SSP - JOURNAL OF CIVIL ENGINEERING Vol.
18, Issue 1, 2023
DOI: 10.2478/sspjce-2023-0011
Harmonic Response of CFRP Tensegrity System in a Suspen-
dome
IfeOlorun Olofin1,2*, Ronggui Liu 2
1
Federal University Oye-Ekiti, Nigeria
Faculty of Engineering, Department of Civil Engineering
2
Jiangsu University, China
School of Civil Engineering and Mechanics
*e-mail: epher2002@yahoo.com
Abstract
Wind-induced excitation causes structure to vibrate which leads to instability. This paper focuses on the
performance of CFRP tensegrity system in a suspen-dome subjected to wind load by assuming such load as
harmonic. A comparison is made with the traditional steel tensegrity system, in order to justify the integrity of
CFRP cable application in a suspen-dome system. A finite element software, namely ANSYS, was implemented
for the simulation by analyzing a physical model of 4 m span and 0.4 m height. Results show that CFRP tensegrity
system has similar performance as steel tensegrity system and can be used as a substitute for steel.
Keywords: suspen dome, tensegrity system, CFRP cable, steel cable, harmonic load
1 Introduction
Researchers have been critical about the dynamic behavior of structural systems in order to
achieve stability and safety [1-4]. Wind loading is usually dominant in structural loading on
roofs of large buildings. It is paramount to fully understand the behavior pattern of the structure
in respect to dynamic effect such as wind load. This kind of load increases the amplitude of the
structures’ vibration along the wind direction.
Wind is a random and dynamic phenomenon that occurs in space and time in structural systems.
The importance of dynamic resonant response to wind for large span roofs is dependent on the
natural frequency of vibration which is in turn dependent on the mass, stiffness properties and
the damping [5]. However, the use of stiffening cables often increases the stiffness of a system
sufficiently to reduce resonant contribution to a minimal proportion [6]. A structural system
such as a suspen-dome [7] must have sufficient strength and adequate stiffness to resist wind-
induced forces. The response of the geometrical structure under wind loading is crucial.
Therefore, the need to understand such performance is required. The structural performance of
wind load can be realistic by assuming such loads as harmonic [8]. Sustained cyclic load
produces harmonic response in a structure. When natural frequency of a structure coincides
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Olofin IO., and Liu R.
with the forcing excitation frequency, resonance occur causing maximum displacement. The
harmonic regime is a representative of many excitation cases encountered in engineering
practice, such as vibration induced by imbalance, and torsional vibration in engines [9].
From history, damage to structures by severe winds has been a fact of life and solution to resist
wind forces is required. With the upcoming development in the utilization of CFRP cable in
space spatial structures [10-14], it is salient to understand the behavior pattern of such structure
induced by wind.
In this paper, harmonic response technique was adopted on ANSYS software for a suspen-dome
prototype constructed with CFRP as the tensegrity system and comparison was made with steel
tensegrity system based on natural frequency and maximum displacement, in order to justify
the implementation of CFRP cables in a suspen-dome.
2 Literature review
In this section, a brief explanation of wind loads on large roofs, harmonic response and theory
of harmonics are presented in some detail.
2.1 Wind flow over large span roofs
Wind is the terminology used to describe air in motion and it is usually applied to the natural
horizontal motion of the atmosphere. A constant flow of wind can suddenly gust to a rush of
air. This sudden variation in wind speed plays an important part in determining a structure’s
oscillation. The flow of wind is not steady and fluctuates in a random fashion. Because of this,
wind loads imposed on buildings are studied statistically [5]. Wind load effect on a large roof
has some significant difference in comparison with its effect on smaller roof; the resonant
effects, although not dominant can be significant. Upward and downward external pressures are
also significant. Domed structures are sensitive to wind load distribution hence the possibility
of critical unbalanced pressure distribution must be considered [6]. When considering dynamic
response of a structure to wind, distinguishing between the resonant response or near the natural
frequency of the structure is necessary. The fluctuating responses at frequencies below the first
or lowest frequency are usually great contributors [6]. However, because of the huge fluctuating
components in wind loading on large roofs, the statistical correlation between pressure
separated by a sizable distance is nanoscopic.
