Rigid Pavement Mechanics Curling Stresses
Major Distress Conditions Cracking Bottom-up transverse cracks Top-down transverse cracks Longitudinal cracks Corner breaks Punchouts (CRCP) 2
Major Distress Conditions Punchouts Cracks Corner Break 3
Major Distress Conditions Cracking Bottom-up transverse cracks Top-down transverse cracks Longitudinal cracks Corner breaks Punchouts (CRCP) Joint Faulting Pumping 4
Major Distress Conditions Faulting Pumping 5
Sources of Stress Wheel Loads Curing Shrinkage Thermal Contraction/Expansion Thermal Curling Moisture Warping 6
Design Considerations Slab Thickness Base Type and Thickness Drainage Joint Spacing Temperature Steel Dowel Bars Tie Bars 7
Beam Bending q o = q b (applied load per unit length) q o k o R R 8
Beam Bending dm V dx 2 dm q 2 o dv dx q o dx Shear and Moment Relationships 9
Beam Bending 1 2 dw 2 R dx 2 dw M EI dx 2 M EI Moment-Curvature Relationships 10
Beam Bending d dx EI d w 2 2 dx 2 2 q o EI 4 dw dx q 4 o Governing Differential Equation for Beam Bending 11
Beams on Elastic Foundations q o = q b (applied load per unit length) q o k o R R 12
Beams on Elastic Foundations q o = q b (applied load per unit length) q o k o p = k o w = k b w (resistance per unit length) Modulus of Subgrade Reaction 13
Beams on Elastic Foundations dm V dx 2 w 2 dv dx k o w q o dx dm ko q o Shear and Moment Relationships 14
Beam Bending 1 2 dw 2 R dx 2 dw M EI dx 2 M EI Moment-Curvature Relationships 15
Beams on Elastic Foundations d dx d w EI kw o dx 2 2 2 2 q o 4 dw EI k w dx q 4 o o Governing Differential Equation for Beam Bending 16
Beams on Elastic Foundations EI 4 dw k 0 4 ow dx x sin cos sin cos w e C x C x e C x C x x 1 2 3 4 k 4 o 4 EI (flexibility) Solution of the Homogeneous Differential Equation 17
Slabs on Elastic Foundations 2 dw M EI Beams dx 2 2 2 w w Mx D 2 2 Slabs x y Moment-Curvature Relationships 18
Slabs on Elastic Foundations D Eh 3 12 1 2 Slab Stiffness 19
Slabs on Elastic Foundations EI 4 d w w 4 dx k o q o Beams 4 4 4 w w w D 2 kw q 4 2 2 4 x x y y Slabs Modulus of Subgrade Reaction Governing Differential Equation for Slab Bending 20
Slabs on Elastic Foundations Radius of Relative Stiffness 4 D k k 4 o 4 4 EI Eh Beam Flexibility 3 12 1 2 k Slab Stiffness Solution of the Homogeneous Differential Equation 21
Stresses Due to Curling
Temperature Curling COOL UPWARD CURLING WARM WARM DOWNWARD CURLING COOL 23
Moisture Warping DRY UPWARD WARPING WET WET DOWNWARD WARPING DRY 24
Reference Axes x y 25
Infinite Slab Curling x y x E E y y E E x Hooke s Law for an Infinite Elastic Plate 26
Slab Bent About y-axis x y 27
Infinite Slab Curling y x y E E 0 y x 0 y x If slab is bent about the y axis, y = 0 28
Infinite Slab Curling x x x x 1 E E E 2 x E 1 x 2 Slab bent about y axis 29
Slab Bent About x-axis x y 31
Infinite Slab Curling x y x E E 0 x y 0 x y If slab is bent about the x axis, x = 0 32
Infinite Slab Curling y y y y 1 E E E 2 y E y 1 2 Slab bent about x axis 33
Infinite Slab Curling Coefficient of Thermal Expansion x y t 2 Strains due to temperature difference t through slab depth 35
Coefficient of Thermal Expansion Aggregate Type Coefficient (10-6 in/in/ o F) Quartz 6.6 Sandstone 6.5 Gravel 6.0 Granite 5.3 Basalt 4.8 Limestone 3.8 Average 5.5 36
Infinite Slab Curling E Et x 1 21 y x 2 2 x Et 21 2 Stresses due to curling about y axis due to temperature difference 37
Infinite Slab Curling E y Et y 1 21 x y 2 2 Et 21 2 Stresses due to curling about x axis due to temperature difference 38
Infinite Slab Curling Et Et xy, 2 2 21 21 Maximum curling stresses in an infinite slab 39
Finite Slab Curling Et Et xyc C, 1 21 2 21 2 2 Curling stresses in the Interior of a Finite Slab 40
Finite Slab Curling 41
Finite Slab Curling 0 0 y x E t xy, C 1 2 Curling stresses at the Edges of a Finite Slab 42
Temperature Gradients Source Slab Thickness Temperature Gradient AASHO Road Test 6.5" 3.2F/in 1.5F/in Arlington Road Test 6" 3.7F/in 9" 3.4F/in Typical Assumption 6 9" 3.0F/in 1.5F/in 43
Example Calculate the curling stresses in a concrete slab 25 12 8 thick subject to a daytime temperature difference of 24 F (i.e., a temperature gradient of 3 F/in). Assume the slab is resting on a foundation with a 200-psi/in modulus of subgrade reaction. 44
Standard Assumptions Assume = 0.15 (use for all PCC) Assume = 5.510 6 in/in/ F (typical) Assume f c = 5000 psi (typical) Estimate Estimate psi Estimate psi Not needed for this problem psi 45