Calculation of C With Rayleighs Model

8/2/2019 Calculation of C With Rayleighs Model

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Dynamic Analysis with Damping for Free Standing Structures usingMechanical Event Simulation

I Giosan*

Key Words: Dynamic Analysis, Damping, Numerical Simulation, Free StandingStructures

This Paper presents a modern, new approach, which accounts for damping, in designingfree standing structures that are subject to dynamic induced loads.The procedures outlined in this paper involve the most advanced engineering simulationconcept Mechanical Event Simulation to investigate, analyze, and optimize verycomplex geometries that are usually the most critical areas for free standing structures.The new design approach proposed in this paper places the system in direct contact withthe surrounding environment and calculates the loads based on this interaction.Using this method the engineer no longer has to approximate the loads and to apply themstatically. The simulated interaction will be a realistic one triggering the fatigue sourceswhich are the cyclic loads.

Introduction.

One of the most amazing achievements in engineering is the practical application of Numerical Simulation.

Leibnitzs dream [1] , almost three centuries ago, of developing a generally applicablemethod that could arrive at a solution to a differential equation of any type became

possible in our time with revolutionary discoveries in the computer and computingdomains.Practically, numerical simulation has been applied successfully for more than 50 years inthe main engineering disciplines.Today, the market offers a large variety of general purpose numerical simulationsoftware, which in the hands of a well trained engineer can become an extremely

powerful engineering tool.The latest developments in computing technology have made it possible to simulate theworld around us, as it really is in a nonlinear fashion.The Mechanical Event Simulation (MES) [2] concept introduced and developed byALGOR Inc. 1 represents a paradigm shift in engineering design. It allows engineers anddesigners to simulate the actual conditions that a mechanical component will experience;that is, the event associated with its application. This is possible because MES accounts for both the interaction of the component with itssurroundings, and the inertial forces generated by the motion of the component itself.To simulate the nonlinear behavior of the (surrounding) real world one of the mostcritical parameters the analyst must account for is the damping coefficient.

* Senior Design Engineer, West Coast Engineering Group Ltd., Vancouver, Canada.1 Algor Inc. Pittsburg located, advanced Finite Element Method software developer.


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Dynamic Analysis with Damping for Free Standing Structures using Mechanical Event Simulation

2

Characterization of damping forces in a vibrating structure has long been an active areaof research in structural dynamics. In spite of significant research, thoroughunderstanding of damping mechanism is not well developed. A major reason for this isthe state variables that govern damping forces are generally not clear, unlike the case for inertia and stiffness forces. There is advanced research results to identify a general model

of damping [3] or the estimation of damping in a random vibrating system [4]. The most common approach is to use viscous damping or Rayleigh damping where it isassumed that the damping matrix is proportional to the mass [ M] and stiffness matrices[K], or:

[ ] [ ] [ ] K M C +=

For large systems, identification of valid damping coefficients and , for all significantmodes is a very complicated task.This paper presents the theoretical basis of event simulation together with a methodologyto incorporate damping for free standing systems with a large degrees of freedom,

including transmission line towers, wind mill, and antenna or high Mast towers.

1. General Derivation of Mechanical Event Simulation Equilibrium Equations.

Event simulation as an engineering methodology is vastly different from the techniquesthat have been taught to engineers since the onset of formal engineering training,

beginning with the Greek Mathematician Archimedes, around 200 BC. Event simulationis the process of engineering, by simulating a physical event in a virtual laboratory. To

perform an engineering analysis using event simulation, a different viewpoint from thatof classical stress analysis is required.

Hooke's law which states that force is a linear function of displacement forms the basis of classical stress analysis and thus, of modern finite element stress analysis.In finite element analysis, the matrix equation {F} = [K]{U} is solved for thedisplacement vector, {U}, from the force vector, {F}, and the stiffness matrix, [K]. Subsequently, the stresses are calculated from the equation { }= E{ }, where { } is thestrain vector, which is a normalized displacement vector. E is Young's modulus thatcorresponds to Hooke's constant, k .This method works well if the analyzed system is always at rest. However, in practicalmechanical or structural engineering the static case would never dictate the design. Thedesign must consider the "worst case scenario" which generally occurs when the systemis in motion, when the forces and thus the stresses are greater than those under static

conditions.This is where virtual engineering enters the design process: it allows us to simulate theentire event, not simply obtain a static solution. A useful by-product of simulating theevent is the forces generated by motion.

