Construction and Design of Rigid Pavement

Design and Construction of Rigid Pavement

Due to frequent maintenance requirements of Flexible pavements Rigid pavement has become a preferred choice by Highway agencies.

If designed and CONSTRUCTED properly by exercising strict supervision and Quality checks the Rigid pavement does offer a service life of above 30 years with minimal maintenance.

Example of Mumbai Pune Expressway completed in year 1999- still 90-95 % of the stretch in good-excellent condition.

We have various options for designing of Rigid Pavements- Design and Construction Practice manuals are as follows- 

Rigid Pavement :- Those Pavements which poses note worthy flexural strength or flexural rigidity. In rigid pavement the stresses are not transferred by aggregate interlock mechanism to the lower layers (as in case of flexible pavement). The rigid pavements are made of Portland cement concrete either plain or reinforced concrete. The plain cement concrete slabs are expected to take up about 4.5 MPa flexural stress. The rigid pavement has a slab action and is capable of transmitting the wheel load stresses through a wider area below. Tensile stresses are developed due to the bending of the slab under the wheel load temperature variations Providing a good base at sub base course layer under the cement concrete slab increase the pavement life considerably and there for workout more economical in the long run. The rigid pavements are usually designed and the stresses are analysed using the elastic theory.

1) IRC 58-2015 – Guidelines for Design of  Plain Jointed Rigid  Pavements for Highways / Rigid Pavements (JPCP), 

2) IRC 101( CRCP)- Guidelines for Design of Continuously Reinforced Concrete Pavement with Elastic Joints

3) IRC SP 118- 2015 – Guidelines for Design and Construction of Continuously Reinforced Concrete Pavement (CRCP)

4) IRC SP 76 - 2014 (TWT)  Guidelines for design of Thin White Topping pavements

5) IRC 15-2017- Standard Specifications and Code of Practice for Concrete Roads 

6) IRC SP-62-2014 – Guidelines for design and construction of cement concrete pavements for low volume roads 

   

Typical section of  Plain Jointed Rigid  Pavements for Highways

                                                        Typical Pavement Section

3D Diagram of Typical Rigid Pavement

Stress Distribution

 Loads and stresses distributed over wider area of Subgrade by Rigidity and Strength of  Pavement Slab

Pressure intensity on subgrade much lower than in case of Flexible pavement (approx. 10 times)

All loads carried by bending action of the Slab.

Stresses in Rigid Pavement :- 

 1) Wheel Load Stresses :- Slab is subjected to Flexural stresses

 2) Temperature Stresses :- 

        

a) Warping Stresses :- (Due to Daily variation of temperature) 

Due to temperature difference between top and bottom of slab, curling stresses similar to bending stresses are induced at bottom or top of the slab.

b) Frictional Stresses :- (Due to Seasonal Variation of temperature) 

Due to contraction of slab due to shrinkage or due to drop in temperature tensile stresses are induced at the middle portion of the slab.

•      IN IRC 58:2015, Typical cross sections of two pavements are shown.

–     (a) Debonding Layer of Polythene Sheet Over DLC

–     (b) Debonding Layer of 30 to 40 mm BC1 Over Cement Treated Subbase Layer

•      Typical type of pavements

–     CRPC:- Complete reinforcement in longitudinal direction

•      Does not require any transverse contraction joint

•      Transverse cracks are acceptable in slab at 0.5 to 1.5m interval held together By designed reinforcement 0.6 to 0.7 % of area for holding cracked concrete together

•      More effective life than JPCP, JRCP

•      Good candidate for asphalt surfacing due to tight crack width and minimal vertical movements between adjacent vertical joints

•      Design life 30-40 years

•      Cab be resurfaced later

•      Costlier than JPCP due to steel

•      The reinforcement in the rigid pavement is designed to prevent opening or widening of cracks, it has no role in contribution of flexural resistance.

CRPC

–     JPCP:-  Jointed Plain Concrete Pavement, No steel  (apart from dowel and tie bar) – Most commonly used in India

•      Uses contraction joints to control cracking

•      Typical Slab size 3.5 to 3.75m X 4.5m

•      Has longitudinal and Transverse joints at Every 3.5 to 4.5 m interval

•      Has dowel bars across Transverse joints

•      Has Tie bars across Longitudinal joints

–     JRCP:-Jointed reinforced concrete pavement, Nominal Steel 

•      Transverse joint spacing longer than that for JPCP 

•      Uses contraction joints and reinforcing steel to control cracks

•      Typical Slab size 3.5 to 3.75m X 9 to 15 m

•      Has dowel bars across Transverse joints

•      Has Tie bars across Longitudinal joints

•      Additional steel mesh

•      Crack observed at Midpanel-held by steel

JRCP

Design of Slab Thickness for Pavement 

(with and without doweled transverse joints. Beta value will be 0.66 for doweled joint and 0.90 for without dowels case)

Typical Layers of the Rigid Pavement

    PQC (Pavement Quality Concrete) Slab – M40 grade and above generally 200 to 350 mm thickness

    Polythene Membrane/Sheet – Separation sheet (200 micron) 

    Dry Lean Concrete Base (DLC) – generally M 10 with thickness 100 to 150 mm thickness as upper Subbase 

    Granular Subbase as drainage layer as lower subbase (permeability 300 m per day) 

    Soil subgrade minimum 8 % CBR

Subgrade

• Theoretically not required if k > 6 or CBR > 10%

• Required for better performance of the pavement by protecting the subgrade for Uniform support

• Should be  of non-erodible material (unaffected  by water)

• Provide resistance to mud pumping, better uniform support and stable working platform

• Very stiff subbase do not conform to the shape of curled concrete slabs leading to non-uniform support.

Materials:

a. Soil. 

b. Moorum. 

c. Gravel. 

d. Mixture of aggregates.

Requirements of materials:

• Material should be free from organic matter and soluble salts. 

• Materials used is non expansive soil. 

