In addition to basic layer thickness and composition calculations, rigid pavement structural design must also consider surface layer joint layout and (for CRCP) reinforcing steel.  This section, taken largely from the 1993 AASHTO Guide, discusses each of these features.

8.1  Joint Design

Joints, which are integral to JPCP and JRCP, and also necessary in CRCP, must be designed to minimize slab cracking, joint deflection, joint stresses and roughness as well as accommodate the intended joint sealant.  Four key design components are manipulated to meet these goals:

• Joint spacing
• Joint orientation
• Joint size
• Load transfer design

8.1.1  Joint Spacing

Joint spacing influences internal slab stresses, which determine how and where a slab cracks, as well as how much a slab will shrink or expand with temperature changes.  Typically, joint spacing decisions must be made on JPCP transverse and longitudinal contraction joints.  Of these, transverse contraction joints involve the most options.  Longitudinal joints are typically spaced at lane edges, which makes them between about 3 and 4.25 m (10 and 14 ft.) apart.  Expansion joints are rarely used any more, and construction and isolation joints are determined by project geometry, field placement and equipment capabilities.

Joint spacing is highly dependent on the local environment, materials and subgrade.  First, expected temperature changes will influence slab curling stresses.  In general, the greater the temperature changes, the shorter the joint spacing should be.  Second, the materials within the PCC slab (the coarse aggregate is of overriding concern) will influence the slab's thermal coefficient.  The higher the thermal coefficient, the more a slab will shrink and expand for a given temperature change.  Generally, slabs made with limestone coarse aggregate have lower thermal coefficients, while slabs made with quartz or sandstone have higher thermal coefficients.  Third, as the slab expands and contracts, the frictional resistance offered by the base material will also influence slab stresses.  In general, the more frictional resistance, the higher the slab stresses.

Joint spacing is also related to slab thickness.  In general, the thinner a slab is, the higher the curling stresses and thus, the shorter the joint spacing should be.  As a general rule-of-thumb, joint spacing should be less than about 24 x slab thickness.  Thus, a 230 mm slab (9 inches) should have joints spaced no more than about 5.5 m (18 ft.) apart.  Also, as a general guide, the ratio of longer side slab length to the shorter side slab length should be kept less than about 1.25.

The FHWA (1990) recommends that the L/l ratio (slab length divided by radius of relative stiffness) not exceed 5.0 when determining the maximum slab length.  Table 6.3 shows some slab lengths resulting from using L/l = 5.0 for a range of normal slab thicknesses.

Table 6.3: Slabs Lengths Resulting from Using an L/l Ratio = 5.0

8.1.2  Joint Orientation

Skewed transverse contraction joints can reduce load transfer joint stresses and may be beneficial in undoweled JPCP.  Typically, joint skew should be limited to a maximum of 1:10 to prevent excessive corner breaks (see Figure 6.20) (FHWA, 1999).

Figure 6.20: Skewed Joint Showing a Corner Break

8.1.3  Joint Size

Joint width and depth are dependent on two separate things.  First, joint depth should be between 1/4 and 1/3 of the total slab depth to ensure crack formation at the joint.  Joints shallower than this may not sufficiently weaken the vertical plane.  Second, joint width is selected to provide an adequate joint sealant reservoir.  Typically, a contraction joint is first sawed very narrow (3 mm (0.125 inches)) to control cracking, then later widened (10 - 15 mm (0.4 - 0.6 inches) wide) to create a joint sealant reservoir (FHWA, 1999).

The proper joint sealant reservoir is determined as follows (FHWA, 1999):

1. Estimate the total joint movement using the slab shrinkage/expansion equation.
2. Determine the reservoir width based on the joint sealant to be used.
   • Hot pour liquid sealant / silicone sealant.  Dependent upon the estimated joint opening, the allowable sealant strain and a sealant shape factor.  The shape factor is used to determine the required depth of sealant.  For example, if the required joint width is 12.5 mm (0.5 inches), and the shape factor is 1:1, then the depth is 12. 5 mm (0.5 inches).

   where:	W	=	required joint width
   	ΔL	=	estimated joint opening
   	S	=	allowable sealant strain (dependent upon the sealant type)
   		=	0.15 to 0.50 for rubberized asphalt (width:depth shape factor of 1:1)
   		=	0.30 to 0.50 for silicone sealant (width:depth shape factor of 2:1)

   • Compression sealant.  The uncompressed seal width (USW) should be selected based upon the anticipated joint openings and the maximum and minimum recommended compression of the seal (generally 0.5 and 0.2, respectively).  The sawcut width is determined based on the anticipated state of compression of the seal at the time of compression, which is based largely on the expected temperature range and the installation temperature.

