Determining the range of allowable axial force for the third-order Beam Constraint Model

Abstract. The Beam Constraint Model (BCM) was developed for the purpose of accurately and analytically modeling nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (10 % of the beam length). The BCM is expressed in the form of Taylor's expansion associated with the axial force. It has been found that the BCM may yield large predicting errors (>  5 %) when the applied axial force goes beyond a certain boundary, even the deflection is still in the intermediate range. However, this boundary has not been clearly identified so far. In this work, we mathematically determine the non-dimensional boundary of the axial force by the condition that the strain energy expression of the BCM is a positive definite quadratic form, and by the buckling condition relate to compressing axial force. Several examples are analyzed to demonstrate the effects of the axial force on the modeling errors of the BCM. When using the BCM for modeling, it is always suggested to check if the axial force is within this boundary to avoid large modeling errors. If the axial force is beyond the boundary, the Chained Beam Constraint Model (CBCM) can be used instead.