• The size of aggregate should be less than 50mm 

• Liquid limit should be less than 50% 

• Plasticity index should be less than 25% 

• MDD should be grater than 1.75gm/cm3

• Soluble sulphates should be less than 0.5% 

• Maximum compacted dry density should not be less than 97%

Settlement Cracks

Longitudinal Cracks

• Strength not the most important criteria 

        higher strength subgrade not reducing crust thickness significantly as in case of flexible pavement 

• Uniformity is most important: no abrupt change in character of material (soft or weak spots) 

        If deflection of slab more than 1.25 mm it will crack 

• Non-expansive to provide uniform support during wetting and drying 

• Resistance to erosion due to slabs deflection under heavy loads

Construction procedure: 

• The site should be cleared off and the top soil consisting of grass ,roots ,rubbish and other organic matter are to be removed.

• After site has cleared the work should be set out .Before spreading the material batter pegs are marked on both sides of an embankment at regular intervals. 

• The selected soil in the loose condition is spread to a uniform thickness using appropriate equipment over a prepared ground. 

• Additional water as required is sprayed so as to obtain the OMC of the soil determine from the laboratory compaction test. 

• The soil with the added water is mixed thoroughly using appropriate equipment so that the water gets distributed in the soil layers uniformly. The mixed soil is spread again to the uniform layer thickness by using graders. 

• The soil layer is compacted by a rolling, by vibratory roller of 80 to 100KN static weight or heavy pneumatic tiered roller. 

• The soil layer is compacted by rolling using the selected equipment so as to obtain the specify density. 

• Bring the proper camber profile of the compacted surface. 

• the soil is spread over the already compacted layer, water added mixed and compacted as mentioned above. 

• The process is repeated until the desired height of the subgrade is archived. 

Subbase - GRANULAR SUB-BASE OR DRAINAGE LAYER:

The GSB course have to serve as an effective drainage layer of the rigid pavement to prevent early failures due to excessive moisture content in the subgrade soil. It also supports the other pavement layers.

Materials:

a. Crushed stone aggregates 

b. Gravel. 

c. Coarse sand. 

d. Crushed slag. 

e. Crushed bricks. 

f. Crushed concrete. 

g. Natural sand 

h. Moorum. 

Requirements of materials:

• A material should not contain organic matter or other 

deleterious constituents. 

• The aggregate size should be less then 75mm.

• Water absorption of the aggregates should be less then 2% 

• Aggregate impact value should less than 40% 

• Liquid limit should be less then 25% 

• Plasticity index should be less then 6%. 

• CBR value should be greater then 30%. 

    a. For high 

        volume roads CBR should be minimum 30% 

    b. For low 

        volume roads CBR should be less then 20%. 

• Gradation: (% passing by weight).

IS Sieves	I	II	III	IV	V	VI
75 mm	100	-	-	-	100	-
53 mm	80-100	100	100	100	80-100	100
26.5 mm	55-90	70-100	55-75	50-80	55-90	75-100
9.5 mm	35-65	50-80	-	-	35-65	55-75
4.75 mm	25-55	40-65	10-30	15-35	25-50	30-55
2.36 mm	20-40	30-50	-	-	10-20	10-25
0.85 mm	-	-	-	-	2-10	-
0.425 mm	10-15	10-15	-	-	0-5	0-8
0.075 mm	<5	<5	<5	<5	-	0-3

Construction procedure: 

The GSB layer is constructed on the top of the prepared subgrade therefore first the surface of the subgrade is checked and grass and vegetation if any are removed. The grade and the cross slope of the top surface of the subgrade are corrected as required. The construction steps are give below: 

• The subbase material is spread to the uniform thickness and specified cross slope using a mortar grader by adjusting the blade of the grader. 

• The moisture content of the material is checked and the additional quantity of water required to bring up to the optimum moisture content is sprinkled at an uniform rate using a truck mounted sprinkler. 

• The water material is mixed properly using machinery such as disc harrows and rotavators. 

• The mixed material is spread to the desired thickness, grade and camber using a mortar grader with hydraulic controls of the blade. 

• The loose GSB layer is compacted by rolling if the compacted thickness of the layer is 100mm or lesser an ordinary smooth wheeled roller may be used. For compacted thickness exceeding 100mm and up to 225mm compaction is don by vibratory rollers of static weight 10 tons or more. 

• Rolling is done starting from the lower edge and proceeded towards the centre of the un divided carriage way or towards the upper edge of the divided carriage way with a minimum 1/3 rd overlap between each run of the roller. The rolling speed is limited to less than 5kmph. 

• Rolling is continued till at least 98% of maximum density of the material is archived. 

• The surface level tolerance will be (+ or -) 6 mm. 

Quality control tests: 

        • Gradation test : 1 tests per 400m3 

        • Atterberg limits: 1 tests per 400m3 

        • Moisture content test before compaction : 1 test per 400m3 . 

        • CBR test : as required. 

        • Deleterious constituents : as required.

Base course: ( Dry lean concrete): 

• The granular base course is generally provided under the CC pavement slab in low volume roads and also in roads with moderate traffic loads. 

• On roads carrying heavy to very heavy traffic loads high quality base course materials such as dry lean concrete are preferred. 

• In the base course of the CC pavement as they are designed for a life of 30 years or more with good maintenance. The CC pavement are expressed to provide a service life of 40 years or even more. 

• The DLC layer provides a uniform support, high K value and excellent working platform for laying the PQC slab with a sensor paver.

• The suppression member is spread on the top of the DLC/ base course before laying the CC pavement slab.

Requirements of material: 

a. Cement: OPC 43, Portland slag cement, Portland Pozzolana cement. If the subgrade is found to consist sulphates more than 0.5% cement shall be sulphate resistance. 

b. Aggregates: 

        a. Coarse aggregate: 

            • Loss angeles abrasion value should be less than 35 % 

            • Combined elongation and flaky index should be less than 35% 

            • Water absorption should be less than 2% 

            • Soundness for 5 cycles : 

                sodium sulphate should be less than 12% and 

                magnesium sulphate should be less than 18% 

        b. Fine aggregate : natural sand/ crushed stone sand. 

        c. Gradation: (% passing by weight).

