Bridge engineers must be familiar with the two lateral stiffness formulae for prismatic column sections; that is, k=3EI/L^3 for pin-fixed boundary condition and k=12EI/L^3 for fixed-fixed condition. These two simple formulae are fairly basic concepts often used in seismic analysis of bridge bents for ductility check and earthquake displacement assessment.
However, 3EI/L^3 or 12EI/L^3 is only applicable for prismatic columns. When column section varies along the height; for instance, a flared column or column supported by a large diameter drilled-shaft, we cannot directly use the two formulae any more. Luckily, nowadays finite-element programs like SAP2000 are popular so that you can simulate any section variations or boundary conditions you can imagine. But, sometimes computer-aid analysis may be even low-efficient than a quick hand-calc. For example, there is a bridge frame with 7 two-column bents all supported by Type-II drilled shafts to be built in a seismic zone, and we need verify the bent/column stiffness are compatible within the frame per Caltrans SDC (Seismic Design Criteria) article 7.1.1 balanced stiffness requirements. What shall we do if using SAP2000? Modeling every bent one-by-one and then push it to get lateral stiffness – good luck with that, you may spend half-a-day on finishing the modeling all 7 bents. In contrast to that half-day task, using the two equations shown in Appendix A and B below, you may just need one hour to get the work done.
As a verification of the two new formulae for single-stepped column, first assuming material and sectional properties E, I, and G are identical from top to toe, then the two complicated formulae are reduced to the simple form 3EI/L^3 or 12EI/L^3. Second, I created a SAP2000 model to perform case study and the SAP results match perfectly with what formulae generate. Please note that 3EI/L^3 or 12EI/L^3 only counts the flexural deformation of column, whereas SAP2000 will automatically consider both the flexural and shear deformations. Shear deformations can become significant in comparison with the flexural deformations when the column height is not significantly larger than the column sectional dimension (Priestley et al, §4.3.1, 1996).
 Priestley, M.J.N., Seible, F., and Calvi, G. M.. . “Seismic Design and Retrofit of Bridges”. John Wiley & Sons, Inc., New York, N.Y.
 Zhu, Boqin, Zhou, Jingou, and Xu, Zheming. . “Mechanics of Structures”. Tongji University Press, Shanghai, China.
 California Department of Transportation (Caltrans), Seismic Design Criteria, version 1.7, April 2013.