Nonlinear analysis and form-finding in GSA Training Course

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Nonlinear analysis and form-finding in GSA

Training Course

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Oasys Ltd

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Using the GSA GsRelax Solver

Trainers: Thomas Li & Sarah Kaethner Duration: 5 hours

Programme:

• What is GsRelax

• Nonlinear analysis

Introduction

Geometric nonlinear analysis P-δ analysis

Buckling analysis

• Element types and features available in GsRelax

• Structures suitable for analysis by GsRelax

• Solution method used by GsRelax

• Example of dynamic relaxation analysis of one degree of freedom problem

• Running GsRelax

• Tips of running GsRelax successfully

• Drawbacks and limitations in using GsRelax

• Exercises

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What is GsRelax

GsRelax is a nonlinear analysis solver in Windows GSA.

Special feature in GsRelax

As a nonlinear analysis solver, GsRelax can take the following special features into account in the analysis:

• Geometric nonlinear effects (automatically considered)

• Geometric stiffness of beam elements (it can be turned on or off, default is on)

• Material nonlinearity (Once nonlinear material is defined for beam, bar, tie & strut elements)

Advantage of using GsRelax

The advantages of using GsRelax solver in structure analysis:

• Since GsRelax analysis does not rely on small displacement assumption and geometric nonlinear effects are always considered, GsRelax can produce more accurate and realistic results compared with linear analysis solver especially when the structure deformations are relatively large.

• Because a vector approach (Dynamic Relaxation) rather than a stiffness matrix method is used in GsRelax analysis, it does not impose any special requirements to the stiffness of the structure, e.g. zero stiffness of some nodes in some directions are allowed in GsRelax analysis. Therefore, GsRelax can analyse virtually any types of structures even a mechanism, for example, normal structural analysis programs cannot cope with the following two special types of structures, but GsRelax will be able to give a solution as that in the real world.

(i) The structure shown in Fig 1a has a zero vertical stiffness at its initial state, normal linear analysis programs cannot solve the problem since the stiffness matrix is singular.

(ii) The structure shown in Fig 1b is a mechanism and it cannot be analysed by linear analysis program. Even some of the nonlinear analysis programs may also have difficulty to give a proper solution for this problem.

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Original

Balanced (a)

Original

Balanced

(b)

Fig 1 Examples of extremely geometric nonlinear problems

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Nonlinear analysis

Introduction

In order to understand GsRelax and interpret its analysis results, it is important to know the differences between linear and nonlinear analysis.

Linear structural analysis is using the following two assumptions and the equilibrium condition of the structure is established at the non-deformed geometry of the structure.

• material stress-strain relationship is linear (material Young's Modulus is constant)

• displacement and strain relationship is linear (small displacement problem)

These two assumptions are acceptable in most cases of structural analysis since majority of the structures (except light-weight structures etc) in practice are quite stiff and the deformations of the structures are relatively small compared with the size of the structure. In these situations, using linear analysis will not result in any significant error of the analysis results. However, if a structure is flexible or the deformation is relative large compared with the size of the structures (as shown in Fig 1) and/or real material property needs to be considered, these two assumptions become invalid and nonlinear analysis should be used.

There are two types of nonlinear problems in structural analysis:

• Geometric nonlinear analysis. In geometric nonlinear analysis, the equilibrium condition of the structure is established at the deformed (real) geometry of the structures, therefore load-displacement relationship is no longer linear even though a linear elastic material is used.

• Material nonlinear analysis. In material nonlinear analysis, the material Young's Modulus is no longer taken as constant and the real or simplified material stress-strain relationship (as shown in Fig 5) should be used.

As far as the accuracy of the analysis result is concerned, nonlinear analysis should always be used.

However, because nonlinear analysis requires longer computing time, linear analysis is frequently used in structure analysis if the two assumptions mentioned above are applicable (deformations are small and stresses are below material yield strength).

Geometric nonlinear analysis

The basic difference between linear and geometric nonlinear analysis is the structure geometry on which the equilibrium condition is established. The example shown in Fig 2 illustrates the implication of linear and nonlinear analyses on the internal forces of the column.

In linear analysis the displacement δ is ignored (taken as zero) when establishing the equilibrium condition, so the forces and moment at the base of the column will be:

Qh M

Q V

P F

=

(1)

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In geometric nonlinear analysis, the equilibrium condition is established on the deformed (real) geometry of the structure (actual δ is considered), so the forces and moment at the base of the column become:

δ

θ θ

P Qh M

P Q

V

Q P

F

+

=

+

=

−

=

) sin(

) cos(

) sin(

) cos(

(2)

P Q

δ

h

M

F V

θ

Fig 2 Simple geometrical nonlinear analysis example

Bear in mind that the horizontal deflection δ cannot be obtained straightforwardly in nonlinear analysis since the element stiffness is changing along with δ. Even for this very simple structure, a number of iterations may still need to obtain the balanced geometry of the structure if the magnitude of horizontal load Q is significant.