IS Sieves	% by weight passing the seive
31.5 mm	-
26.5 mm	100
19 mm	75-95
9.5 mm	50-70
4.755 mm	30-55
2.36 mm	17-42
600 micon	8-22
300 micon	7-17
150 micon	2-12
75 micon	0-10

c. Water: It should be free from oil ,salts, acids, and vegitable matter. 

d. Storage of materials: Place the material with slope such that rain water should be drained off . 

e. Proportioning of the material for the mix: Aggregate and cementitious material ratio of 15:1 

f. Moisture content: The moisture content should be +2% keeping in view effectiveness of compaction archived. 

g. Cement content: The cement content should not be less than 150kg/m3 . 

h. Concrete strength : The average compressive strength of 5 cubes shall not be less than 10Mpa at 7 days. Single cube compressive strength should be minimum 7.5Mpa at 7 days.

Construction procedure: 

    Batching and mixing: 

        The batching plant shall be capable of proportioning the material by weight. The plant should have higher capacity by 25% than the proposed laying arrangements. 

    Transporting : 

        A plant mix lean concrete shall be discharged immediately from the mixer. The concrete shall be transported by tipping trucks. And they should be a continues supply of the material. To feed the laying equipment to work at a uniform speed and in an uninterrupted manner. 

    Placing: 

        Dry lean concrete shall be placed by a paver with electronic sensor on the drainage layer. The equipment shall be capable of laying the material in one layer in an even manner with out segregation. Dry lean concrete shall be placed and compacted across a full width. 

    Compaction: 

• The compaction should be carried out immediately after the material is lied and levelled, rolling shall be continued on the full width. 

• The minimum dry density obtained shall not be less than 98%. 

• Spreading compacting and finishing not to exceed 90min when temperature is 25 to 30 degree Celsius. And 120 min if less than 25 degree Celsius. 

• It is desirable to stop concreting when the temperature is above 35 degree celsius . 

• Double drum smooth wheeled vibratory rollers of minimum 80-100KN static weight are suitable for rolling dry lean concrete. 

    Joints: 

Construction and longitudinal joints shall be provided. 

    Curing: 

Curing may be done by covering the surface by gunny bags which shall be kept wet continuously for 7 days by sprinkling water. 

Surface level tolerance should be (+ or -) 5mm. 

    Quality control test: 

• Quality of cement : 1 test per 5 tonnes 

• Compressive strength : as required 

• Water content : 2 test per 500m2 

PQC- PAVEMENT QUALITY CONCRETE REQUIREMENTS 

    PQC is special concrete which has to provide Structural as well as Functional performance(wear and tear, durability) 

    The high quality CC mix with high flexural strength is used for the construction of PQC slab of the CC pavement so PQC should never be lower than M 35 Grade.

    More than 90 to 95 % of the Concrete road performance will depend on PQC 

    Concrete Flexural Strength 40 – 45 Kg/Sqcm (M40 to M 45 Grade) 

Flexural strength Test

    Slump 20-25 mm for Paver work, 40-45 mm Manual laying 

    Cement Content – Min cement/cementitious material content 360 kg/m3 & Max not including mineral admixtures shall be 450 kg/m3 

    Use of Superplasticizers required for workability 

    Water Cement Ratio – 0.38 

    Flyash/ GGBS addition

        Flyash25 % by mass of cementitious materials 

        GGBS – 50 % by mass of cementitious materials

The slab prevents the infiltration of excess surface water in to the sub-base.

Requirements of material:

 

a. Cement: Ordinary Portland cement 43 & 53 grade, Portland slag cement, Portland pozzolana cement. 

b. Chemical admixtures: Chemical admixtures are permitted to improve workability of concrete and setting time. 

c. Silica fumes: Silica fumes are used as an admixture in the proportion of 3 to 10 percent of cement. 

d. Fibres: Fibres are used to reduce the shrinkage cracking and post cracking. The fibres may be steel fibres or polymer synthetic fibre. With a diameter of 10 micron to 100 micron ad length 6 to 48 mm and suggested dosage should be 0.6 to 2kg/cm3 .

e. Aggregates:

    ➢ Coarse aggregate: 

        • It should contain clean, hard, strong, dense, non porous and durable pieces of crushed stone or crushed gravel. Requirements: 

        • Abrasion value should be less than 35% 

        • Combined EI and FI should be less than 35% 

        • Water absorption should be less than 2% 

        • Soundness for 5 cycles sodium sulphate should be less than 12%, and magnesium sulphate should be less than 18%. 

    ➢ Fine aggregates: 

        • fine aggregates shall consist of clean natural sand or crushed stone sand or a combination of two. It should be free from soft, clay, organic and other matters. 

• Gradation: (% passing by weight).

IS Sieves	% by weight passing the seive
31.5 mm	100
26.5 mm	85-95
19 mm	68-88
9.5 mm	45-65
4.755 mm	30-55
600 micon	6-30
150 micon	5-15
75 micon	0-5

f. Water: 

    It should be clean, free from oil, salts, acid and vegitable matter. 

g. Steel: 

    •Dowel bars:- mild steel bars 

    •Tie bars :- HYSD bars. 

h. Joint fillers: Joint filler board for expansion joints up to 20 to 25 mm thickness. 

i. Joint sealing compound: The joint sealing compound shall be of hot poured, elastomeric type or cold poly sulphide, silicon. 

j. Storage of materials: Materials should be placed with slope such that rain water should be drained off. 

k. Proportioning for concrete: The mix design is based on IS:10262. 

l. Cement content: The cement content should be 360 kg/m3 . And we should not be less than 310 kg/m3 when blended with fly ash of 20%. 

m. Concrete strength: The flexural strength of the concrete should not be less than 4.5Mpa. 

n. Preparation of base: Clean DLC with mechanical broom or air compressor. 

o. Separation member: A Separation member shall we used between the concrete slab and the subbase. Separation member with PVC sheet 125 micron thick is used. 

p. Form work: Fixed form are side form type and slip form type. 

q. Joints: 

    • Longitudinal joints: tie bars 

    • Transfers joints : dowel bars

Construction procedure: 

Batching and mixing: 

    Batching and mixing of the concrete shall be done at a central batching and mixing plants with automatic controllers. Plant should have higher capacity by 25% as the propelled laying arrangements. 

