Ship FE Models Equilibrum

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Method for the generation of the boundary condition

for a balanced Finite Element Model

Dr.-Ing. Marius PopaHull Drawings Approval Engineer - Germanischer Lloyd Romania

ABSTRACT

The scope of this paperwork is to propose a method for the achieving of the static

equilibrium of the forces loading a Finite Element Model.

Due to the specificity of the authors interest the work is focused mainly on FE Models

used for ships structure analyze.

The basic assumption is that a tool for the generation of the unitary sectional forces on

the nodes of the model end sections is available.

The method provides a practical way to compute the loads factor for a.m. unitary

sectional forces in order to balance the forces on the model.

1. Introduction

Boundary conditions are necessary to be used in order

to solve a Finite Element Model.

Up to now the common boundary conditions were the

suppressing or the prescribing of the rotations or the

displacements of the nodes.

In general - in case of prescribing of the

displacements of the nodes - specially on boundary

sections nodes the values prescribed are first

computed as results of other large but rough models.

Unfortunately this means a double effort: once for theanalyze / generation / computation / results

interpretation of the large rough model and twice

for the analyze / generation / computation / results

interpretation of the in detail model.

In case of suppressing the nodes displacements were

supported or constrained based on the modelpeculiarities as geometry / structure / loads. As an

example a seldom case of FEM structure / loads

peculiarity is the symmetry. However the boundary

conditions resulted from this type of assumption are

not all the time realistic. In general the symmetry

assumption is made for models that could be

considered as regular parts of regular structures (as in

case of bulk carriers cargo area). Other assumptionmade in this situation is that this model could be

analyzed distinct to general longitudinal bending of

the hull. The results of these computations are so

called local stresses or second order stresses. In

order to have a complete analyze of the structure

these stresses are to be superpose with the

longitudinal strength stresses as first order stresses

(see Germanischer Lloyd Rules for Seagoing Ships

Hull Structures S.8.8 Direct calculation of bottom

structure).

This model is not proper for analyzing the cargo areas

that couldnt be considered as homogenous divided or

in the case when the model is considered to carry also

the global longitudinal strength.

Other possibility to have more realistic boundary

conditions is to use a balanced model. In this case the

boundary condition consist of forces loading the

boundary nodes. Only in few nodes not necessary

on boundary sections have to be suppressed in order

to have a consistent model equations system. These

nodes are usually located in the neutral areas andspring elements element are used in order to

suppressed the displacements. In a proper balanced

model the forces resulted in spring elements have to

be close to zero in the range of numeric rounding

error of the loads.As it is known the difficulty in this third possibility is

to obtain the proper and as far is possible realistic

loading situation for the boundary nodes.

As is already stated the scope of this work is to

propose a method able to solve this aspect.

2. Mathematical model

The object of the examination is a part of ships hullwhich will be nominated further as model.

It is assumed that the ships hull is in static

equilibrium. The scope of this study is to find a

method to achieve the static equilibrium for the model

too.

It is assumed that the stresses resulted from the action

of a force loading the model are in linear relation to

the corresponding force so the model is assumed

linear.

THE ANNALS OF DUNAREA DE JOS UNIVERSITY OF GALATI

FASCICLE XI SHIPBUILDING, ISSN 1221-4620

2004

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It is assumed that the model is a coarse Finite

Element Model. It is assumed too that the in case of a

coarse model our interest is for the average values ofthe stresses so the effect of the stress distribution

along the supporting width could be ignored.

The model is load by an external force noted E. This

force is generated by own weight, cargo or ballast

loads or external water pressure (static pressures ordynamic pressures). The general force E has 6

components Ek with k= 16.

The model is separated from the ships hull which is

in static equilibrium. Stress will occur on models

transversal ends in order to achieve equilibrium.

It is assumed that the stress distribution are linear

superposition of classical beam theory stress

distributions as uniform distribution for axial forces,

linear distribution for bending moments or Saint-

Venant theory stress distribution for torsion. The

stress distributions are noted Sij with i= 12 (aft /

fore end) and j= 1..6 (spatial directions).

According to the beam theories each classic stressdistribution could be computed taking into account a

general force and end section geometric

characteristics. The generalized forces are noted Fij

(i= 12, j= 16) and could be observed further as

unknown values to be computed instead of the stressdistributions Sij.

In order to achieve static equilibrium the resultant

sum of all generalized forces loading the model has to

be null irrespective of the reference point.

