A closed-form nonlinear model for spatial Timoshenko beam flexure hinge with circular cross-section

Dan WANG, Jie ZHANG, Jiangzhen GUO, Rui FAN

School of Mechanical Engineering and Automation, Beihang University, Beijing 100083, China

KEYWORDS

Abstract Beam flexure hinges can achieve accurate motion and force control through the elastic deformation. This paper presents a nonlinear model for uniform and circular cross-section spatial beam flexure hinges which are commonly employed in compliant parallel mechanisms. The proposed beam model takes shear deformations into consideration and hence is applicable to both slender and thick beam flexure hinges.Starting from the first principles,the nonlinear strain measure is derived using beam kinematics and expressed in terms of translational displacements and rotational angles. Second-order approximation is employed in order to make the nonlinear strain within acceptable accuracy. The natural boundary conditions and nonlinear governing equations are derived in terms of rotational Euler angles and subsequently solved for combined end loads. The resulting end load-displacement model,which is compact and closed-form,is proved to be accurate for both slender and thick beam flexure using nonlinear finite element analysis. This beam model can provide designers with more design insight of the spatial beam flexure and thus will benefit the structural design and optimization of compliant manipulators.

1. Introduction

Compliant mechanisms, on account of their inherent advantages such as high precision,zero backlash,long operation life and design simplicity, have been widely used in a variety of applications, e.g. micro-alignment assembly,1micropositioning stage,2-5and force/acceleration sensor.6,7One of the most commonly used flexure elements is a spatial beam flexure hinge (SBFH), as illustrated in Fig.1(b), which is usually mounted between two rigid components of the host mechanism shown in Fig.1(a).

Since a beam flexure hinge transmits motions and forces through its elastic deformation, a large body of investigations have focused on the load-displacement model of flexure beam.These studies provided the research community with several applicable methods, such as the finite-element-method(FEM) based solution,8,9the Pseudo-Rigid-Body model(PRBM),10compliance-based matrix method11,12and the constraint based spatial-beam constraint model (SBCM).13-16

Fig.1 A compliant mechanism and its flexure element.

The FEM based model is probably the most accurate computational method to estimate the performance of a SBFH,but it is usually denounced for its low efficiency and lack of parametric design insight. The recently reported PRBM reduces the spatial beam to a general spatial spring model which is accurate for deriving the displacement of a beam over a large range of deformation. However, this PRBM is only valid for a certain loading case and may be invalid when a six-component general load is concerned. Moreover, the PRBM fails to capture the compliance contributions of the bending displacements to the axial and twisting displacements.For example, Hao et al.15provided an analytical formulation for the spatial beam flexure,for which load-displacement relation for the two bending directions were modeled in decoupled bending planes,respectively.Therefore,this model is incapable of acquiring the intercoupling between the two bending directions.

In Awtar’s recent work,13,14a closed-form nonlinear spatial beam model SBCM is given in matrix form. The SBCM is developed based on the Euler’s deformation assumption and can capture the coupling between the two bending directions,the load stiffening effect, the trapeze effect coupling and the kinematic and elastokinematic nonlinearities, which all result from the axial force and twisting moment. Furthermore, the SBCM is proved to be accurate when the transverse displacement is within 10% of the total beam length. In the previous work, authors17applied the SBCM to the design of a 6-DOF compliant parallel manipulator, obtaining accurate kinematic solutions as compared to the nonlinear finite element analysis results.

However,most of the existing beam formulations and models are based on Euler’s deformation assumption valid only for long, slender spatial beams. As for the behavior of thick beams, Timoshenko beam theory is more effective since the shear distortion of the beam’s cross-section cannot be ignored anymore.For example,in the application of load bearings,the host compliant mechanism employing SBFHs needs to be designed in high stiffness while maintaining the product’s scale.Hence the related beam flexures should also be designed with higher stiffness and may be no more slender.In this case,shear distortion of the beam’s cross-section cannot be ignored anymore and most of the existing beam models will be unavailable.Timoshenko beam is a classic but not dated theory,leading to many interesting research topics: Selig and Ding18derived a screw form of Hooke’s law for Timoshenko beams;Rahmani and Pedram19studied the vibration properties of a functionally graded nano-beam with nonlocal Timoshenko beam theory; in a study of fully compliant bistable mechanisms, Chen and Ma20developed an improved planar beam constraint model based on Timoshenko beam theory, which took the shear effects into consideration and significantly enhanced the prediction accuracy over the beam model.