Transporting: 

    Transporting is done by transit mixer and dumper. Hauling and placing of concrete: Spreading, compacting and finishing not to exceed 90 min when temperature is 20 to 30 degree Celsius . 120 min if les than 25 degree Celsius. and work shall not proceed and reject when temperature is high. 

Compaction: 

    Compaction is done by screed vibrators the compaction should be carried out immediately after the material is laid and levelled. 

Finishing: 

    Finishing is done by flat and finishers. 

Texture: 

    Texture is done by trimming and brushing. 

Curing: 

    Covering the surface by gunny bags, pounding, sprinkling water continuously for 28 days and the surface tolerance should be (+ or -) 5mm.

Quality control tests: 

    • Quality of cement 1 test per 5 tonnes. 

    • Aggregate gradation 2 test per day 

    • Water absorption 2 test per day 

    • Soundness test 1 test per each source 

    • Compressive strength for 2 cubes for 150 m3 

    • Flexural strength for 2 beams for 150 m3 . 

    • Slump test 1 test per each load. 

    • Deleterious constituents as required.

Joints in Rigid Pavement – Transverse/ Contraction Joint 

Typical Transverse contraction Joint

    Concrete shrinks with strength gain 

    Shrinkage tensile stress induce random cracks 

    Transverse joints are typically provided at interval of 4 to 5 m 

    Transverse joints create weak planes and control location of cracks

 Full depth Transverse cracks due to late joint cutting

    Transverse joints are created by saw cutting the concrete pavement (1/3rd depth) 

    Transverse joints are filled by approved sealant after joint widening to avoid entry of the dirt/soil in the joint 

    Some amount of movement is accommodated at the transverse joint by providing the dowel bar with plastic sheathing 

Typical Transverse Joint with Dowel bar

Joints in Rigid Pavement – Longitudinal Joint

Typical Longitudinal Joint

    Longitudinal joint is required if pavement width is above 4.5 m 

    Longitudinal joints are typically provided at interval of 3.5 m to 3.75 m (each lane) 

    Longitudinal joints are provided for release of thermal stresses 

    Tie Bars are provided at the Longitudinal joints 

    Longitudinal joints are created by saw cutting the concrete pavement (1/3rd depth) or by just keeping cold joint 

    Minimum 150 mm length at center of Tie rod shall be painted to avoid corrosion.

Painted Tie Rods

    Longitudinal joints are filled by sealant to avoid entry of the dirt/soil in the joint 

    No movement is accommodated at the longitudinal joints

Opened longitudinal joint if tie Bars are not provided

Construction Joint 

    Construction joint is provided at end of days work 

    It has dowel bars with plastic sheathing 

    Construction joint is placed at location of the Transverse Joint 

    Generally the steel channel/ heavy Bulkhead is provided to support the concrete at construction joint

Sealants

1) Hot Poured Rubberized Bitumen Sealant

Contraction Joint

Longitudinal joint

Expansion joint

Note :- There is no need to provide Expansion joints at regular intervals but they are essential where cement concrete pavement is designed to abut with structures like bridges. It may sometime be necessary to provide more number of expansion joints in succession in such locations to release the pressure. Expansion joints against culverts, underpasses etc. having RCC box structure should normally be avoided by taking the PQC over the deck of such structures. To deal with the lack of compaction in the vicinity of structures and subsequent settlement, RCC approach slabs must be provided on both sides abutting with DLC layer. Wherever, PQC is taken over the deck of structures, underpasses etc., saw cut transverse construction joints must be provided just above the deck and approach slabs on both sides to avoid full depth transverse cracks in PQC.

2)  Cold Poly-Sulphide Sealant
3) Cold Silicon Sealant

Load transfer across Joints

Load Transfer Mechanism across joints

In above Drawing shear resistance offered by Aggregate Interlock at the cracked interface is neglected, whereas it can offer shear resistance.

Dowel Bars

    Dowel bars are provided at the Transverse/ Contraction joints for load transfer from one slab to another 

Slab Faulting if Dowel bars not provided

    Dowel Bars are always Mild steel bars 

    Diameter ranges between 25 mm to 38 mm 

    Dowel Bars are provided with Plastic sheathing to accommodate movement at the transverse joint 

Full depth transverse crack due to misalignment of Dowel bars

    One end is free and other end is fixed 

    Special steel chairs are provide to accommodate the dowel bars during construction (semi mechanized) 

    Dowel bars are placed at mid depth of the slab

Tie Bars 

    Tie bars are provided to tie the longitudinal joint 

    Tie bars are typically 12 mm to 16 mm tor steel bars to hold together two lanes of concrete road 

    Tie bars help in the thermal stress release 

    Tie bars are not provided with any plastic sheeting 

    Bonding with concrete is essential 

Separation membrane/Polythene Sheet

        200 micron thick transparent Or white

 

        LDPE material 

        Not manufactured from Recycled plastic/ carbon Black 

        Minimum 4 m wide sheet 

        Overlap of 300 mm 

        Nailed in DLC to avoid crease

Design Of Rigid Pavement as per IRC:58

1) Type of pavement considered

        Carriageway

        Shoulders :- Tied concrete shoulders ? (yes/no)

        Transverse joint spacing (m)

            Spacing of Contraction Joints, (L_c)

                Case 1) When reinforcement is not provided

$${L_c}={{(2 \times 10^4){S_c}}\over {w.f}}$$

Where, 

          L_c = Spacing of Contraction joint (m)

