The Membrane Element

The membrane element implemented in ASE can be used only for calculations with large deformations with SYST... PROB TH3. It is activated with the mate- rial input AQUA NMAT MEMB or with an input in SOFiMSHA...QUAD...NRA=2 (QUAD only with membrane action). The program configuration levels ASE1-3 are necessary for the material non-linear element and ASE4 for compression

failure. Properties

The membrane element described here is implemented in the FE program ASE of the SOFiSTiK AG. It has following properties:

• The membrane element processes only membrane internal forces and mo- ments (NX, NY, NXY).

• It bears arbitrary large strains and rotations.

• It bears large twists and transmits the membrane forces from the twist into the right direction (here forces are available perpendicular to the thought element centre area).

• It is possible to use three-noded or four-noded elements for it. • A prestress can be defined (also orthotropic).

• Stress modifications can be suppressed for the formfinding. • It failures for compression (adjustable).

• Orthotropic material properties can be considered (linear-elastic approximation). For anisotropic poisson’s ratio see example

membrane_poisson_ratio.dat

Figure 2.25: Nodal forces at twisted membrane element

The stiffness of the membrane element consists of the normal strain stiffness in the element plane and of a initial stress stiffness from the prestress of the element.

K = K₀+ Ks (2.46)

Input of the Membrane element

If the element formulation NRA=2 (see SOFiMSHA-QUAD) is set immediately, the element is marked as membrane. Otherwise a normal QUAD element can be defined as membrane with a non-linear material input AQUA NMAT

Non-linear properties can be activated in AQUA with AQUA NMAT MEMB P1 P2.

P1 Yield strength for tension

maximum tensile strength in kN/m in warp and fill direction, practical e.g. for geo-textiles

The input P1=0.0 is taken as ’no input’. The old input P1=1.0 is not considered

(P1 was used formerly with another meaning.) P2 Factor for compression survey

P2=1.0 The membrane can sustain the compression. P2=0.0 The membrane cannot sustain the compression. (only reasonable after formfinding)

P2=0.1 Intermediate values are possible, the elastic modulus is reduced correspondingly for the

compression strains.

Nonlinear warp-fill material law

In ASE a "‘Nonlinear yarn-parallel warp-fill behaviour"’ is implement according to: Dr. Cédric Galliot + Dr. Rolf Luchsinger ’A simple non-linear material model for PVC-coated polyester fabrics’ Tensinews Newsletter Nr. 18 April 2010

The definition of additional parameters AQUA NMAT MEMB P3+P4 activates this law. The nonlinear behavior is expressed as a stress-strain relation. This means that for a given stress sigma-w, sigma-f (w=warp direction, f=fill direction) it gives a corresponding nonlinear strain eps-w, eps-f. The values of the stress- strain matrix depend on the ratio of sigma-w to sigma-f using the factors γ_{ und} γƒ: γ_= σ_ Æ σ 2+ σƒ 2 γ_ƒ = σ_ƒ Æ σ 2+ σƒ 2 (2.47) Stress-strain relation:   ε_ ε_ƒ  =   1 E_(γ) −νƒ E_(γ) −νƒ E_(γ) 1 Eƒ(γƒ )     σ_ σ_ƒ   (2.48)

with the stress-ratio depending warp and fill stiffness: E_(γ_) = ΔE_  γ_{− p}1 2  + E1:1_{ (2.49)} Eƒ(γƒ) = ΔEƒ  γƒ − 1 p 2  + E1:1_{ƒ (2.50)}

Special Features for the System Input

The system should be already defined, if possible, three-dimensionally with boundary arches. The boundary cables can be introduced then with full stiff- ness in the first formfinding step, because they have already the right length. The three-dimensional input has also the advantage, that the span cables and columns can be already input in the three-dimensional system. Then the still inaccurate form of the membrane is smoothed via”shrinkage”of the membrane - see formfinding.

Only for systems with high reference point it is reasonable to input the system at first two-dimensionally, because the input is significantly simpler here. The membrane can be hoisted then at marked points via nodal point displacements. Mesh selection

Automatically generated meshes are unproblematic for systems without high reference points. They should be avoided at high reference points. At high reference points a radially and tangentially oriented mesh is numerically more stable and optically more beautiful due to the often orthotropic prestress.

Mesh macros

Pregenerated macros can be used for high reference points. Macros which are read in such a way are optimized for the registration of the stress conditions at the high reference points and delivers a good geometry for high reference points (The distance of the inner elements is selected deliberately near in the initial system, because they are stretched due to the hoisting during formfinding). The macros are placed in the plan, adjusted to the size (stretched) and the remaining membrane area is closed with a normal element mesh.