In order to have some indications when geometric nonlinear effect needs to be considered, the force and moment ratios from linear and nonlinear analyses for this simple example are drawn in Fig 3 in which the following values are used in this calculation:

m h

P Q

0 . 10

/

= α =

It can be seen from Fig 3 that the difference between linear and nonlinear analysis is very small when shear force/axial force ratio α is relatively large and/or the deflection δ is relatively small. This suggests that:

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• If the axial force (destabilising force) of an element is relatively small, the geometric nonlinear effect will be insignificant. For example, if the axial force is less than half of the shear force, the relative difference of shear force and moment between linear and nonlinear analyses is less than 5%, and the difference of axial force between linear and nonlinear analysis is less than 20% for this example.

• If the lateral deformation of a member is relatively small, the geometric nonlinear effect is also insignificant, for example, if the deflection δ is less than 0.05 m (h/200, a typical code requirement), the nonlinear effect becomes very small and the relative deference between linear and nonlinear results is smaller than 1%, so it is negligible in engineering practice.

0.80 0.83 0.85 0.88 0.90 0.93 0.95 0.98 1.00

0.00 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 Horizontal to vertical load ratio (Q/P)

non-linear to linear analysis axial force ratio

δ = 0.05 δ = 0.2 δ = 0.5 δ = 1.0

(a) Column axial force ratio

1.00 1.05 1.10 1.15 1.20 1.25 1.30

non-linear to linear analysis shear force ratio

δ = 0.05 δ = 0.2 δ = 0.5 δ = 1.0

(b) Column shear force ratio

Fig 3 Numerical example of column internal force ratios between nonlinear and linear analyses (1)

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1.00 1.05 1.10 1.15 1.20 1.25 1.30

non-linear to linear analysis moment ratio δ = 0.05 δ = 0.2 δ = 0.5 δ = 1.0

(c) Column moment ratio

Fig 3 Numerical example of column internal force ratios between nonlinear and linear analyses (2)

P-δ analysis

P-δ analysis in GSA is a two-step analysis, the first analysis is a linear analysis from which element forces and moments can be obtained. Using the element forces and moments obtained from the first step analysis, geometric stiffness matrix of elements can be obtained. Then the tangent stiffness matrix at the initial geometry is calculated by adding the geometric stiffness matrix to the normal linear stiffness matrix. The second step analysis is then conducted using the tangent stiffness matrix (the sum of linear and geometric stiffness matrices). The displacements, forces and moments etc from the second analysis are taken as P-δ analysis results. In this way, the geometric nonlinearity is approximately considered in this analysis, i.e. the effect of element forces on their stiffness is considered. Bear in mind that even though geometric stiffness is considered in P-δ analysis, the analysis is still based on non-deformed geometry, i.e. the equilibrium is still established at non-deformed geometry, therefore, if displacements are relatively large, P-δ analysis will not be applicable.

In order to have some indications of the differences of the three analyses (linear, P-δ and nonlinear analyses), the simple model shown in Fig 2 has been analysed using the three analysis methods assuming that:

2

5 .

10 0 . 1

80 2 . 0

0 . 10

400

m kN EI

kN P

Q

m h

kN P

×

=

(10)

The numerical results from the three analyses are given in the following table:

Numerical results from the three analysis methods Axial force

(kN)

Shear force (kN)

Moment (kN.m)

linear 400.0 80.0 800

P-δ 400.0 93.3 946.1

nonlinear 397.3 92.5 921.1

Note: the structure used is shown in Fig 2 and the parameters are defined above

From this example it can be seen that, even though P-δ analysis does not consider geometric nonlinear effect precisely, it can produce results quite close to that obtained from nonlinear analysis. Since P-δ analysis is simple and efficient in terms of computing time, it is frequently used in structural analysis to approximately consider geometric nonlinear effect if the deformations of the structures are still relatively small.

Buckling analysis

Buckling analysis is to seek the potential maximum load capacity of a structure before it collapses (buckled).

It may be divided into the following 3 categories.

(i) Linear buckling analysis (also known as Eigen value or modal buckling analysis, e.g. GSA modal buckling analysis )

In this analysis, the material is assumed to be linear and the actual deformations of the structure are not the results of this analysis. As it is known, the element (structure) stiffness is not only dependent on its material properties and geometry, but also dependent on the forces in the elements, e.g. if a column is subjected to compression, its bending stiffness will be reduced compared with the same column without axial force. If the axial force is equal to the Euler load capacity of the column, the bending stiffness of the column will become zero. Based on this principle, for a multiple degree of freedom system, the buckling loads will be the loads that make the general stiffness matrix singular.