          S_c = Allowable stress in tension in cement concrete.

           f = Coefficient of friction ~ 1.5

            W = Unit weight of cement concrete (Kg/cm³)

        Lane width (m)

        Transverse Joints have dowel bars?   (yes/no)

2) Design Traffic Estimation 

        Design Period (years) 

        Total Two-way Commercial Traffic (cvpd) in the year of completion of construction 

        Av. Annual rate of growth of commercial traffic (expressed as decimal) 

        Cumulative No of Commercial vehicles during design period (two-way), A 

        Average No of axles per commercial vehicle, B 

        Cumulative No of Commercial Axles during design period (two-way), C = A*B 

        Proportion of traffic in predominant direction (For 2-lane 2-way highways use a value of 1.0), D 

        Lateral Placement factor (0.25 for 2-lane 2-way. For multilane highways the value is 0.25 X C), E 

        Factor for selection of traffic for BUC analysis (for six-hour period during day), F 

•       Bottom Up Cracking (BUC)

–      10 am to 4 PM

        Factor for selection of traffic for TDC analysis (for six-hour period during day), G 

•       Top Down Cracking (TDC)

–      0 am to 6 am

        Design axle repetitions for BUC analysis (for 6 hour day time traffic), H = B*E*F 

        Proportion of vehicles with spacing between front and the first rear axle less than the spacing of             transverse joints, I 0.55

        Design axle repetitions for TDC analysis (for 6-hour night time traffic), J = B*E*G*I 

        Proportion of Front single (steering) Axles, K1 

        Proportion of Rear single Axles,K2 

        Proportion of tandem Axles, K3 

        Proportion of Tridem Axles, K4 = (1-K1-K2-K3) 

Westergards and Bradbury equation for curling stresses

    Warping stress at interior region (σ_i)

$$\sigma = {{C.E.{\alpha}.Δt}\over 2}$$

C is the correction factor depending in L\l ratio

$$C={\left[{{C_x}+{\mu}{C_y}}\over {1-{\mu}^2}\right]}$$

C_x = Coefficient based on (L_x/l) in desired (X) direction

C_y = Coefficient based on (L_y/l) in right angle to X direction

μ = Poisson's ratio 〜0.15

Lx/l or Ly/l	Cx or Cy
4	0.6
8	1.1
12	1.02

 L_x & L_y are the dimensions of the slab considering x & y direction along the length & width of slab

Warping stress at Edge region (σ_e) will be maximum of

$$\sigma_e = {{E.{\alpha}.Δt}\over 2}{\,}{C_x}$$

or

$$\sigma_e = {{E.{\alpha}.Δt}\over 2}{\,}{C_y}$$

Warping stress at Corner region (σ_c)

$$C={{E \alpha T}\over {3({1- \mu})}}{\sqrt{a\over l}}$$

Westergards Stress equation

    Stress at Interior loading 

$${S_i} = {{{0.316 P}\over {h^2}}\left[4.{log_{10}}{\left({l\over b}\right)}+1.069\right]}$$

    Stress at Edge loading

$${S_e} = {{{0.572 P}\over {h^2}}\left[4.{log_{10}}{\left({l\over b}\right)}+0.359\right]}$$

Stress at Corner loading

$${S_c} = {{{3 P}\over {h^2}}\left[{1-\left({{a\sqrt 2}\over l}\right)^{0.6}}\right]}$$

Where,

        h = slab thickness (cm)

        a =radius of contact area (cm)

        P = wheel load (Kg)

        b = radius at resisting section (cm)

radius at resisting section (b) when a < 1.724 h

$$b=\sqrt{1.6.a^2 + h^2}-0.675h$$

radius at resisting section (b) when a >1.724 h

b = a

Combination of Stresses

1) Critical Combination during Summer

        i) Stress for edge/interior regions at bottom = Load stress + warping stress of day time - Frictional stress

        ii) Stress for corner region at top = Load stress + warping stress at night

2) Critical Combination during Winter

        i) Stress for edge/interior regions at bottom = Load stress + warping stress of day time + Frictional stress

        ii) Stress for corner region at top = Load stress + warping stress at night

3) Pavement Structural Details

       Modulus of subgrade reaction of subgrade (k), MPa/m

$$k = {P\over \delta}$$

P = pressure required for δ deflection (Kg/cm2)

δ = deflection (cm)

        Thickness of Granular Subbase, mm

        Thickness of Dry Lean Concrete subbase, mm

        Effective modulus of subgrade reaction of foundation, MPa/m

        Unit weight of Concrete, kN/m³

        28-day Flexural strength of cement concrete, MPa

        Max. day-time Temperature Differential in slab, ⁰C (for bottom-up cracking)

        Night-time Temperature Differential in slab, ⁰C (for top-down cracking) = day-time diff/2 + 5

        Trial Thickness of Concrete Slab, m

        Load Transfer Efficiency Factor for TDC analysis, 

Beta = 0.66 for dowel Joints, 

Beta = 0.90 for joints without dowels

        Elastic Modulus of Concrete, Ec (MPa)

        Poisson's Ratio of Concrete, μ

        Radius of relative stiffness, l 

$$l={{\left[{{Eh^3}\over {12k{(1-{\mu}^2)}}}\right]}^{1\over 4}}$$

l =radius of relative stiffness , cm

E = modulus of elasticity of cement concrete (kg/cm2)

μ =  Poisson's Ratio of Concrete = 0.15

h = slab thickness (cm)

k = Modulus of subgrade reaction of subgrade (kg/cm3)

4) Design Axle Load Repetitions for Fatigue Analysis

    a) For Bottom-up Cracking Analysis

        Front single (steering) Axles = H * K1

        Rear single Axles = H * K2

        Tandem Axles = H * K3

        Tridem Axles = H * K4

   b) For Top-Down Cracking Analysis

        Front single (steering) Axles = J * K1

        Rear single Axles = J * K2

        Tandem Axles = J * K3

        Tridem Axles = J * K4

            The subgrade is usually considered as a Winkler foundation, also known as dense liquid foundation. In Winkler model, it is assumed that the foundation is made up of springs supporting the concrete slab. The strength of subgrade is expressed in terms of modulus of subgrade reaction, k, which is defined as the pressure per unit deflection of the foundation as determined by plate load tests. The k-value is determined from the pressure sustained at a deflection of 1.25 mm. 

