## Introduction

Dynamic relaxation is a method commonly used for form finding of shell structures. It was introduced by Alistair Day [1] and solves the nodal position for each node in the structure in such a way that nodal equilibrium is fulfilled

,

in which is the sum of all the applied external loads and the support reaction if the node is restrained, and the internal force in member ending in node .

As long as the position is incorrect a *out of balance* or *residual* force appears at each node

.

By introducing a small time step and a fictitious mass at each node, we can trace the structure’s (fictitious) motion in time as it moves towards the equilibrium position.

To update the position for node , the following three operations are performed consecutively

,

,

,

where and are the acceleration and velocity of the node, respectively, and is a damping parameter introduced to ensure numerical stability.

## Fictitious mass

The mass is fictitious and adjusted to adjusted to optimise the rate of convergence [2] and usually taken as

,

where is the largest direct stiffness occuring at the node during the simulation. Further strategies on choosing the fictitious mass is discussed in [3], including using different masses in different directions, effectively turning the mass in to a vector .

## Axial stiffness

If there only exists axial stiffness in the system, the internal force in member is found from

,

where and are the tension in member and the *current* length, respectively. is a unit vector in the direction from node to node .

It should be noted that so far we have discussed pure static equilibrium. No assumptions have been made regarding the material properties of the members which might be linear or non-linear elastic or be subject to creep or plastic deformation. The structure may be statically determinate or indeterminate or even a mechanism, provided that it is in equilibrium. The structure may have undergone a large deformation from some initial state.

In order to determine the form found geometry we need further information regarding the tensions and their relationship with the current lengths . The simplest relationship is linear elastic,

,

in the case of a member with unstressed length . The constant in which is equal to the Young’s modulus times the cross-sectional area of the member. However, during form finding we can postulate any relationship between tension and length, including inextensible members whose length cannot change and members with a constant tension.

## References

[1] A. S. Day. An introduction to dynamic relaxation. *The Engineer*, 219:218–221, 1965.

[2] M. R. Barnes, “Form finding and analysis of tension structures by dynamic relaxation,” *International Journal of Space Structures*, vol. 14, no. 2, pp. 89–104, 1999.

[3] M. Rezaiee-pajand, M. Kadkhodayan, J. Alamatian, and L. Zhang. A new method of fictitious viscous damping determination for the dynamic relaxation method. *Computers & Structures*, 89 (9–10):783–794, 2011.