Abstract A set of equations for helical and sinusoidal buckling under an ideal condition, where weight and frictional effects are not considered have been derived based on experimental and FEA data. An experimental method was developed canceling out frictional effects and FEA was carried out independently on both types of buckling mode. A simple axial force PC program for drill string has been utilized to predict buckling behavior, where the proposed equations in the present paper are applied to modify an axial force distribution that considers the frictional effect due to buckling deformation history, i.e. process and weight effect. Introduction Excessive axial compression force on the drill string causes sinusoidal and/or helical buckling, which gives rise to such problems as drill string failure, mechanical damage on tubulars or bit, stick slip damage and lockup. Due to the longer horizontal or high deviated sections being drilled the industry has become more aware of the problem and the need to resolve it. The mathematical expression showing the relations between mode and axial force for helical / sinusoidal buckling have been previously proposed based on theoretical studies. In general it has been the norm to use the simple formulas describing the critical limits of axial force leading to buckling, not as a function of mode, because the mathematical expression showing the relations between mode and axial force are too complicated to apply, and it would appear that the critical force can be well predicted by the formulas. Apart from the work on these formulas there has been no further serious attempt to study their applicability experimentally, and the effects of frictional force between well bore and drill string have not been taken into account in many cases. Recently, R. C. McCann and P. V. R. Suryanarayana have carried out some experiments employing rod specimens with a specially developed apparatus in order to discuss the reliability of these formulas for both buckling modes and taking friction into account. Within this present study, the mathematical expression showing the relations between mode and axial force for helical and sinusoidal buckling formations are discussed with the experimental data base obtained from laboratory tests canceling out frictional effects with developed jigs. The mathematical expression showing the mathematical expression for both buckling modes are simplified to a set of brief equations under deal conditions. The new mathematical expressions are applied to a PC program of an axial force model to predict drill string deformation during buckling process under the condition where frictional and weight effects exist. Data base obtained from FEA shown in the present paper also supports the mathematical expression showing the relations between mode and axial force. The paper describes experimental and FE results to derive the prototype of buckling strength equations for the ideal condition, and discusses the prediction method of a buckling deformation history. 2. Theoretical Background The mathematical expression showing the relation between mode and axial force in constrained buckling phenomena depends on type of buckling, i.e. helical and sinusoidal buckling. The mathematical expressions are too complicated to express in simple formulas, so that it would have been quite usual to employ the simpler formulas describing the critical limit for both buckling formations. The representative conventional buckling formulas are summarized as follows: P. 175