        As k-value is influenced by test plate diameter, the standard test is to be carried out with a 750 mm diameter plate. IS:9214, “Method of Determination of Modulus of Subgrade Reaction of Soil in the Field” may be referred to for guidance in this regard. 

        A frequency of one test per km per lane is recommended for assessment of k-value. If the foundation changes with respect to subgrade soil, type of subbase or the nature of formation (i.e. cut or fill) then additional tests may be conducted. 5.7.3.2 Though 750 mm is the standard plate diameter, smaller diameter plate can be used in case of homogeneous foundation from practical consideration and the test values obtained with plates of smaller diameter may be converted to the standard 750 mm plate value using equation.

k₇₅₀ = kᵩ (1.21 f +0.078)

Φ = plate diameter, metre 

k_Φ = modulus of subgrade reaction (MPa/m) with plate diameter Φ metre 

k₇₅₀ = modulus of subgrade reaction (MPa/m) with plate diameter of 750 mm (k)

Axle Load Spectrum Data

Rear Single Axle			Rear Tandem Axle			Rear Tridem Axle
Load Group (KN)	Mid-Point of Load Group (KN)	Frequency (%)	Load Group (KN)	Mid-Point of Load Group (KN)	Frequency (%)	Load Group (KN)	Mid-Point of Load Group (KN)	Frequency (%)

 Fatigue Damage Analysis

Rear Single Axle

Rear Tandem Axle

Rear Tridem Axle

Expected Repetitions (ni)

Flexural stress (MPa)

Stress Ratio (SR)

Allowable Repetiotions (Ni)

Fatigue Damage (ni/Ni)

Expected Repetitions (ni)

Flexural stress (MPa)

Stress Ratio (SR)

Allowable Repetiotions (Ni)

Fatigue Damage (ni/Ni)

Expected Repetitions (ni)

Flexural stress (MPa)

Stress Ratio (SR)

Allowable Repetiotions (Ni)

Fatigue Damage (ni/Ni)

 Fatigue Damage Analysis

Bottom-up Cracking Fatigue analysis for day time (6 Hour) and positive temperature differential
Rear Single Axle					Rear Tandem Axle
Expected Repetitions (ni)	Flexural stress (MPa)	Stress Ratio (SR)	Allowable Repetiotions (Ni)	Fatigue Damage (ni/Ni)	Expected Repetitions (ni)	Flexural stress (MPa)	Stress Ratio (SR)	Allowable Repetiotions (Ni)	Fatigue Damage (ni/Ni)

 Expected Repetitions (ni)

Frequency (%) in that Load Group * Number of that type of Axles (Applicable for BUC & TDC)

Flexural stress MPa

Single axle – Pavement with tied concrete shoulders

(a)    k ≤ 80 MPa/m

S = 0.008 – 6.12 (γh² /kl² ) + 2.36 Ph/(kl⁴ ) + 0.0266 ΔT

(b)  k > 80 MPa/m, k ≤ 150 MPa/m

S = 0.08 – 9.69 (γh² /kl² ) + 2.09 Ph/(kl⁴ ) + 0.0409 ΔT

(c)  k > 150 MPa/m

S = 0.042 + 3.26 (γh² /kl² ) + 1.62 Ph/(kl⁴ ) + 0.0522 ΔT

Single axle – Pavement without concrete shoulders

(a)    k ≤ 80 MPa/m

S = - 0.149 - 2.60 (γh² /kl² ) + 3.13 Ph/(kl⁴ ) + 0.0297 ΔT

(b)  k > 80 MPa/m, k ≤ 150 MPa/m

S = - 0.119 - 2.99 (γh² /kl² ) + 2.78 Ph/(kl⁴ ) + 0.0456 ΔT

(c)  k > 150 MPa/m

S = - 0.238 + 7.02 (γh² /kl² ) + 2.41 Ph/(kl⁴ ) + 0.0585 ΔT

Tandem axle – Pavement with tied concrete shoulders

(a)    k ≤ 80 MPa/m

S = - 0.188 + 0.93 (γh² /kl² ) + 1.025 Ph/(kl⁴ ) + 0.0207 ΔT

(b)  k > 80 MPa/m, k ≤ 150 MPa/m

S = - 0.174 + 1.21 (γh² /kl² ) + 0.87 Ph/(kl⁴ ) + 0.0364 ΔT

(c)  k > 150 MPa/m

S = - 0.210 + 3.88 (γh² /kl² ) + 0.73 Ph/(kl⁴ ) + 0.0506 ΔT

Tandem axle – Pavement without concrete shoulders

(a)    k ≤ 80 MPa/m

S = - 0.223 + 2.73 (γh² /kl² ) + 1.335 Ph/(kl⁴ ) + 0.0229 ΔT

(b)  k > 80 MPa/m, k ≤ 150 MPa/m

S = - 0.276 + 5.78 (γh² /kl² ) + 1.14 Ph/(kl⁴ ) + 0.0404 ΔT

(c)  k > 150 MPa/m

S = - 0.3 + 9.88 (γh² /kl² ) + 0.965 Ph/(kl⁴ ) + 0.0543 ΔT

In Excel sheet formula for Rear Single Axle can be given as below in Excel attached in link

