Calculating Dimensional Change
All materials undergo dimensional change as a result of temperature variation above or below the installation temperature. The extent of expansion or contraction is dependent upon the coefficient of linear expansion for the piping material. These coefficients are listed below for the essential industrial plastic piping materials in the more conventional form of inches of dimensional change, per ° F of temperature change, per inch of length. They are also presented in a more convenient form to use. Namely, the units are inches of dimensional change, per 10° F temperature change, per 100 feet of pipe.

The formula for calculating thermally induced dimensional change, utilizing the convenient coefficient (Y), is dependent upon the temperature change to which the system may be exposed – between the installation temperature and the greater differential to maximum or minimum temperature – as well as, the length of pipe run between directional changes or anchors points.
Also, a handy chart is presented below, which approximates the dimensional change based on temperature change vs. pipe length
L=Yx(Tl-T2)/10 X L/100
L = Dimensional change due to thermal expansion or contraction(in)
Y = Expansion coefficient (See table above)
(in/10°/100 ft)
(Tl-T2) = Temperature differential between the installation temperature and the maximum or minimum system temperature, whichever provides the greatest differential (° F).
L = Length of pipe run between changes in direction (ft.)

EXAMPLE 1:
How much expansion can be expected in a 200 foot straight run of 3 inch PVC pipe that will be installed at 75° F when the piping system will be operated at a maximum of 120° F and a minimum of 40° F?
L=( 120-75)/10×200/100=0.360×4.50×2.0=3.24 in.

Note: Temperature change ( T) from installation to the greater of maximum or minimum limits.
To determine the expansion or contraction for pipe of a material other than PVC, multiply the change in length given for PVC in the table above by 1.2667 for the change in CPVC, by 1.6667 for the change in PP, or by 2.6333 for the change in PVDF.

Calculating Stress
If movement resulting from thermal changes is restricted by the piping support system or the equipment to which it is attached, the resultant forces may damage the attached equipment or the pipe itself. Therefore, pipes should always be anchored independently at those attachments. If the piping system is rigidly held or restricted at both ends when no compensation has been made for thermally induced growth or shrinkage of the pipe, the resultant stress can be calculated with the following formula.
St = EC (Tl-T2)
St = Stress (psi)
E = Modulus of Elasticity (psi) (See table below for specific values at various temperatures)
C = Coefficient of Expansion (in/in/ ° F x 105)

(see physical property chart on for values)
(Tl-T2) = Temperature change (° F) between the installation temperature and the maximum or minimum system temperature, whichever provides the greatest differential.

The magnitude of the resulting longitudinal force can be determined by multiplying the thermally induced stress by the cross sectional area of the plastic pipe.

EXAMPLE 2:
What would be the amount of force developed in 2″ Schedule 80 PVC pipe with the pipe rigidly held and restricted at both ends? Assume the temperature extremes are from 70° F to 100° F.

St = EC (Tl – T2)
St = EC (100 – 70)
St = (3.60 X 105) X (3.0 X 10-5) (30) St= 324 psi

The Outside and Inside Diameters of the pipe are used for calculating the Cross Sectional Area (A) as follows: (See the Pipe Reference Table for the pipe diameters and cross sectional area for specific sizes of schedule 80 Pipes.)
A= TT/4(OD2-ID2) =3.1416/4(2.3752-1. 9132)
=1.556 in2
The force exerted by the 2″ pipe, which has been restrained, is simply the compressive stress multiplied over the cross sectional area of that pipe.

F = St x A
F = 324 psi x 1.556 in.2
F = 504 lbs.

Managing Expansion/Contraction in System Design
Stresses and forces which result from thermal expansion and contraction can be reduced or eliminated by providing for flexibility in the piping system through frequent changes in direction or introduction of loops as graphically depicted on this page.

Normally, piping systems are designed with sufficient directional changes, which provide inherent flexibility, to compensate for expansion and contraction. To determine if adequate flexibility exists in leg (R) (see Fig. 1) to accommodate the expected expansion and contraction in the adjacent leg(L) use the following formula:
R = 2.877v’D L SINGLE OFFSET FORMULA
Where: R = Length of opposite leg to be flexed (ft.)

D = Actual outside diameter of pipe (in.)
L = Dimensional change in adjacent leg due to thermal expansion or contraction (in.)

Keep in mind the fact that both pipe legs will expand and contract. Therefore, the shortest leg must be selected for the adequacy test when analyzing inherent flexibility in naturally occurring offsets.

EXAMPLE 3:
What would the minimum length of a right angle leg need to be in order to compensate for the expansion if it were located at the unanchored end of the 200 ft. run of pipe in Example 1 from the previous page?
R = 2.877v3.500 X 3.24 = 9.69 ft.
Flexibility must be designed into a piping system, through the introduction of flexural offsets, in the following situations:

1. Where straight runs of pipe are long.
2. Where the ends of a straight run are restricted from movement.
3. Where the system is restrained at branches and/or turns.

Several examples of methods for providing flexibility in these situations are graphically presented below. In each case, rigid supports or restraints should not be placed on a flexible leg of an expansion loop, offset or bend.

An expansion loop (which is fabricated with 90° elbows and straight pipe as depicted in Fig. above) is simply a double offset designed into an otherwise straight run of pipe.

The length for each of the two loop legs (R’), required to accommodate the expected expansion and contraction in the pipe run (L), may be determined by modification of the SINGLE OFFSET FORMULA to produce a LOOP FORMULA, as shown below:
R’ = 2.041 v’ D L LOOP FORMULA

EXAMPLE 4:
How long should the expansion loop legs be in order to compensate for the expansion in Example 1 from the previous page?
R’ = 2.041 y 3.500 X 3.24 = 6.87 ft.

Minimum Cold Bending Radius
The formulae above for Single Offset and Loop bends of pipe, which are designed to accommodate expansion or contraction in the pipe, are derived from the fundamental equation for a cantilevered beam – in this case a pipe fixed at one end. A formula can be derived from the same equation for calculating the minimum cold bending radius for any thermoplastic pipe diameter.
RB= DO (0.6999 E/SB – 0.5)
Where: RB= Minimum Cold Bend Radius (in.)
DO = Outside Pipe Diameter (in.)
E * = Modulus of Elasticity @ Maximum Operating Temperature (psi)
SB * = Maximum Allowable Bending Stress @ Maximum Operating Temperature (psi)

*The three formulae on this page provide for the maximum bend in pipe while the pipe operates at maximum long-term internal pressure, creating maximum allowable hydrostatic design stress (tensile stress in the hoop direction). Accordingly, the maximum allowable bending stress will be one half the basic hydraulic design stress at 73° F with correction to the maximum operating temperature. The modulus of elasticity, corrected for temperature may be found in the table in the second column of the preceding page.

EXAMPLE 5:
What would be the minimum cold radius bend, which the installer could place at the anchored end of the 200 ft. straight run of pipe in Examples 1 and 3, when the maximum operating temperature is 100° F instead of 140°F?
RB = 3.500 (0.6999 X 360,000/ 1/2 X 2000 X 0.62 – 0.5) =1,420.8 in. or 118.4 ft