The choosing of reference point is arbitrary but has

to take into account the numerical aspects in order to

decrease the dimension of the values computed

during the mathematical operations. In this way also

the accumulation of the numerical errors could be

reduced.

However the final sum results of the equilibrium

equation could not be mathematical null. According

to author experience - in general is a very small

values - at least 6 numerical orders less than the other

variable involved. Few displacements restrictions are

necessary still necessary in order to solve the FEM

numerical system.

In general these restrictions are imposed as spring

elements in a limited number of nodes. For a usual

vertical bending local loads superposition problem

the author choose 4 nodes as far as possible close to

the neutral axis of expected main stress distributions

(see example).

The spring elements will carry the residual

equilibrium forces. Practically these forces are cvasi-

null and represent possible numeric errors

accumulated during approximations and mathematic

computations.

Due to this fact the values of the forces in thesespring elements are a good indicator of the

equilibrium method efficiency.

In order to build the equilibrium equations related to

the origin point each generalized force could be

reduced to a 6 elements vector.

Each force Fij j= 13 could be reduced to vector

Fijk where Fijj=Fij; Fijk=0 j= 13 k= 1...3 and kj

and Fijk k = 4..6 is the moment of Fij in relation to

the origin point and the direction k.

For the moments Fij j= 46 the 6 elements vector is

Fijj=Fij; Fijk=0 j= 46 k= 1...6 and kj.

Using the 6 elements vectors reductions to origin

point the equilibrium equations became:

[1] k2..1i 6..1j

EFijk= = =

with k= 16

Theoretical this system is a diagonal system.

However from authors practical experience resulted

that as a result of the accumulation of possible

numerical errors the exact diagonal form is not

achieved in all cases. The explanations for the

occurrence of these residual values will be provided

further see comment for below point 4.

However the residual values are in general few

dimensional order less than the main values (basically

the diagonal line values) so these terms becameextremely small during pivoting step.

For solving this system the author propose following

steps:

1) The end forces Fij could be normalized usingstandard values (or unit values) noted Fij0 so

Fij= lij*Fij0. The unknown values are now lij

which are known as load factors. Fij0 represent

an arbitrary values for the generalized end force

Fij. These values have to be choose in such a way

to minimize the numerical errors occurring

solving a.m. system in which the unknown values

are defined above as load factors lij.

2) The standard end forces F0ij are transformed inclassic stress distributions on end section i. Therelation between F0ij and the stress distribution is

linear.

3) The standard stress distribution is reduced asforces in end section nodes. Taking into account

the assumptions for coarse model The stress

reduction is based on the stress in node observed

as average stress and the area of the elements

which are in incidence to the node. The relation

between F0ij and each force in nodes is linear

too.

4) The nodal force distribution due generalizedforce F0ij could now be reduced in relation to

origin point to a 6 elements vector F0ijk k= 16.

Please observe the comments on F0ijk on theend of the paragraph.

5) The equilibrium equation [1] became now:[2] kij

2..1i 6..1j

El*ijk0F = = =

where k= 16

The system [2] has to be solve taking into account the

unknown values the load factors lij i= 1...2 / j= 16.

FASCICLE XI THE ANNALS OF DUNAREA DE JOS UNIVERSITY OF GALATI

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Comments on point 4

Theoretically the 6 elements vector F0ijk has to have

following form:- for j= 13 (stress distribution due to the unit end

forces)

F0ijk = 0 for kj, k= 13; F0ijj= F0ij

F0ijk = the moment of F0ij related to origin

point and direction k- for j= 46 (stress distribution due to the unit endmoments)

F0ijk = 0 for k= 13

F0ijk = 0 for kj, k= 46; F0ijj= F0ij

However according to the authors practical

experience some terms which theoretical have to be

null are not. This situation is a result of the hypothesis

regarding the stress distribution reduction to the

forces on nodes (closely related to the coarse model

assumptions).According to the classic theories the sum of the stress

distributions are null on the directions other than the

direction of the generalized forces which generate

them:[3] 0dA

C

= where

C is the section, is the stress, dA is the elementary

area from section C where is located the stress .

For the purpose of the computation of formula [3] are

used numerical approximations as:

[4] 0A* ii

mi = where

i is the node from the section C, mi is the stress in

node i assumed as average stress, Ai is haft of theelement areas which are incident in the a.m. node.