=IF(B$8="yes",IF(E$9<=80,0.008-(6.12*E$10*E$14*E$14/(E$9*E$18*E$18))+(2.36*H9*E$14/(E$9*E$18^4))+0.0266*E$12,IF(AND(E$9>80,E$9<=150),0.08-(9.69*E$10*E$14*E$14/(E$9*E$18*E$18))+(2.09*H9*E$14/(E$9*E$18^4))+0.0409*E$12,0.042+(3.26*E$10*E$14*E$14/(E$9*E$18*E$18))+(1.62*H9*E$14/(E$9*E$18^4))+0.0522*E$12)),IF(E$9<=80,-0.149-(2.6*E$10*E$14*E$14/(E$9*E$18*E$18))+(3.13*H9*E$14/(E$9*E$18^4))+0.0297*E$12,IF(AND(E$9>80,E$9<=150),-0.119-(2.99*E$10*E$14*E$14/(E$9*E$18*E$18))+(2.78*H9*E$14/(E$9*E$18^4))+0.0456*E$12,-0.238+(7.02*E$10*E$14*E$14/(E$9*E$18*E$18))+(2.41*H9*E$14/(E$9*E$18^4))+0.0585*E$12))) 

Similarly it can be written for other cases also.

Stress Ratio

Flexural stress/(28-day Flexural strength of cement concrete * 1.1)

Relation between fatigue life (N) and stress ratio (SR)

1) N = Unlimited for SR < 0.45

2) When 0.45 < SR < 0.55 $$N={\left[4.2537\over {SR-0.4235}\right]^{3.268}}$$  

3) When SR > 0.55 $${log_{10}}N={\left[{0.9718-SR}\over {0.0828}\right]}$$ 

Bottom-up Cracking

      Pavement with tied concrete shoulders for single rear axle

      Pavement without concrete shoulders for single rear axle

      Pavement with tied concrete shoulders for tandem rear axle

      Pavement without concrete shoulders for tandem rear axle

Fatigue Damage Analysis

Top-Down Cracking Fatigue analysis for Night-time (6 Hour) and Negative Temperature Differential
Rear Single Axle
Expected Repetitions (ni)	Flexural stress (MPa)	Stress Ratio (SR)	Allowable Repetiotions (Ni)	Fatigue Damage (ni/Ni)
Rear Tandem Axle(Stress computed for 50% of axle load)
Expected Repetitions (ni)	Flexural stress (MPa)	Stress Ratio (SR)	Allowable Repetiotions (Ni)	Fatigue Damage (ni/Ni)
Rear Tridem Axle(Stress computed for 33% of axle load)
Expected Repetitions (ni)	Flexural stress (MPa)	Stress Ratio (SR)	Allowable Repetiotions (Ni)	Fatigue Damage (ni/Ni)

Top-down Cracking

      Pavements with and without dowel bars having front steering axle with single tyres and the first axle of the rear axle unit (single/tandem/tridem) placed on the same panel. 

F_cr =0.7*√F_ck

Total Bottom-up Fatigue Damage due to single and tandem axle loads = X

Total Top-Down Fatigue Damage  = Y

Sum of CFD for BUC &  TDC= X +Y

Condition --> SUM OF CFD FOR BUC AND TDC< 1

if above condition satisfies then the pavement is safe from large scale cracking.

Design for Bonded Pavement Option 

Subgrade CBR (%)=

Granular Subabse Thickness (mm) =

Effective k-value from Tables 2 and 3 (MPa/m) =

Relationship between K-value and CBR value for Homogeneous soil subgrade
Soaked CBR	2	3	4	5	7	10	15	20	50	100
K-value	21	28	35	42	48	55	62	69	140	220

                                                Table 2 of IRC:58-2015

For k of   K' MPa/m  and for 

K-Value of Subgrade (MPa/m)	Effective k (MPa/m) of Untreated Granular Subbase of Thickness (mm)			Effective k (MPa/m) of Cement Treated Subbase of Thickness (mm)
	150	225	300	100	150	200
28	39	44	53	76	108	141
56	63	75	88	127	173	225
84	92	102	119

Table 3 of IRC:58-2015

K-Value for Dry Lean Concrete subbase
K- Value of Subgrade(Mpa/m)	21	28	42	48	55	62
Effective k for 100 mm DLC, (Mpa/m)	56	97	166	208	278	300
Effective k for 100 mm DLC, (Mpa/m)	97	138	208	277	300	300

 Table 4 of IRC:58-2015

For concrete pavements laid over a bituminous subbase, the k-value can be adopted from IRC:SP:76. k-values for different combinations of DLC subbase (with DLC having minimum 7-day compressive strength of 7 MPa) thicknesses laid over granular subbase consisting of separation and drainage layers can be adopted from Table 4.

Doweled Joint and  Tied Concrete Shoulders, Slab Thickness (m) = 

Trial Slab thickness (m) over DLC, h1

Provide DLC thickness (m), h2 

Elastic Modulus of Pavement Concrete (MPa), E1

Elastic Modulus of DLC (MPa), E2

Poisson's Ratio of Paving Concrete, m1

Poisson's Ratio of DLC, m2

Depth to Neutral axis, m 

$$d={{0.5({h_1}^2)+({{E_2}\over {E_1}}){h_2}({h_1}+0.5{h_2})}\over {{h_1}+({{E_2}\over {E_1}}){h_2}}}$$

Flex Stiffness of design Slab

 Flexural stiffness of a slab of thickness, h, is given as

$${{EI}\over ({1-{{\mu}^2}})}={{E{h^3}}\over {12({1-{{\mu_1}^2}})}}$$

Flex Stiffness of Partial Slab Provided

$$\text{Flexural stiffness of PQC} = {{{E_1}\left(\left({{h_1}^3\over {12}}\right)+{h_1}(d-0.5{h_1})^2\right)}\over {({1-{{\mu_1}^2}})} }$$

Flex Stiffness of DLC

$$\text{Flexural Modulus (DLC)} ={{{E_2}\left[ {{\left({E_2}\over {E_1}\right)}\over 12}{h_2}^3+{\left({E_2}\over {E_1}\right){h_2}\left({h_1}+{{h_2}\over 2}-d\right)^2}\right]}\over {({1-{{\mu_2}^2}})}}$$

Total Flexural Stiffness Provided =  Z1

which is more than the Flexural Stiffness of the Design Slab =   Z2

Hence, Provide a Slab of thickness (m)  

Slab thickness (h1) over DLC layer may be obtained by iteratively changing h1 and matching the design stiffness with the combined stiffness provided.