The theory behind Mechanical Event Simulation is based on the general finite elementequilibrium equations clearly presented in 1982 by Bathe [5] .


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Later on, Weaver and Timoshenko [6] , Inman [7] and Hutton [8] had major contributionsin modeling, and incorporating the damping into the free vibration mathematical models.The derivation of MES equations will be presented further on.

In fig. 1.1 a general three-dimensional body is shown in equilibrium, under external

surface forces f S

, body forces f B

and concentrated forces F i

.These forces include all externally applied forces and reactions and have threecomponents corresponding to the three coordinate axes:

;;;

=

=

=i

z

i y

i x

i

S z

S y

S x

S

B z

B y

B x

B

F F F

F f f f

f f f f

f (1.1)

Figure 1.1 General three -dimensional body.

The displacements of the body from the unloaded configuration are denoted by U , where

[ ]W V U U T = (1.2)


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The strains corresponding to U are,

zz yy xx zz yy xxT = (1.3)

And the stresses corresponding to are,

zx yz xy zz yy xxT = (1.4)

To express the equilibrium of a body the principle of virtual displacement will be used.This principle states that equilibrium of a body requires that for any compatible, smallvirtual displacements imposed onto the body, the total internal virtual work is equal to thetotal external virtual work:

++=i

iT iS

V

T S B

V

T

V

T F U dS f U dV f U dV (1.5)

The left term is the internal virtual work and is equal to the actual stresses goingthrough the virtual strains (that correspond to the imposed virtual displacements),

= zz yz xx zz yy xx

T (1.6)

The external work is equal to the actual forces f S , f B and F i going through the virtualdisplacement U , where

= W V U U T (1.7)

The superscript S denotes that surface displacements are considered and superscript idenotes displacements at the point where concentrated forces F i are applied.It should be emphasized that the virtual strains used in (1.5) are those corresponding to

the imposed body and surface virtual displacements, and that these displacements can beany compatible set of displacements that satisfy the geometric boundary conditions. Theequation (1.5) is an expression of equilibrium, and for different virtual displacements,correspondingly different equations of equilibrium are obtained. However, equation (1.5)also contains the compatibility and constitutive requirements if the principle is used in theappropriate manner; namely, the considered displacements should be continuous andcompatible and should satisfy the displacement boundary conditions, and the stressesshould be evaluated from the strains using the appropriate constitutive relations. Thus, the

principle of virtual displacements embodies all requirements that need be fulfilled in theanalysis of a problem in solid and structural mechanics. Finally, it can be noted thatalthough the virtual work equation has been written in (1.5) in the global coordinatesystem X , Y , Z of the body, it is equally valid in any other system of coordinates.


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Now, it will be detailed how the principle of virtual displacements is used effectively, asa mechanism to generate finite element equations that govern the response of a structureor continuum. Response of the general three-dimensional body shown in Fig.1.1 will beconsidered.In the finite element analysis we approximate the body in Fig.1.1 as an assemblage of

discreet finite elements, with the elements being interconnected at nodal points on theelement boundaries. The displacements measured in a local coordinate system x, y, z within each finite element are assumed to be a function of the displacements at the Nfinite element nodal points. Therefore, for element m we have

( )( ) ( )( )U z y x H z y xu mm)

,,,, = (1.8)

where H (m) is the displacement interpolation matrix, the superscript m denotes element mand is a vector of the three global displacement components U i, V i, and W i at all nodal

points, including those at the supports of the element assemblage.

[ ] N N N T W V U W V U W V U U ,,...,,,, 222111= )

(1.9a)

More generally, relation (1.9a) can be written as

[ ] N T U U U U ...21= )

(1.9b)It is understood that U i may correspond to a displacement in any direction which may not

be aligned with a global coordinate axis, and U i may also signify a rotation when beams, plates or shell finite elements are considered.

