The Mixed-Body Model: A Method for Predicting Large Deflections in Stepped Cantilever Beams

Beams with stepped cross sections are common, like the stepped cantilever beam shown in Fig. 1(a), and there has been extensive research regarding their behavior. This research has occurred in deflection of stepped shafts [1,2], static deflection and vibrational analysis [3], vibrational analysis [4–6], electrostatic microbeams for microelectromechanical systems (MEMS) [7,8], telescoping beams for space applications as demonstrated in Fig. 2 [9,10], and energy harvesting [11,12]. Stepped compliant beams show promise for use in medical implants. A biomedical application, not disclosed in this work, is the primary motivation for this research.

Another instance of stepped beams comes through stepping of material properties. Beams with the same cross section along the length could behave similarly to that of a stepped cross section through stepping down in material stiffness at a discrete point (see Fig. 1(b)). As manufacturing methods expand, multi-material beams and mechanisms are becoming increasingly feasible [13–15]. Efficient modeling of beams with steps in geometry and/or material properties undergoing large deflections could enable advances in compliant mechanisms and mechanism design.

Past research is generally focused on small deflections where linear equations are accurate. The transition from small deflections to large deflections is not trivial, even with a simple cantilever beam. To determine analytical expressions for large deflections, elliptic integrals are often necessary [16]. Models, such as the pseudo-rigid-body model (PRBM) [17], have been developed to simplify the analysis and make design of large-deflection beams more feasible, opening up the study of compliant mechanisms. However, many of these approaches generally apply to beams with constant material properties and constant moments of inertia of the cross section [18–20]. It is not uncommon for compliant flexures to be anchored into bodies that are stiffer than the flexure but are not stiff enough to be assumed to be fully rigid. An example of this could be telescoping members (see Fig. 2) where initial members could be considered as relatively stiff, even within the range of deflections such that traditional linear approaches apply, while later members deflect enough to induce geometric nonlinearities.

Nonlinearities come in two forms: material nonlinearities and geometric nonlinearities. Material nonlinearities occur when the stress and strain of the materials are not linearly related, such as plastic deformation or nonlinear material properties. Geometric nonlinearities occur when the deflections change the problem, such as with large deflections. This is well understood in the area of compliant mechanisms [17]. When subsequent discussion of nonlinearities occurs in this paper, it will be discussing the geometric nonlinearities of large bending deflections. Discussion of the linear solution refers to Castigliano’s second theorem.

Work in compliant mechanisms has created well-defined models to describe the motion and stresses of cantilever beams undergoing deflections that cause geometric nonlinearities. In all these cases, the attachment or “wall” where the beam attaches is assumed to be perfectly rigid. Of particular interest in this work is the case when the wall attachment cannot be assumed as perfectly rigid and non-negligible deformation and angle change occurs at the transition from a stiffer segment to a segment with a reduced stiffness. A common example of this is the transition region of an electrical cord into the wall plug. While the plug could be considered as approximating rigid motion, manufacturers often include a stiff segment extending from the plug as a transition from the plug to the cord. The cord is designed for large deflections, while the transition segment is often far more rigid. Models to understand this transition and the stress reduction provided by the stiff segment could improve the design of consumer electronics.

The chain algorithm [17], the beam-constraint-model [21,22], and chained beam-constraint-model [23] provide ways to incorporate differing geometries and material properties when modeling cantilever beams. These models discretize beams into a number of elements (smaller beams) whose deflection is sequentially solved to determine the overall deflection of the beam. These solutions provide valid approaches but often incorporate computational expenses similar to other numerical approaches such as finite element analysis (FEA) or are limited in the deflection range that can be analyzed.

Here, we present an analytical model to analyze stepped cantilever beams undergoing large deflections. The resulting model shows promise for use in design because it can be used to quickly analyze many design choices.