Boundary cables

Boundary cables should be always defined with the desired final curvature at an arch during input in the plan - see chapter ”Free Cable Edges defined in the

Figure 2.26: A boundary cable left with small, right with large prestress

Figure 2.27: Left a too small, right a too large prestress

Initial System with Radius”. Mixed systems

If the membrane should be calculated together with other structural member- s (walls, pylons, girders), the input is mostly urgently necessary with three- dimensional initial system.

Prestress and Formfinding

As in outline mentioned in chapter ”Overview”, the prestress is decisive for the formfinding. Different membrane forms can be generated with different pre- stressing states.

This phenomenon becomes especially clear for boundary cables: If a boundary cable is more prestressed for a given membrane prestress, a larger cable radius will result and thus a smaller pass of the boundary cable:

At high reference points a too large prestress ties up the ”neck”:

The user has to be known the desired form at the beginning. The pass of the boundary cable should be used already during the system input. The input FE mesh should include therefore the boundary cable curvature.

Soap skin

In a soap skin an isotropic prestress is available in all points of the membrane. This prestress is determined about the surface tension of the liquid for the gen- uine soap skin.

The strain stiffness disappears here in the mathematical model. The equilibrium results only from the three-dimensional equilibrium of the isotropic stresses. The stiffness of the membrane results to:

K = Kσ (2.51)

The stiffness keeps the membrane in its form perpendicularly to the membrane area. Thought points are freely movable in the plane of the membrane area. For the genuine soap skin the phenomenon is visible at the blurring of the points (bubbles) on the skin surface.

The in all directions constant prestress is input in ASE with the record GRP ... PREX,PREY (acts on all element types, also on cables, beams ...).

Constant orthotropic prestress

The direction of effective span is often dominating in one direction for rectangular membrane areas. Then it is desired to set a larger prestress in this direction than perpendicularly to it. Nevertheless the prestress is of the same size in all points, if also orthotropically.

Figure 2.28: Orthotropic prestress - in longitudinal direction larger than in trans-

verse direction

The orthotropic constant prestress is input in ASE either with the record GRP

... PREX,PREY in local element direction or with the record HIGH with a high reference point distance >_{999 m in global direction.}

Orthotropic high reference point prestress

If genuine high reference points are available for membranes, the orthotropic prestress is often desired with a fixed ratio of tangential/radial prestress in order to avoid a large tying up of the membrane at the high reference point. A radial stress which increases to the high reference point is necessary for that.

A such axisymmetric stress state of the high reference point is generated with

HIGH. The ratio of the tangential to the radial prestress is input with the item PTPR. In dependence on PTPR the stress increases much or not so much to the high reference point.

Example of a high refence point at X = 5.0 m, Y = 0.0 m:

HIGHX 5.0 Y 0.0 PR1 20 PTPR 0.4 produces: sig-r in distance of 1m = 20.00 kN/m

sig-t in distance of 1m = 8.00 kN/m (0.4*20) and due to equilibrium reasons in distance of e.g. 10 m: sig-r in distance of 10m = PR1*1/r*eˆ(PTPR*ln(r))

= 20*1/10*exp(0.4*ln(10)) = 5.02 kN/m

sig-t in distance of 10m = 2.01 kN/m (0.4*5.02) see example file: high_point.dat

input: HIGHXM YM ZM NX NY NZ PR1 PTPR NOG

As a default an input for a high reference point has an effect for all QUAD el- ements, also for elements which are not a membrane. For mixed systems the prestress is allocated therefore with NOG to the corresponding group. It is also possible to input some high reference points per group. The program generates then the average value from the inputs in each element in dependence on the distance to the different high reference points. In the following example there are four high points and one low point in a membrane area. The tangential part PT- PR may not be too large for the high reference points, because the membrane constricts itself and tears off. The factor PTPR is input therefore different for the five high reference points in this example.

Figure 2.29: Orthotropic high reference point prestress with some high reference

If the distance is larger than 1000 m, the constant prestress is assumed with a stress in direction to the high reference point always of PR1 and a stress per- pendicularly to it always of PTPR·PR1 and therefore without radial reduction. The advantage of this input is the simple definition of skewed prestress inde- pendently on the direction of the local element coordinate systems! The stress in the QUAD elements results from the global directions!

Elastic skin

A membrane can be defined from the beginning with the real stiffness and can be hoisted from the plane initial system at the high reference points or at the boundary cables. It results then large stresses in direction to the high reference points in dependence on the material properties. They can be scaled, however, by using this state with the group factors FACL+FACS. The use of a elastic skin formfinding is described in chapter ”Unstable Membrane Forms”.

