When a piping system changes its condition from installation to operating, it will expand or contract depending on the service fluid temperature. If the piping system is provided with restraints, it will try to expand or contract against its restraints resulting in internal forces, moments and stresses.

In this article, we will discuss how to calculate the axial force generated in a piping system anchored at both ends. Let us consider a piping system as shown in figure below which is anchored at one end and the other end is free to move. Assuming that the pipe is heated due to the service fluid, the free end will expand by an amount which is dependent on the linear thermal expansion rate and the length of the pipe.

Thermal axial expansion △ = α L ..........(a)

where

△ = thermal axial expansion of the unrestrained pipe in mm

α = linear thermal expansion rate of the material (from ambient to operating temperature) in mm/mm

L = length of pipe in mm

However, if the other end of the pipe is restrained, a compressive load is generated in the pipe. The magnitude of the forces on the anchors can be calculated as the force required to compress the pipe back to its original length after it has been allowed to expand freely.

We know Modulus of Elasticity (E) = \(\large\frac{Stress (F/A)}{Strain(\Delta/L)}\)..........(b)

where

F = compressive axial load on pipe in N

A = cross-sectional area of pipe in mm^{2}

E = modulus of elasticity of pipe material in N/mm^{2}

The axial force required to compress the pipe back to its original length can be calculated by rearranging equation (b) in terms of F:

E = \(\large\frac{F*L}{A*\Delta}\) from equation (b) above

Thus F = \(\large\frac{E*A*\Delta}{L}\) ..........(c)

Substituting △/L = α in above equation (c) gives the axial force as

Compressive Axial Load (F) on pipe = AEα ..........(d)

Note that the value of E changes with material of pipe as well as the temperature.

ProblemFind the axial force and axial stress on a 6-inch NPS ASTM A106 Gr B pipe of standard wall thickness operating at 320°C. Consider installation temperature as 20°C.

**Solution**

For a temperature change of 300°C, the expansion rate of carbon steel pipe is 4.02 mm/m (Refer Table C-1M of ASME B31.3).

Thus α = 0.00402 mm/mm

The cross sectional area of 6-inch standard wall thickness pipe is calculated as A = 3600mm^{2} (Use this calculator).

The value of modulus of elasticity (E) for carbon steel pipe = 182.6 x 10^{3} MPa (N/mm^{2}) (Refer Table C-6M of ASME B31.3)

Substituting the values in equation (d) above gives

Compressive Axial Load (F) = 3600 * 182600 * 0.00402 = 26,42,587 N = 2642 kN

The compresive axial load is significantly large and hence it is imperative that straight pipe with anchors at two ends is not an acceptable configuration. It is recommended that flexibility be provided in the piping system to avoid failures in the piping system due to high loads and stresses. The flexibility in the piping system can be achieved in several ways. It could be achieved by either providing expansion loops between the two anchors or by offsetting the anchor such that an L-shaped configuration is achieved. Any pipe length at right angle to the thermal expansion is capable of absorbing the thermal expansion and reducing the load on the anchors.

It may be noted that in the straight run configuration there is no bending stress, since no moment is produced in the axial run. However there is a possibility of the pipe buckling under the load which can be calculated using the Euler formula.