Large deflections are common in the analysis of compliant mechanisms or mechanisms that gain some or all of their motion through the bending or flexing of material [17]. The flexibility of a beam is in large part determined by the moment of inertia of the cross section of the beam. In stepped beams, the moment of inertia changes with each subsequent step. In small deflections, Castigliano’s second theorem is commonly used to determine endpoint deflection of these stepped beams [24]. But expanding to large displacements, another approach needs to be considered. By combining traditional small-deflection theory and the models used to predict motion in large-deflection compliant mechanisms, a method is presented here to predict the behavior of stepped cantilever beams undergoing s. The combining of small and large deflection analyses for stepped cantilever beams has been studied previously for numerical solutions [25]. In this work, we consider stepped beams with a single step from a larger moment of inertia near the wall and a smaller moment of inertia towards the tip.

Although small-deflection assumptions simplify calculations and have many applications, they become more inaccurate as the deflections increase. Because the deflections of compliant mechanisms often exceed the valid range of the small-deflection assumptions, large-deflection theory must be used and we have to step back to Eq. (1). The use of elliptic integrals provides an analytical solution for beam-endpoint locations and angles but the complexity of these solutions make them impractical to use in many cases and only numerically solvable in others [16,26–28]. To simplify the design and analysis of compliant mechanisms, the PRBM was developed.

The PRBM approximates the motion of a large-deflection cantilever beam by approximating the compliant beam with rigid links connected by a pin joint, called a characteristic pivot, with a torsion spring. The motion of the end of the rigid link approximates the motion of the end of the flexible beam [17]. The PRBM has found wide application in the design of mechanisms and structures with large deflections [20,29–37]. Due to the low computational expense of the PRBM, it provides a computationally inexpensive alternative to numerical approaches for design and optimization [38].

Presented here are the equations for a cantilever beam with a pure vertical load at the end. The equations used in defining the PRBM in Eq. (6) are the framework for the mixed-body model (MBM). The governing principles behind the parameters used in the PRBM are well known and can be found in greater detail in Ref. [17], including other loading cases and boundary conditions.

This section details the development of the MBM for stepped cantilevered beams and the accompanying steps taken to validate the model. This includes a comparison to assuming rigidity in the stiffer segment of the beam, the use of FEA, physical testing, and comparisons to numerically solving a boundary value problem (BVP) describing large deflection. For the development of the model, a purely vertical force at the tip of the stepped cantilevered beam is used, as shown in Fig. 4.

The MBM was developed through the combination of the traditional small-deflection assumptions and the use of the PRBM. The small-deflection assumptions are to be applied to the segment of the stepped beam that is attached to the wall. This cross section is thicker than the subsequent step and therefore would be more stiff. The subsequent step is then assumed to have a stiffness low enough to undergo large deflections. The PRBM is applied to this segment. The MBM is shown in Fig. 6 and will be referenced throughout the formulation of the model.

Note that the small-deflection assumption makes no change to the endpoint deflection in the x-direction, contributing the entire length L_s to the value of a. The flexible segment deflection in the x-direction again includes a term to account for the stiff segment angle. The MBM with endpoint deflections and angles labeled is shown in Fig. 6.

We now have two equations (Eqs. (16) and (29)) and two unknowns (F and Θ _f) and can therefore solve for the force and Θ_f which can be used in Eq. (17) to solve for the second endpoint location a. The maximum stress can then be determined using the conditions stated in Sec. 2.2.1.

For force loads, i.e., F, is given, Eq. (16) has two unknowns, b and Θ _f. Here, we can use Eq. (29) to solve for Θ_f directly and use that value in Eq. (16) to solve for b and Eq. (17) to solve for a. Once again, the maximum stress can then be determined using the conditions stated in Sec. 2.2.1.

The governing differential equation for the large deflection of a cantilever beam with end loads consisting of both forces and a moment can be solved numerically as a BVP. In this formulation, bending is considered to be the dominant deformation with negligible shear and axial deformations. A numerical solution to this problem for constant cross sections is shown in Ref. [39]. This procedure can be extended to accommodate for variable cross sections.

