8 forcederivations-20
TRANSCRIPT
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VOL. V - PART 2
FULL-INTEGRALSample Hand Calculations and Derivations
pF Total lateral pile force required to resist rotation of superstructure (kips)
)cos( δ θ −=
qW P Full passive force across the width of the bridge (kips)
Passive Force per foot across backwallq
δ cos⋅= PPn Component of “P” normal to the backwall (kips)
δ sin⋅= PPt Component of “P” tangential to backwall (kips)
δ Angle of Wall Friction
θ Skew angle of bridge
δ θ −α 2
2
1 H K q ps ⋅⋅⋅= γ Resultant of passive force at a point along the backwall. (kips/LF)
H Height of integral abutment (ft)
pK Coefficient of passive earth pressure
sγ Unit weight of soil (pcf)
θ sin' ⋅= L L Moment arm for normal passive forceθ cos" ⋅= L L Moment arm for tangential passive force
)"()"()'(0 LF LP LP M pt n A ⋅−⋅−⋅==Σ
DATE: 11May2007
SHEET 1 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION FULL INTEGRAL FILE NO. 20.07-1
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VOL. V - PART 2
""'0 LF LP LP pt n ⋅−⋅−⋅=
"'" LP LP LF t n p ⋅−⋅=⋅
"
"'
L
LP LPF t n p
⋅−⋅= θ sin' ⋅= L L θ cos" ⋅= L L δ sin⋅= PPt δ cos⋅= PPn ; ; ;
)cos(
)cossin()sincos(
θ
θ δ θ δ
⋅
⋅⋅⋅−⋅⋅⋅=
L
LP LPF p
θ
θ δ θ δ
cos
)cossin()sincos( ⋅⋅−⋅⋅=
PPF p
)cos( δ θ −=
qW P
θ
θ δ θ δ
δ θ cos
)cos(sin)sin(cos
)cos(
⋅−⋅⋅
−
⋅=
W qF p )sin()cossinsin(cos δ θ θ δ θ δ −=⋅−⋅
)cos()sin(
)cos( θ δ θ
δ θ ⋅−⋅
−⋅= W qF p
θ
δ θ
cos
)tan( −⋅⋅=
W qF p
DATE: 11May2007
SHEET 2 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION FULL INTEGRAL FILE NO. 20.07-2
For skew angles, δ θ ≤ θ δ = 0= p R, assume , therefore .
0cos
)0tan(
=⋅⋅
= θ
W q
θ
θ θ
cos
)tan( −⋅⋅=
W qF p
Field observations to date indicate superstructure rotation with skew angles as low as 5º. Itappears that the shear force at the backwall/backfill interface is not always mobilized. Therefore,
it is recommended that the interaction angle of friction, δ between the soil and backwall be set tozero. Therefore, it can be assumed that the Pt and Pn forces are not mobilized, but only Pn remains. Therefore, these modifications can be made to the following equation:
)"()"()'(0 LF LP LP M pt n A ⋅−⋅−⋅==Σ
Or written as:
)"()'(0 LF LP M p A ⋅−⋅==Σ
"'0 LF LP p ⋅−⋅=
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VOL. V - PART 2
'" LP LF p ⋅=⋅
"
'
L
LPF p
⋅=
DATE: 11May2007
SHEET 3 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION FULL INTEGRAL FILE NO. 20.07-3
θ sin' ⋅= L L θ cos" ⋅= L L;
)cos(
)sin(
θ
θ
⋅
⋅⋅=
L
LPF p
θ cos
qW P =
)(cos
)sin(
θ
θ ⋅= P
F p
θ
θ
θ cos
sin
cos⋅
⋅= W q
F p
θ
θ
cos
tan⋅⋅=
W qF p
To ensure conservatism in the design, it is suggested that the above equation be used in allcases.
It should be noted that the force calculated using these equations is the theoretical force requiredto restrain all lateral movement. If no lateral restraint is intended or required, this force will beconsiderably smaller. Lateral restrain should be considered when transverse displacements caninterfere with performance, or adjacent structures (e.g. parallel bridges, MSE walls, utilities, etc.).
To determine the forces on the piles when transverse displacements are not restrained usesoftware such as COM624, L-PILE, or other similar software for analysis.