The non null values are errors which occur from the

numerical computation of formula [3] usingnumerical formula similar to [4].

The steps proposed above are a possibility. As an

alternative the steps could be as follows:1. Forces normalization by standard unit ends

forces: Fij= lij*F0ij.

2. The standard unit end forces are reduced to theorigin point: F0ij= (F0ijk) k= 16.

3. The equilibrium system [2] is solve taking intoaccount the loads factors lij as unknown values.

4. The end forces Fij are computed as lij*F0ij5. The end forces Fij are transformed in standard

stress distributions.

6.

The standard stress distributions are transformedin nodal loads in order to be included in the

Finite Element Model.

As could be observed the numerical errors are

transferred into the step 5 and are introduced in

equilibrium equations on the end of the balancing

process.

In this way all the numerical errors are residual values

in equilibrium equations. These residual values as

residual generalized forces are transferred in the

spring elements requested by the solving of FEM

system.

The residual forces in spring elements couldcompromise the stress distribution in areas nearby

spring elements - situation which is not

recommended.

3. Solving the equilibrium system

The equilibrium system [2] could be transformed in:

[5] k6...1j

jk02j2jk01j1 E)F*lF*l( =+=

, k= 16

The system has 6 equations and 12 unknown values

load factors - l1j, l2j j= 16 and couldnt be solve

without the reduction of the unknown values to 6

(initial conditions).

The easiest possibility to solve the system is to

assume that the end load factors corresponding to the

same direction j are equal l1j= l2j.This assumption could be accepted for the direction

which are a priory assumed as without significant

influence on the study. For example in case of a

model which is in study for vertical general bending

this hypothesis could be made for the directions 1

(axial force - thrust), 4 (torsion) and 6 (bending

around vertical axis horizontal bending). Basically

this assumption could be made in all cases for

direction 1 axial forces. However for the study of a

heeled model or for the study of a model in

transverse wave this assumption for directions 4 and 6

could not be made anymore.

Other possibility is use for one end and for a directionan already known values for the end force. In this

situation the best example for this possibility is the

study of the vessel in still water condition. The still

water vertical bending moment (BM) and the still

water sheare force could be achieved from the

computation of the loading case.

In general the initial conditions which are used for a

study could be a combination of the hypothesis

above. The problem of the system equation solving is

open for further comments. The author is open to

receive such comments or suggestions.

4. Example - Description

As example is presented a computation for a long mid

ship area in still water condition. In this situation the

main aspects are related to the general strength

vertical longitudinal bending.

The Finite Element Model was generated and solved

using Germanischer Lloyd Computer Aid Design

software Poseidon - official version 4.0.

THE ANNALS OF DUNAREA DE JOS UNIVERSITY OF GALATI FASCICLE XI

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In order to make easiest any further checks a very

simple box structure was considered.

The dimensions of the box are: L= 100.0 m / B= 10.0m / H= 10.0 m / CB= 1.0

This box has the normal frame spacing of a= 1.0 m.

In this assumptions a frame numbering could be

assigned assuming Frame 0 at longitudinal ordinate

x= 0.0 m. The most forward frame is Frame 100 at x=100.0 m.

The box has transversal rings (web frames) at each 2a

spacing and transversal watertight bulkheads at each

10th

frame (Frames 0, 10, 20, 90, 100).The transversal section topology consist of followings

elements: shell / main deck / a central line

longitudinal bulkhead and a double side longitudinal

bulkhead.

From the same reason as above a very simplestructure was modeled. Following two pictures are

relevant.

Fig. 1 Box structure simple frame / web frame and transversal bulkhead

Fig. 2 Loading scheme

S h e a r F o r c e [k N ]

-1 0 0 0 0

-8 0 0 0

-6 0 0 0

-4 0 0 0

-2 0 0 0

0

2 0 0 0

4 0 0 0

6 0 0 0

8 0 0 0

1 0 0 0 0

1 5 9 1 3 1 7 2 1 2 5 2 9 3 3 3 7 4 1 4 5 4 9 5 3 5 7 6 1 6 5 6 9 7 3 7 7 8 1 8 5 8 9 9 3 9 7 1 0 1

B e n d i n g M o m e n t [k N m ]

-2 0 0 0 0 0

-1 8 0 0 0 0

-1 6 0 0 0 0

-1 4 0 0 0 0

-1 2 0 0 0 0

-1 0 0 0 0 0

-8 0 0 0 0

-6 0 0 0 0

-4 0 0 0 0

-2 0 0 0 0

0

1 5 9 1 3 1 7 2 1 2 5 2 9 3 3 3 7 4 1 4 5 4 9 5 3 5 7 6 1 6 5 6 9 7 3 7 7 8 1 8 5 8 9 9 3 9 7 1 0 1

Fig. 3 Still water Shear Force (SF - kN) and Bending moment (BM- kNm)


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The actual sectional characteristics are: Wbottom =

2.299 m3

and Wdeck = 2.340 m3.