The numerical solution to the problem can be found using a BVP solver such as matlab’s bvp5c function. A piecewise cubic hermite interpolating polynomial (pchip) can be convenient to represent the moment of inertia and modulus of elasticity functions. After solving numerically for θ(s) or z₁(s), numerical integration can be used with Eq. (30) to find the coordinates of the beam. To represent a step in cross section, five points were used in the pchip function, the endpoints of the beam, the location of the step, and the location of the step ±0.01L_t. For the fixed endpoint and the location −0.01L_t from the step, the stiff values of I_s and E_s were used. For the loaded end and location +0.01L_t from the step, the flexible values of I_f and E_f were used. At the step location, the average value of I and E was used. The solution was calculated through a mesh of points along the entire beam length. A total of 150 points were used along the beam length in the analysis.

This process can be used to help validate the MBM as will be shown later on.

To validate the MBM, a large selection of force and displacement analyses were performed using FEA. Each analysis varied in some geometric value, loading condition, or material property. The parametric study parameters for the analyses are given in Tables 1 and 2 and an example image of an output from the FEA program is given in Fig. 11. All possible combination parameters from Tables 1 and 2 were analyzed. ansys 19.1 was used for the analysis. Geometric nonlinearities were included in all analyses. In total, 14,580 analyses were performed for each force and deflection loading condition on rectangular cross sections. The results from the FEA were used to compare the model predictions. The moduli of elasticity were picked to approximate three common materials. These materials and the accompanying tensile yield strengths used in the analysis are given in Table 3. If a material exceeded the yield strength in the FEA results, that sample was removed from the subsequent comparison to the MBM.

Circular cross-sectional stepped beams were also analyzed to show application in non-rectangular cross sections. The parametric study parameters for the analyses are given in Table 2. Six hundred and forty eight analyses were performed for both force and deflection loading conditions on circular cross sections.

To analyze the effect of steps in material properties, material changes at the step were introduced, as can be seen in the elastic moduli used shown in Tables 1 and 2. These cases had both a step of geometry and material properties. The case of just a step of material with no geometry change was also tested with the first step being the Ti-64 simulated material and the second step being the simulated polyetheretherketone (PEEK) material. A selection of lengths, ratios, and dimensions similar scale to those used in Tables 1 and 2 was used in the validation.

For each of the FEA models, the beam was meshed with 75 divisions along each step length, resulting in 150 divisions along the beam, as was used in the BVP. The cross section was divided into a 10 × 10 grid of elements for each division.

Since there is not an analytical solution to compare against, optimization of the model parameters γ, $cΘ$⁠, and $KΘ$ used in the MBM was performed against both the FEA and the BVP solutions to see the influence on the respective error. The optimal values for γ, $cΘ$⁠, and $KΘ$ for the MBM are then provided for best approximating the FEA and BVP solutions. The optimization was performed by minimizing the absolute value of the sum errors of each of the measured values (angles, deflections, forces to deflect, stress) for the rectangular stepped geometry case. Therefore, the optimized parameters apply to all evaluated loadings, material properties, and geometries tested, not individual cases. This was done in an effort to provide a universal set of parameters for the best results regardless of the scenario. matlab’s constrained optimizer function, fmincon, was used with an active-set algorithm.

In addition to computational analysis, physical prototypes were constructed and tested to validate the MBM. Aluminum 6061-T6 beams were cut using wire- electrical discharge machining (EDM). Three sample sets were made, with the step at 25%, 50%, and 75% of the total length with ten samples in each set. All 30 samples were 6.5 mm wide with a stiff segment height of 3 mm and a flexible segment height of 1 mm. A 2 cm long 5 mm high clamping area was manufactured into the beam as well as a millimeter diameter hole at the end to attach a mass. The total length of each beam or the length from the clamping area to the loading hole was 15 cm. Each of the three sample sets were loaded with calibrated masses attached by a metal wire. The loads were selected to approach the yielding stress and obtain a large endpoint deflection for each beam. As such, for the 25/75 stiff/flexible beams a 200 g load was used, for the 50/50 beams 400 g was used, and for the 75/25 beams 700 g was used. The wire used to attach the masses had a mass of less than 1 g. Pictures were taken (resolution of 6000 × 4000 pixels taken from a distance to result in approximately 30 pixels/mm) of the loaded and unloaded case for each sample and pixel locations were used to determine the deflections and angles under load. An example of images taken can be seen in Fig. 12.