The general structural stiffness matrix also contains two parts as shown below

[ ] [ ]K = K e +[K( )F ]g (3)

where:

[K]_e is the linear stiffness matrix without considering the effect of element internal force (the stiffness matrix used in linear analysis)

[K(F)]_gis the geometric stiffness matrix which is a function of structural geometry and element internal forces {F}, but not the element material properties. [K]_g will be used below to represent this matrix for simplicity reason.

F or {F}, the load vector.

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Since buckling load capacity is not known before the analysis, it is assumed that the buckling loads are λ{F}, where λ is the load factor to be determined in linear buckling analysis. After introducing load factor λ, the general stiffness matrix becomes:

[ ] [ ]K = K e+λ[ ]K g (4)

As mentioned above, when the applied loads λ{F} are equal to the structure buckling load capacity, the stiffness of the structure will be zero. In matrix form, it is the determinant of the general stiffness matrix becomes zero, this gives:

det

(

[ ]K e ⁺λ[ ]K g

)

⁼⁰ (5)

This is an eigen value problem and only load factor λ is unknown and it can be solved, so the buckling load capacity of the structure will be λ{F}. Since structure deformations are ignored and the matrices are all built at the non-deformed geometry (it implies that the structure has no deformations before the buckling), therefore, this analysis is called as linear buckling analysis. The buckling load capacity obtained from this analysis is the upper bound of the structural capacity against buckling. This analysis only gives relative deformations of different parts of a structure (mode shapes) and the real magnitudes of the deformations are unknown from this analysis.

(ii) Elastic nonlinear buckling analysis, e.g. GsRelax automatic load increment analysis with element slenderness (geometric stiffness) being considered.

In this analysis, the structural stiffness matrix contains three parts:

[K( )δ ] [ ]= K e+[K( )δ ] [ ]L + K g ⁽⁶⁾

In which [K(δ)]_L is the stiffness matrix arisen from large deformations, it is a function of nodal displacements {δ}. The meaning of the other two stiffness matrices is the same as before.

Since [K(δ)]L is a function of {δ}, [K(δ)] will be also a function of {δ}, therefore equilibrium equation [K(δ)]{δ} = {F} is no longer linear in terms of {δ}, therefore it is impossible to solve the equation straightforwardly like linear buckling analysis using eigen solver and iteration or other nonlinear solution technique is needed to solve this nonlinear equilibrium equation.

GsRelax uses dynamic relaxation method to solve the problem, so GsRelax does not explicitly use the above stiffness matrix equation, but the principle is the same. GsRelax considers geometric nonlinear effect by updating the nodal coordinates and the element forces at each iteration (cycle) according to current deformations of the elements. Based on the current element forces and deformations, the bending stiffness of the beam elements is updated, furthermore, the nodal coordinates are updated according to actual nodal deformations and the equilibrium conditions are checked at the deformed geometry of the structure, so large deformation and element force effects on element stiffness have both been considered in the analysis.

This analysis will give actual deformation of the structure under the given loads. To obtain the maximum (buckling) load capacity, automatic load increment can be used, so a load-deformation curve can be drawn and the maximum load capacity can be evaluated.

(iii) Elastic-plastic nonlinear buckling analysis (e.g. GsRelax automatic load increment analysis with material nonlinearity being considered)

This analysis is similar to elastic nonlinear buckling analysis except that material plasticity is also considered. According to the nonlinear characteristics, iterative solution techniques are also required. For estimating the real ultimate load capacity of a structure, this is the most appropriate analysis method.

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The schematic load deflection curves for the three buckling analyses are shown in Fig 4. It shows that the real load-deflection curve cannot be obtained from linear buckling analysis. The elastic- plastic nonlinear buckling analysis gives the lowest (most real) load capacity of a structure.

Load

Displacement Load capacity from linear

buckling analysis

Load deflection curve of elastic nonlinear buckling analysis

Load deflection curve of elastic-plastic nonlinear buckling analysis

Fig 4 Load deflection curves of three buckling analyses

Material nonlinear analysis

When the stress level in an element is likely to exceed the material yield strength and plastic behaviour is acceptable (i.e. non-brittle material such as steel), material nonlinear analysis should be considered. Real material stress-strain relationships are very complicated as shown in Fig 5a. In structural analysis, simplified stress-strain relationships are frequently used to simplify the analysis procedure. Figs 5b-5e shows some of the simplified stress-strain curves used in finite element analysis and design. GsRelax can only consider Elastic perfectly plastic stress-strain relationships at present.

Tension only and compression only materials are the extreme cases of material nonlinearity. Cable, Tie and Fabric elements are examples of tension only elements, and Strut elements are compression only elements.

Material nonlinear analysis is easier to understand than geometric nonlinear analysis and it just limits the stress level in the elements based on the strain level and stress-strain curve used, e.g. in linear analysis, there is no limit to a beam section moment capacity and it is proportional to the applied loads, but in material nonlinear analysis, the beam section moment capacity will be limited by the material strength and section plastic modulus. Similar to geometric nonlinear analysis, iterative analysis procedure is also required when doing material nonlinear analysis.

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ε σ

0

(a) Real steel material stress-strain relationship

ε σ

0

E ε

σ

0 E

E’ = E/50

ε σ

0 E

E’ = E/50

ε σ

0 E (b) Elastic-perfectly

plastic (Fablon uses this)

(c) Elastic and strain hardening

(d) Elastic-plastic and strain hardening (e) Multi-linear stress-strain relationship

Fig 5 Real and assumed material stress-strain relationships

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Element types and features available in GsRelax

Elements

• Bar element (tension and compression)

A Bar element has axial stiffness only and can sustain both tension and compression forces. If elastic- plastic material is defined for the bar elements, the maximum tensile and compressive forces in the bar will be limited to be A_bf_y (A_b – bar cross section area, f_y – bar material yield strength). Bar element can only be stretched or squashed and no bending effect is taken into account, so a bar element will never buckle no matter how large the axial force.

• Strut element (compression only)

A Strut element has only axial compressive stiffness and can only sustain compressive forces. If elastic- plastic material is defined, the compressive force of the strut is limited to be A_bf_y. The same as bar elements, the strut element will also not buckle as bending effects are ignored.

• Tie element (tension only)

A Tie element has only axial tensile stiffness and can only sustain tensile forces. If elastic-plastic material is defined, the tensile force of the tie is limited to be A_bf_y.

• Cable element

Cable element is similar to Tie element and it can only take tensile force. If cable element is used individually, i.e. not linked with another cable element with the same property, it will be the same as tie element. However the intended use of cable elements is to make up a Sliding-Cable. A Sliding-Cable is a chain of cable elements with the same property number and joined at the common nodes as shown in Fig 6a. The Sliding-Cable shown in Fig 6a can be defined in a number of ways. The following two tables are two examples of defining a Sliding-Cable as shown in Fig 6a.

Example 1 Definition of the Sliding-Cable shown in Fig 6a

Element record Type property 1^st node 2^nd node

1 Cable 1 1 5

2 Cable 1 5 7

3 Cable 1 7 2

4 Cable 1 2 3

Example 2 Definition of the Sliding-Cable shown in Fig 6a

element number Cable a 7 5

element number Other element element number Other element

element number Other element

a - Cable element property number which should be the same for all the cable elements belong to a Sliding-Cable A sliding-cable made up of a number of cable elements is considered as a single element in GsRelax analysis. The sliding-cable can freely slide over the middle nodes of the sliding-cable as if the sliding- cable is jointed to the middle nodes by a pulley. As a result, the tensile forces of all the Cable elements in a sliding-cable will be the same. If the final total length (the sum of the deformed length of all the cable elements) of a sliding-cable becomes shorter than its unstressed total length, the sliding-cable will be out of action and the axial force in all the cable elements will be zero.

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There is no limit for the numbers of sliding-cables in a structure model, but each sliding-cable should have a unique property number. All the cable elements in a sliding-cable should have the same property number and it should be different from that of other sliding-cables. This is necessary for GsRelax to recognise each of the individual cable elements as a leg of a sliding-cable. A sliding-cable cannot be discontinued or bifurcated as shown in Fig 6b. The property of sliding-cable is defined by unit length stiffness which is equal to EA (E - material Young’s modulus, A – cable cross section area). The tensile force in the cable is calculated from:

(8) in which:

L₀ – sum of the unstressed length of all the legs (cable elements) of a sliding-cable L – sum of the deformed length (nodal distance) of all the legs of a sliding-cable

=

Cable element

1

5 7

2 3

(a) Sliding-Cable

=

Cable element

(b) unacceptable sliding-cable definition Fig 6 Construction of sliding-cable









≤

− >

=

0 0 0

0

L L if 0

L L if L EA

L L F

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• Beam element

In GsRelax analysis there are two options for computing beam section bending stiffness, they are:

(i) Element slenderness is included by checking “Element slenderness included” box in the analysis wizard

(ii) Element slenderness is ignored by unchecking “Element slenderness included” box in the analysis wizard

If the first option is selected (the default option), the effect of element forces on its bending stiffness will be considered in the analysis, i.e. geometric stiffness is considered. If this effect needs to be ignored, the second option can be used.

If material yield strength is defined, the moment and axial force that can be sustained by a beam section will be limited to its plastic moment capacity and axial load capacity in the analysis. Beam section plastic moment capacity is equal to the material yield strength times the section’s plastic modulus if there is no axial force on the beam. If a beam section is subjected to both bending moment and axial force, the interaction of the moment and axial force is considered, so the moment capacity will be smaller than that given by the yield strength times the plastic modulus of the section. The plastic moment capacities of a beam in one direction will also be reduced if there is moment in other direction, e.g. plastic moment capacity M_yy will be reduced if moment M_zz is not zero. However, shear force- plastic moment capacity interaction is not ignored, i.e. the plastic moment capacity will not be reduced if there are shear force in the beam.