The above analysis highlights the need for a general spatial beam model that can be used for a general spatial beam flexure both thick and slender. This paper starts from the basis of mechanics, and derives a general nonlinear model applicable for a thick beam(5≤slenderness ratio ＜20)within a small displacement range. It is noteworthy that the proposed model is accurate when the beam displacement is within 10% of beam length and rotary angle less than 0.1 rad.Moreover,the model proposed here can be utilized as more accurate deflected elements in chained spatial beam model to describe the large spatial deflection of flexible beams,as described in Chen and Bai’s work.21A further research on this application will be discussed in the future works.

Accordingly, the remainder of this paper is organized as follows. Section 2 presents the second-order approximation of the nonlinear strain which is subsequently used in Section 3 to derive the generalized natural beam boundary conditions.In Section 4, the end load-displacement relation for a slender SBFH is derived. In Section 5, a novel end loaddisplacement model that can be used for thick SBFH is deduced based on the model in Section 4. Then the proposed models are validated and compared with finite element analysis(FEA) method in Section 6. Finally, conclusions are drawn in Section 7.

2. Nonlinear strain formulation

For a spatial beam suffering a general end-loading, the induced deformation can be completely defined with respect to the related cross-section that is perpendicular to the centroidal axis prior to deformation. In general, the beam deformation may be classified into five groups, including translation and rotation of the cross-section, rotation of the deformed cross-section due to the shear force, in-plane distortion of the cross-section, in-plane dilation/contraction of the cross-section and the out-of-plane warping of the crosssection.

The analysis in this paper is performed far below the elastic limit of the beam material and no external load is applied on the lateral surfaces of the beam, and thus in-plane dilation/-contraction of the cross-section arises only due to the Poisson’s effect and is usually negligible. The SBFH concerned in this analysis is a cylindrically constructed beam with circular cross-section. Hence the out-of-plane warping of the crosssection can also be safely ignored. Translation and rotation of the cross-section, according to the Euler’s assumption, are vital to bending deformation and hence are surely kept in this analysis. The shear deformation that occurs in the bending plane of the SBFH can be safely ignored when the beam body is long, uniform and slender.14However, the SBFH in this paper is not always slender and therefore the shear deformation is taken into consideration according to Timoshenko and Goodier’s beam theory.22The in-plane distortion induced by beam torsion can be nonzero for a spatial beam flexure with combined end-loads and is also included in this analysis.

In Fig.2,a SBFH of circular cross-section subjected to general end-loading is illustrated. Translation, rotation and the shear deformation of the cross-section are shown; however,the in-plane distortion of the cross-section, which is included in this analysis, is not shown in Fig.2 for the purpose of simplification.

Fig.2 Spatial kinematics of SBFH.

As shown in Fig.2, coordinate frame XYZ is the fixed global Cartesian coordinate system and point Q denotes the center of a cross-section at distance x0from the fixed end of the SBFH before deformation. When a general end-loading [FXL, FYL, FZL, MXL, MYL, MZL] is applied at the free end of the SBFH, point Q translates to Qdwith mutually perpendicular translations [u, v, w] and rotations[θd, β, α]. Point Qdlies on the deformed centroid axis of the SBFH. The three Euler angles [θd, β, α] can be used to describe the relationship between the fixed frame XYZ and the local frame XdYdZdwith a transformation matrix T.The deformed axis Xdis located at point Qdand tangent to the deformed centroid axis of the SBFH while the two perpendicular axes Ydand Zdare defined within the deformed cross-sectional plane at point Qd. For an Euler beam which is long and slender, shear deformation at cross-section Qdcan be ignored because of the equivalency of two related terms, -β(α) and dv/dx (dw/dx). However, when the SBFH is thick, according to Timoshenko’s beam theory, the shear distortion can no longer be neglected, as shown in Fig.2.In this paper, YZX-typed Euler angles are adopted for calculation simplification of the shear deformation.

Since the load equilibrium should be established in the deformed beam configuration while all the translational deformations and end loads are completely defined in the fixed coordinate frame, the transformation matrix is employed in this analysis to relate the local coordinate frame XdYdZdand fixed frame XYZ.With the Euler angles expressed in Fig.2,the final transformation matrix T can be written as where TX,TZand TYindicate rotations around X-, Z- and Yaxis respectively.