The errors of the model predictions as compared to the other analysis methods are given here, comparing both the rigid assumption model and the MBM. A discussion on computational expense is also provided. Benefits and limitations of the MBM identified during the comparison are presented.

In the comparison of the FEA results to the MBM, seven variables were compared. In the case of deflection loads, of the 14,580 samples with rectangular cross sections, 3997 samples were under the yield stress and used in the comparison. The errors can be found in Table 4. The average error between the model prediction over the 3997 samples along with the standard deviation (SD) is reported. The maximum absolute value error value is also reported. The model was then optimized to the FEA solution and the errors are again reported using the optimized values of γ, $KΘ$⁠, and $cθ$⁠. For the rigid assumption, four of the seven parameters (those parameters not involving the stiffened segments of the beam) are compared. An optimization was also performed and reported to see if the rigid assumption could simply be used if optimizing the parameters in the PRBM.

For force loads, of the 14,580 samples with rectangular cross sections, 2115 samples were under the yield stress and deflection criteria. For all 2115 force-loaded samples with rectangular cross sections, the MBM was compared to the BVP solution. The BVP solution was also compared to the FEA results. A summary of the errors is shown in Table 5. As in the case of deflection loads, average error between the model prediction over the 2115 samples under the yield and deflection criteria is reported, along with the SD and maximum absolute value error term.

For circular cross sections under deflection loads, of the 648 samples, 108 samples were under the yield stress and used in comparison. Under force loads, of the 648 samples, 462 samples were under the yield stress. Error ranges were similar in the case of circular cross sections as those found in the rectangular cases. The optimization was run with just the rectangular solutions but the optimal values still provided dramatic reductions in errors (approximately 4% average to approximately 0.5% average) similar to those seen in rectangular solutions.

For just steps in material properties at a constant cross section, 180 samples were analyzed for force and deflection loads. Thirteen and 38 of the cases, respectively, remained under yield. The error ranges for the cases tested remained below the values for stepped geometries.

From Tables 4 and 5, we can see that the MBM closely approximates the finite element solution in all cases. When the values of γ, K_Θ, and $cθ$ are optimized to the FEA solution, there is little difference between the two solutions. The rigid assumption has the potential for large errors, especially in parameters such as determining force, angle of deflection, stress in deflection load cases, stress in force load cases, and vertical deflection, b. When optimizing the rigid assumption to the FEA, little improvement is seen but the solution remains less accurate than the MBM. The MBM compares to the BVP better than the FEA results by a small margin. The BVP solution was also run at a higher refinement (1000 points along the length) to see if the higher refinement reduced the errors relative to the MBM and FEA solutions. The higher refinement did bring the solutions towards the FEA and MBM by reductions in the errors of approximately 1% in the average values and 2–4% in the maximum errors but at an increase of about 10x to the computational cost. Optimizing for the BVP provided little improvement, indicating that the original PRBM values are close to optimal for this case. In optimizing both to the BVP and the FEA solutions, γ and $KΘ$ were reduced in both cases. $cθ$ made no change for the BVP as θ_f was not calculated, but for the FEA optimization, it deceased from 1.2385 to 1.2196.

Table 6 includes the optimized values and the PRBM values for comparison. As the values for both γ and $KΘ$ are reduced and those for γ are so similar, it is likely that the true optimal values for these parameters are also reduced from the pure vertical parameters in the PRBM. These results can show trends while an analytical solution would be required to find the true optimal values.

In both loading conditions, as expected, the linear solution maintains a high accuracy at lower deflections. This accuracy begins to diverge as deflection increases. This can be seen in the large maximum errors and large SDs from the average in Tables 4 and 5. This becomes clear when plotting the error in key results such as F for displacement loads and b for force loads against angular deflection at the endpoint, θ_f. This is shown in Fig. 13. The MBM decreases in error under larger deflections.