If an explicit beam section property is defined (e.g. only A, Iy, Iz are defined), the estimated section plastic modulus (given below eqn (9) will be used in calculating plastic moment capacity.

(9) in which:

A – beam cross section area

I_pp – second moment of area about principal axes (local y or z)

• 3 noded triangular element (Tri3 in GSA)

The triangular element can be used:

(i) to model fabric part of a structure if Fabric property is selected for the element, fabric material property can have both tension and compression stiffness or have tension stiffness only

(ii) as plane stress element if plane stress property is defined or (iii) as plane strain element if plane strain property is defined

Out of plane bending cannot be considered for triangular element in GsRelax analysis. Since a linear displacement function is used for 3-node triangular element, the strain and stress are constant within an element. If the stress gradient is steep at some part of the structure, a fine mesh layout is needed to obtain accurate results.

• 4 node quad element

AIpp

9 . 0

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In GsRelax, the 4 node quad element is subdivided into four triangular elements by introducing a dummy node at the centre of the quad element. The application of quad elements is the same as 3-node triangular elements in GsRelax.

• Spring element

In GsRealx analysis, both linear and nonlinear spring elements can be used. To use nonlinear spring elements, nonlinear spring curves need to be defined. Nonlinear spring curves define the relationships of force-displacement and moment-rotation of the spring elements in a particular direction.

• Link element

This is a 2-node element and it can have the following properties, which define the directions of the 2 nodes being linked:

All XY Plane YZ Plane ZX Plane Pin All Pin XY Plane Pin YZ Plane Pin ZX Plane Tension Compression Bar

Link elements with properties rather than Tension, Compression and Bar are exactly the same as 2-node rigid constraints (explained later).

Link elements with Tension, Compression or Bar properties are specifically designed for GsRelax (nonlinear analysis). These link elements are used to maintain the original distance of the two nodes of the link elements. The first node of the link is defined as master node and the second node as the slave node. The master node can be restrained, but the slave node cannot be restrained or be a slave to other links.

Link elements with Tension properties will prevent the two nodes from moving apart but the two nodes can move closer. Similarly, link elements with Compression properties will prevent the two nodes from moving closer but they can move apart. Link elements with Bar properties will maintain the nodal distance of the link element, i.e. they cannot not move apart or move closer.

• Spacer element

The Spacer element is only used in soap-film form-finding analysis and it will be ignored in all other analyses even though they may exist in the model. Spacer element is only used to make up a SPACER that is a super element used in GsRelax. As its name indicates, SPACER is used to maintain or adjust the nodal spacing as desired in the form-found structure. A SPACER is composed of two or more spacer elements that are jointed at the common node as shown in Fig 7. The way of defining a SPACER is similar to that for Sliding-Cable, except that the direction of spacer elements in the SPACER is not arbitrary. This is because that the order of the nodes of the spacer elements is important when spacer leg length type is 'ratio'. The nodes for spacer elements should be defined so that they can be jointed together to form a SPACER by a head-to-tail sequence, in other words, the local X axis of all the spacer elements belong to a SPACER should be in the same direction. The SPACER shown in Fig 7 can be defined in a number of ways. The following two tables are two examples for defining the SPACER shown in Fig 7.

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Example 1 Definition of the SPACER shown in Fig 7a

1 Spacer 1 1 5

2 Spacer 1 5 7

3 Spacer 1 7 2

4 Spacer 1 2 3

Example 2 Definition of the SPACER shown in Fig 7a

Element record Type Property 1^st node 2^nd node

element number Spacer A 1 5

element number Other element - - -

a - Spacer element property number which should be the same for all the spacer elements in a SPACER

There are three types of SPACERS and they can be specified in the spacer property table and they are:

• Geodesic spacer: its main use is to control the nodal spacing within 2D element surface

• Free spacer: its main use is to control the nodal spacing along a Bar, Tie or Strut elements

• Bar spacer: its use is similar to Free spacer, but it acts exactly as Bar elements as explain below and the nodal spacing can be controlled more precisely. The precision can be defined in GSA specification.

SPACER may be considered as a chain of Tie elements except that the spacer forces to the node in the suppressed directions will be ignored. The suppressed directions vary according to the spacer type. For Geodesic SPACER, the suppressed direction (one only) is the normal direction defined by the average normal direction of the surrounding 2D elements connected to the node. For Free SPACER, the suppressed directions (two directions) are in the plane perpendicular to the average direction of the two spacers connected to the node. The average direction of the two spacers is the direction from the first node of the first spacer to the second node of the second spacer. Since the force components of Geodesic and Free SPACERS in the suppressed directions have been suppressed (ignored) when seeking equilibrium of the SPACER nodes, these two SPACERS do not affect the shape of the surface of the form-found structure; it only affects the position of the nodes on the surface or along the spacers. There is no suppressed direction for Bar SPACER, so a Bar SPACER works in exactly the same way as bar element except that its length will be adjusted during form-finding analysis according to the leg length type.

