)

Allowable Tensile Capacity:

\[T_{allow} = \frac{T_{ult}}{FS_T}\]

• Typical \(FS_T\) = 1.5 to 1.8

Pullout Capacity

Pullout Resistance:

\[Q_u = \pi D L q_u\]

• \(Q_u\) = ultimate pullout resistance (lb or kN)
• D = drill hole diameter (ft or m)
• L = unbonded length of nail (ft or m)
• \(q_u\) = ultimate bond stress (psf or kPa)

Allowable Pullout Resistance:

\[Q_{allow} = \frac{Q_u}{FS_{PO}}\]

• Typical \(FS_{PO}\) = 2.0

Design Forces

Maximum Nail Force:

\[T_{max} = K_a \gamma z S_h S_v\]

• \(S_h\) = horizontal spacing of nails (ft or m)
• \(S_v\) = vertical spacing of nails (ft or m)
• Similar to MSE reinforcement

Surcharge Loads

Uniform Surcharge

Lateral Pressure from Uniform Surcharge:

\[\Delta \sigma_h = K_a q\]

• \(\Delta \sigma_h\) = additional lateral pressure (psf or kPa)
• q = uniform surcharge load (psf or kPa)
• Acts uniformly over height of wall

Resultant Force from Uniform Surcharge:

\[P_q = K_a q H\]

• Acts at H/2 above base

Line Load Surcharge

Lateral Pressure from Line Load (Boussinesq):

\[\sigma_h = \frac{2Q}{\pi H} \frac{m^2 n}{(m^2 + n^2)^2}\]

• Q = line load intensity (lb/ft or kN/m)
• m = x/H (dimensionless)
• n = z/H (dimensionless)
• x = horizontal distance from load to wall (ft or m)
• z = vertical distance below top of wall (ft or m)

Point Load Surcharge

Lateral Pressure from Point Load (Boussinesq):

\[\sigma_h = \frac{3P}{2\pi z^2} \frac{x^2 z}{(x^2 + z^2)^{5/2}}\]

• P = point load (lb or kN)
• x = horizontal distance from load (ft or m)
• z = vertical distance from load (ft or m)

Water Effects

Submerged Conditions

Effective Unit Weight:

\[\gamma' = \gamma_{sat} - \gamma_w\]

• \(\gamma'\) = effective (submerged) unit weight (pcf or kN/m³)
• \(\gamma_{sat}\) = saturated unit weight (pcf or kN/m³)
• \(\gamma_w\) = unit weight of water = 62.4 pcf or 9.81 kN/m³

Active Earth Pressure (Submerged Backfill):

\[\sigma_a = K_a \gamma' z + \gamma_w z\]

• First term: effective stress contribution
• Second term: hydrostatic pressure

Total Active Force (Submerged):

\[P_a = \frac{1}{2}K_a \gamma' H^2 + \frac{1}{2}\gamma_w H^2\]

• Soil pressure acts at H/3 above base
• Water pressure acts at H/3 above base

Seepage Effects

Seepage Force per Unit Volume:

\[j = i \gamma_w\]

• j = seepage force per unit volume (pcf or kN/m³)
• i = hydraulic gradient (dimensionless)

Hydraulic Gradient:

\[i = \frac{\Delta h}{L}\]

• \(\Delta h\) = head loss (ft or m)
• L = flow path length (ft or m)

Modified Unit Weight (Horizontal Seepage):

\[\gamma_{mod} = \gamma' \pm i\gamma_w\]

• Plus sign when seepage is downward
• Minus sign when seepage is upward

Seismic Design

Mononobe-Okabe Method

Seismic Active Earth Pressure Coefficient:

\[K_{AE} = \frac{\cos^2(\phi' - \theta - \beta)}{\cos \theta \cos^2 \beta \cos(\delta + \beta + \theta)\left[1 + \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' - \theta - i)}{\cos(\delta + \beta + \theta)\cos(i - \beta)}}\right]^2}\]

• \(K_{AE}\) = seismic active earth pressure coefficient (dimensionless)
• \(\theta\) = seismic inertia angle = \(\tan^{-1}(k_h/(1-k_v))\)
• \(k_h\) = horizontal seismic coefficient (dimensionless)
• \(k_v\) = vertical seismic coefficient (dimensionless)
• i = backfill slope angle (degrees)

Seismic Inertia Angle:

\[\theta = \tan^{-1}\left(\frac{k_h}{1 - k_v}\right)\]

• Typically \(k_v = 0\) for design
• \(\theta = \tan^{-1}(k_h)\) when \(k_v = 0\)

Total Seismic Active Force:

\[P_{AE} = \frac{1}{2}K_{AE} \gamma H^2 (1 - k_v)\]

• \(P_{AE}\) = total seismic active force (lb/ft or kN/m)
• Acts at 0.6H above base (higher than static case)

Incremental Seismic Force:

\[\Delta P_E = P_{AE} - P_a\]

• \(\Delta P_E\) = additional force due to seismic loading (lb/ft or kN/m)
• Often assumed to act at 0.6H above base

