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Rigid Pavement Mechanics
Curling Stresses
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Sources of Stress
Autogenous (Curing) Shrinkage
Thermal Contraction/Expansion
Thermal Curling
Moisture Warping Frost Heave
Expansive Soils Wheel Loads
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Major Distress Conditions
Fatigue Cracking
Bottom-up transverse cracks
Top-down transverse cracks
Longitudinal cracks Corner breaks
Joint Faulting (JPCP and JRCP)
Pumping (JPCP and JRCP)
Punchouts (CRCP)
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Design Considerations
Slab Thickness
Base Type and Thickness
Joint Spacing
Temperature Steel Dowel Bars
Drainage
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Beam Bending
qo = q b (applied load per unit length)
qo
ko
R R
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Beam Bending
Shear and Moment Relationships
odV
qdx
dM Vdx 2
2 od M
qdx
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Beam Bending
Moment-Curvature Relationships
EI
2
2
1 d wR dx 2
2
d w
M EI dx
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Beam Bending
Governing Differential Equation for Beam Bending
2 2
2 2
o
d d wEI q
dx dx
4
4 o
d wEI q
dx
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Beams on Elastic Foundations
qo
ko
qo = q b (applied load per unit length)
p = ko w = k b w (resistance per unit length)
Modulus of Subgrade Reaction
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Beams on Elastic Foundations
Shear and Moment Relationships
ookd
xw
Vq
d
dM Vdx 2
2 o od M
kd w qx
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Beam Bending
Moment-Curvature Relationships
EI
2
2
1 d wR dx 2
2
d wM EI
dx
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Beams on Elastic Foundations
Governing Differential Equation for Beam Bending
2 2
2 2
o o
d d wEI q
dxk w
dx
4
4 o o
d wE q
xk
dwI
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Beams on Elastic Foundations
Solution of the Homogeneous Differential Equation
1 2 3 4sin cos sin cos
x x
w e C x C x e C x C x
4
4 o
k
EI
4
4 0 od w
EI k wdx
(flexibility)
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Slabs on Elastic Foundations
Moment-Curvature Relationships
2
2 d wM EI
dxBeams
2 2
2 2
x
w wM D
x ySlabs
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Slabs on Elastic Foundations
Slab Stiffness
3
212 1
EhD
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Slabs on Elastic Foundations
Governing Differential Equation for Slab Bending
Modulus of
Subgrade Reaction
4
4 o okdEI
xww q
dBeams
4 4 4
4 2 2 42
w w wD kw q
x x y y
Slabs
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Slabs on Elastic Foundations
Solution of the Homogeneous Differential Equation
4
4 ok
EIBeams
3
44
212 1
D Eh
k kSlabs
Radius of
Relative
Stiffness
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Stresses Due to Curling
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Temperature Curling
UPWARD CURLING
DOWNWARD CURLING
WARM
WARM
COOL
COOL
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Reference Axes
x
y
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Infinite Slab Curling
Hookes Law for an Infinite Elastic Plate
y xy
E E
yxxE E
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Slab Bent About y-axis
x
y
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Infinite Slab Curling
If slab is bent about the y axis, y = 0
y x
0 y xy E E
0 y x
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Infinite Slab Curling
Slab bent about y axis
2
1
xx
E
21 x x xxE E E
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Slab Bent About y-axis
xx
2
1
xx
E
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Slab Bent About x-axis
x
y
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Infinite Slab Curling
If slab is bent about the x axis, x = 0
x y
0 yxx E E
0 x y
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Infinite Slab Curling
Slab bent about x axis
2
1
y
y
E
21 y y yyE E E
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Slab Bent About x-axis
y
y
21
y
y
E
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Infinite Slab Curling
Strains due to temperature difference t through slab depth
2
x y
t
Coefficient ofThermal Expansion
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Coefficient of Thermal Expansion
AggregateType Coefficient(10-6 in/in/oF)
Quartz 6.6
Sandstone 6.5
Gravel 6.0
Granite 5.3
Basalt 4.8
Limestone 3.8
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Infinite Slab Curling
Stresses due to curling about y axis due to temperature difference
2 2
2
1 2 1
2 1
xx
y x
E E t
E t
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Infinite Slab Curling
Stresses due to curling about x axis due to temperature difference
2 2
2
1 2 1
2 1
y
y
x y
E E t
E t
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Infinite Slab Curling
Maximum curling stresses in an infinite slab
, 2 2
2 1 2 1
x y
E t E t
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Finite Slab Curling
Curling stresses in the Interior of a Finite Slab
, 1 22 2
2 1 2 1
x y
E t E tC C
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Finite Slab Curling
L
C
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Finite Slab Curling
Curling stresses at the Edges of a Finite Slab
, 12
x y
E tC
0 0 y x