Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms

Abstract

There are several design equations available for calculating the torsional compliance and the maximum torsion stress of a rectangular cross-section beam, but most depend on the relative magnitude of the two dimensions of the cross-section (i.e., the thickness and the width). After reviewing the available equations, two thickness-to-width ratio independent equations that are symmetric with respect to the two dimensions are obtained for evaluating the maximum torsion stress of rectangular cross-section beams. Based on the resulting equations, outside lamina emergent torsional joints are analyzed and some useful design insights are obtained. These equations, together with the previous work on symmetric equations for calculating torsional compliance, provide a convenient and effective way for designing and optimizing torsional beams in compliant mechanisms.

1 Introduction

A compliant mechanism achieves its mobility through the deflections of its compliant elements [1]. In most compliant mechanisms, the compliant elements are designed to produce motion through bending deflections [2,3,4]. In fact, torsional deflections could be another valuable source for obtaining mobility in compliant mechanisms. There have been successful designs utilizing torsional deflections, for example, a split-tube flexure based on the torsion of an open-section hollow shaft was presented [5], a revolute joint comprised of two crossed torsion plates which shows a good performance in resisting axis drift was presented [6], torsional micromirrors were proposed for optical switches and optical displays [7, 8], a torsional micro-resonator was fabricated for mass sensing [9], lamina emergent torsional (LET) joints were devised to facilitate the design of compliant mechanisms that can be fabricated from a planar material but have motion that emerges out of the fabrication plane [10,11,12] and were employed in precision adjustment mechanisms [13], torsion hinges were proposed as surrogate folds in origami-based engineering design [14, 15], and torsional beams were successfully used for achieving static balancing of an inverted pendulum [16].

There are several design equations available for calculating the torsional compliance and maximum torsion stress of a rectangular cross-section beam. However, before using these equations, one of the two dimensions (i.e., the thickness and the width) of the cross-section must be defined as the wider of the two dimensions [17]. This situation might be troublesome and error-prone during the design phase because we always do not know which dimension is larger in advance. This is especially true for an optimization design process considering that the two dimensions of the torsion beam(s) may change greatly during the design iteration process including the relative size of the two dimensions. In our previous work [17], general compliance equations that are symmetric with respect to the two dimensions were obtained to facilitate the design of torsional beams in compliant mechanisms. These equations had been used in characterizing parasitic motions of compliant mechanisms [18, 19] and spatial deflections modeling [20,21,22]. However, there still lacks an equation for predicting the maximum stress in torsional beams that is symmetric with respect to the two dimensions.

To complement previous work [17], this paper is going to address this absence. The organization of this paper is as follows: Section 2 presents a brief summary on various equations for predicting the maximum stress in torsional beams; two maximum stress equations that are symmetric with respect to the two dimensions of the cross-section are formulated in Section 3; Section 4 offers some design insights for outside lamina emergent joints using the proposed equations; and Section 5 has concluding remarks.

2 Various Equations for Calculating Maximum Shearing Stress

For torsion of rectangular sections, the stress at each corner is zero, and the maximum shearing stress $\tau_{\text{max} }$ occurs at the middle points of the longer sides (the points most remote from the centroid of the cross section), as illustrated in Figure 1. It can be obtained using the membrane analogy by assuming w ≥ t [23]:

Figure 1

Stress distribution on a cross-section of a twist beam (w ≥ t)

$$\tau_{\text{max} } = G\theta t\left[ {1 - \frac{8}{{\uppi^{2} }}\sum\limits_{n = 1,3,5 \ldots }^{\infty } {\frac{1}{{n^{2} \cosh (n\uppi w/2t)}}} } \right],$$

(1)

where θ is the angle of twist per unit length. The infinite series on the right side converges rapidly. Because the torque as a function of θ is given as [23]

$$T = \frac{{G\theta wt^{3} }}{3}\left[ {1 - \frac{192t}{{\uppi^{5} w}}\sum\limits_{n = 1,3,5 \ldots }^{\infty } {\frac{\tanh (n\uppi w/2t)}{{n^{5} }}} } \right].$$

(2)

Dividing Eq. (1) by Eq. (2) yields the maximum shearing stress as a function of the torque in the form

$$\tau_{{\max} } = \frac{3T}{{wt^{2} }}Q,$$

(3)

where Q (denoted as Q _s for this infinite series expression) is a constant whose value depends only on ratio t/w (t/w ≤ 1):

$$Q_{s} = \frac{{1 - \frac{8}{{\uppi {}^{2}}}\sum\limits_{n = 1,3,5 \ldots }^{\infty } {\frac{1}{{n^{2} \cosh (n\uppi w/2t)}}} }}{{1 - \frac{192}{{\uppi {}^{5}}}\frac{t}{w}\sum\limits_{n = 1,3,5 \ldots }^{\infty } {\frac{\tanh (n\uppi w/2t)}{{n^{5} }}} }}.$$

(4)

For narrow rectangular sections (t/w<0.1), Q approximately equals 1.