Input of the Prestress for Different Groups Definition in different groups

For membrane analyses the system has to be got already in the first step an information about a prestress in the elements, because otherwise the system is unstable - the stiffness is zero perpendicularly to the membrane without pre- stress! A load prestress is still not considered for the system stiffness. The prestress has to be input therefore with GRPor HIGH.

The different elements of the structure like: • membrane areas

• boundary cables • structural cables

• pylones and other beams

• massive support elements (concrete walls ...)

are defined in different groups and can get thus different prestresses from GRP

and HIGH.

If different radii in boundary cables should be kept exactly (formfinding also for boundary cables), then also the boundary cables should be defined in differ- ent groups. If the boundary radii were already input graphically (is absolutely recommended), it is possible to refrain the exact input of the boundary cable prestress, because the boundary cable force results from the radius and the membrane prestress during the formfinding.

Criteria for the Input of the Prestress

Free cable edges (free membrane edges reinforced with cables) should be input already as arch with the desired curvature radius in the initial system. The radius is preset in any case by the architect. If the boundary cable is defined as line in the initial system and the final edge circle should be determined by the program, then impermissible element angles are often available due to the distorsions. The iterations are much faster and clearer, if the edge arch has approximately the final position already in the initial system.

The prestress which should be input for the boundary cable results to: cable force = membrane force radius P = n ·r

It is to be noted, that physical impermissible inputs do not arise. Unconsistent inputs can arise especially at the connection points of cables. In the following example an equilibrium is possible without an angle of the cable forces, because P1 >_P2+P3+P4.

cable 1

cable 2

cable 3

Figure 2.30: Illogical preset cable prestress can not be right

2.13.3 Formfinding

System Definition - Two Options

The initial structure can be defined with two options for the formfinding:

• Definition of a three-dimensional initial system with at first plane partial ar- eas: The boundary points of the structure are input three-dimensionally. The remaining areas are defined e.g. as folded structure. The program takes over the formfinding of the inner area.

• Definition of a plane initial system: The structure is input two-dimensionally. At arbitrary points the structure is”hoisted”then at support nodes.

Three-dimensional Initial System

Three-dimensional initial system without cable edges Example angle, example file simple_angle.dat.

A system is defined three-dimensionally from two planes (folded structure). All boundary points are supported, the lower edge is free.

Figure 2.31: Three-dimensional initial system - angle

The membrane prestress is defined isotropic with GRP ... SIGX SIGY in kN/m during the formfinding step. Because the strains should not lead to stress mod- ifications due to the formfinding, the element stiffness is set almost to 0: GRP

FACS=1E-10. The QUAD elements with the material number 1 are defined as membrane elements ( AQUA NMAT 1 MEMB). ASE input:

PROG ASE

HEAD Formfinding for Three-dimensional Initial Systems

SYST PROB TH3 $ for geomatrical non-linear iterations $

GRP 0 FACS 1E-10 PREX 10 PREY 10 $ prestress definition 10 kN/m $

LC 1 TITL Formfinding $ formfinding without further load $

END

A load case with real 1.0-times stiffness should be follow after each formfind- ing load case for the check of the formfinding in order to guarantee that possible constraints do not lead to impermissible differences during formfinding (see con- straints during formfinding CTRLFIXZ 1).

PROG ASE

HEAD Compensation of Possible Residual Forces

SYST PROB TH3 PLC 1 $ uses the primary load case 1 $

GRP 0 FACS 1 $ elemets with full stiffness, stresses $

LC 2 $ are used from load case 1 (see record GRP) $

END

The iterations are necessary due to the effects from third-order theory. The vertical force parts (sinus(α_{) 6= α) change due to the large displacements. In}

addition the element geometries change also in part considerably. The first ASE calculations ends successfully after 9 iterations:

Iteration 1 Residual 1.889 energy 22.6089 Step 1-1 f= 1.000 Update nonlinear stiffness

Iteration 2 Residual 0.239 energy 30.7733 Step 2-1 f= 1.487 Iteration 3 Residual 0.222 energy 32.4090 Step 3-1 f= 1.814 Update nonlinear stiffness

Iteration 4 Residual 0.134 energy 32.7557 Step 4-1 f= 1.838 Iteration 5 Residual 0.017 energy 32.6185 Step 4-2 f= 0.604 Iteration 6 Residual 0.008 energy 32.6450 Step 5-1 f= 0.607 Update nonlinear stiffness

Iteration 7 Residual 0.003 energy 32.6701 Step 6-1 f= 1.178

The convergence has to be checked by the user. Indeed the programs prints a warning in the case of inadequate convergence, but it saves the results never- theless.

The result of the formfinding of load case 1 is shown in the following picture. The load case 2 does not deliver any modifications. The check of the formfinding does not show disturbances.