2.2 Harmonic Assessment
The response of the single degree of freedom systems to harmonic excitation is a consequential
subject matter in structural dynamics. Apart from the fact that excitations are encountered in
engineering system, for example force caused by unstable revolving machinery, the reaction of
structures is understood based on harmonic excitation. This provides an insight to what degree
the system will respond to any other type of force [13]. Forced harmonic vibration theory has a
functional implementation in earthquake engineering because of its structural dynamics.
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2.2.1 Theorem
Within a structural system any sustained cyclic load will produce harmonic response[15-16] as
illustrated in Figure 1.
Figure 1: A system subjected to harmonic excitation [16]
A harmonic force is represented as [15]:
𝑃(𝑡) = 𝑃0 𝑠𝑖𝑛 𝜔 (𝑡) (1)
where Po is the amplitude, ω is the exciting frequency and t is the period of excitation.
The governing differential equation for forced harmonics vibration for damped system is given
as [17]:
𝑀𝑈̈ + 𝐶𝑈̇ + 𝐾𝑈 = 𝑃0 𝑠𝑖𝑛 𝜔 (𝑡) (2)
Solving the Equation (2) for initial condition, 𝑈 = 𝑈(0)𝑎𝑛𝑑 𝑈̇ = 𝑈̇(0) and the complimentary
solution is the free vibration response given as:
𝑈𝑐 = 𝑒 −𝜉𝜔𝑛𝑡 (𝐴 𝑐𝑜𝑠 𝜔𝐷 𝑡 + 𝐵 𝑠𝑖𝑛 𝜔𝐷 𝑡) (3)
𝑃 𝜔
𝑈̇0 ( 0 )( )
𝑘 𝜔𝑛
𝐴 = 𝑈0 , 𝐵 = 𝜔 − ( 𝜔 2
) (4)
𝑛 (1−( ) )
𝜔𝑛
where
𝜔𝐷 = 𝜔𝑛 √1 − 2𝜉, (5)
The solution for Equation (3) is given as:
𝑈𝑃 (𝑡) = 𝐶 𝑠𝑖𝑛 𝜔 𝑡 + 𝐷 𝑐𝑜𝑠 𝜔 𝑡 (6)
where
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Olofin IO., and Liu R.
𝑃0 (1 − 𝜉 2 )
𝐶 = ( )[ ] (7)
𝑘 (1 − 𝜉 2 ) + 2𝜉(𝑟)2
−𝑃0 2𝜉𝑟
𝐷= [(1−𝜉2)2+(2𝜉𝑟)2 ] (8)
𝑘
The total response is given as:
𝑈𝑡 = 𝑒 −𝜉𝜔𝑛𝑡 (𝐴 𝑐𝑜𝑠 𝜔𝐷 𝑡 + 𝐵 𝑠𝑖𝑛 𝜔𝐷 𝑡) + 𝐶 𝑠𝑖𝑛 𝜔 𝑡 + 𝐷 𝑐𝑜𝑠 𝜔 𝑡 (9)
3 Model Verification
A small-scaled suspen-dome was designed according to Chinese specification [18-20] with a
span of 4 meters and a rise of 0.4 meters, as shown in Figure 2. The tensegrity system is made
of the struts, radial and hoop cables. The tension members were constructed with CFRP.
Figure 2: Model and reticulated single layer plan
Figure 3: Construction and deflection dial indicator layout
With the successful construction of the prototype as illustrated in Figure 3, experimental static
performance was compared with numerical analysis via ANSYS software to validate the
application of CFRP cables as a tensegrity system in a suspen-dome. Contrast between
experimental and numerical findings for CFRP and steel cables was established. Figure 4(a)
illustrates the maximum stresses generated on the single reticulated layer at high imposed load.
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Similarly, Figure 4(b) shows the maximum displacement at specified nodes obtained for
analysis.