A polynomial fit of Q _s as a function of t/w given in Eq. (4) [24]:

$$Q_{\text{r}} = 1 + 0.6095\frac{t}{w} + 0.8865\frac{{t^{2} }}{{w^{2} }} - 1.8023\frac{{t^{3} }}{{w^{3} }} + 0.91\frac{{t^{4} }}{{w^{4} }}.$$

(5)

Substituting into Eq. (3) results in the following expression for $\tau_{\text{max} }$:

$$\tau_{{\max}} = \frac{3T}{{wt^{2} }}\left( {1 + 0.6095\frac{t}{w} + 0.8865\frac{{t^{2} }}{{w^{2} }} - 1.8023\frac{{t^{3} }}{{w^{3} }} + 0.91\frac{{t^{4} }}{{w^{4} }}} \right).$$

(6)

Pilkey [25] approximated 1/Q using the following polynomial expression:

$$\frac{ 1}{{Q_{\text{p}} }} = 1 - 0.63\frac{t}{w} + 0.25\frac{{t^{2} }}{{w^{2} }},$$

(7)

which leads to another expression for $\tau_{\text{max} }$:

$$\tau_{{\max} } = \frac{3T}{{wt^{2} }}\frac{ 1}{{\left( {1 - 0.63\frac{t}{w} + 0.25\frac{{t^{2} }}{{w^{2} }}} \right)}}.$$

(8)

The following expression employing a linear approximation of Q [26] is often used by researchers [10]:

$$Q_{l} = 1 + 0.6\frac{t}{w},$$

(9)

and we have

$$\tau_{{\max} } = \frac{T}{{w^{ 2} t^{2} }}\left( { 3w + 1.8t} \right) = \frac{3T}{{wt^{2} }}\left( {1 + 0.6\frac{t}{w}} \right),$$

(10)

which approximately equals the first two terms of Eq. (6).

Figure 2 compares Q, Q _r , Q _p and Q _l . By denoting the errors as:

Figure 2

Curves of different expressions for Q

$$E_{r} = \frac{{Q_{r} - Q_{s} }}{{Q_{s} }},$$

(11)

$$E_{p} = \frac{{Q_{p} - Q_{s} }}{{Q_{s} }},$$

(12)

$$E_{l} = \frac{{Q_{l} - Q_{s} }}{{Q_{s} }}.$$

(13)

Figure 3 shows that Q _r has an maximum error less than 0.3%, the maximum error of Q _p is less than 2%, while Q _l may result in an error up to 4.5%.

Figure 3

Error comparison of different expressions for Q

When using the expressions for $\tau_{\text{max} }$ given in Eqs. (1), (6), (8) and (10), one of the two dimensions of the cross-section must be defined as the wider of the two dimensions because they assume that w ≥ t. That is to say, if w (width) is no larger than t (thickness), this expression for $\tau_{\text{max} }$ needs to be changed by switching w and t. This situation might be troublesome and error-prone during design phase (we always do not know which one is bigger in advance).

3 General Equations with Symmetric Relation of t and w

A two step procedure was taken to obtain a symmetric and accurate expression for maximum torsion stress $\tau_{\text{max} }$.

First, a symmetric base expression is formulated by revising Eq. (9) as:

$$Q_{\text{c}} = 0.8 + 0.8\frac{t}{w},$$

(14)

which yields another equation for $\tau_{\text{max} }$:

$$\tau_{{\max} } = \frac{3T}{{wt^{2} }}\left( {0.8 + 0.8\frac{t}{w}} \right).$$

(15)

As shown in Figure 3, this equation may result in an error up to 20% (the error is defined as $E_{c} = (Q_{c} - Q_{s} )/Q_{s}$).