As the transformation matrix T varies along the deformed centroid axis of the beam, the curvature matrix of the beam can be obtained by calculating the derivative of matrix T with respect to the deformed X-coordinate variable Rnin the following form:

A useful relation,which will benefit the simplification of the strain formulation, can be derived from the curvature matrix given in Eq. (2) and can be described as

Obviously, nonlinear curvatures in Eq. (3) are directly derived from the transformation matrix without any approximation, and thus they are theoretically more accurate than their counterpart in previous works.13,14,23

The position vector of an arbitrary point [x, y, z] on the undeformed SBFH can then be translated to the deformed beam configuration using the transformation matrix T in the following manner:

where Rdand R0are the same point on the deformed and undeformed SBFH, respectively.

With the beam deformation completely defined,the Green’s strain measure at any point within the deformed beam configuration can be retrieved as

where du, dv and dw are small increments along the three undeformed coordinate axes X, Y and Z.

Using Eqs. (1)-(6), the complete form expression for nonlinear strain with respect to the concerned deformations can be obtained.The full form of the nonlinear strain formulation is too complicated for both of computation and strain analysis.Hence appropriate simplifications will be applied in this analysis.The beam deformation concerned in this analysis is small,i.e.the transverse displacements are confined to be within 10%of the beam length L and the rotational displacements are less than 0.1 rad. Over this displacement range, the derivative u′ is of the order 0.01 and the v′ and w′ are of the order 0.1. Therefore,the results derived using Eqs.(1)-(6)can be simplified by truncating the high-order terms in order to approximate the result to the second order. On the other hand, the following approximation can also be applied to the results considering the small rotations:

Thus,the final expression for nonlinear strain of the SBFH can be retrieved in a second-order form as

The strain formulation given in Eqs.(7)-(9)is referred to as the second-order approximation in the rest of this paper. The first three terms in the axial strain, given in Eq. (7), capture the elastic extension under bending moments. The remainder in Eq.(7)reveals the influence of bending and torsion on axial strain, by the aid of beam curvatures given in Eq. (3). Shear strains given in Eq. (8) and Eq. (9) each consists of two parts:shear strain arises from pure bending moment and shear strain related to shear distortion.It is evident that shear strains given in Eqs. (8) and (9) are more accurate in solving the issue of a Timoshenko beam.Strain components εXXand εZZare associated with in-plane contraction and arise only due to the Poisson’s effect in the absence of lateral forces. When the out-ofplane warping is ignored, they will equal zeros and thus not appear in the final expression of the nonlinear strains. Moreover, the strain component εYZalso equals zeros along with the neglect of the cross-section warping.

Consequently, the final expression for nonlinear strain,approximated to the second order, will subsequently be used to derive the general closed-form nonlinear model of the SBFH.

3. Load equilibrium and nonlinear analysis of SBFH

In the view of load equilibrium, the moments at any crosssection of the beam can be obtained by integrating the force acting on an infinitesimal area of the cross-section multiplied by its perpendicular distance from the centroid axis. Since the in-plane distortion and the out-of-plane warping of the cross-section are ignored, the bending moments related to the target cross-section can be simply derived as

where E,A and I are the elastic modulus,the area and the area moment of the circular cross-section, respectively; MYdand MZdare bending moments about the deformed axes Ydand Zd, respectively.

The twist in the deformed beam, which occurs in the deformed circular cross-section plane, can be obtained by

where G is the shear modulus and the torsion constant J can be expressed as

Eqs. (10)-(12) can be expressed in a matrix form as

The moments in the deformed beam configurations can be transformed to the fixed frame using the transformation matrix T. Substituting matrix T into Eq. (13), and along with Eq. (3), the relationship between Euler angles and moments can be expressed in the fixed frame XYZ as

Then the following relations can be obtained:

It can be seen that Eqs.(16)and(17)are independent of θd,and hence can be solved in the absence of the first equation.Compared with Eq. (17),Eq. (16) has an extra nonlinear term which may largely complicate the solving process.However,it is evident that the nonlinear term in Eq. (16) is two orders of magnitude smaller than the remaining terms and can be dropped in this analysis. The simplified equations are listed below:

The two expressions on the right side of Eqs. (18) and (19)are simply the effective bending moments in the Y and Z directions approximated to the second order. Furthermore, Eqs.(18) and (19) actually reveal the cross coupling effect between the two bending directions. However, it should be noted that the three equations that are already derived, i.e. Eqs. (15),(18) and (19), are still not enough to formulate the loaddisplacement relations of the SBFH which is generally a sixinput-and-six-output nonlinear model. As a result, three more equations are required to provide additional relations between the translational displacements and the general end-loading.

