Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms
Chen and Howell Chin. J. Mech. Eng.
Symmetric Equations for Evaluating Maximum Torsion Stress of Rectangular Beams in Compliant Mechanisms
Gui‑Min Chen 0 1 2
Larry L. Howell 0 1
0 Department of Mechanical Engineering, Brigham Young University , Provo, UT 84602 , USA
1 Department of Mechanical Engi‐ neering, Brigham Young University , Provo, UT 84602 , USA
2 State Key Laboratory for Manufacturing Systems Engineering, Xi'an Jiaotong University , Xi'an 710049, Shaanxi , China
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 crosssection (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.
Compliant mechanism; Maximum torsion stress; Rectangular beam; Lamina emergent joint
A compliant mechanism achieves its mobility through
the deflections of its compliant elements [
]. In most
compliant mechanisms, the compliant elements are
designed to produce motion through bending
]. 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
], a revolute joint comprised of two crossed
torsion plates which shows a good performance in
resisting axis drift was presented [
], torsional micromirrors
were proposed for optical switches and optical displays
], a torsional micro-resonator was fabricated for mass
sensing , 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 [
and were employed in precision adjustment mechanisms
], torsion hinges were proposed as surrogate folds in
origami-based engineering design [
], and torsional
beams were successfully used for achieving static
balancing of an inverted pendulum .
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 [
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
], 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 [
] and spatial deflections modeling [
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 [
], 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
2 Various Equations for Calculating Maximum
For torsion of rectangular sections, the stress at each
corner is zero, and the maximum shearing stress τmax 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 [
8 ∞ 1
τmax = Gθ t1 − π2 n=1,3,5... n2 cosh(nπw/2t) , (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 [
Gθ3wt3 1 − 1π952wt
n5 . (2)
Dividing Eq. (1) by Eq. (2) yields the maximum
shearing stress as a function of the torque in the form
τmax = wt2 Q,
where Q (denoted as Qs for this infinite series expression)
is a constant whose value depends only on ratio t/w (t/w
1 − π82
1 − 1π952 wt
n=1,3,5... n2 cosh(nπw/2t)
For narrow rectangular sections (t/w<0.1), Q
approximately equals 1.
A polynomial fit of Qs as a function of t/w given in Eq.
which leads to another expression for τmax:
1 − 0.63 wt + 0.25 wt22
The following expression employing a linear
approximation of Q [
] is often used by researchers [
Ql = 1 + 0.6 ,
and we have
τmax = w2t2 (3w + 1.8t) = wt2
1 + 0.6
which approximately equals the first two terms of Eq. (6).
because it is symmetric with respect to t and w due to
|log(t/w)| = |log(w/t)|.
By fitting the results of Ec using a quadratic/quadratic
rational function, the following compensation function
is found to reduce the maximum error to 0.4% (see
Figure 4). This leads to the following expression forτmax:
This compensation function reduces the maximum
error to 0.07% (see Figure 4) and leads to the following
expression for τmax:
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.
4.1 Torsional Segment: Stiffness vs. Stress
The stiffness of each torsional segment is expressed using
the symmetric equation obtained in Ref. [
wt22 + 2.609 wt + 1
= GL · kc, (22)
where kc is the torsional stiffness constant [
] that is
solely determined by the dimensions of the cross-section:
7 t2 + w2
As to the maximum stress, we define a stress constant as
which is also solely determined by the dimensions of the
If using meters as the length dimensions (e.g., t and w),
the units for kc and τc 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
Tc = kc3τc4 = 0.7738tw(t + w)4 ·
t2 + w2 3
expressed by the Bi-BCM (the first bending mode) [
in which fo and po are the normalized axial and transverse
forces, respectively, defined as
and yo is the normalized displacement defined as
Tc 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 Tc as a function of t/w. For
a LET joint design, we always expect kc to be small so as
to lower the torsional stiffness in the desired direction for
reducing actuation effort, and τc to be small to increase
the maximum allowed rotation angle. Tc reaches its
maximum, 2.2747, at t/w = 0.658 and t/w = 1.52. Tc reaches
its local minimum, 1.4942, at t/w = 1 (square
cross-section). Tc 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.
