1. iesl/pub/guide
1 ENGINEER
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

2. ENGINEER 2
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.

3. 3 ENGINEER
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

4. ENGINEER 4
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

5. 5 ENGINEER
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