By applying load equilibrium in the deformed beam configuration, moments at any point Q can be represented as

where MX, MYand MZdenote the moments at point Q that has a distance×from the fixed end of the SBFH. The general end-loading [FXL, FYL, FZL, MXL, MYL, MZL], as depicted in Fig.2, will keep constant during the static analysis.

Substituting Eq.(20)into Eqs.(18)and(19)and truncating the high-order terms, we can derive the following equations:

Redundant constant values in Eq.(21)can be eliminated by differentiating Eq.(21)with respect to×and the results can be written in matrix form:

where α′′and β′′are the second derivatives of α and β respectively.

Eq. (22) has a pair of coupled but linear equations. The coupling arises from the axial twisting mXl, which may contribute additional loads in two bending directions.In addition,this equation has two desired unknowns (β and α) and two undesired unknowns(w′and v′),which make it mathematically unsolvable. Two supernumerary relations are still needed to relate the desired unknowns and undesired unknowns. On the one hand, when the spatial beam is long and slender,desired unknowns (β and α) are approximately equal to the two undesired ones (w′ and v′) within a small displacement range.14,18On the other hand, when the spatial beam is short and thick, a certain relationship can also be found between the desired and undesired unknowns according to Timoshenko’s beam theory.17,24

It is noteworthy that Eq.(22)provides a constraint relation which can be extended to governing equations for both long slender beams and short thick beams within a small displacement range.

4. End load-displacement relations of a slender SBFH

An Euler beam refers to the long,slender and uniform beam in this analysis. Within a small displacement range and the second-order approximation, the differences between the two Euler angles, β and α, and the two derivatives, -w′ and v′,are quite small, as shown in Fig.3. Therefore, the two derivatives can be safely replaced by the two Euler angles based on Eq. (22), which can be rewritten as

Thus the beam governing equations can be obtained as

Fig.3 Spatial kinematics of beam deformation: slender beam.

The general solutions for the two Euler angles β and α can be derived by solving Eq. (24) and given by

Then,the transverse displacements w and v can be obtained by integrating Eq. (25) and expressed in a simplified form:

where ci(i=1-8) are the constants of integration.

The geometric boundary conditions to Eq. (24) can be described as

Referring to Eqs. (20), (23), (25) and (26), end loaddisplacement relations at the free end of the slender SBFH can be determined in a compact form as shown in Eq. (28).

In this 4×4 matrix K, components are actually transcendental expressions and too complex to be given in full form.Therefore, Taylor series expansion is applied here to matrix K in terms of mXLand fXLin order to retrieve a concise form.The final expression of the transcendental components of matrix K, dropping third- and higher-order terms, can then be given by

The results illustrated in Eqs.(28)and(29)capture essential stiffness characteristics of the SBFH under general end-loading and within the displacement range of interest. The stiffness matrix K will become the linear elastic stiffness associated with the four transverse bending displacements when the twisting moment mXLand axial force fXLare zeros. Other inconstant terms in Eq.(29)capture the load-stiffening in these directions in the presence of mXLand fXL. Furthermore, the approximated stiffness matrix K can produce an acceptable accuracy of less than 1% error while making a notable decrement on computation burden and providing more design insight.

In order to derive the load-displacement relation along X direction, the displacement component ulshould be expressed in terms of bending displacements, i.e., β, α, w and v. As shown in Fig.3, the axial force at point Q along the deformed centroid axis equals

Since the translational and rotational displacement in this analysis are both small, axial force in Eq. (30) has ignorable difference with the force along X-axis of the fixed frame.Thus the normalized axial displacement in the fixed coordinate frame XYZ can then be obtained by integrating Eq. (30) with respect to x

where kSLis the normalized elastic stiffness and has the following form:

By substituting Eqs.(3),(25)and(26)into Eq.(31),the displacement in X direction can be expressed in transcendental form. For the sake of brevity, the derived transcendental expression is also expanded in terms of mXLand fXL.By truncating the second-and higher-order terms,the final expression for total axial displacement of the beam’s free end can be stated in the following compact form:

where

The first term in Eq. (32) captures the normalized elastic extension of the spatial beam while kSLis the normalized elastic stiffness in the X-direction.Constant terms of series expansion in Eq. (33) capture the purely kinematic contributions of the transverse forces and bending moments to the axial stretching,while the remaining inconstant terms reveal the elastokinematic coupling between the transverse displacements and the axial extension.

