Front matter for Theory Manual for the Load Displacement Compatibility Method (LDC) for Design of Column-Supported Embankments

Other Titles of Interest

Contributors

Acknowledgments

Preface

Notation

A

Unit cell area

A_c

Area of column in a unit cell; the cross-sectional area of the column or cap

A_rectangular

Unit cell area in a rectangular array

A_s

Portion of the unit cell area occupied by soil at the elevation of the reinforcement

A_s1

Tributary area of unit cells along span c₁

A_s2

Tributary area of unit cells along span c₂

A_triangular

Unit cell area in a triangular array

a

Equivalent square width of the column or pile cap

a_s

Area replacement ratio; the ratio of the area of column in a unit cell, A_c, to the unit cell area, A

AR

Arching ratio used in the concentric arches and Hewlett and Randolph models

AVG, $\overline{x}$

Average parameter value

B

Ratio of the area of soil in a unit cell to the perimeter of the column

BAT

Bilinear adapted Terzaghi method for modeling arching mobilization

c

The total cohesion or adhesion

c′

Effective cohesion

c₁, c₂

Clear spans used in the parabolic method, over which the vertical load is uniformly distributed; c₁ ≤ c₂

c₁, c₂, c₃, c₄, c₅, c₆, c₇

Coefficients used to predict the arching ratio

C_εc

Compression ratio; the slope of the rebound portion of the consolidation curve on a log-stress versus strain diagram

C_εr

Recompression ratio; the slope of the virgin portion of the consolidation curve on a log-stress versus strain diagram

c_v

Coefficient of consolidation

CA

Concentric arches model

COV

Coefficient of variation

d, d_c

Column diameter

d_cap

Pile cap diameter

DS_b,d

Largest differential settlement at the base of the embankment

DS_b,d,yield

Displacement at which arching becomes fully mobilized

d_w

Depth below the base of the embankment to the groundwater table

dz

Vertical thickness of an increment used in the incremental form of the adapted Terzaghi (AT) method

d₁

Maximum deflection of the reinforcement across the c₁ span

d₂

Maximum deflection of the reinforcement across the c₂ span

$d{\sigma }_z^{\prime }$

Effective pressure of the increment used in the incremental form of the adapted Terzaghi (AT) method

E

Value of efficacy used in the calculation of the SRR according to the HR model

E_a

Young's modulus of the arching soil

E_c

Young's modulus of the column and pile cap

E_s

Young's modulus of the sand layer

F_r (T)

Infinite series used in calculating the Janbu degree of consolidation, U_r

f’_c

Compressive strength of concrete

f_s

Shape factor for the strain distribution used in time rate of consolidation calculations

GGB

GeogridBridge

GRC

Ground reaction curve

H, H_emb

Total embankment height

H_a

Height of arching used in the incremental form of the adapted Terzaghi method

H_cap

Height of the pile cap

H_col

Length of the column

H_crit

Critical height of the embankment; the minimum embankment height at which differential settlement at the base of the embankment will not produce differential settlement at the surface of the embankment

H_dr

Maximum drainage path

H_q

Height above arching used in the incremental form of the adapted Terzaghi method

HR

Hewlett and Randolph model

i

Reference to the current increment used in the incremental form of the adapted Terzaghi (AT) method

J

Combined long term tensile stiffness of the geosynthetic across each span between columns

J₁

Tensile stiffness of the geosynthetic along the c₁ span

J₂

Tensile stiffness of the geosynthetic along the c₂ span

K

Lateral earth pressure coefficient for evaluating load transfer by soil arching or drag force

K_o

At-rest earth pressure coefficient

K_p

Rankine passive earth pressure coefficient

k

Coefficient relating DS_b,d to the average of the midspan deflections, generally equal to 1

k₁

Coefficient for span c₁ used in the parabolic method

k₂

Coefficient for span c₂ used in the parabolic method

LB

Lower bound of a distribution used in an MCS

LDC

Load displacement compatibility

LTP

Load transfer platform

LTM

Load transfer mat

M_c

One-dimensional elastic compression modulus of the column and pile cap

M_s

One-dimensional elastic compression modulus of the sand

MCS

Monte Carlo Simulation

m

Slope of the deflected shape of the reinforcement

N

Number of trials to be run in an MCS

n

Total number of increments used in the incremental form of the adapted Terzaghi (AT) method

OCM

Overconsolidation margin

OCR

Overconsolidation ratio

P_a

Atmospheric pressure

P_exceed

Probability of serviceability failure

$P_{\text{bottom}}^{\prime }$

Effective normal force acting upward on the bottom of the increment used in the incremental form of the adapted Terzaghi (AT) method

$P_{\text{top}}^{\prime }$

Effective force acting on top of an increment because of overburden used in the incremental form of the adapted Terzaghi (AT) method

p

Perimeter of the column or pile cap

P

Preload during construction

q

Postconstruction surface loading

RF_CR,5%

Reduction factor for creep at 5% strain

RF_CR,ult

Reduction factor for creep at the ultimate strain

RF_D

Reduction factor for durability

RF_ID

Reduction factor for installation damage

r

Coefficient that alters the shape of the strain profile used in the time rate of consolidation calculations