The box is loaded according to the loading scheme inFig. 2. The fluid is assumed to have a density of 1.019

t/m2 and the draught achieved in still water is 4.05 m.The still water bending moment and shear force

distribution are in accordance to Fig. 3. For the

middle area the extreme BM value is 198751 kNm.For this value the expected longitudinal bending

stresses for bottom and deck are 86.5 N/mm2

respectively -84.9 N/mm2.

A Finite Element Model was generated by Poseidon

between Fr. 14 and 86. For aft and fore end

transversal sections (Fr. 14 and Fr. 86) the boundary

condition 2 unit force / St. Venant unit force / unit

moment was used. According to Poseidon users

manual the 2 boundary condition means:

Quote

Nodal forces in separate load groups acting on the

hull cross-section according to the stress distribution

for the relevant direction. For the XX-direction nodal

forces according to St.Vernant torsional stress

distribution are calculated.

Unquote.According to the mathematic model below the

boundary conditions generated on ends by Poseidon

are equivalent to the standard unit forces noted F0ij, i=

1...2, j= 16.

For a better understanding the Figures 4 shows thenodal loads of the unit forces on global directions 3, 4

and 5 for aft end (vertical forces shear / torsion and

general longitudinal bending).

Thirteen global load cases (GLC) were generated.

First twelve load cases are the standard unit forces

assumed with load factor 1.

Last load case are the external loads - in this case the

still water pressure and the cargo load the static

pressure of the fluid in tanks.

The Poseidon FEM solver provide the sum of the

forces on the spatial direction for all load cases. This

sums are show in table 1:

Fig. 4 Nodal loads generated by the standard units loads on the directions 3, 4 and 5

GLC Foce [kN] Moment [kNm]

No. x y z xx yy zz

1 10003.4 0.0 0.0 0.0 50505.7 0.0

2 0.0 -10196.4 0.0 50170.8 -0.6 -142749.8

3 0.0 0.0 -10064.7 2.6 140906.0 0.3

4 0.0 0.4 1.0 -6498.7 -13.6 5.1

5 28.4 0.0 0.0 0.0 101282.4 0.0

6 0.0 0.0 0.0 0.0 0.0 -100867.6

7 10003.4 0.0 0.0 0.0 50505.7 0.0

8 0.0 -10196.4 0.0 50171.0 -3.9 -876891.5

9 0.0 0.0 -10064.7 2.6 865564.0 1.8

10 0.0 0.4 1.0 -6498.7 -83.4 31.6

11 28.4 0.0 0.0 0.0 101282.4 0.0

12 0.0 0.0 0.0 0.0 0.0 -100867.6

13 0.0 0.0 -10985.7 0.0 549282.5 0.0

Table 1 Sum of the forces for the Standard End Unit Forces and External Forces

The values in table 1 are the terms of the system [2].

The rows in table are the terms of the equilibrium

equation for each of the 6 directions. For the row k,

k= 1..6 the first 6 values are the terms F01jk. The


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next 6 values are F02jk and the last value is the

component of the external force on direction k Ek.

As could be observed the system matrix is not exactlydiagonal. In this example for direction 4 and 5

(torsion and vertical moment) some residual forces

are achieved. In case of direction 4 (torsion) the

residual forces are generating also residual moment

on directions 5 and 6.However dimension of the residual is at least 3rd

order

less that the dimension of the main term of the

equilibrium equation.

For solving the equilibrium system followings initial

condition are used:

- Direction 1 axial force - trust: equal loadfactors l11=l21. The still water condition

assume that the influence of the trust force is

not significant so the possible axial forces

are extremely small.

- Direction 2 lateral force: equal load factorsl12=l22. The still water condition assume

that the model has transversal symmetry asgeometry and load so the lateral force is null.

In this situation the equal load factors

assumption has no influence on the model

behavior.

