Introduction
Pipe buckling is a phenomenon of large deformation in which a cylindrical pipe buckles due to external loading. In the literature, it is referred to as the Euler buckling of a pipe. The buckling phenomenon is an important consideration for designing and operating piping systems, especially in industrial settings where unexpected pipe failure due to buckling may lead to property damage or loss of life. Two principal theories are employed in pipe buckling analysis. The first theory is based on the elastic properties of the pipe material and the second theory is based on the energy principles of plastic-ity.
Elastic analysis of pipe buckling
In general, when a pipe is subjected to an axial compressive load, its cross-section deforms into an elliptical shape, due to Poissons effect. If the load is increased, buckling may occur, depending upon the material properties of the pipe and the applied load. Assuming that the pipe is subjected to a compressive force per unit length F, then the critical buckling force F_c can be found from Eulers equation, which is given by:
F_c = frac{pi^2 EI}{(KL)^2}
where E is Young’s modulus, I is the cross-sectional moment of inertia and L is the unstrained length of the pipe. K is a dimensionless factor depending on boundary conditions at the ends of the pipe.
For a full circle pipe, K = 4.0 and for a pipe with free or fixed-ended boundary conditions, K = 2.0 or 3.2 respectively. If the applied force is larger than the critical buckling force, F > F_c, then the pipe buckles and a large strain is developed in the pipe.
Energy based analysis of pipe buckling
Although the elastic analysis provides a good approximation of the buckling load, the buckling mechanism is better understood by analyzing the pipe based on energy principles. The energy theory of plasticity is used to predict stability and ultimate failure of the pipe system. In the energy theory of plasticity, the strain energy U stored in a deformed body is estimated using:
U=frac{1}{2} E int varepsilon dV
where varepsilon is the strain, E is the Young’s modulus and V is the volume of the body. The critical buckling load can be determined from the condition that the total strain energy of the system is a minimum. The critical buckling load is then equal to the load at which the strain energy reaches its minimum value. To analyze the stability of a pipe, two different energy terms are considered, namely bending and stretching energies.
Conclusion
To summarize, pipe buckling is a phenomenon in which a cylindrical pipe deforms due to an external compressive force. Two separate theories are applied in pipe buckling analysis, namely the elastic and energy based approaches. The elastic theory is based on the material properties of the pipe and Euler’s equation. The energy based approach makes use of the energy theory of plasticity to predict stability and ultimate failure of the pipe system. The critical buckling load can be determined from the condition that the strain energy of the system is a minimum.