As mentioned above, the component of Geodesic spacer forces in the normal direction of a 2D element surface is suppressed. Geodesic spacers will only move the internal nodes of SPACER within the plane defined by the surface normal vector passing through the node. Therefore the final SPACER path will be the shortest path on the surface linking the two end nodes of the SPACER. Free spacers only move internal nodes in the tangent direction defined by the two legs of the SPACER connected to the node. If we assume the surface normal at node A of the SPACER shown in Fig 7b is in the global Z direction (out of the paper), neither Geodesic or Free SPACERs will move node A in the Z direction even though the spacer leg forces have a component in this direction. If the SPACER is a Free spacer, the SPACER will move node A only in the X1 direction. If the SPACER is a geodesic spacer, the SPACER will move node A in both X1 and Y1 direction. Since the surface normal for an internal node is required by geodesic spacers, the internal node of a geodesic spacer must be surrounded by triangle or quad

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elements, otherwise, GsRelax cannot compute the normal direction and it is considered to be a data error.

Bar spacer acts as Bar element and there is no force component to be suppressed. The difference between Bar spacer and Bar element are:

• Bar spacer is only active in soap-film form-finding analysis and it will be ignored in other analysis

• Bar element property is defined by section property, but Bar spacer property is defined by form- finding property of soap-film 1D.

• Bar element has constant unstressed length (e.g. initial length if no pre-stress) in the analysis, but Bar spacer’s unstressed length will be altered automatically during GsRelax analysis in order to meet the nodal spacing requirements.

SACER can control nodal spacing in four different ways which is called spacer leg length type in spacer property table. The four leg length types are:

• Proportional: the final nodal spacing will be proportional to their original nodal spacing.

• Ratio: the final nodal spacing will form a geometrical series. If the ratio is 1.0, the final nodal spacing will be equal. The initial spacing of the spacer elements is irrelevant to and has no effect on the final nodal spacing

• XY plane projected ratio: the projected length of the nodal spacing on to the XY plane of the specified Axis will form a geometrical series. If the ratio is 1.0, projected length of the nodal spacing will be equal. The same as leg length type “Ratio”, the initial spacing of the spacer elements has no effect on the final nodal spacing

• X axis projected ratio: the projected length of the nodal spacing on to the X axis of the specified Axis will form a geometrical series. If the ratio is 1.0, projected length of the nodal spacing will be equal.

The same as leg length type “Ratio”, the initial spacing of the spacer elements has no effect on the final nodal spacing

The same as Sliding-Cables, a SPACER should not be discontinued or bifurcated as shown in Fig 6b.

spacer element 1

5 7

2

3

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(a) Composition of a SPACER

X Y

θ θ

X1 Y1

A

(b) The action of SPACER on the internal node (drawn in XY plane)

Fig 7 Example of SPACER

Features

• Joint

The displacements and/or rotations in the specified direction of the second node (slave node) of a joint will be the same as those of the first node (master node)

• Applied displacements

The nodes specified will be given an imposed initial displacement or rotation. If the relevant direction of the node is not restrained, the node will be free to move in that direction after the applied displacement is imposed and analysis is started. The final position of the node is dependent on the equilibrium condition. If the relevant direction of the node is restrained, the node will be moved to the new position and stay there.

• Nodal settlement

The nodes specified will be given an imposed initial displacement or rotation. The settlement can only be applied to a node with restraint in the relevant directions.

• Rigid constraint

Two or more nodes are constrained rigidly in all or some defined directions, one node is defined as the master node and the rest of the nodes are slave nodes. Rigid constraints can be applied in all directions or applied to a specified plane. The master node can be constrained, but slave nodes cannot be constrained or be a slave of another constraint. A Rigid constraint can be considered as a rigid body or rigid plane depending on the type of its property. The rigid constraint can have the following properties in GsRelax analysis:

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All rigid constraint applied to all directions (a rigid body) XY Plane rigid constraint applied to XY plane only (a rigid plane) YZ Plane rigid constraint applied to YZ plane only (a rigid plane) ZX Plane rigid constraint applied to ZX plane only (a rigid plane)

Pin All same as All, except that the slave node will not take any moment.

Pin XY Plane same as XY Plane, except that the slave node will not take any moment.

Pin YZ Plane same as YZ Plane, except that the slave node will not take any moment.

Pin ZX Plane same as ZX Plane, except that the slave node will not take any moment.

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Structures suitable for analysis by GsRelax

Any structures where the geometric nonlinear effect is significant or the nature of the structure (such as those shown in Fig 1) requires nonlinear analysis or material plasticity need to be considered, GsRelax analysis should be used. To be specific, the following analyses should be conducted by GsRelax.