(a) (b)
Figure 4: Contrast of maximum values for stress and displacement
In addition, forces generated internally within the cables had similar values. It can be concluded
that the digital analysis using ANSYS achieved a similar trend with the experimental findings
with an excellent accuracy [21]. With positive findings obtained for the static investigation, it
is paramount to further investigate the integrity of the structure, especially in dynamics which
is a common practice. Although dynamic analysis is computationally extensive, complex and
expensive compared with static analysis, numerical simulations are essential to observe the
structure’s performance before experimentation on the structure, in order to establish
guidelines.
4 Numerical Technique
This section explains in detail the Finite Element Analysis (FEA), material properties utilized,
and loading conditions.
4.1 Finite Element Method
With the vast development of software, the calculation of the behavior of suspen-dome
application can be realistic and economical. The ability of CFRP structure to withstand critical
loading can be evaluated by computational method. The two materials (steel and CFRP cables)
are subjected to finite element analysis. Basic assumptions for the structure include cables are
elastic and the tensegrity members are pin-jointed. The complexity of the suspen-dome structure
allows results for three-dimensional FEM (Finite Element Method) analysis to be more
comprehensive and reliable than results of empirical formula. The analysis was conducted with
ANSYS 10 finite element software package [22]. The algorithm for the analysis is illustrated in
Figure 5.
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Figure 5: Simulation Algorithm
4.2 Model and Mesh
The finite element digital model is illustrated in Figure 6. Fixed boundary condition was
assigned at the edge of the model. Beam 188 was incorporated for the single reticulated layer
while link 10 was incorporated for the cable system. A mesh of 1mm x 1mm was utilized. CFRP
cable was modeled as an anisotropic linear-elastic material and an isotropic elastic-plastic
material was assumed for the mechanical behavior of steel cables.
Figure 6: Finite element model
4.3 Material Properties
The mechanical properties of the materials used for the simulation are shown in Table 1.
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Table 1: Material properties for the suspen-dome members
Member Elastic Density (kg/m3)
Modulus (MPa)
Steel Tensegrity 1.8x105 78.5
System
CFRP 1.6x105 16.0
Tensegrity
system
Single 2.05x105 78.5
reticulated layer
Material properties used in the analysis are subjected to certain loads which have an impact on
the structural behavior of the suspen-dome system.
4.3.1 Dimensions
The dimension for the tension members is illustrated in Table 2.
Table 2: Illustrates the dimensions of the tension members
Cable ϕ Radial (mm) ϕ Hoop (mm)
Steel 8 10
CFRP [21] 5 7.9
The length for the hoop cable was 2.6 m and radial cable was 44 mm.
4.4 Loading
Modal analysis is the most fundamental of all types of dynamic analysis; it allows a given design
to avoid resonant vibration and gives the engineer an idea of how the design will respond to
dynamic load. The wind load coefficient was computed based on the codal provisions from the
relevant standard GB50009-2012 [18]. The wind load on the structure was assumed to be
sinusoidal and applied at nodes. All loads are assumed to have similar frequency and the
maximum amplitude is identified by the static load intensity. Damping of 5% is considered for
the models through the dynamic analysis.
5 Results and Discussion
Mode superposition method was employed for the analysis of harmonic response. The system
frequency response domain is often utilized for experimental identification for a dynamic
system. Prior to the method, modal analysis was carried out as a pre-requisite by using reduced
method for mode extraction. Secondly, the system was assumed to be excited through external
loads, namely loads that had time variation. An assumption made was that forces applied for a
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time period were enough for transients to vanish [13], so that the only response is a forced
motion.
5.1 Modal Analysis
The structure’s natural frequencies were obtained because when the structure excites at one of
these frequencies, and the resonance occurs, this prevents the structure from fulfilling its desired
function. Hence, resonance should be avoided because vibration generates dynamic stress and
strain which causes fatigue and failure of a structure [23].
Modal analysis was computed and the natural frequencies and modal periods evaluated. Six
mode values were extracted as illustrated in Figures 7 and 8.
