Buckling of Euler Columns with a Continuous Elastic Restraint via Homotopy Analysis Method

Table 1

The stability criteria for continuously restrained elastic columns.

Column	Stability criteria
C-F column	𝛼𝑆^2+𝑇^2−2𝑆^2𝑇^2cos𝑇cos𝑆−𝛼(𝑆^2+𝑇^2)+(𝑆^4+𝑇^4)+𝑆𝑇[2𝛼−(𝑆^2+𝑇^2)]sin𝑇sin𝑆=0
P-P column	sin𝑇=0
C-P column	𝑇cos𝑇sin𝑆−𝑆sin𝑇cos𝑆=0
C-C column	2𝑆𝑇[cos𝑇cos𝑆−1]+(𝑆^2+𝑇^2)sin𝑇sin𝑆=0
C-S column	𝑇sin𝑇cos𝑆−𝑆cos𝑇sin𝑆=0