8.1.4  Load Transfer Design

Dowel bars for load transfer must typically be designed into all medium to high volume rigid pavements.  In general, aggregate interlock becomes ineffective at a joint width of approximately 0.9 mm (0.035 inches) and is generally unable to accommodate typical slab edge stresses at transverse joints associated with medium to high traffic loading (FHWA, 1990).

The FHWA (1990) recommends the use of dowel bars.  Further it recommends that they have a minimum diameter of 1/8 the pavement thickness, but not less than 32 mm (1.25 inches).  Typical designs use 460 mm (18 inch) long dowel bars at 305 mm (12 inch) on center spacing, placed at slab mid-depth. 

8.2  Reinforcing Steel Design

In CRCP and JRCP, reinforcing steel is used to hold tightly together any cracks that may form.  Cracks formation depends upon temperature, moisture and base material friction.  As the slab cools and loses moisture, it will contract.  This contraction is resisted by friction with the base material.  If this frictional force becomes greater than the tensile strength of the PCC, the slab will crack and the tensile stresses will be transferred to the embedded reinforcing steel.  Thus, in order to prevent excessive crack widths, the reinforcing steel must be designed to accommodate these stresses without significant elongation.  The amount of steel is typically expressed as a percentage of the slab cross sectional area.  This section, taken largely from the 1993 AASHTO Guide, briefly discusses the design process for JRCP and CRCP.

8.2.1  JRCP Reinforcing Steel Design

JRCP reinforcing steel design is a straightforward process that depends on the following three factors:

1. Slab length.  This has a large effect on the maximum PCC tensile stresses developed within the slab.  As the slab length increases, the contact area with the base material increases, which increases the total resisting frictional force, resulting in higher tensile stresses as the slab contracts and/or loses moisture.
2. Steel working stress.  This is usually taken to be 75% of the steel yield stress.  The steel working stress must be great enough to resist the frictional forces developed during slab contraction. 
3. Friction factor.  This represents the frictional resistance between the bottom of the slab and the top of the base material.  It is like a coefficient of friction.  Table 6.4 shows the 1993 AASHTO Guide recommended frictional factors.

Table 6.4: Recommended Friction Factors
(from McCullough, 1966 as referenced in AASHTO, 1993)

Taking the above three factors into account, the following equation is used to determine the amount of reinforcing steel as a percentage of slab cross-sectional area:

This JRCP design procedure is also used to design CRCP transverse reinforcing steel.

8.2.2  CRCP Reinforcing Steel Design

CRCP reinforcing steel design is used to determine the amount of longitudinal steel that will satisfy the following three limiting criteria:

• Crack spacing.  To minimize crack spalling, the maximum spacing between cracks should be less than 2.5 m (8 ft.).  To minimize the potential for punchouts, the minimum spacing between cracks should be 1.07 m (3.5 ft.).
• Crack width.  To minimize spalling and water penetration, the allowable crack width should not exceed 1 mm (0.04 inches).  Small crack widths are essential to CRCP performance.
• Steel stress.  This is usually taken to be 75% of the steel yield stress to prevent any plastic deformation, although studies have shown that many CRCP pavements have performed adequately even though their steel stress was calculated to be above yield stress (Majidzadeh, 1978 as referenced in AASHTO, 1993). 

One longitudinal steel design procedure is given by the 1993 AASHTO Guide:

1. Solve the following three limiting criteria equations for the percentage of steel required (yes, they appear difficult, but the 1993 AASHTO Guide contains nomograph solutions).  Note that crack spacing (x) should be solved using input values of x = 2.5 m (8 ft.) to determine a minimum amount of steel required to keep the maximum crack spacing less than 2.5 m (8 ft.), and x = 1.07 m (3.5 ft.) to determine a maximum amount of steel required to keep the minimum crack spacing greater than 1.07 m (3.5 ft.).  Crack width and steel working stress solutions will give a minimum amount of required steel.

where:	ft	=	PCC tensile stress at 28 days
		=	ratio of the steel thermal coefficient (5 x 10-6 in./in./°F) to PCC thermal coefficient
	φ	=	steel bar diameter
	σw	=	wheel load stress
	P	=	cross-sectional amount of steel as a percentage of cross-sectional slab area
	Z	=	PCC shrinkage coefficient
	ΔT	=	design temperature drop (between high and low expected temperatures)

2. The solutions to step 1 will provide the minimum (P_min) and maximum (P_max) required percentage of reinforcing steel. If P_max > P_min then the design is feasible and can continue. If not, the design inputs need to be modified and the above equations recalculated.
3. Determine the number (N) of reinforcing bars required:

4. Determine the design number of reinforcing bars (N_design) such that it is a whole number between N_min and N_max.

Transverse steel can then be designed using the JRCP procedure to define the amount of steel required and the following equation to determine the reinforcing bar spacing: