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5th International Congress on Computational Mechanics and Simulation, 10-13 December 2014, India�

NUMERICAL FORM-FINDING OF A TENSILE MEMBRANESTRUCTURE USING DYNAMIC RELAXATION METHOD

SUBHRAJIT DUTTA1 and SIDDHARTHA GHOSH2

1Department of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076,India.

E-mail: [email protected] of Civil Engineering, Indian Institute of Technology Bombay, Mumbai 400076,

India.E-mail: [email protected]

Abstract

Tension structures are increasingly in demand due to their ability to span large distances with

elegance and structural efficiency. Tensile membrane structures are designed to carry the

external actions through the in-plane membrane prestress and the anticlastic surface curvature.

The determination of optimal form of the structure which satisfies the constraints of both

architectural and mechanical requirements is a challenge for the designer. This paper is focused

on the form-finding analysis of a tensile membrane structure using dynamic relaxation with

kinetic damping. The dynamic relaxation method is a robust numerical scheme for solving

nonlinear structural mechanics problems. A nonlinear finite element analysis with large

displacement and small strain formulation is described here and the numerical implementation

of the form-finding analysis is illustrated.

Keywords: Form-finding; dynamic relaxation; anticlastic; prestress; kinetic damping, tensile

membrane.

Introduction

For the past few decades, fabric architecture is extensively used in permanent construction of

structural components. These structures are light weight and span large distances without any

intermediate support. Due to the membrane action of stresses in these structures, they are not

designed for flexure which enhances the structural efficiency. From an architectural viewpoint,

the free-form structure provides a large variety of flexible shapes from outside and form of the

space from inside. In the initial stage of design, the geometry of the structure is not known apriori, and a form-finding analysis is performed to obtain an optimal shape. Form-finding is

a crucial step in the design process. For a particular combination of initial prestress and the

type and configuration of boundary supports, a unique shape of the structure is obtained. The

obtained anticlastic form coupled with the desired surface prestress resists the external wind

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c©2014 ICCMS Organisers. Published by Research Publishing. All rights reserved.ISBN:978-981-09-1139-3 || doi:10.3850/978-981-09-1139-3 371

and snow loads that the structure is typically subjected to. Physical models developed for form-

finding have their drawbacks and thus robust computational tools must be developed in order

to carry out the membrane structural analysis. This paper presents a combined computational

tool involving nonlinear finite element formulation along with dynamic relation algorithm for

the analysis of tensile membrane structures.

Analysis methodology

The equilibrium shape of the membrane structure is determined based on the initial prestress

applied on the membrane. In practice, the initial prestress value is taken to be the target

prestress (σd) which is desired to be obtained after the form-finding process. Thus the problem

of form-finding can be viewed as an initial equilibrium problem for a given set of input

parameters. For a to-be-designed membrane structure, the basic parameters of interest are: (a)

surface topology, (b) surface geometry, (c) geometry boundary condition and (d) initial stress

distribution 2. Among these parameters, the input parameters for analysis are surface topology,

geometry boundary condition and initial prestress. In this analysis, initially a trial surface

geometry of the structure is assumed, which is not in equilibrium condition. The geometry

boundary conditions are specified for the analysis and constraints are also introduced on

symmetry and anti-symmetry requirements for the simplicity of analysis. Membrane structures

exhibit high geometric nonlinearity and hence proper element formulation is necessary to

simulate the nonlinear behaviour. The 3-noded constant strain triangular (CST) element is

chosen for analysis in this case since the CST element is based on nonlinear pseudo-cable

analogy with linear strain function and small strain assumption. Many researchers have

worked on computational form-finding techniques and the most popular ones are: (a) Transient

stiffness method, (b) Force density method, and (c) Dynamic relaxation method. These form-

finding analysis methods for membrane structures are compared in past research studies 3. The

dynamic relaxation method was introduced by Day (1962) 4. Dynamic relaxation is regarded

as a robust technique for form-finding analysis. Programs developed for designing tensile

fabrics in industries uses dynamic relaxation algorithm mostly 5. Dynamic relaxation method

along with kinetic damping is adopted for the present analysis. The CST element formulation

along with dynamic relaxation solution procedure provides a stable and efficient computation

tool for membrane analysis.

Finite element for analysis

The 3-noded plane constant strain triangular element has two translational degrees of freedom

(u, v) per node within a local coordinate system as shown in Figure 1(a). Figure 1(b) shows

the element with side L3 parallel to X-axis and having one degree of freedom θ per node in

the local X,Y coordinate system and the properties of the element are described with respect

to this reference coordinate. The element sides (i = 1, 2, 3) and angles (θi = 1, 2, 3) are

numbered in counter-clockwise manner. The local Z coordinate direction is defined normal to

the surface. If the element side strain is denoted by εi, (i = 1, 2, 3) and the orthogonal strain

as {εx, εy, γxy} then from the assumption of small strain model, we can write,

εi = εx cos2 θi + εy sin

2 θi + γxy cos θi sin θi (1)

5th International Congress on Computational Mechanics and Simulation 1285

The strain-displacement relationship matrix and the total stiffness matrix (elastic and geometric

stiffness) are formuated based on the small strain large displacement model. The transfor-

mation of matrices needs to be performed from the local element coordinates to the global

coordinates. Finally the nodal displacements and velocities are obtained due to the out-of-

balance force (i.e. initial prestress in the case of form-finding) using the dynamic relaxation

method.

1

X

Y

X

3

2

Y

12

θ2

θ3

θ1

L3

L1L2

u1

v1 u2

v2

u3

v3

3

Fig. 1. (a) A general CST element; (b) Element with side L3 parallel to X-axis

Dynamic relaxation with kinetic damping

Dynamic relaxation is an explicit numerical technique for the solution of static structural

analysis problems. The method is based on the principle that any system which is subjected

to an out-of-balance force will come to rest only when the system is in equilibrium. The

structure is discretized and the mass (fictitious) of the continuum is assumed to be lumped at

nodal points. The system of lumped masses oscillates about the equilibrium position under the

influence of out-of-balance forces. With time, under the influence of damping, it comes to rest.

Since we are concerned about the final equilibrium state and not the history of motion of the

system, the mass and damping parameters are controlled artificially.

According to D’Alembert’s principle, the dynamic equilibrium of a system in motion takes

the form:

Pij = Kδij + Cδ̇ij +Mij δ̈ij (2)

Pij −Kδij = Cδ̇ij +Mij δ̈ij (3)

Rij = Mij δ̈ij + Cδ̇ij (4)

where, the subscript ij refers to the ith node in the jth direction, j can take the values 1 to 3

corresponding to the global coordinates {x, y, z} respectively. The remaining parameters used

in the Equations 2 and 4 are defined below:

Pij is the external load vector acing at the nodal points, this term includes the surface prestress

in form-finding analysis;

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Kij is the total nodal stiffness of the system;

C is the viscous damping coefficient, (C = 0 for the kinetic damping case);

Mij is the fictitious nodal mass, which is obtained from nodal stiffness;

Rij is the residual nodal force;

δ̈ij is the nodal acceleration;

δ̇ij is the nodal velocity; and

δij is the nodal displacement.

Dynamic relaxation with kinetic damping is a more stable and efficient numerical scheme as

compared to the viscous damping for analysis of tension structures 9. In this method, the

system is allowed free undamped vibration about the equilibrium position and the kinetic

energy is monitored. At the equilibrium position, the kinetic energy of the system is maximum.

Kinetic damping uses the principle of minimization of total energy (kinetic and potential) as

the system approaches to equilibrium configuration. The velocity of the system increases with

time and it is brought to zero whenever there is a peak in kinetic energy. Then, the iteration is

started again from the current configuration. With the viscous damping coefficient equal to be

zero, Equation 4 gives

Rij = Mij δ̈ij (5)

The acceleration (δ̈ij) can be written as the change in velocity over a time interval of Δt using

finite difference approximation, such that,

δ̈ij =δ̇t+Δt/2ij − δ̇

t−Δt/2ij

Δt(6)

substituting Equation 6 to 5, we get the recurrence Equation for velocities and displacements

as

δ̇t+Δt/2ij = δ̇

t−Δt/2ij +Rij

Δt

Mij(7)

δt+1ij = δtij + δ̇t+1

ij Δt (8)

Implementing the numerical stability of the algorithm, the time increment Δt is chosen such

that,

Δt ≤√

2Mij

Kij(9)

Mij ≥ Kij

2Δt2 (10)

substituting Equation 10 to 7, the nodal velocities are obtaines as

δ̇t+Δt/2ij = δ̇

t−Δt/2ij +Rij

2

ΔtKij(11)

Finally, the current nodal displacements are given by Equation 8. During each iterative cycle,

the kinetic energy for the current configuration U t+Δt/2 is compared with that of the previous

configuration, denoted as


U t+Δt/2 =1

2

n∑i=1

3∑j=1

Mij(δ̇t+Δt/2ij )2 (12)

U t−Δt/2 =1

2

n∑i=1

3∑j=1

Mij(δ̇t−Δt/2ij )2 (13)

The algorithm uses Equations 5, 11 and 8 to form an iterative loop. The iteration continues

until the out-of-balance force is less than the specified residual ε.

