The non-linear displacement method

Among the first computer methods applied to the solution of the initial equilibrium problem was the non-linear displacement method, which is based on the large dis-placement finite element technique used for analysis of structural behaviour under external loads. As the same program can be used for both the initial equilibrium problem and the load analysis, this approach is quite common. Nonetheless, there are some serious disadvantages associated with this technique. This section will be divided into two subsections: the first one dealing with cable nets and the second one with membranes.

The non-linear displacement method may be summarised as follows. First, an ele-ment mesh in equilibrium with a prescribed force distribution is established in the horizontal plane. A three-dimensional form of the mesh is created by displacing the support points almost vertically until they attain their prescribed positions, Fig-ure 3.2. An iterative algorithm, e.g. the Newton-Raphson method, is used to obtain the equilibrium configuration of the deformed structure.

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3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

Cable nets

Argyris et al. [4] were among the first to use the non-linear displacement method to solve the initial equilibrium problem for cable nets. Their method was developed in order to find the form of the cable roofs at the 1972 Olympics in Munich. Straight bar elements were used to represent the cables.

A method similar to the non-linear displacement method has been used by Barnes [8].

This method is an application of the dynamic relaxation method, where an initially out-of-balance structure is allowed to undergo damped vibrations until a steady equilibrium shape is obtained.

The displacements of the fixed nodes may give rise to an unfavourable force dis-tribution in the net, when actual material properties are used. Therefore, when the fixed nodes have reached their final positions, a force adjustment procedure is applied to the net. In this procedure the original unstrained lengths of the ele-ments are recomputed in such a manner that the desired force values are obtained.

For a straight cable element satisfying Hooke’s law this is straightforward as the total lengths of the elements before and after adjustment must be the same, i.e.

L⁰+ ∆L⁰ = L⁰ + ∆L⁰. It leads to the following relation:

L0 = L0+ ∆L0

1 +  = L0+ ∆L0

1 + T /AE. (3.4)

After this adjustment step the structure is no longer in equilibrium. Therefore, some more iterations are needed to re-impose equilibrium. But, these iterations will not change the final force distribution very much, so it will be close to the desired one. Another way to keep control over the forces is to use a very small modulus of elasticity for the cables, but then the control over the cable lengths is lost. With the procedure outlined above control of both the forces and cable lengths is possible.

Figure 3.2: The principles of the non-linear displacement method. Reproduced from [16]

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

Membrane structures

In principle, the application of the non-linear displacement method to membrane structures does not differ very much from the case of cable nets. Some general ap-proaches used for finding the reference configuration of membranes will be discussed.

First, a finite element suitable for membrane representation has to be selected. Ow-ing to the great geometric non-linearity of membrane structures, it is preferable to use a dense mesh of primitive elements rather than a coarse mesh made up of higher order elements [77, 115]. Then, a stress distribution has to be chosen. Generally, two different approaches are used:

• the minimal surface approach, or

• the nonuniform stress approach.

The simplest choice for an initial equilibrium shape is the minimum surface config-uration, which is characterised by a state of isotropic tensile stress. In order to find the minimal surface, it is assumed that the flat membrane has a very small modulus of elasticity and is in the isotropic prestressed state [115]. For other members in the structure, such as beams or bars, actual material properties should be used [106].

Due to the small modulus used, the specified stresses in the membrane will only slightly change, even though large deformations occur during the displacements of the fixed nodes [115].

The advantages of the minimum surface are its aesthetically pleasing shape and the associated uniform tensile stress. However, in some cases the minimal surface configuration cannot satisfy all the architectural and structural requirements [115].

Since the mean curvature for minimum surfaces is zero, such surfaces are rather flat and these have poor load bearing capacities. A nonuniform stress approach has to be used. In this, a very small modulus of elasticity is still used for the membrane, but as the name of the approach implies, the initial stresses are no longer specified uniformly. Following the same procedures as for the minimal surface approach, the final configuration should be in equilibrium with the nonuniform prestress. Several trial calculations are usually needed to find a satisfactory equilibrium shape. Hence, it is not obvious how to choose the nonuniform stresses [114].

For structures, where it is difficult to specify nonuniform initial stresses, an alterna-tive approach can be used. This approach, which is based on elastic deformations, is similar to that of Argyris et al. [4] for cable nets. In this approach, the actual modulus of elasticity of the membrane is used. As for the cable net, the deformed equilibrium configuration will have nonuniform and possibly large stresses. At this stage, the stresses due to deformation are removed by a stress adjustment algorithm and only the initial stresses are retained [115].

It was explained above that the way to keep the stresses within the elements con-stant during displacement is to assign a very small modulus of elasticity to the elements. However, in many cases the small elasticity has undesirable consequences, such as numerical instability and divergence of the solution [65]. These problems

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3.2. LITERATURE REVIEW OF INITIAL EQUILIBRIUM SOLUTION METHODS

stem from the assumption of small strains made in the derivation of the membrane elements. For an ill-chosen initial surface, i.e. in most cases a horizontal surface, this assumption is often violated because gross changes in the element geometry take place during the displacements of the fixed nodes. A way to avoid numerical instability is to choose a mathematically defined initial surface close to the final shape. Similar convergence problems were also reported in [36].

Another problem is that for surfaces which exhibit high curvatures, elements gather in certain regions of the surface and leave the remaining regions represented to a lesser accuracy. It is suggested that a suitable element arrangement in complex cases should be chosen with the aid of a physical model [66].

Recently, Bletzinger [13] used a method called the updated reference strategy, which is a numerical continuation method, to solve the initial equilibrium problem of mem-branes with minimal surfaces. This technique had to be used because of the occur-rence of a singular stiffness matrix, which excludes the use of the ordinary Newton-Raphson algorithm. The stiffness matrix is singular when the nodal displacements are tangential to the membrane surface. To understand that, consider a plane, which obviously is a minimal surface. The surface area of that plane does not change if the geometry of discretization is changed, e.g. by small tangential displacements within the plane. Hence, the area variation of in-plane displacements is zero. A special case of the updated reference strategy is the force density method. As the updated reference strategy is claimed to be absolutely robust, it is perhaps the best non-linear displacement method available for the initial equilibrium problem.

There are some drawbacks of the non-linear displacement method applied to the initial equilibrium problem. Both the final shape and the stresses in the structure are difficult for the designer to control. It is not an easy task to specify a desirable force distribution [44]. If actual material values are used it is possible for some el-ements in the structure to end up in compression [4]. The specification of material properties (fictitious or real) represents unnecessary additional decision making for the designer. In addition, the computations involved in this method are time con-suming for large structures [44]. An advantage is that the program used to solve the initial equilibrium problem can also be used for further load analysis.

The non-linear displacement method may be summarised as follows [44]. The vari-ables specified by the designer are:

• structural topology,

• boundary conditions, and

• material properties.

The problem unknowns are:

• structural geometry, and

• internal force distribution.

CHAPTER 3. THE INITIAL EQUILIBRIUM PROBLEM

The following additional constraint is placed on the solution:

• an initial force distribution may be specified.