The twisting displacement in the deformed beam configuration is expressed in a derivative form in Eqs. (15)-(17). The beam twisting θlin the fixed frame XYZ can be expressed as25

By substituting Eqs. (15)-(17) and Eq. (26) into Eq. (34),the twist angle at the free end of the SBFH can be expressed in transcendental functions of mXLand fXL. To allow further design insight and computational simplification, all transcendental functions derived should be expanded in terms of mXLand fXL. In the load and displacement ranges of interest,acceptable approximations, which will incur an error of less than 1%, can be derived by truncating the second- and higher-order terms. The resulting simplified expression for the twist angle of the SBFH can be given in the following form:

where

The first term in Eq. (35) is only dependent on the twisting moment and therefore a purely elastic component of θl. Constant terms in c41, c32, c23and c14are free of any end load and only depend on the transverse displacements and bending angles, thus capturing the purely kinematic contributions of the transverse loads to the twist angle. The remaining inconstant terms with respect to mXLand fXLare load-dependent and will incur the elastokinematic coupling between the transverse displacements and the beam twisting.

In Eqs. (28), (32) and (35), the end load-displacement relations for a long, slender spatial beam with combined endloading are presented, where the shear deformation of the cross-section are ignored according to Euler’s assumptions.However, slender SBFHs will cause superfluous compliances which are not always expected in flexure hinge design. For instance, a beam hinge designed for higher load bearing may have a slenderness ratio less than 10 in order to obtain extra stiffness enhancement, in which case the shear deformation,as captured by Timoshenko and Goodier,22will become prominent and cannot be ignored. Accordingly, loaddisplacement relations for a stubby beam flexure hinge will be subsequently developed based on the analysis above.

5. End load-displacement relations of a thick SBFH

Fig.4 Spatial kinematics of beam deformation: thick beam.

In the deformed configuration of a stubby SBFH as shown in Fig.4,shear deformations that are caused by transverse forces are non-ignorable.Applying load equilibrium at the free end of the deformed beam configuration, shear distortions can be approximately related to transverse forces in the fixed coordinate frame XYZ by

where μ is the shear coefficient for a solid circular crosssectional spatial beam, as expressed by26

with ν being the Poisson’s ratio.

By substituting Eq. (37) into Eq. (22), natural boundary conditions for the thick SBFH can be rewritten as

Since terms on the right side of Eq. (38) are all constant,governing equations for the thick SBFH are identical with those given in Eq. (24), and meanwhile the general solutions for the two Euler angles β and α are also the same as those shown in Eq. (26).

Using Eq.(37)and Eq.(25),the transverse displacements w and v are given by

Referring to Eqs. (20), (26), (27) and (39), the end loaddisplacement relations are obtained at the free end of a thick SBFH, as shown in Eq. (40).

where

The expressions in Eq. (41), which are derived by Taylor series expansion, indeed give more insight into the effects of the axial load and twisting moment on the four transverse displacements. It can be seen that terms relating to mXLand fXLbecome zeros and the resulting stiffness matrix turns to the purely linear stiffness matrix in the absence of axial load and twisting moment. In each bending direction, there exists a prominent load-stiffening component associated with the first order of axial load fXLwhile components with the secondorder fXLterms cast relative small influences on transverse displacements. Furthermore, the stiffness matrix components relating to third- or higher-order mXLand fXLterms always have very small coefficient and thus are removed from Eq.(41).

Next, the displacement along X-axis can be divided into two parts. The first part indicates the purely elastic deformation caused by the elastic stretching of the SBFH arc-length due to the axial force. The second part is induced by the two transverse displacement, indicating the geometric constraint of beam-arc length conservation. Substituting Eqs. (25) and(29) into Eq. (31), we can state the total axial displacement at the free end of SBFH in the following compact form:

where

Note that matrix components given in Eq. (43) can also be divided into two groups. Terms in the first group accord with their corresponding ones in Eq. (33), showing the kinematic and elastokinematic contributions of the axial load and twisting moment to the displacement in the X-direction. It can be seen that the coefficients of terms fXLand mXL, except kSL,are usually small, which indicates that the two transverse and the two rotational displacements exert relative small effect on the axial displacement.Other terms in Eq.(33),which relate to constant ψ, provide essential corrections to compensate the displacements caused by shear distortions.Generally speaking,constant ψ is quite small for a slender Euler beam and negligible in the slender beam model.However,while the beam is no longer slender,i.e.the slenderness ratio is less than 20,the constant ψ can no more be neglected and the terms with respect to constant ψ will become prominent.