S_b,c

Settlement at the base of the embankment directly over the column

S_b,d

Settlement at the base of the embankment at a distance s′ from the edge of the nearest column or pile cap

S_c

Settlement at the embankment surface directly over the column

S_d

Settlement at the embankment surface at a distance s′ from the edge of the nearest column or pile cap

S_s

Total uniform settlement at the embankment surface

S_s,pc

Total postconstruction settlement at the embankment surface

S_u

Undrained strength

SD, σ

Standard deviation

SGRC

Simplified ground reaction curve

SGRC-ACA

Simplified ground reaction curve estimated using the approximate concentric arches model

SGRC-AHR

Simplified ground reaction curve estimated using the approximate Hewlett and Randolph model

SRR

Stress reduction ratio

SRR_emb

Stress reduction ratio associated with load transfer by arching within the embankment

SRR_fndn

Stress reduction ratio associated with load transfer by arching within the embankment and tension developed in the geosynthetic

SRR_peak

Value of SRR_emb at peak arching

SRR_term

Value of SRR_emb at terminal arching

s′

Greatest distance a location within the unit cell can be from the edge of the closest column, as represented using a circular cross section

s₁, s₂

Center-to-center spacing between the columns. s₁ ≤ s₂

T_v

Dimensionless time factor used in the time rate of consolidation calculations

T

Tension in the geosynthetic reinforcement

T₁

Tension developed within the reinforcement across the c₁ span per unit width

T₂

Tension developed within the reinforcement across the c₂ span per unit width

T_x,1

Horizontal component of T₁

T_x,2

Horizontal component of T₂

T_z,1

Vertical component of T₁

T_z,2

Vertical component of T₂

T_5%,LT

Long-term tensile strength at 5% strain

T_5%,ST

Short-term tensile strength at 5% strain

t

Time available for consolidation prior to post-construction surface loading

U_r

Janbu degree of consolidation

U₀

Terzaghi degree of consolidation

UB

Upper bound of a distribution used in an MCS

UGRC

User-defined ground reaction curve

V

Vertical shear force between moving and fixed portions of the increment used in the incremental form of the adapted Terzaghi (AT) method

W′

Effective weight of the increment used in the incremental form of the Adapted Terzaghi (AT) method

w₁

Magnitude of the uniform vertical line load across the c₁ span

w₂

Magnitude of the uniform vertical line load across the c₂ span

z_np

Depth below the embankment surface to the neutral plane

α

Factor used to scale undrained strength to soil-column adhesion in α-Method

γ

Material unit weight

γ_total

Total unit weight

γ′

Effective unit weight

γ_water

Unit weight of water

δ

Interface friction angle

Δσ_z

Change in total stress in compressible layer owing to excavation or filling

ε_s

Strain at the top of the clay layer

ε_d

Difference between the strain at the top and the bottom of the clay layer

ε_s,ult

Ultimate strain in a sand increment

ε₁

Average strain in the geosynthetic reinforcement along span c₁

ε₂

Average strain in the geosynthetic reinforcement along span c₂

ν_a

Poisson's ratio of the arching soil

ν_c

Poisson's ratio of the column and pile cap

ν_s

Poisson's ratio of the sand layer

${\sigma }_c^{\prime }$

Vertical stress in the column

σ_col,geobot

Stress acting on the columns owing to the embankment load considering both soil arching and the tensioned geosynthetic (if present) for load transfer

σ_col,geotop

Stress acting on the columns owing to the embankment load considering exclusively load transfer through soil arching

σ_net

Average net vertical pressure carried by the reinforcement

${\sigma }_o^{\prime }$

Initial vertical effective stress in the foundation soils based on the effective overburden stress prior to loading from the embankment or surface loading and including any stress adjustment

${\sigma }_p^{\prime }$

Preconsolidation stress

${\sigma }_s^{\prime }$

Final vertical effective stress in the soil after embankment construction and application of surface considering the effects of vertical load transfer

${\sigma }_{s,i}^{\prime }$

Vertical stress at the base of each foundation soil increment above the neutral plane

σ_soil,geobot

Stress acting on the foundation soil owing to the embankment load considering both soil arching and the tensioned geosynthetic (if present) for load transfer

σ_soil,geotop

Stress acting on the foundation soil owing to the embankment load considering exclusively load transfer through soil arching

${\sigma }_{z,a}^{\prime }$

Vertical effective stress calculated after embankment construction without considering any load transfer to the columns; equivalent to the average vertical stress within the unit cell with load transfer to the columns

${\sigma }_{z,\text{adjust}}^{\prime }$

Adjustment to the effective vertical stress based on the profile before embankment and surface loading to obtain the initial vertical effective stress

${\sigma }_{z,b}^{\prime }$

Effective vertical stress in the soil based on the profile before the loading from the embankment and surface loading and without any stress adjustment

${\sigma }_{z,\text{base}}^{\prime }$

Vertical effective stress imposed on the base of the embankment without considering load transfer

φ′

Effective internal friction angle