Figure 2.32: Result of the formfinding -three-dimensional initial system angle

For orthotropic prestress other forms which are all free form areas result in de- pendence on the prestress condition:

Figure 2.33: V-long/V-lat=1:5 V-long/V-lat=1:2 orthotropic prestress -three- dimensional initial system angle-

Corresponding input files: PROG ASE

HEAD Angle with Orthotropic Prestress

SYST PROB TH3

GRP 0 FACS 1E-10

HIGH 9999 0 PR1 10 PTPR 0.2

$ PR1 = prestress radial in a distance of 1m from high reference point $

$ PTPR = prestress ration tangential/radial $

LC 1 TITL Formfinding END

Free Cable Edges defined in the Initial System with Radius Example stand roofing, example file roof.dat

If possible, a cable radius should be considered already during the graphical input. That means the cable should be input in an arch (see chapter ”Boundary cables”).

Following system was generated three-dimensionally as folded structure with plane partial meshes during a graphical input. The cable edges are displaced only horizontally in the plane at a circle:

Figure 2.34: Stand roofing - initial system plane left and angular picture right

Group classification:

group 1: membrane expected membrane force X-Y=10-5 kN/m group 2: edge cable left expected radius = 16 m

group 3: edge cable right expected radius = 46 m

Here the cable radius is preset instead of the cable force. The membrane pre- stress should have 10 kN/m in x direction, however, only 5 kN/m in y direction! Thus a first estimated cable force of P = n ·r with a membrane force n=10 kN/m perpendicular to the cable results (group 2: N = 16m· 10 kN/m = 160 kN).

Because the cable radius is not to be modified significantly, the cable elements are considered with their normal stiffness (GRP ... FACS 1.0) during the calcu- lation. A cable force modification is possible thereby. Here it is important, that the radius of the input is kept approximately in the final result (specification of the architect).

Otherwise the membrane should be kept the stress. The membrane stiffness is set therefore as usual with GRP... FACS 1E-10:

PROG ASE

HEAD Formfinding

CTRL CABL 0 $ without inner cable sag of the single cable $

SYST PROB TH3

GRP 1 FACS 1E-10 PREX 10 PREY 5 $ membrane 10 KN/m - 5 KN/m2 $

GRP 2 FACS 1 PREX 160 $ cable N= p*r = 10*16 = 160KN $

GRP 3 FACS 1 PREX 460 $ cable N= p*r = 10*46 = 460KN $

LC 1 DLZ 1 TITL ’Formfinding with DL’ END

The dead load is used simultaneously. The form is searched therefore for the loading prestress + dead load. Only the elimination of possible constraint forces is done again in a following calculation in load case 2:

PROG ASE

HEAD Compensation of Possible Residual Forces with FACS=1.0

SYST PROB TH3 PLC 1 $ uses primary load case 1 $

GRP - FACS 1 $ elements now with full stiffness, stresses $

LC 2 DLZ 1 TITL ’end of formfinding FACS=1.0’ END

Because the displacement picture is not different for load case1 and 2, only the final result of load case 2 is shown here:

Figure 2.35: Found form with prestress + dead load

Free cable edges defined straightly in initial system Example angle, example file simple_angle2.dat.

Such a process should be avoided, because the QUAD elements are deformed possibly impermissible during the deformation of the boundary cable. This dis- tortion and rotation of the QUAD elements is very unfavourable for orthotropic prestress, because the local coordinate system of the elements and the direction of the orthotropic prestress are turned.

Following example should demonstrate nevertheless the possibility of the formfinding for cable edges which are input straightly. The first example

simple_angle.datis so modified, that a upper boundary is defined as free edge (without support conditions) and a boundary cable is generated at the boundary nodes. The membrane is defined in group 0 and the cable in group 1.

The iteration is very fast for the system and the result is reasonable, because boundary cable curvature does not distort the QUAD elements. The cable radius is resulted always according to following formula:

or r = P / n = 8 kN / 2 kN/m = 4 m

Figure 2.36: Free cable edge - result of the formfinding

Plane Initial System

Plane initial system without high reference point Example file innenhof.dat

Without additional elements like columns it is possible to define systems in plane and to hoist at corners. Only corner nodes in the plane, boundary cables with desired edge radii as well as meshes which are hooked in are generated here. The system is simple hoisted then at the corner nodes about the support dis- placements. The membrane becomes mostly a soap skin prestress which is input with GRP ... PREX,PREY. The boundary cables have mostly a fixed ra- dius. The first estimation of the prestress of the boundary edges results from the membrane force multiplied by this radius.

Figure 2.37: Patio - left plane initial system - right result of the formfinding

Plane initial system with high reference point

The plane system input is very advantageous for systems with high reference

In document ase_1_sofistik analysis (Page 63-84)