- Direction 3 vertical forces shear force:the value on the aft end is assumed known

from the analyze of the still water load case.

F13= -5572 kN according to SF distribution.

- Direction 4 torsion: equal load factorsl13=l23. The still water condition assume

that the model has transversal symmetry as

geometry and load so the torsion moment is

null. In this situation the equal load factors


behavior.

- Direction 5 vertical bending moment: thevalue on the aft end is assumed known from

the analyze of the still water load case. F15=

-39004 kN according to BM distribution.

- Direction 6 horizontal bending moment:equal load factors l16=l26. The still water

condition assume that the model has

transversal symmetry as geometry and load

so the horizontal bending moment is null. In

this situation the equal load factors


behavior.

Solving the equilibrium equation system the loadfactors are achieved.

The final Global Load Case is the combination of the

Standard End Units Forces multiplied with the load

factors computed above and the external loads (load

factor 1.0).For this case the equilibrium equations are cvasi-null.

Only residual values of 0.4 and 0.9 kNm are

achieved for the directions 5 and 6 but these residual

values are considered extremely small in relation to

the forces involved.

In order to solve the FEM system spring elements are

introduced in 4 nodes.

Fig. 5 Spring elements

The nodes are located in the mid area of the model

Fr. 49 and Fr. 51 - as far is possible in the neutral axis

of the main loads. In the example model two nodes

are on side close to the vertical bending neutral axis -

loaded with springs on vertical displacement. Other

two nodes are on bottom and main deck close to the

center line - loaded with springs on axial and lateraldisplacements.

Running the FEM the forces in spring elements are

maximum 0.1 kN for x direction and 1.1 kN for y and

z direction. These forces are the result of the residual

moments indicated above.

5. Example - results

As results the deformed view and the general stress

map will be presented.

Fig. 6-1 Deformation View on top


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Fig. 6-2 Deformations Lateral view

Fig. 6-3 Deformation General view

Fig. 6-4 Deformation General view / the model is not hide


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Fig. 7-1 Stress Equivalent Von Misses Stresses Deck and Longitudinal bulkhead in CL

Fig. 7-2 Stress Equivalent Von Misses Stresses Bottom and Shell

Ordinary elements as far as possible not affected by

the local stresses were selected for the comparison togeneral longitudinal bending stress as where

computed previous for the midship area The stresses

in main deck in a average element area. The

longitudinal stresses determined for main deck are

about -77 -82 N/mm2

estimated -85 N/mm2 For

bottom the stresses are about +89+93 N/mm2

estimated 86.5 N/mm2.

The difference are about 3 8 N/mm2 which means

maximum 10%.

The differences between theoretical longitudinal

bending stresses and computed longitudinal stressesare due to the increased participation of the lower

flange material.

Due the water load assumed as uniformly

distributed load - the effective width of the bottom

longitudinal is usually assumed greater than the

effective width of the unloaded main deck

longitudinal.

In actual case this assumption justify the increased

participation of the material of the lower flange.


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Fig. 8-1 Stresses parallel to longitudinal direction View on top - deck

Fig. 9-2 Stresses Paralel to longitudinal direction - bottom

6. References

Germanischer LloydRules Chapter 1 - Guidelines for StrengthAnalyses of Ship Structures with the Finite Element Method,

Volume V. - Analysis Techniques, Part 1 Strength and Stability

, Edition 2001Germanischer Lloyd - Poseidon Help, Poseidon Revision 4.0 august 2004

Paper received at 15.09.2004

7. AbstractScopul lucrarii este de a propune o

metoda practica de echilibrare statica a

fortelor care actioneaza asupra unui

Model de Element Finit

Datorita interesului specific al autorului

metodata este prezentata pentru un Model

de Elemente Finite specific analizelor

structurilor navale.


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Se presupune de la inceput ca se dispune

de o unealta (un algoritm automatizat sau

un software) prin care pot fi calculate /generate fortele nodale pe sectiunile de

capat ale modelului datorate

solicitarilor unitare de baza (forta si

moment unitary pentru toate cele sase

directii spatiale).

Metoda ofera o cale practica pentru calcularea

factorilor de incarcare pentru fortele unitarea

sectionale mai sus numite astfel incat sa seobtina echilibrarea fortelor care actioneaza

asupra modelui.

Fig. 9-1 Main Stress View on top - deck

Fig. 9-2 Main Stress View on side shell


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