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Analysing floating structures and dynamic systems by using the concept of virtual supports and by software

in static mode.

Eng.W.K.R.Peiris,Eng.Harsha Kumarasinghe Abstract: In general a mathematical analysis of a structure for its displacement and stress-strain with loads cannot be carried out if adequate coordinates of fixed points in the structure required to define all the points of the structure are not given. Such structures may be named as floating structures and in practice these appear very often. These are very much observed in ancient buildings. The concept of virtual supports provides a method to analyze such structures without influencing its original stress-strain conditions. This paper illustrates the method by using an actual example (roof structure of kulasinghe auditorium at NERDC) and discusses how this concept could be used to analyse dynamic systems by using software under static mode. Keywords: floating structure, virtual supports, static mode

1. Introduction In general stress-stain analysis a body under internal and external forces will be analysed. In this case 2nd & 3rd laws of Newton, stress-strain (sheer & direct) relationships of the material and geometrical compatibility conditions will be taken in to consideration. In compatibility internal materialistic compatibility (ie absence of internal discontinuity such as cracks, separated shear plans etc.) and external boundary conditions will be considered. External forces acting on the body could be devided in to two parts viz (i) forces-movements acting through constrains (ii) forces-movements acting freely on the body.

2. General stress-stain equations These stress equations are written by applying 2nd & 3rd laws of Newton for an infinitesimal part of the body. The equations relating displacements of the points in the body and direct and shear straining and rotation of the infinitesimal part will be written by considering the geometry (with continuity). For any point there exist 06 independent stress components viz 6x, 6z, 6y (direct stresses in x, y, z directions respectively). Zzy, Zyz, Zzx (shear strains) in 3 places

Figure 1- Stresses on infinitesimal element

If gravity is in Z direction and the acceleration of the point in fx i + fy i + fz k and density 0 at the point is ρ Applying p=mf to a infinite smile part of the point will yield following stress relations,

If the displacement of the point is U,V,W (in 3 directions respectively).Direct strain,

Shear strain,

Y

Z

Y

σz

σy

σx

C B

A

Zyz

Zxz C

X

Zzy Zxy

B X

Z

Zzy

Zyx

A

Y

X

Z

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Rotations of the point in 3 directions respectively.

Now for a system (body) with linear elastic relation and noting the relationship.

Where, E = Young‟s modules

σ = Shear Modules

Now if the strains field (direct & shear) induced in the body due to external and internal forces (and due to acceleration) are known, direct & sheer strain field could be determined.

3. Determination of displacements, Stress-strain under static conditions When a body (structure) is subjected to external forces/moments (from supports, constraints, free forces) it will be deformed and a stress-strain field will be developed. Now if the supports does not rigidly fix the structure with reference to a frame fixed with inertial frame (in most cases the earth) the displacements of the structure will be ambiguous, i.e. With the same force/moment system and for the same stress strain field different displacement fields may occur. In other words this presents a situation where boundary values for displacements are not sufficient to exactly solve the partial

differential equations related to displacements. These phenomena could be verified by using

the theory of virtual work for deformable structure. Now for a structure without adequate supports, let us apply few more supports (linear and / or angular restraints) to the structure such that no forces and/or moments will be transferred to the structure through these under the given loading systems. (This zero force/moments conditions should be verified simply by statics-moments equation) and the structure will be fixed to the reference frame, and hence the displacement field will be deterministic. Whether the structure is fixed to the frame of reference could be determine by checking the existence of force/moments values at supports so as to keep the structure in equilibrium under any external force/moments condition that means the supports (original and newly applied) fixes the structure to the frame. Now these newly added supports (virtual supports) will provide a method to determine displacements unambiguously without altering the original force, moment, stress-strain systems in the structure. In other words the virtual supports provide the required boundary condition for the system to solve the displacement field. For a particular totally/semi floating type structure with respective loading system there exist infinite number of virtual support system and each support system will results in different displacements (fields) but with same stress-strain field. 3.1 Totally floating, semi floating and totally supported structures

Totally floating type: If a structure (body) does not have any supports and it will be in equilibrium only with external forces/moments (ie the external forces/moments should be in equilibrium) such structure could be known as a totally floating structure.

Semi floating type: If a structure has few supports which are unable to fixed the structure to the frame to withstand any external force/moment condition such structure could be known as semi floating type structure.

Totally supported type: If a structure has sufficient supports to fixed it to the frame, ie these supports can with stand any external force/moment condition such structure could be known as totally supported type structure.

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3.2 Introducing virtual supports

Let us consider the totally floating body (structure) let us support it at points (1,0,0), (0,0,0), (1,1,1) to restrict linear displacements.