Figure 7: Natural frequency response (red = steel, purple = CFRP)
Figure 8: Modal period response (green = steel, blue = CFRP)
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The displacement of the whole structure with steel tendons was 0.021 mm and that of CFRP
was 0.029 mm at first modal period. The value for CFRP was 0.118 sec which is not a high
value indicating that the whole structure is stiff enough. From the first six modes, it can be
observed that the integrity of the roof with CFRP cables is very good and has similar behavior
as that of steel cables. Similarly, Figures 9 and 10 illustrate the sixth mode for the structural
system; both CFRP (Carbon Fiber Reinforced Polymer) and steel had similar structural shape.
Figure 9: Sixth mode for CFRP
Figure 10: Sixth mode for steel
Frequencies obtained from the analysis were relatively low. The fundamental frequency of
CFRP tensegrity system from the modal analysis was approximately 15.404 Hz, as shown in
Figure 9, which is sufficiently similar to steel (Figure 10) with frequency of 17.329 Hz. The
difference was 12% compared to steel, which still falls into the acceptable criteria within design.
This satisfies the assertion of Cheng and Lau [24] which states that the modal frequencies and
order don’t affect the motion of the cable. Substituting steel cable with CFRP has a stiffening
effect on the structure.
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5.2 Harmonic Response
The harmonic response was performed using mode superposition method on ANSYS. The
response was obtained from the modal analysis in previous section and by considering the center
node of the structural system to draw out conclusive findings for design process. The wind load
on the structure was assumed to be sinusoidal and applied at nodes. All loads are assumed to
have similar frequency and the maximum amplitude is identified by the static load intensity.
The first mode and lowest frequency value was used in performing the harmonic response
analysis from the modal solution because it is associated with wind and wave forces [6]. Due
to the symmetrical nature of the structure any node or element can be represented by one of the
typical nodes or elements. However, the node at the mid-section of the structure was considered
because the effect is concentrated at the central part of the single reticulated layer. Figures 11-
16 illustrate the typical response curves for the center node of the structure for both CFRP cables
and steel cables at x, y, and z axis.
Figure 11: Typical response curve for the centre node (steel) x-axis
Figure 12: Typical response curve for the centre node (CFRP) x-axis
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Figure 13: Typical response curve for the centre node (steel) y-axis
Figure 14: Typical response curve for the centre node (CFRP) y-axis
It can be observed that the peak frequency occurred at 0.2 Hz. At the x-axis, the maximum
displacement was 0.016 mm for CFRP and 0.0135 mm for steel, at the y-axis, the maximum
displacement was 0.0115 mm for steel and 0.0225 mm for CFRP, and at the z-axis, the
maximum displacement for CFRP was 0.0128 mm and that of steel was 0.0087 mm. When one
cable is at resonance, the other is at a different frequency which acts to dampen the resonance
vibration, hence the total vibration amplitude is kept small for both steel and CFRP, thanks to
the nature of tensegrity systems. It was also observed that all amplitudes reduced as natural
frequencies increased. Internal forces in the cables and bars are not significant under vertical
frequency for both materials.
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Figure 15: Typical response curve for the centre node (steel) z-axis
Figure 16: Typical response curve for the cenre node (CFRP) z-axis
6 Conclusion
Numerical analysis of the tensegrity system model was proposed to predict the dynamic
mechanical response of the suspen-dome. From the simulated results, the following conclusions
are drawn:
• CFRP is highly stable and has a good damping behavior.
• The vibration of CFRP cable is similar to that of steel, validating its exceptional
mechanical properties of high stiffness-to-weight ratio and less curvature under gravity
load. Such behavior resists vibration.
From a technical point of view, the harmonic response was assumed as wind induced response,
the behavioral pattern of the suspen-dome with CFRP tensegrity system correlated well with
that of steel system irrespective of the natural frequency difference. This is a realization that
CFRP cables can resist wind forces and the novel material is suitable for designing of suspen-
domes.
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