A compensation function, f(v), can be used to reduce the error, where

$$\tau_{{\max} } = \frac{2.4T}{{w^{2} t^{2} }}\left( {w + t} \right)f\left( v \right).$$

(16)

The variable in the compensation function is chosen to be

$$v = \left| {\log \frac{t}{w}} \right|,$$

(17)

because it is symmetric with respect to t and w due to |log(t/w)| = |log(w/t)|.

By fitting the results of E _c using a quadratic/quadratic rational function, the following compensation function

$$f\left( v \right) = \frac{{ 1. 2 7 1v^{ 2} + 0. 2 8 2 9v + 0. 0 4 9 8}}{{v^{ 2} + 0. 2 7v + 0. 0 4 9 6}}$$

(18)

is found to reduce the maximum error to 0.4% (see Figure 4). This leads to the following expression for $\tau_{{\max} }$:

Figure 4

Errors of using compensation function f (v) and f′(v)

$$\tau_{{\max} } = \frac{{T\left( { 2. 4w + 2.4t} \right)}}{{w^{2} t^{2} }}\frac{{ 1. 2 7 1v^{ 2} + 0. 2 8 2 9v + 0. 0 4 9 8}}{{v^{ 2} + 0. 2 7v + 0. 0 4 9 6}}.$$

(19)

Even better fitting results may be achieved with a 4th degree polynomial numerator and 4th degree polynomial denominator:

$$f^{\prime}\left( v \right) = \frac{{ 1. 2 3v^{ 4} - 0. 7 1 6 7v^{ 3} + 0. 5 8 9 4v^{ 2} - 0. 2 3 7 6v + 0. 2 2 9 5}}{{v^{ 4} - 0. 6 7 7 8v^{ 3} + 0. 6 9 4 1v^{ 2} - 0. 3 4 2 7v + 0. 2 2 9 3}}.$$

(20)

This compensation function reduces the maximum error to 0.07% (see Figure 4) and leads to the following expression for $\tau_{\text{max} }$:

$$\tau_{\text{max} } = \frac{{T\left( {w + t} \right)}}{{w^{2} t^{2} }}\frac{{ 2.952v^{ 4} { - 1} . 7 2v^{ 3} + 1.4146v^{ 2} { - 0} . 5 7 0 2v + 0. 5 5 0 8}}{{v^{ 4} { - 0} . 6 7 7 8v^{ 3} + 0. 6 9 4 1v^{ 2} { - 0} . 3 4 2 7v + 0. 2 2 9 3}}$$

(21)

In the following section, we will demonstrate the use of this general (width-thickness independent) equation for designing lamina emergent torsional (LET) joints.

4 Outside LET Joint: Design Considerations

Figure 5 shows an outside LET joint fabricated from a planar sheet of material whose modulus of elasticity and modulus of rigidity are E and G, respectively. The joint consists of two parallel sets of torsional segments connected by connecting segments in bending. The geometric parameters of the torsional segments are shown in Figure 5, and I = wt³/12 represents the area moment of inertia in the sheet plane.

Figure 5

a Outside LET, b its torsional deflection, and c unconstrained parasitic deflection

4.1 Torsional Segment: Stiffness vs. Stress

The stiffness of each torsional segment is expressed using the symmetric equation obtained in Ref. [17] as

$$K_{t} = \frac{{2Gt^{3} w^{3} \left( {1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17} \right)}}{{7L\left( {t^{2} + w^{2} } \right)\left( {\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1} \right)}} = \frac{G}{L} \cdot k_{c} ,$$

(22)

where k _c is the torsional stiffness constant [23] that is solely determined by the dimensions of the cross-section:

$$k_{c} = \frac{{2t^{3} w^{3} }}{{7\left( {t^{2} + w^{2} } \right)}} \cdot \frac{{1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17}}{{\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1}}.$$

(23)

As to the maximum stress, we define a stress constant as

$$\tau_{c} = \frac{{\tau_{{\max} } }}{T}{\kern 1pt} = \frac{ 2. 4w + 2.4t}{{w^{2} t^{2} }} \cdot \frac{{ 1. 2 7 1v^{ 2} + 0. 2 8 2 9v + 0. 0 4 9 8}}{{v^{ 2} + 0. 2 7v + 0. 0 4 9 6}},$$

(24)

which is also solely determined by the dimensions of the cross-section.