External linear forces are

External moments are And this external load system will be reduced to a force F & moments M at (0,0,0) Where,

and

Figure 2 – Virtual supports on structure

Let us take forces at (0,0,0) =

Let us take forces at (1,1,1) =

Let us take forces at (1,0,0) = Than

[A][X] =[B]

Now the determinant of [A] ≠ 0 therefore for any value of Fx,Fy,Fz,Mx,My,Mz (that is for any systems of external forces/moments).There are unique values for fx1, fy1 ,fz1, fx2, fy2, fz3 that is this support will keep the structure stable under any load system. Also if all Fx,Fy,Fz,Mx,My,Mz

are zero (that is the system of forces are in equilibrium) all the values of fx1, fy1 ,fz1, fx2, fy2, fz3 will be zero therefore no influence to the structure. Therefore the above support system is a “virtual support” for the structure with external forces/moments in equilibrium acting on the structure. Now with this virtual supports, displacement, stress-strain fields of the structure for any force/moment systems (Which are in equilibrium) acting on the structure could be determined. The same procedure is applied for semi floating type structures also. 3.3 Analysing of stress-strain with software

When softwares are used to analyse stress strain generally initially it is required to introduce the supports. Therefore, for totally & semi floating type structure the above said virtual supports has to be introduced, otherwise the software will not work. 3.4 Analysing the roof structure of Kulasinghe Auditorium at NERDC (A semi floating type structure)

The structure is a semi floating type. Therefore the displacement (linear/rotational) boundary conditions are not adequate to determine displacement, stress-strain field. If could only withstand vertical loads (weights of structure and hanging objects).Now if we introduce following supports at „0‟

a) A support restricting linear moments in X &Y directions.

b) A support restricting angular moments in Z direction.

Now if various forces and moments are applied to the structure and if it could be represented at „0‟ as.

Fx,Fy,Fz (Forces) Mx,My,Mz (Moments)

And under the loads forces on suppots restricting linear movements X & Y directions are fx,fy respectively and moments on support restricting angular moments in Z directions is mz.

Z Mi

(1,1,1)

fy2 fx2

fy1 fx1

fz1

(0,0,0)

(1,0,0)

fz3

Y

X

Pi

ri

f11 f15

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Figure 3 – Schematic diagram of roof structure with virtual supports

Let vertical forces at existing supports are fo , f1,…..f15 Now

Now it is clear from equations [6] –[11] that for any valve of Fx,Fy,Fz,Mx,My,Mz..There exists values for fx,fy,fx & mz. Therefore the introduced supports will fix the structure in order to provide adequate boundary conditions for displacement. Now in actual force system only vertical forces exists (weight of structure & hanging objects) Therefore

Fx =Fy =0 fx=fy=0 Mz = 0 =mz

Therefore no forces, moments will be applied to the structure through supports. Therefore the stress-strain field in the structure will not change. Therefore this introduced supports has full filled the conditions required for a virtual support. Now the structure is analysed for stress-strain by using these suppots with ABAQUS softwear

Figure 4 -Von misses stress of the structure

The displacement will be respect to the OXYZ frame.

3.5 Analyzing dynamic systems with virtual supports A dynamic system could be converted to a static system by applying inertial forces as external forces. Then these systems most oftenly fall into totally or semi floating type static structures. Then by introducing appropiate virtual supports stress-strain of the object could be analysed by using basic equations or by software for static mode. If the initial forces mifi , where fi is the acceleration of point mass mi ) the dynamic system could be considered as a static system. Most of the dynamic systems when converted to static systems will fall into floating/semi floating types. Therefore by introducing proper virtual supports the systems could be arranged as static systems.

fi = fi1 + fi2

Where, fi mi with respect to the initial frame fi1 = Acceleration of point mass only due to the motion of the frame fixed with the virtual supports (i.e. the structure is considered as rigid) fi2 = Acceleration of point mass with respect to the frame fixed with virtual supports. For most of practical cases fi2 can be considered as zero. Therefore by using kinematic and dynamic analysis fi1 could be determined and therefore the initial forces could be determined and hence the stress-strain analysis of the system.

Figure 5 – shows a thin disk (vertical plane) rotating on a frictionless shaft by an external force (0.01N) on the circumference.

Y

Z

X f3

f0

F

r

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Figure 6 – shows the descreterized systems with inertial forces and applied virtual supports.

Figure 7 – Shows the Von misses stresses obtain from ABAQUS software

4. Conclusions The concept of the virtual support provides a method to analyse floating/semi floating type structures for their displacement, stress-strain fields. The paper has introduced a mathematical method in determining of a proper system of virtual supports for a structure with a particular loading systems and how the displacement field of the deformed structure could be determined with respect to the introduced virtual support frame. Also this concept has been used to determine stress-strain field of dynamic systems as well.

References 1. D.F.M Perera,Virtual supports for rigid

foundation, 2. E.P.Nikishkov, Introduction to the finite element

method, University of Aizu-Wakamatsu 965-8580,Japan,2004 lecture notes,

3. Energy method in structural analysis, www.facweb.iitkgp.ernet.in

4. Structural analysis,Chapter 09, www.site.iugaza.edu.ps