Purely kinematic terms in Eqs.(33)and(43)generally dominate the axial displacements in terms of magnitude, but they make no contribution to the axial stiffness. On the contrary,the elastokinematic terms, although with small magnitude,have prominent influences on the axial compliance. The correction terms in Eq.(43),which are usually small in magnitude,play a vital role in keeping Eq. (42) accurate while solving the axial displacement for a thick SBFH. But higher-order terms with respect to mXL, fXLand ψ are dropped due to their insignificant contribution.

Finally,solutions given in Eqs.(3),(25)and(39)are substituted into Eq. (31), which upon integration provides the solution for the twisting of the stubby SBFH. Similarly, the solution contains a 4×4 matrix and each nonzero element is a transcendental function of the axial load fXLand twisting moment mXL. After applying the Tayler series expansions to the matrix element and truncating the high-order terms, the final expression for the twisting displacement of the thick SBFH can be written as

where

The first term in Eq. (44) is the purely elastic component.The kinematic and elastokinematic terms in Eq. (44) are exactly the same as their counterparts in Eq. (36), while the correction terms are complicated in form and usually have relative small magnitude. It can be seen in Eq. (45) that elastokinematic terms with respect to mXLbecome the most important contributors while the purely kinematic terms are not as important as those in Eq. (43). Elastokinematic terms with respect to axial load fXLcan also be found in Eq. (45),which reflect the effect of axial forces on torsion. In general,this term has very small effect on beam twisting, but can become significant under high axial load and therefore is not ignorable.

Consequently,a closed-form load-displacement model for a thick SBFH can be established using Eqs. (40), (42) and (44).Compared to the load-displacement model for slender SBFH in Section 4, this model is improved by taking into consideration of the shear distortion which is usually concerned in the Timoshenko’s beam theory. The two load-displacement models proposed above are similar in the expression form and identical in execution. In addition, the model developed for the thick SBFH can be easily transformed into the model for the slender SBFH by setting constant ψ to zero.In other words,load-displacement model established here can be used as a general beam model for all circular cross-sectional spatial beam flexure hinges,whether it is slender or thick.This model is subsequently referred to as the spatial Timoshenko beam model(STBM).

6. Model validation via finite element analysis

The results derived in the previous sections are validated via nonlinear FEA software ABAQUS. Simulations are performed in two groups with designated beam parameters given in Table 1.In these simulations,established load-displacement solutions to both slender and thick SBFHs are verified by applying designed combined loads to the free end of the beams. In these simulations, the 2-node cubic beam elementB33 is adopted to model the slender SBFH, while the 2-node linear beam elements B31 are chosen for simulations of the thick SBFH. All simulations are performed in ABAQUS software with large deformation switch turned on.

Table 1 Geometric and material parameters of SBFHs.

Table 2 Simulation load spectrums.

In order to keep the analysis within the displacement range of interest, combined loads applied to the end of tested SBFH are carefully designed in magnitude as well as load history.Since there are large differences existing between structural compliance of the two sorts of beam flexures, end loads are also quite different in magnitude, as shown in Table 2.

For each of the two loading histories,end displacements of the tested SBFHs are firstly obtained numerically by solving the STBM and then compared with the results that are obtained by ABAQUS software. Comparison results are plotted in Figs.5 and 6 with respect to simulation time, where the prefix ‘‘Num” and ‘‘FEA” denote results derived from the STBM and simulation respectively in this paper.

The comparison results given in Figs.5 and 6 illustrate that the presented load-displacement models are good approximations to the FEA finite element model within the interested displacement range. Another noteworthy fact is that the slender SBFH and the thick SBFH exhibit similar behavior under the given load histories.

As shown in Figs. 5 and 6, increments of the transverse forces FYLand FZLhave little effect on the axial displacement,along which direction the SBFH has the maximum structural stiffness. This is also accordance with the small coefficients that are given in Eqs. (33) and (43). The twisting moment,which arises at the fourth second of the load history,has a relative small impact on the transverse displacements. In the last second, the bending moment MZLleads to a prominent increment on the displacement v, which is mostly caused by the purely kinematic terms of large magnitude listed in Eqs. (29)and (41).