Although all nodal point displacements are listed in , it should be realized that for agiven element, only the displacements at the nodes of the element affect the displacementand strain distribution within the element. With the assumption on the displacements in(1.8) the corresponding element strains can be evaluated,

( )( ) ( )( )U z y x B z y x mm)

,,,, = (1.10)where B(m) is the strain-displacement matrix and the rows of B(m) are obtained byappropriately differentiating and combining rows of the matrix H (m).The purpose of defining the element displacements and strains in terms of the completearray of finite element assemblage nodal point displacements is to automate theassemblage process of the element matrices to the structure matrices, using (1.8) and

(1.10).The stresses in a finite element are related to the element strains and the initial stresses,using

( ) ( ) ( ) ( )m I mmm C += (1.11)


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where C (m) is the elasticity matrix of element m and I(m) are the element initial stresses.The material law specified in C (m) for each element can be that of an isotropic or anisotropic material and can vary from element to element.Using the assumption on the displacements within each finite element as expressed in(1.8), equilibrium equations that correspond to the nodal point displacements of the

assemblage of the finite elements can be derived. First, (1.5) will be rewritten as a sumof integrations over the volume and areas of all finite elements,

( )

( )

( ) ( ) ( )

( )

( ) ( )

( )

( )

( ) ( )

++

=

i

iT i

m

mmS

S

T mS

m

mm B

V

T m B

m

mm

V

T m

F U dS f U

dV f U dV

m

mm

(1.12)

where m=1,2,, k and k =number of elements. It is important to note the integrations in(1.12) are performed over the element volumes and surfaces.If substituted in (1.12) for the element displacements (1.8), strain (1.10), and stresses(1.11), the following will be obtained.

( )( )

( ) ( ) ( )

( )( )

( ) ( )

( )( )

( ) ( )

( )( )

( ) ( )

+

+

+

=

F dV B

dS f H

dV f H

U U dV BC BU

m

mm I

V

m

m

mmS

S

mS

m

mm B

V

m B

T

m

mmm

V

mT

m

T

m

T

m

T

m

T

) ) )

(1.13)

Surface displacement interpolation matrices H S(m) are obtained from the volumedisplacement interpolation matrices H (m) in (1.8) by substituting the element surfacecoordinates, and F is a vector of the externally applied forces to the nodes of the elementassemblage.To obtain from (1.13) the equations for the unknown nodal point displacements, thevirtual displacement theorem will be invoked by imposing unit virtual displacements inturn at all displacement components. In this way we have

I U T = )

( I =identity matrix), and denoting the nodal point displacements by U , i.e. letting U ,from this point on, the equilibrium equations of the element assemblage corresponding tothe nodal point displacements are

R KU = (1.14)where


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C I S B R R R R R ++= (1.15)The matrix K is the stiffness matrix of the element assemblage,

( )( )

( ) ( ) ( ) ( ) ==m

m

m

mmm

V

m K dV BC B K m

T (1.16)

The load vector R includes the effect of the element body forces,

( )( )

( ) ( ) ( ) ==m

m B

m

mm B

V

m B RdV f H R

m

T (1.17)

the effect of the element surface forces,

( )( )

( ) ( ) ( ) ==m

mS

m

mmS

S

mS S RdS f H R

m

T

(1.18)

the effect of the element initial stresses,

( )( )

( ) ( ) ( ) ==m

m I

m

mm I

V

m I RdV B R

m

T (1.19)

and the concentrated load,

F RC = (1.20)It can be noted that the summation of the element volume integrals in (1.16) expresses thedirect addition of the element stiffness matrices K (m) to obtain the stiffness matrix of thetotal element assemblage. In the same way, the assemblage body forces vector R B iscalculated by directly adding the element body force vectors R B(m); and similarly RS , R I ,and RC are obtained. Therefore, the formulation of the equilibrium equations in (1.14)carried out above includes the assemblage process to obtain the structure matrices fromthe element matrices, usually referred to as the direct stiffness method.Equation (1.14) is a statement of the static equilibrium of the element assemblage. Inthese equilibrium considerations, the applied forces may vary with time, in which casethe displacements also vary with time and (1.14) is a statement of equilibrium for anyspecific point in time. However, if the loads are applied rapidly, inertia forces must beconsidered. Using dAlamberts principle, the element inertia forces can be simplyincluded as part of the body forces. Assuming the element accelerations are approximatedin the same way as the element displacements in (1.8), the contribution from the total

body forces to the load vector R is

( )

( )

( ) ( ) ( ) ( ) ( ) =

=m

m B

m

mmmm BT

V

m B RdV U H f H R

m

&& (1.21)

where f B(m) no longer includes inertia forces, lists the nodal point accelerations and (m) is the mass density of element m. The equilibrium equations are, in this case,


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R KU U M =+&& (1.22)

where Rand U are time dependent. The matrix M is the mass matrix of the structure,

( ) ( ) ( )

( )

( ) ( ) ==m

m

m

m

V

mmm M dV H H M m

T (1.23)

In actual measured dynamic response of structures it is observed that energy is dissipatedthrough vibration, which in vibration analysis is usually taken into account byintroducing velocity-dependent damping forces. Introducing the damping forces asadditional contributions to the body forces is obtained in (1.21),

( )

( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) =

=m

m B

m

mmmmmm BT

V

m B RdV U H k U H f H R

m

&&& (1.24)

Where is a vector of the nodal point velocities, and k (m) is the damping property parameter of element m.The equilibrium equations are, in this case,

R KU U C U M =++ &&& (1.25)

where C is the damping matrix of the structure,

( ) ( ) ( )( )

( ) ( ) ==m

m

m

m

V

mmm C dV H H k C m

T (1.26)

In practice it is difficult if not impossible to determine for general finite elementassemblages the element damping parameters, in particular because the damping

properties are frequency dependent. For this reason, the matrix C is in general notassembled from element damping matrices. Instead, it is constructed using the massmatrix and stiffness matrix of the complete element assemblage together withexperimental results on the amount of damping.

Equation (1.25) is the basic equation of virtual engineering; note how it models thecombination of motion, damping and mechanical deformation. If the stresses are still of interest, they can be calculated at any time during the analysis by application of theformula { } = E{ }, where { } (the strain vector) is easily obtained from the displacementvector { U }.

2. Change of Basis to Modal Generalized Displacements.

In order to transform the equilibrium equations into a more effective form for directintegration the following transformation on the finite element nodal point displacementsU , will be used


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( ) ( )t t PX U = (2.1)

Where P is a square matrix and X(t) is a time-dependent vector of order n. Thetransformation matrix P is still unknown and will have to be determined. The componentsof X are referred to as generalized displacements. Substituting (2.1) into (1.25) and pre-multiplying by P T , we obtain

( ) ( ) ( ) ( )t t t t R X K X C X M ~~~~ =++ &&&

(2.2)

Where:

=

==

=

R P R

KP P K

CP P C

MP P M

T

T

T

T

~

~

~

~

(2.3)

The above transformation is obtained by substituting (2.1) into (1.8) to express theelement displacements in terms of the generalized displacements,

( )( ) ( ) ( )t mm PX H t z y xu =,,, (2.4)

and then using (2.4) in the virtual work equation (1.12).The objective of the transformation is to obtain new system stiffness, mass, and dampingmatrices, K , M , and C , which have a smaller bandwidth than the original system matrices,and the transformation matrix P should be selected accordingly. In addition, it should benoted that P must be non-singular (i.e., the rank of P must be n) in order to have a uniquerelation between any vectors U and X as expressed in (2.1)Theoretically, there can be many different transformation matrices P , which wouldreduce the bandwidth of the system matrices. However in practice, an effectivetransformation matrix is established using the displacement solutions of the free vibrationequilibrium equations with damping neglected,

0=+ KU U M && (2.5)With solution having the form

( )0sin t t U = (2.6)

Where is a vector of order n, t the time variable, t 0 a time constant, and a constantidentified to represent the frequency of vibration (rad/sec) of the vector .


