17 - curling stresses in concrete slabs

8/10/2019 17 - Curling Stresses in Concrete Slabs

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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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Shear and Moment Relationships

ookd

xw

Vq

d

dM Vdx 2

2 o od M

kd w qx


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Beam Bending


EI

2

2

1 d wR dx 2

2

d wM EI

dx


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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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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


2

2 d wM EI

dxBeams

2 2

2 2

x

w wM D

x ySlabs


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Slab Stiffness

3

212 1

EhD


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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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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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If slab is bent about the y axis, y = 0

y x

0 y xy E E

0 y x


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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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If slab is bent about the x axis, x = 0

x y

0 yxx E E

0 x y


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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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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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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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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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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

17 - curling stresses in concrete slabs

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