Seismic Passive Earth Pressure Coefficient:

\[K_{PE} = \frac{\cos^2(\phi' + \theta - \beta)}{\cos \theta \cos^2 \beta \cos(\delta - \beta + \theta)\left[1 - \sqrt{\frac{\sin(\phi' + \delta)\sin(\phi' + \theta - i)}{\cos(\delta - \beta + \theta)\cos(i - \beta)}}\right]^2}\]

Seismic Coefficients

Horizontal Seismic Coefficient:

\[k_h = \frac{F_{PGA} \times S_{DS}}{2}\]

• \(F_{PGA}\) = site coefficient for peak ground acceleration
• \(S_{DS}\) = design spectral response acceleration parameter
• Factor of 2 provides conservatism for allowable stress design
• Consult AASHTO or building code for specific requirements

Anchored Systems

Ground Anchor Capacity

Ultimate Anchor Capacity (Cohesionless Soil):

\[Q_u = \pi D L \sigma_v' K \tan \phi'\]

• \(Q_u\) = ultimate anchor capacity (lb or kN)
• D = anchor diameter (ft or m)
• L = anchor bond length (ft or m)
• \(\sigma_v'\) = average effective vertical stress along bond length (psf or kPa)
• K = lateral earth pressure coefficient for anchor

Ultimate Anchor Capacity (Cohesive Soil):

\[Q_u = \pi D L \alpha c_u\]

• \(\alpha\) = adhesion factor (dimensionless, typically 0.5 to 1.0)
• \(c_u\) = undrained shear strength (psf or kPa)

Allowable Anchor Capacity:

\[Q_{allow} = \frac{Q_u}{FS}\]

• Typical FS = 2.0 to 3.0

Anchor Spacing and Layout

Anchor Load (Tributary Area):

\[T = \sigma_h S_h S_v\]

• T = anchor force (lb or kN)
• \(\sigma_h\) = lateral earth pressure at anchor level (psf or kPa)
• \(S_h\) = horizontal spacing between anchors (ft or m)
• \(S_v\) = vertical spacing between anchor rows (ft or m)

Drainage Considerations

Drainage Design

Filter Criteria (Terzaghi):

\[\frac{D_{15(filter)}}{D_{85(base)}} < 5\]="" \[\frac{d_{15(filter)}}{d_{15(base)}}=""> 5\] \[\frac{D_{50(filter)}}{D_{50(base)}} < 25\]="">

• First criterion prevents piping
• Second criterion ensures permeability
• Third criterion provides additional safety
• \(D_{xx}\) = particle size at which xx% is finer

Drainage Layer Flow Capacity:

\[q = kiA\]

• q = flow rate (ft³/s or m³/s)
• k = coefficient of permeability (ft/s or m/s)
• i = hydraulic gradient (dimensionless)
• A = cross-sectional area of flow (ft² or m²)

Deflection and Deformation

Wall Deflection Estimates

Cantilever Wall Deflection (Approximate):

\[\delta_H \approx \frac{H}{200} \text{ to } \frac{H}{500}\]

• \(\delta_H\) = horizontal deflection at top of wall (ft or m)
• Depends on wall stiffness, soil properties, and construction

Anchored Wall Deflection:

\[\delta_H \approx \frac{H}{500} \text{ to } \frac{H}{1000}\]

• Generally smaller than cantilever walls
• Depends on anchor prestress and stiffness

Ground Movement Behind Wall:

\[\delta_v \approx \frac{\delta_H}{2} \text{ to } 2\delta_H\]

• \(\delta_v\) = vertical settlement at ground surface (ft or m)
• Zone of influence typically extends 1H to 2H behind wall

Special Conditions

Reinforced Backfill

Apparent Cohesion from Geogrid:

\[c_{apparent} = \frac{T_f}{S_v}\]

• \(c_{apparent}\) = apparent cohesion contribution (psf or kPa)
• \(T_f\) = tensile force in geogrid at failure (lb/ft or kN/m)
• \(S_v\) = vertical spacing of geogrid (ft or m)

Broken Back Walls

Equivalent Height Method:

\[H_{eq} = H_1 + H_2 \frac{\gamma_2}{\gamma_1}\]

• \(H_{eq}\) = equivalent height for uniform backfill (ft or m)
• \(H_1\) = height of lower backfill zone (ft or m)
• \(H_2\) = height of upper backfill zone (ft or m)
• \(\gamma_1, \gamma_2\) = unit weights of respective zones (pcf or kN/m³)

Temperature Effects

Thermal Stress in Struts:

\[\Delta T = \alpha L \Delta T E\]

• \(\Delta T\) = additional stress due to temperature change (psi or MPa)
• \(\alpha\) = coefficient of thermal expansion (in/in/°F or mm/mm/°C)
• L = length of member (ft or m)
• \(\Delta T\) = temperature change (°F or °C)
• E = modulus of elasticity (psi or MPa)
• For steel: \(\alpha \approx 6.5 \times 10^{-6}\)/°F or \(11.7 \times 10^{-6}\)/°C