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VOL. V - PART 2
SEMI-INTEGRALButtress Force Derivations
p R Buttress force required to resist rotation of superstructure (kips)
)cos( δ θ −=
qW P Full passive force across the width of the bridge (kips)
δ cos⋅= PPn Component of “P” normal to the backwall (kips)
δ sin⋅= PPt Component of “P” tangential to backwall (kips)
δ Angle of Wall Friction
θ Skew angle of bridge
δ θ −α
2
2
1 H K q ps ⋅⋅⋅= γ Resultant of passive force at a point along the backwall. (kips/LF)
H Height of integral backwall (ft)
pK Coefficient of passive earth pressure
sγ Unit weight of soil (pcf)
θ tan2
⋅=W
C Moment arm for buttress force
θ sin' ⋅= L L Moment arm for normal passive force
θ cos" ⋅= L L Moment arm for tangential passive force
DATE: 11May2007
SHEET 4 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION SEMI-INTEGRAL FILE NO. 20.07-4
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VOL. V - PART 2
)()()"()'(0 C RC L R LP LP M p pt n A ⋅−+⋅−⋅−⋅==Σ
)2("'0 C L R LP LP pt n +⋅−⋅−⋅=
"')2( LP LPC L R t n p ⋅−⋅=+⋅
)2(
"'
C L
LP LP R t n
p+
⋅−⋅= θ tan2
⋅=W
C θ sin' ⋅= L L θ cos" ⋅= L L δ sin⋅= PPt ; ; ; ;
δ cos⋅= PPn
)tan2
(2
)cossin()sincos(
θ
θ δ θ δ
⋅+
⋅⋅−⋅⋅⋅=
W L
LP LP R
p
)cos( δ θ −=
qW P
)tan(1
)cossin()sincos(
θ
θ δ θ δ
⋅+
⋅⋅−⋅⋅=
L
W
PP R
p
)tan(1
)cos(sin)sin(cos
)cos(θ
θ δ θ δ
δ θ ⋅+
⋅−⋅⋅
−
⋅=
L
W
W q R
p )sin()cossinsin(cos δ θ θ δ θ δ −=⋅−⋅
)tan(1
)sin(
)cos(θ
δ θ
δ θ ⋅+
−⋅
−
⋅=
L
W
W q R
p
)tan(1
)tan(
θ
δ θ
⋅+
−⋅⋅=
L
W
W q
R p
DATE: 11May2007
SHEET 5 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION SEMI-INTEGRAL FILE NO. 20.07-5
For skew angles, δ θ ≤ θ δ = 0= p R, assume , therefore .
0
)tan(1
)0tan(=
⋅+
⋅⋅=
θ L
W
W q
)tan(1
)tan(
θ
θ θ
⋅+
−⋅⋅=
L
W
W q R p
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VOL. V - PART 2
Field observations to date indicate superstructure rotation with skew angles as low as 5º. Itappears that the shear force at the backwall/backfill interface is not always mobilized. Therefore,
it is recommended that the interaction angle of friction, δ between the soil and backwall be set tozero. Therefore, it can be assumed that the P and Pt n forces are not mobilized, but only Premains. Therefore, these modifications can be made to the following equation:
)()()"()'(0 C RC L R LP LP M p pt n A ⋅−+⋅−⋅−⋅==Σ rendering it as so:
)()()'(0 C RC L R LP M p p A ⋅−+⋅−⋅==Σ
)2('0 C L R LP p +⋅−⋅=
')2( LPC L R p ⋅=+⋅
θ tan2
⋅=W
C )2(
'
C L
LP R
p+
⋅=
DATE: 11May2007
SHEET 6 of 6
INTEGRAL / JOINTLESS BRIDGESFORCE DERIVATIONS
LATERAL FORCE DERIVATION SEMI-INTEGRAL FILE NO. 20.07-6
θ sin' ⋅= L L ;
)tan2
(2
)sin(
θ
θ
⋅+
⋅⋅=
W L
LP R p
θ cos
qW P =
)tan(1
)sin(
θ
θ
⋅+
⋅=
L
W
P R
p
)tan(1
sin
cos θ
θ
θ ⋅+
⋅⋅
=
LW
W q R
p
⎥⎦
⎤⎢⎣
⎡⋅+
⋅⋅=
)tan(1
)tan
θ
θ
L
W
W q R p
To ensure conservatism in the design, it is suggested that the above equation be used in allcases.