The physical test results were obtained using pixel locations and therefore values of a, b, a_s, b_s, θ_f, and θ_s could be obtained. These could then be compared to the FEA, BVP, and MBM solutions. The MBM used the pure vertical PRBM values for γ, $KΘ$⁠, and $cθ$⁠. These model solutions were compared to the average of the physical sample set solution. Inherent error is expected to be introduced in the physical results from the measurement methods. These were recorded as follows. The samples were measured using calipers to ensure manufacturing resulted in the desired dimensions. The resolution of the calipers was ±0.1 mm. The pixel/millimeter measurement introduced a second resolution error of approximately ±0.04 mm. As the loading was in the direction of gravity, the self weight of the beam also provides an additional load, altering endpoint locations up to an additional 0.5 mm. The cumulative measurement uncertainties plus the SD of the sample set were then added and used to create a percent error range due to measurement. The model errors when compared to the average physical test result, along with the percent error range from measurement, are shown in Table 7.

From Table 7, we see that the MBM performs similarly to the other two methods of analysis. It strays outside the percent measurement error range for b, a_s, and θ_f but in each case by less than a percent. Using optimized parameters for γ, $KΘ$⁠, and $cθ$ creates error results similar to the FEA case, removing the instances that stray outside the measurement error range except in the case of a_s which remains a tenth of a percent outside the measurement error. This further indicates that the optimal parameters for the model are close to those obtained in the FEA and BVP optimizations.

The primary benefit of the MBM is a simple, computationally inexpensive way of analyzing large deflections in stepped cantilever beams. The MBM provides a robust approach to different geometries and material properties. While the equations are more involved, the computational expense is similar to that of the PRBM. Additionally, the computational expense of the MBM is much less than both FEA or the BVP presented above. The times to run the FEA solution on ansys averaged approximately 1.6 s per evaluation, BVP on matlab timed at approximately 4.5 s per evaluation, and the MBM (coded in matlab) at approximately 0.04 s per evaluation using the same computer and processor. This results in evaluations 40 × faster than FEA and 112 × faster than the BVP.

The maximum and average errors in the MBM are equal or lower for all the cases when compared to the rigid assumption predictions and are more robust to all large-deflection situations. The stress prediction was often over-predicted by a small percentage. While this is a limitation in analyzing failures, it can be a benefit in design by having slightly conservative stress predictions. A limitation here is the model does not account for the stress concentration at the step. A stress concentration factor could be added to the stress prediction when specific geometry is known.

As an approximate analytical solution to the problem, the MBM introduces some slight error into the results. These errors are typically lower than 5% and often much lower when compared to the other methods of analysis. While this error could be a limitation in fine-tuned design, the computational speed of the analysis lends better to optimization problems where large numbers of computations are required.

When comparing against the linear solution, the optimized MBM parameters showed equivalent performance at low deflections while also providing a similar amount of accuracy at high deflections where the linear assumption breaks down. At low deflections, the increased complexity of the model does not justify replacement of standard models such as Castigliano’s theorem but the wide range of accuracy of the MBM does allow for use cases that span both large and small deflections without necessitating transitions between models. An advantage of the MBM is the ability to easily accommodate both changes to moment of inertia and material properties at the step. This could extend the usage of the MBM to cases where the moment of inertia does not change but the beam material properties change from a stiff to a flexible member at a discrete point along the length.

Limitations to the MBM are primarily seen in inaccuracies which begin to be introduced in very high deflections of the flexible beam. This is observed in the PRBM and results in limits on accuracy for force and position predictions. As θ_f approaches π/2, the pure vertical load results in less of a bending load and more tensile and shear loading, resulting in an exponential increase in error. Therefore, these limits are values of Θ_f that define when the error exceeds a given value. For the MBM, a similar limit can be observed. However, given the contribution of θ_s, these values will be lower than those limits in the PRBM. For the optimized parameters, the error can be kept below the maximum errors shown in Tables 4 and 5 for force and position predictions when limits of approximately 85% of those given by the PRBM are used. Therefore, for force and end-position, respectively, the MBM remains under the maximum errors of 2.42% and 1.55% when Θ_f does not exceed approximately 0.87 and 0.96 radians (50 deg and 55 deg).