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
4.2 LET Joint: Torsional Stiffness vs. Compressive/Tensile
Ideally a LET joint would have low torsional stiffness
while maintaining high stiffness in the other directions
]. 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 Po = 0
(because the connecting segments is floating) as
illustrated in Figure 5. The transverse force Fo, can be
A well-designed LET joint should have large Rs so as
to achieve low torsional stiffness but provide good
constraint in the compression/extension direction. The right
By substituting Eqs. (27) and (28) into Eq. (26), the
parasitic compression/extension stiffness of the whole LET
joint can be obtained as:
K y = Fyy = 22YFoo = 12LE3I = ELw3t3 .
The equivalent torsional compliance of the outside LET
joint along the x-axis [
k1k2 + k1k5 + k2k5
k3k4 + k3k6 + k4k6
where the stiffnesses of the torsional segments are
k1 = k2 = k3 = k4 = Kt ,
while the connecting segments in bending can be
considered stiff because they are short, i.e.,
k5 = k6 = ∞.
Then we have
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 ν ranges from 0.2 to 0.5 [
], thus the range
of the second term E/G = 2(1+ν) 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 Rs 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 Rs dramatically increases as w/t increases.
Considering Tc shown in Figure 6, w/t > 2.86 is suggested if
parasitic motion along the compression/extension direction is
required to be small.
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
Figure 7 Rs as a function of t/w
For the purpose of comparison, we calculated τmax
using Eq. (36) by assuming α = 0.1 rad.
The results of Kαx, K y and τmax for the 5 designs are
listed in Table 2. Among the 5 joints, Design 3 is the best
design, with the smallest Kαx, the largest K y, and the
lowest τmax. Design 2 has the same values for Kαx and
τmax as Design 3, but has the lowest
compression/extension stiffness. Design 1 has modest values for K y and
τmax but the largest Kαx. Design 4 and Design 5 are the
worst designs because they have the largest Kαx and τmax.
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 [
], 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.
GC and LLH conceived the idea, GC carried out the calculation, and GC
and LLH drafted the manuscript. All authors read and approved the final
Gui‑Min Chen is a professor at School of Mechanical Engineering, Xi’an Jiaotong
University, China. He received his PhD, MS and BS degrees from Xidian
University, China. He was a visiting professor at Brigham Young University, US. His
research interests include compliant mechanisms and their applications. He
serves as an associate editor for the Journal of Mechanisms and Robotics.
Larry L. Howell is a professor and associate dean of the Department of
Mechanical Engineering, Brigham Young University, US, where he also holds a
University Professorship. Prior to joining BYU in 1994 he was a visiting profes‑
sor at Purdue University, a finite element analysis consultant for Engineering
Methods, and an engineer on the design of the YF‑22 (the prototype for
the U.S. Air Force F‑22). He received his PhD and MS degrees from Purdue
University and his BS from Brigham Young University. He is a licensed profes‑
sional engineer and the recipient of a National Science Foundation CAREER
Award, a Theodore von Kármán Fellowship, the BYU Technology Transfer
Award, the Maeser Research Award, several best paper awards, and the ASME
Mechanisms & Robotics Award. He is a Fellow of ASME, associate editor for
the Journal of Mechanisms and Robotics, past chair of the ASME Mechanisms
& Robotics Committee, past co‑ chair of the ASME International Design Engi‑
neering Technical Conferences, and a past Associate Editor for the Journal of
Mechanical Design. Prof. Howell’s technical publications and patents focus on
compliant mechanisms, including origami‑inspired mechanisms, microelec‑
tromechanical systems, and medical devices. He is the author of the book
Compliant Mechanisms published by John Wiley & Sons.
Supported by National Science Foundation Research of the United States
(Grant No. 1663345), National Natural Science Foundation of China (Grant No.
51675396), and Fundamental Research Fund for the Central Universities (Grant
The authors declare that they have no competing interests.
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