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

The Beam Constraint Model (BCM), developed by

It was demonstrated that the maximum error of the first-order BCM is less
than 5

For compliant mechanisms that use flexible elements experiencing axial-force
dominant loads, compressive force

The determination of the range of allowable axial force includes finding the
lower axial force boundary

The rest of this paper is organized as follows. Section

A simple beam flexure subject to combined force and moment loads.

Beam characteristic coefficients of the BCM matrices

Figure

Note that the values of coefficients in Eqs. (

The strain energy stored in the deflected beam (Eq.

A compliant parallelogram mechanism, as shown in Fig.

A fixed-guided beam in compliant parallelogram mechanism.

Geometric parameters of the compliant parallelogram mechanism.

A beam flexure in compliant parallelogram mechanism.

Since the compliant parallelogram mechanism has a symmetric configuration,
only a single beam of the mechanism is modeled. As shown in
Fig.

A finite element model of the compliant parallelogram mechanism was built
with the ANSYS software where the beam was meshed into 200 elements with Beam 188.
The results are also plotted in Fig.

Comparion of the load-displacement relationship of the compliant parallelogram mechanism.

In order to show the effect of the axial tensile force on the modeling error
of the third-order BCM, we compare its results to those obtained using CBCM.
Considering the effectiveness and efficiency, the CBCM results were obtained
by dividing the beam into 3 elements

Comparision of normalised axial force and error of the compliant parallelogram mechanism.

When the compressive force applied at the end of a flexible beam exceeds a
certain value

The bistable mechanism shown in Fig.

It should be demonstrated that for both examples the non-dimensional
transverse displacements perpendicular to the beam are strictly limited
within

A bistable mechanism.

Considering that the geometry and loading of the bistable mechanism are
symmetrical, a single fixed-guided limb is chosen to analyze.
Figure

Schematic of the bistable mechanism.

Parameters of the examples.

Comparison of the load-displacement relationship of Example I.

Comparision of normalized axial force and error of Example I.

Modelling the deflected beam of Example I using an FEA model built with the
ANSYS software (the beam was meshed into 200 elements with Beam 188), the FEA
force-displacement relationship is also plotted in Fig.

The problem in Example I can be easily solved and the axial force at each
displacement step was obtained by numerically solving the CBCM
equations

As for Example II, we also model it using both the third-order BCM and FEA
methods. The force-displacement relationship can be obtained as shown in
Fig.

Comparion of the load-displacement relationship of Example II between FEA and the BCM.

Comparision of normalized axial force and error of Example II.

In this work, we determine the range of the allowable axial force of the
third-order BCM. The upper bound

Three examples are analyzed to demonstrate the effects of the axial force on
the modeling errors of the third-order BCM, and the CBCM with 3 elements were
used in this work to guarantee the accuracy of the results so that they can
be used as the exact solutions for the purpose of comparison. Firstly, the
case that the axial tensile force is beyond the upper boundary of the
allowable force is given showing the effectiveness of

For the non-dimensional transverse displacements within

It should be acknowledged that the CBCM expands the range of the allowable
axial force for the BCM through discretizing the beam

All the data used in this manuscript can be obtained on request from the corresponding author.

The authors declare that they have no conflict of interest.

The authors gratefully acknowledge the financial support from the National Natural Science Foundation of China under Grant No. 51675396, and the Fundamental Research Funds for the Central Universities under No. JB170403/K5051204021. Edited by: Xianwen Kong Reviewed by: two anonymous referees