Numerical example

The membrane structures considered for this analysis exercise is a conic structure. The details

of the structure is shown in Figure 2. The following information has been provided for the

exercise:

(1) Type and configuration of the boundary supports (fixed edges, point support etc.). Here,

both the square base and the head ring are fixed.

(2) Fabric orientation direction: In this analysis, the principal directions (radial and circum-

ferential) coincide with the fabric yarn (warp and fill) directions.

(3) Fabric initial prestress is applied in warp and fill directions.

(4) Modulus of elasticity, E = 600 kN/m; Poisson’s ratio, ν = 0.4 and membrane thickness =

1 mm.

(5) Initially the membrane is subjected to principal stress state only, i.e. shear stress is assumed

zero.

CST element is used to discretize the membrane structure and the surface topology is defined

for the trial shape to begin the form-finding analysis. The aim of the analysis is to find the

equilibrium prestress in principal directions.

Results and discussion

In the present analysis, the membrane initial prestress value is taken as 4:4 kN/m (warp:fill)

in both the yarn directions. For this prestress force, the maximum principal stress (σ1) varies

in the range of 3.5 kN/m to 10 kN/m generally as shown in Figure 3 with the maximum value

of 16 kN/m. So, the equlilibrium surface obtained is not a minimal surface which has equal

stress values in all the directions. Physically, we can interprete that due to the square base

and the circular head ring. There is a variation in stresses as encountered after analysis. The

minimum principal stress for most of the boundary elements are zero while for other elements

the average value is 0.582 kN/m from Figure 4 which indicates that the membrane surface

is in the state of compression owing to probability of wrinkiling. The form-found surface

for the quarter membrane structure is shown in Figure 5. According to a recent round robin

exercise performed for analysis of membranes structure, the same structure was analysed and

the maximum principal stress (σ1) varied from 5.1 kN/m to 11.4 kN/m and the minimum from

1.6 kN/m to 3.2 kN/m for a target stress of 4kN/m in both the yarn directions. Hence, the results


5 m

4 m

14 m

HEAD RING

ELEVATION

FIXED BASE

PLAN

X

Y

14 m

Fig. 2. Conic structure (plan and elevation view)

obtained after the present analysis for initial stress value 4:4 kN/m are in good agreement with

those presented in the round robin exercise of membrane structures 7. However, it should be

noted that the initial prestress values do not reach close to the target stresses in both directions.

0 50 100 150 200 250 300 350 4000

2

4

6

8

10

12

14

16

18

Element number

σ1(kN/m)

Max. principal stress (σ1)Target prestress (σd = 4 kN/m)

Fig. 3. Element maximum principal stress for initial stress value 4:4 kN/m (warp:fill)


0 50 100 150 200 250 300 350 400−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

Element number

σ2(kN/m)

Min, principal stress (σ2)Target prestress (σd = 4 kN/m)

Fig. 4. Element minimum principal stress for initial stress value 4:4 kN/m (warp:fill)

12

34

56

78

0

2

4

6

80

1

2

3

4

5

6

7

Fig. 5. Conic surface after form-finding

Conclusion

In this paper, the numerical form-finding analysis of tensile membrane structures with dynamic

relaxation method is illustrated lucidly. Nonlinear geometric analysis is performed with


efficient finite element formulation for realistic modeling of flexible membranes. The CST

element formulation along with dynamic relaxation solution procedure provides a stable and

efficient computation tool for membrane analysis. The ease of implementation of the method

can be extensively used for computer implementation of form-finding analysis of tension

structures. The results also indicate that there is scope of optimizing the initial prestress in

order to achieve final stresses close to the target.

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3. Lewis, W. J. (2008), “Computational form-finding methods for fabric structures,” Proceedings of the

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4. Day, A. S. (1965), “An introduction to dynamic relaxation,” The Engineer, 219, 218–221.

5. Wakefield, D. S. (1999), “Engineering analysis of tension structures: Theory and practice,” Engineer-

ing Structures, 21(8), 680–690.

6. Veenendaal, D. and Block, P. (2012), “An overview and comparison of structural form finding

methods for general networks,” International Journal of Solids and Structures, 49(26), 3741–3753.

7. Gosling, P. D., Bridgens, B. N., Albrecht, A., Alpermann, H., Angeleri, A., Barnes, M., Bartle,

N., Canobbio, R., Dieringer, F., Gellin, S., Lewis, W. J., Mageau, N., Mahadevan, R., Marion, J. -

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8. Lewis, W. J. (2003), “Tension Structures Form and Behaviour,” Thomas Telford Publishing, London.

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