Abstract
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.
Highlights
The Beam Constraint Model (BCM), developed by Awtar et al (2007) a decade ago, offers a parametric and closed-form model for accurately and analytically capturing nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections
As for Example II, we model it using both the thirdorder BCM and FEA methods
We determine the range of the allowable axial force of the third-order BCM
Summary
The Beam Constraint Model (BCM), developed by Awtar et al (2007) a decade ago, offers a parametric and closed-form model for accurately and analytically capturing nonlinear behaviors of a planar beam flexure over an intermediate range of transverse deflections (typically when transverse motion is less than 10 % of the beam length). It was demonstrated that the maximum error of the first-order BCM is less than 5 % for the non-dimensional transverse displacements (i.e., rotation θ and translation y = Y /L, as illustrated in Fig. 1) within ±0.1, intermediate deflection range, and the normalized axial force (p = P L2/(EI )) within ±10 (Awtar et al, 2007). In this paper, by using the conditions: (1) the positive definite quadratic condition of the strain energy expression of the BCM; (2) the characteristic that the BCM can only capture the first buckling mode (the deflected beam carrying no more than one inflection point), we mathematically derive the upper and lower bounds of the allowable axial force for the third-order BCM, corresponding to the maximum tensile and compressive forces that the BCM can take, respectively.
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