Maximum Temperature Differentials for Concrete slabs

Zone	Stages/Regions
		150 mm	200 mm	250 mm	300 to 400 mm
I	Hilly regions of Uttaranchal, West Bengal, Jammu & Kashmir, Himachal Pradesh and Arunachal Pradesh	12.5	13.1	14.3	15.8
II	Punjab, U.P., Uttaranchal, Gujarat, Rajasthan, Haryana and North M.P, excluding hilly regions.	12.5	13.1	14.3	15.8
III	Bihar, Jharkhand, West Bengal, Assam and Eastern Orissa, excluding hilly regions and costal areas	15.6	16.4	16.6	16.8
IV	Maharashtra, Karnataka, South M.P., Chhattisgarh, Andhra Pradesh, Western Orissa and North Tamil Nadu, excluding hilly regions and costal areas.	17.3	19	20.3	21
V	Kerala and South Tamil Nadu, excluding hilly regions and coastal areas	15	16.4	17.6	18.1
VI	Coastal areas bounded by hills	14.6	15.8	16.2	17
VII	Coastal areas unbounded by hills	15.5	17	19	19.2

Design of Dowel Bars

Design Parameters      

       

Design wheel load = 98 percentile axle load is X tonnes, the wheel load therefor is X/2000 kg (dual wheel load)

       

Percentage of load transfer = a%    

       

Slab Thickness h =  ---  mm     

       

Joint Width, z = ---   mm     

       

Assumed Dia of dowel bar in cm =  ----  mm    

       

Permissible bearing stress in concrete is calculated as shown in equation below

$${F_b}={{(101.6-{b_d}){f_{ck}}}\over {95.25}}$$
    

 Fb = -----      

       

Assumed spacing between dowel bars =   ----- mm   

       

First dowel is placed at a distance =    ----- mm from the pavement edge   

       

Assumed length of dowel bar =  ----- mm   

       

Dowel bars upto a distance of 1.0 X radius of relative stiffness, from the point of load application are effective in load transfer.

       

No. Of Dowel bars participating in load transfer when wheel load is just over the Dowel bar close to the edge of the slab

 = ----  Dowels      

       

Assuming the load transferred by the first dowel is P1 and assuming that the load on dowel bar at a distance of l from first dowel.

to do so, one can write simple VBA code as below to calculate system load

Sub System_load()

Dim Counter As Integer

Dim mynum As Integer

Dim i As Integer

Dim j As Integer

    Counter = -1

    mynum = 0

    Do Until Range("B68").Value = mynum * Range("B61").Value

        mynum = mynum + 1

        Counter = Counter + 1

        

        i = (Range("B68").Value - (Range("B61").Value * Counter))

        j = j + i

        If mynum * Range("B61").Value > Range("B68").Value Then Exit Do

    Loop

    Range("D73").Value = j / Range("B68").Value

    Range("D73").Select

End Sub

Above code is applicable to Following excel sheet only, as it is written as cell specific.

IRC58-2015 Rigid Pavement Design Sheet  

to b zero, the total load transferred by dowel bar system   = -----  Pt           

load transferred by outer dowel bar Pt  = -------   Kg            

Check for Bearing stress      

Maximum bearing stress (Fbmax) between the concrete and dowel bar is obtained from equation

$${F_{bmax}}={{k_{mds}{P_t}(2+\beta Z)}\over {4 \beta^3 EI}}$$

Moment of Inertia of Dowel =  I  mm⁴

$$I = {{\pi D^4}\over 64}$$

Relative stiffness of the bar embedded in concrete, mm-1 (β)

$$\beta =\root {4}\of {{k_{mds}{b_d}}\over {4EI}}$$

Relative stiffness of Dowel bar embedded in concrete = (β)  

       

Bearing stress in Dowel bar   = F_bmax < F_b then Safe 

Design of Tie bars          

Coefficient of friction, f = -----     

Density of Concrete = 24000 N/mm³     

 Allowable tensile stress in Plain Bars =  -----   

Kg/cm2 (As per IRC:21 - 2000)     

Allowable tensile stress in Deformed Bars =  ------   

Kg/cm2 (As per IRC:21 - 2000)     

Allowable Bond stress for Plain tie bars =  ------   

 Allowable Bond stress for Deformed tie bars = -----   

  Assumed dia of tie bar in mm  -----   

  Spacing and Length of plain Bar     

 Area of steel Bar per meter width of joint to resist     

the frictional force at slab bottom (As) =   = ------   cm2/m   

Assuming the Diameter of  12 mm, the cross sectional area  A = -----   cm2    

Perimeter of tie Bars (P) =  P = ------   cm      

Spacing of tie Bars =   = ------   cm     

Provide at a Spacing of  -------cm C/C     

Length of tie Bar =   = -------   cm      

length to be increased for loss of Bond(cm)   = -----  cm     

Tolerence in Placement (cm)   = -----   cm     

Therefor length is     = ------  cm    Say -------  cm    

 Spacing and Length of Deformed Bar     

 Area of steel Bar per meter width of joint to resist     

the frictional force at slab bottom (As) =   = ---------   cm2/m   

 Assuming the Diameter of  ----- mm, the cross sectional area  A = ------   cm2    

 Perimeter of tie Bars (P) =  P = ------   cm      

 Spacing of tie Bars =   -------   cm      

Provide at a Spacing of  ----- cm C/C     

Length of tie Bar =   --------   cm      

length to be increased for loss of Bond(cm)   = -------   cm     

 Tolerance in Placement (cm)   = -------   cm     

Therefor length is     = -------  cm    Say -------   cm    

Design for rigid pavements having traffic less than 450 CVPD as per IRC:SP:62-2014

Input values in Red only