• Cable networks

• Fabric structures

• Flexible structures

• Form-finding

• Searching ultimate load capacity of a structure

• Evaluate the axial load capacity of an individual beam element or a chain of beam elements in a structure

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Solution method used by GsRelax

The solution technique used in GsRelax is Dynamic Relaxation. Dynamic relaxation is an analysis method for nonlinear static analysis. In dynamic relaxation analysis it is assumed that the loads are applied on the structure suddenly, so the structure is excited to vibrate around the equilibrium position and eventually come to rest on the equilibrium position because of damping. In order to simulate the vibration, mass and inertia are needed for each of the free nodes. In dynamic relaxation analysis, artificial mass and inertia are used that are constructed according to the nodal translational stiffness and rotational stiffness. As it is known, if there is no damping applied to the structure, the oscillation of the structure will not stop. Therefore, damping is required to allow the vibration to come to rest. Two types of damping are used in GSA – Kinetic Damping &

Viscous Damping. By applying one or both of these damping, the vibration will gradually come to rest at the equilibrium position and this will be the solution given by dynamic-relaxation analysis.

Using the structure shown in Fig 8 as an example, the effect of viscous damping on the analysis process of the dynamic relaxation is shown in Fig 9. It shows that the oscillation of the structure eventually come to rest at the static equilibrium position if damping is applied.

Kinetic damping is unrelated to conventional concepts of damping used in structural dynamic analysis. It is an artificial process to reduce the magnitude of the vibration in order to make the analysis converge quickly.

It is known that the structure's kinetic energy reaches maximum at the static equilibrium position if the structure has only one degree of freedom or the vibration of a multiple degree of freedom system is in a single mode. According to this, the structure's kinetic energy is monitored in the analysis at each time increment.

Once the kinetic energy for the current time increment is smaller than that at previous increment, it is known that the peak structure kinetic energy has been passed, this implies that the equilibrium position of the nodes have also been passed, therefore the true equilibrium position of the nodes could be found somewhere between current displacements and displacements in previous two iterations. Therefore, it is necessary to stop the vibration and re-position the nodes corresponding to the maximum kinetic energy. Assuming the relationship between structural kinetic energy and time is parabolic as shown in Fig 10, the time at which the kinetic energy is peaked can be calculated. Knowing the time for peak kinetic energy and the nodal displacements at current and previous two iterations, the optimum (most close to equilibrium) nodal positions can be calculated, then artificially shifting the nodes to this optimum position to finish applying kinetic damping. After shifting the nodes to the optimum positions, the analysis will restart again by resetting the time, speed and acceleration to zero. Since it is unlikely that a multiple degree of freedom system will vibrate in a single mode, it is impossible to find the equilibrium position by applying kinetic damping once or twice (in one or two iterations). Nevertheless, previous experience has shown that the use of kinetic damping is very efficient in searching for the equilibrium position in dynamic relaxation analysis.

equilibrium position, maximum speed position, acceleration = 0 position max kinetic energy position most unbalanced position,

zero speed position, max acceleration position

vibration F

Fig 8 Vibration of the structure

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Deflection

Time no damping

damping smaller than critical

damping larger than critical

equilibrium deflection

Fig 9 Viscous damping effect on the vibration of structure

Time E^k (kinetic energy)

E^k_i+1 E^k_i

E^k_i-1

t_i-1 t_i t_i+1

E^k_max

t_max

Fig 10 Kinetic energy of structure

According to the above principles, the following steps are adopted in GsRelax analysis

• Compute equivalent nodal forces and moments according to applied loads. In this process, member loads are converted into nodal force or moments. These are the forces that initiate vibration.

• Construct dummy mass and dummy inertia for the active nodes (unrestrained nodes). GsRelax constructs the dummy mass and dummy inertia according to the translational and rotational stiffness of the node. The principles for constructing the dummy mass and dummy inertia are:

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(i) They should be small enough for the convergence to be reasonably fast

(ii) They should be large enough to prevent nodes from shifting too much in one cycle and significantly overshooting the point of equilibrium and the displacements diverge wildly instead of converging

It has been found that, in most circumstances, the best estimation of the dummy mass and dummy inertia can be given by the following equation:

M = 2 K (10)

In which:

M – nodal dummy mass or dummy inertia

K – nodal translational stiffness or rotational stiffness

• Compute the accelerations, velocities and displacements for each node at each cycle.

• Compute a new nodal position and rotation for each node at each cycles, update the nodal stiffness and member force imposed on the nodes.

• Check the force and moment residuals at each node at the current position.

If no residual exceeds the limit, the analysis is considered to have converged and the final position is considered as the equilibrium position of the structure.

If any residual is not satisfied, the analysis is continued to the next step.