If using meters as the length dimensions (e.g., t and w), the units for k _c and $\tau_{c}$ are m⁴ and m⁻³, respectively. To compare the twist performance of rectangular cross-sections with different aspect ratios (i.e., t/w), we define the following non-dimensionalized term called the torsional aspect-ratio factor:

$$\begin{aligned} T_{c} = k_{c}^{3} \tau_{c}^{4} &= \frac{{0.7738tw\left( {t + w} \right)^{4} }}{{\left( {t^{2} + w^{2} } \right)^{3} }}{\kern 1pt} {\kern 1pt} \cdot \\ & \quad \frac{{\left( {1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17} \right)^{3} }}{{\left( {\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1} \right)^{3} }} \cdot \\ & \quad \frac{{\left( { 1. 2 7 1v^{ 2} + 0. 2 8 2 9v + 0. 0 4 9 8} \right)^{4} }}{{\left( {v^{ 2} + 0. 2 7v + 0. 0 4 9 6} \right)^{4} }}, \\ \end{aligned}$$

(25)

T _c is also symmetric with respect to t and w because it is only determined by the width-thickness ratio of the cross-section. Figure 6 plots T _c as a function of t/w. For a LET joint design, we always expect k _c to be small so as to lower the torsional stiffness in the desired direction for reducing actuation effort, and $\tau_{c}$ to be small to increase the maximum allowed rotation angle. T _c reaches its maximum, 2.2747, at t/w = 0.658 and t/w = 1.52. T _c reaches its local minimum, 1.4942, at t/w = 1 (square cross-section). T _c is also smaller than 1.4942 for t/w < 0.35 and t/w > 2.86. In general, cross-sections of 0.35 <  t/w < 1 and 1 < t/w < 2.86 are suggested to be avoided for the torsional segments in LET joints.

Figure 6

T _c vs. t/w

It is obvious that increasing the length can significantly decrease F, which is preferred if the space is allowed. If L is fixed, there is a local minimum at t/w = 1. However, there are two maxima at t/w = 1.5 and t/w = 0.67 (these geometries are feasible and preferred both for design and manufacture), which are suggested to be avoided in design.

4.2 LET Joint: Torsional Stiffness vs. Compressive/Tensile Stiffness

Ideally a LET joint would have low torsional stiffness while maintaining high stiffness in the other directions [10]. However, a LET joint is susceptible to undesired motion when compressive/tensile load is applied because the torsional segments are placed into bending, as illustrated in Figure 5(c).

When an outside LET joint is subject to a compressive/tensile load, the torsional segments can be treated as fixed-guided segments with the axial force P _o = 0 (because the connecting segments is floating) as illustrated in Figure 5. The transverse force F _o , can be expressed by the Bi-BCM (the first bending mode) [27] as:

$$f_{o} = \frac{{9p_{o}^{2} + 312p_{o} + 720}}{{16p_{o} + 480}}y_{o} = \frac{{3y_{o} }}{2},$$

(26)

in which f _o and p _o are the normalized axial and transverse forces, respectively, defined as

$$p_{o} = \frac{{P_{o} L^{2} }}{4EI} = 0, \quad f_{o} = \frac{{F_{o} L^{2} }}{4EI},$$

(27)

and y _o is the normalized displacement defined as

$$y_{o} = \frac{{2Y_{o} }}{L}.$$

(28)

By substituting Eqs. (27) and (28) into Eq. (26), the parasitic compression/extension stiffness of the whole LET joint can be obtained as:

$$K_{\Delta y} = \frac{{F_{y} }}{{\Delta_{y} }} = \frac{{2F_{o} }}{{2Y_{o} }} = \frac{12EI}{{L^{3} }} = \frac{{Ewt^{3} }}{{L^{3} }}.$$

(29)

The equivalent torsional compliance of the outside LET joint along the x-axis [10] is:

$$K_{\alpha x} = \frac{{k_{1} k_{2} k_{5} }}{{k_{1} k_{2} + k_{1} k_{5} + k_{2} k_{5} }} + \frac{{k_{3} k_{4} k_{6} }}{{k_{3} k_{4} + k_{3} k_{6} + k_{4} k_{6} }},$$

(30)

where the stiffnesses of the torsional segments are

$$k_{1} = k_{2} = k_{3} = k_{4} = K_{t} ,$$

(31)

while the connecting segments in bending can be considered stiff because they are short, i.e.,

$$k_{5} = k_{6} = \infty .$$

(32)