The MBM is also limited to cases where the stiffness in the segment attached to the wall is greater than that in the second segment as the majority of bending should occur in the second segment. The current model has only been validated in pure vertical loads. Further validation will be needed to ensure accurate predictions when additional loads are applied, such as horizontal loads or moments.

In this work, a method for predicting large deflections of stepped cantilever beams is presented. This method, the MBM, builds on a mixture of small-deflection assumptions and the PRBM for large deflections. The model uses a combination of the equations provided by both methods of analysis and creates a model that can predict the endpoint locations, stress, and force to deflect a stepped beam under vertical end loads. The MBM can be used for both step changes in geometry or material properties or combinations of both cases. The model was compared to FEA results and maintained a low average error over a large sample of loadings, geometries, and material properties. The model has less error than a simple assumption that the first step in the beam is rigid. The MBM also provided significantly improved results over linear methods for large deflections while remaining accurate in small deflections. The MBM also compared well with numerical BVP solutions and testing of physical prototypes. The validation procedures evaluated a large number of loading cases, geometries, and material properties. In seeking a model that can be universally applicable to changes in these areas, optimization over all tested scenarios to find model parameters that best fit all cases was performed. This further reduced the average error between the MBM and numerical solutions. The maximum error and SD in results between changes in loading cases remained small, showing a model that is applicable to large ranges regardless of the scenario. Future work could analyze the specific effect of given loading cases, geometries, and material properties on the model accuracy and could result in model parameters optimized to specific scenarios for lower average error.

The MBM provides a simple method for analyzing stepped cantilever beams undergoing large deflections, which can be helpful in design where many iterations are often necessary. This shows promise for use in the design of medical implants, which motivated the development of the MBM. A primary motivation for this work, a compliant-mechanism-based medical device, is not disclosed here but is a part of ongoing work. Other areas of future work will include expanding the model to other loading conditions such as mixed vertical and horizontal loading in the fixed-free case, a fixed-guided end condition, precurved beams, and pure end-moments which would increase the applicability of the MBM. PRBMs exist for these loading conditions and future research will ensure that the assumptions used in the load case presented here also provide accurate solutions for these other cases.

This paper is based on work supported by the U.S. National Science Foundation through NSF [Grant No. 1663345] and a grant by Zimmer-Biomet in conjunction with Cincinnati Children’s Hospital Medical Center.

There are no conflicts of interest.

The authors attest that all data for this study are included in the paper.

a =

endpoint position in the horizontal direction

b =

endpoint position in the vertical direction

c =

distance from the neutral axis to the most extreme beam fiber

h or Height =

cross-sectional dimension of a segment perpendicular to the axis of bending

l =

lowercase l indicates the length of a segment considered to be undergoing large deflections

r =

radius

s =

position along a beam, measured along the neutral axis from the wall

w or Base =

cross-sectional dimension of a segment along the axis of bending

E =

modulus of elasticity

F =

force

I =

moment of inertia of the cross section

L =

uppercase L indicates the length of a segment considered to be undergoing small deflections

M =

moment

P =

force that acts perpendicular to the unbent beam

$cθ$ =

non-dimensional parameter used in modeling angles

$KΘ$ =

non-dimensional parameter used in modeling torques

L_sγ =

length of the beam from the wall to the characteristic pivot

M_ts =

theoretical moment of the torsional spring at the pseudo-rigid characteristic pivot

nP =

force that acts parallel to the unbent beam

γ =

non-dimensional parameter used in modeling lengths

δ =

deflection

θ =

angle of the actual segment at a given point

Θ =

angle of the pseudo-rigid link

ν =

Poisson’s ratio

σ =

stress

σ_y =

yield stress

s =

a value relative to the first, stiffer segment attached to the wall

f =

a value relative to the second, more flexible segment further from the wall