• Compute the total kinetic energy of the structure. If the kinetic energy at a cycle overshoots the maximum, it is considered that the equilibrium position has been passed (see Figs 8 & 10). Therefore, all nodes will be re-positioned (apply kinetic damping) so that they are closer to the equilibrium position. Reset the speed and acceleration to be zero and let the structure start to vibrate again from the new position.

• After analysis has been converged, the element forces, moments and stresses are calculated according to the final equilibrium position of the nodes.

A flowchart of the GsRelax solution procedure is provided as an appendix to these notes.

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Example of dynamic relaxation analysis of one degree of freedom

The structure used in this example is shown in Fig 11 and the idealised model is shown in Fig 12. The residual limit is set to be 1.0 N in this example and the analysis procedure is as follows:

q = 10 kN/m

L = 6 m L = 6 m

Fig 11 The Structure

EA = 72,000 kN

F = 60 kN

Fig 12 The Idealised Structure

• At initial state

(i) Equivalent nodal force (also the unbalanced force at the node)

∆F=0.5(2qL) = 0.5×2×10×6 = 60 kN = 60,000 N

(ii) Stiffness

K_x = (2EA/L)= 2×72,000,000÷6= 24,000,000 N/m K_y = 0.0

(iii) Dummy mass (value equal to 2K) Since K_y = 0.0

M = 2K_x/3 = 16,000,000 kg (GsRelax uses this Mass if the relevant stiffness equals zero, divided by three since it is only in x direction)

(iv) Acceleration

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y″o = F/M = 60,000/16,000,000 = 0.00375 m/t²

• At first cycle

y′₁ = y′_o + y″_o∆t = 0+0.00375 = 0.00375 m/t (∆t = 1.0 assumed in GsRelax)

y₁ = y_o + y′_o∆t + 0.5y″_o∆t² = 0+0+0.5×0.00375 = 0.001875 m L₁ = √(L²+y₁²) = 6.0000003 m (bar length at cycle 1) F₁ = (L₁–L)EA/L = 3.6 N (force in bar at cycle 1)

K₁ = sin²(θ)EA/L + 2cos²(θ)F₁/L = 2.3 N/m (vertical stiffness at cycle 1)

∆F₁ = F – 2 sin(θ)F₁ = 60,000 N (residual force) KE₁ = 0.5 M (y′1)² = 112.5 kg m²

y″₁ = ∆F₁÷M = 60,000÷16,000,000 = 0.00375 m/t²

where: θ is the angle between original and deformed bar in mRad

• The calculation of the following cycles are summarised in the following Table

Numerical results of dynamic relaxation analysis (Fig 12)

Cycle y' y Li θ K Fb ∆F KE y'' Note

0 0.0000 0.0000 6.0000 0.00 0.0 0 60000 0.000 0.003750 initial

1 0.0038 0.0019 6.0000 0.31 2.3 4 60000 112.500 0.003750

2 0.0075 0.0075 6.0000 1.25 37.5 56 60000 450.000 0.003750

3 0.0112 0.0169 6.0000 2.81 189.8 285 59998 1012.498 0.003750

4 0.0150 0.0300 6.0001 5.00 600.0 900 59991 1799.974 0.003749

5 0.0187 0.0469 6.0002 7.81 1464.7 2197 59966 2812.299 0.003748

6 0.0225 0.0675 6.0004 11.25 3036.9 4556 59898 4048.986 0.003744

7 0.0262 0.0919 6.0007 15.31 5624.9 8439 59742 5508.627 0.003734

8 0.0300 0.1200 6.0012 19.99 9591.6 14392 59425 7187.826 0.003714

9 0.0337 0.1518 6.0019 25.30 15352.3 23041 58834 9079.407 0.003677

10 0.0374 0.1873 6.0029 31.21 23370.2 35085 57810 11169.632 0.003613

11 0.0410 0.2265 6.0043 37.73 34148.5 51287 56130.5 13434.195 0.003508 12 0.0445 0.2692 6.0060 44.84 48217.1 72453 53504.1 15832.822 0.003344 13 0.0478 0.3154 6.0083 52.52 66111.7 99407 49563.5 18302.526 0.003098 14 0.0509 0.3648 6.0111 60.72 88341.2 132940 43865.2 20749.968 0.002742 15 0.0537 0.4171 6.0145 69.40 115341.9 173744 35903.2 23044.104 0.002244 16 0.0559 0.4719 6.0185 78.48 147414.6 222318 25139.3 25011.329 0.001571 17 0.0575 0.5286 6.0232 87.87 184644.4 278846 11059.7 26436.724 0.000691 18 0.0582 0.5864 6.0286 97.42 226805.3 343049 -6736.8 27076.317 -0.000421

19 0.0578 0.6444 6.0345 106.98 273257.7 414019 -28418.3 26685.812 -0.001776 KE19<KE18

Reset 0.0000 0.5935 6.0293 98.59 232244.0 351346 -9166.5 0.000 0.000000 restart