Then we have

$$K_{\alpha x} = \frac{{2Gt^{3} w^{3} \left( {1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17} \right)}}{{7L\left( {t^{2} + w^{2} } \right)\left( {\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1} \right)}}.$$

(33)

For the purpose of comparing the compression/extension stiffness to the torsional stiffness of LET joints for different design parameters, the following stiffness ratio term is defined:

$$\begin{aligned} R_{s} &= \frac{{K_{\Delta y} }}{{K_{\alpha x} }} = \frac{7}{2} \cdot \frac{E}{G} \cdot \frac{1}{{L^{2} }} \cdot \\ & \quad \frac{{\frac{{t^{4} }}{{w^{4} }} + 2.609\frac{{t^{3} }}{{w^{3} }} + 2\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1}}{{1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17}}. \\ \end{aligned}$$

(34)

A well-designed LET joint should have large R _s so as to achieve low torsional stiffness but provide good constraint in the compression/extension direction. The right side of Eq. (34) is divided into four terms, among which the first is a constant, the second is material-related term, the third is inversely proportional to the square of the length of the torsional segments, while the last monotonically increases with t/w. For isotropic materials, the Poisson’s ratio $\nu$ ranges from 0.2 to 0.5 [28], thus the range of the second term E/G = 2(1+$\nu$) is from 2.4 to 3. However, one can use composite materials to obtain larger E/G (e.g., multi-layer structures). The third term indicates that R _s can be significantly increased by decreasing the length of the torsional segments. However, decreasing the length of the torsional segments will increase the torsional stiffness (as indicated by Eq. (22)) and further decrease rotational range of the joint. The fourth term is plotted in Figure 7 as a function of w/t, which shows that R _s dramatically increases as w/t increases. Considering T _c shown in Figure 6, w/t > 2.86 is suggested if parasitic motion along the compression/extension direction is required to be small.

Figure 7

R _s as a function of t/w

4.3 Design Examples

This section provides a few LET joint designs to demonstrate the design considerations in the previous section. The parameters of 5 designs are listed in Table 1. These designs have equal or almost equal cross-section area. Figure 6 marks these designs on the T _c curve.

Table 1 A few LET joint designs

Considering

$$T = K_{\alpha x} \alpha ,$$

(35)

where α is the torsional deflection angle of the joint, substituting it into Eq. (19) yield

$$\begin{aligned} \tau_{\text{max} } & = \frac{0.689G}{L} \cdot \frac{{tw\left( {w + t} \right)}}{{w^{2} + t^{2} }} \cdot \frac{{ 1. 2 7 1v^{2} + 0.2829v + 0.0498}}{{v^{2} + 0.27v + 0.0496}} \cdot \\ & \begin{array}{*{20}c} {} & {} \\ \end{array} \frac{{1.17\frac{{t^{2} }}{{w^{2} }} + 2.191\frac{t}{w} + 1.17}}{{\frac{{t^{2} }}{{w^{2} }} + 2.609\frac{t}{w} + 1}} \cdot \alpha . \\ \end{aligned}$$

(36)

For the purpose of comparison, we calculated $\tau_{\text{max} }$ using Eq. (36) by assuming $\alpha = 0.1$ rad.

The results of $K_{\alpha x}$, $K_{\Delta y}$ and $\tau_{\text{max} }$ for the 5 designs are listed in Table 2. Among the 5 joints, Design 3 is the best design, with the smallest $K_{\alpha x}$, the largest $K_{\Delta y}$, and the lowest $\tau_{\text{max} }$. Design 2 has the same values for $K_{\alpha x}$ and $\tau_{\text{max} }$ as Design 3, but has the lowest compression/extension stiffness. Design 1 has modest values for $K_{\Delta y}$ and $\tau_{\text{max} }$ but the largest $K_{\alpha x}$. Design 4 and Design 5 are the worst designs because they have the largest $K_{\alpha x}$ and $\tau_{\text{max} }$.

Table 2 Calculated results

5 Conclusions

This work presented closed-form symmetric equations for calculating maximum torsion stress of a rectangular cross-section beam. Together with the symmetric equations in our previous work [17], these equations are independent of the relative magnitude of the two dimensions (i.e., the thickness and the width) of the cross-section, thus are more convenient and effective for designing and optimizing torsional beams in compliant mechanisms.

These equations were utilized to analyze outside lamina emergent torsional joints and some useful design insights were obtained and described.