The accuracy of the proposed models can be proved by error distribution plotted in Fig.7 and the maximum relative discrepancies are listed in Tables 3 and 4.

For the slender SBFH, absolute transverse errors along Yand Z-axis have obvious change during the load history,while the corresponding discrepancy of other four displacements are all small in magnitude and fluctuating around zero. Fig.7(a)shows that: from the first second, the error along Y-axis (v)begins to climb; the error along Z-axis (w) also starts to increase from the next second;these two errors reach a plateau from the fourth to fifth second and become maximum at the last second, but the corresponding relative errors are still within 1.2%,as shown in Table 3 and Fig.7(a).The maximum relative error of the other four displacements lies in the rotational displacement about the Y-axis,where the extreme value is 4.3%.

Fig.5 Simulation results of slender SBFH.

Fig.6 Simulation results of thick SBFH.

Fig.7 Discrepancy between numerical and FEA results.

Table 3 Maximum relative discrepancies between FEA and STBM: slender beam.

Table 4 Maximum relative discrepancies between FEA and STBM: thick beam.

For the thick SBFH, magnitudes of the errors along two transverse axes and the rotational error about Y-axis are relatively large and cannot be neglected. The following trends can be observed from Fig.7(b): the error along Y-axis responds from the first second and reaches the local maximum value (in magnitude) at second two; other discrepancies start to enhance during second two and second three; all the displacement errors turn into a stable state from the fourth and fifth second and the transverse errors along Y- and Z-axis reduce to a low level; all the errors experience dramatic increase and reach the maximum in the last load step. As shown in Table 4, the two maximum transverse discrepancies are 0.04 mm and 0.033 mm, while the corresponding relative discrepancies are both 1.2%. The relative discrepancy about Y-axis is 4.6%, which is much larger than the other five components; however it is still accurate to be within 5% with respect to FEA and is acceptable in practice.

It can be seen from Fig.7 that large deviations occur when the two transverse forces start to act and plateau at the end of the third second. In physical terms, it can be deduced that the large deviations mostly stem from approximation errors of the coupling effect between the two transverse axes.With the same reason,the two transverse displacement errors start to increase with the introduction of the bending moment about Z-axis in the sixth second.

7. Conclusions

In this paper, a closed-form load-displacement model for circular cross-sectional spatial beam is developed. Compared with the existing beam models, the proposed one takes shear distortion of the beam cross-section into consideration and hence can be used to solve problems related to thick beams(5≤slenderness ratio ＜20).

Starting from the first principles, the nonlinear strain measure is derived using beam kinematics and expressed in terms of translational displacements and rotational angles. Natural boundary conditions and governing equations for a constraint spatial beam are then formulated. Based on the Euler’s deformation assumption,the load-displacement model for a slender SBFH is subsequently derived in terms of axial load and twisting moment. The proposed slender-beam model can capture most of the geometric nonlinearities of a SBFH, including load-stiffening in the bending directions, coupling between the bending directions, the kinematic and elastokinematic component in the axial displacement and twisting angle, and the coupling between the axial and torsional directions. However, the slender-beam model is only available to slender beams for which shear distortion in the beam cross-section can be neglected.

For the sake of generality,we develop a novel model STBM by introducing a shear distortion term into the slender-beam model. The proposed model STBM is developed based on Timoshenko’s beam theory and featured by the appropriate engineering approximations. This makes the final STBM suitable for both the thick beam and the slender beam while reducing the computational load to a manageable level.Finally,the STBM is used to solve the load-displacement relations for a slender beam and a thick beam. Results from the STBM are fully compared with those derived using nonlinear FEA and the comparison shows that the proposed STBM is always accurate and within 5% in any displacement direction of the two beams.

The main contribution of this paper exists in the proposition of the STBM which will facilitate the structural design and optimization of compliant mechanisms that employ SBFHs. The STBM can capture most of the main geometric nonlinearities that affect the mechanical behavior of a spatial beam that is either slender or thick, and hence the analytical model will benefit the designer with more parametric design insight. In addition, the derivation process of the STBM can be extended and used in modeling a beam with any other bisymmetrical cross-section with two identical principal bending moments of area.

Acknowledgement

This study was supported by the National Natural Science Foundation of China (No. 51305013).