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Substituting (2.6) into (2.5) will be obtained the generalized eigenproblem, from which and must be determined,

M K 2= (2.7)

The eigenproblem in (2.7) yields n eigensolutions ( 12 , 1 ), ( 22 , 2 ),( n2 , n ), wherethe eigenvectors are M -orthonormalized; i.e.,

===

ji ji

M jT i ;0

;1 (2.8)

and

223

22

21 ...0 n (2.9)

The vector i is called the ith-mode shape vector, and i is the corresponding frequency of

vibration (rad/sec). It should be emphasized that (2.5) is satisfied using any of the n displacement solutions i sin i(t-t 0 ), i=1,2,n.Defining a matrix whose columns are the eigenvectors i and a diagonal matrix 2 which stores the eigenvalues i2 on its diagonal; i.e.,

[ ]

=

=

2

22

21

2

21

.

.

.

;,...,,

n

n

(2.10)

the n solutions to (2.7) can be written as

2= M K (2.11)

Because the eigenvectors are M -orthonormal, we have

I M

K T

T

== 2

(2.12)

Using

( ) ( )t t X U = (2.13)

the equilibrium equations that correspond to the modal generalized displacements will beobtained


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( ) ( ) ( ) ( )t T

t T

t T

t T R X K X C X M =++ &&& (2.14)

Considering (2.12),

( ) ( ) ( ) ( )t T

t t T t R X X C X =++ 2&&& (2.14a)

3. Computation of Rayleigh damping coefficients for large systems.

The orthogonal transformation (2.14a) is valid only when the damping matrix is proportional with the mass and stiffness matrix [ M ] and [ K ].

[ ] [ ] [ ] K M C += (3.1)It is for this reason that the damping in the form, shown in equation (3.1), isadvantageous as on orthogonal transformation the damping term in equation (2.14a)reduces to:

+

++

=

2

22

21

...0

.....

.....

0..0

0..0

n

T C

(3.2)

The general form of the equilibrium equations of the finite element system in the basis of the eigenvectors i , i=1,,n was given in (2.14a), which shows the equilibrium equationsdecouple and the time integration can be carried out individually for each equation,

providing the damping effects are neglected. Considering the analysis of systems inwhich damping effects cannot be neglected, we still would like to deal with decoupledequilibrium equations in (2.14a).Considering that:

ijii jT i C 2= (3.3)

where: i is a modal damping parameter and ij is the Kronecker delta ( ij =1 for i=j, ij =0 for i j),

damping effects can readily be taken into account in mode superposition analysis.

Therefore, using (3.3) it is assumed that the eigenvectors i , i=1,,n, are also C-orthogonal and the equations in (3.14a) reduce to n equations of the form:


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( ) ( ) ( ) ( )t r t xt xt x iiiiiii =++ 22 &&& (3.4)

Comparing equations (2.14a) and (3.4) and using (3.2) can be inferred that:

+=

+=+=

2

2222

2111

2

............................

............................

2

2

nnn

(3.5)

When the system has two degrees of freedom equation (3.5) reduces to:

+=

+=2222

2111

2

2

(3.6)

And to find the values of and , one has to solve the system of equations (3.6).

However, while resolving a system having large degrees of freedom, the analyst can havedifficulty obtaining the values of Rayleigh coefficients, which shall be valid for all the n degrees of freedom or shall be valid for all significant modes.An iterative solution is possible and this can be obtained possibly from the best-fit valuesof and in a particular system.Chowdhury and Dasgupta [9] developed a method through which one can arrive at theunique values of Rayleigh coefficients, and they will also be valid for systems having alarge degree of freedom.

4. The calculation of and Rayleigh coefficients for n-degrees of freedom free standing systems.

Based on equations (3.2) and (3.3) the orthogonal transformation reduces the dampingmatrix [C] to the form:

22 iii += (4.1)

This can be reduced to the form:

22i

ii

+= (4.2)


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From equation (4.2) it can be observed that the damping ratio is proportional to thenatural frequencies of the system. A typical plot of the right term of equation (4.2) is asshown in fig. 4.1.

Damping Ratio vs. Natural Frequency

0

0.05

0.1

0.15

0.2

0.25

0 2 4 6 8 10 12 14 16

Natural Frequency - (Hz)

D a m p i n g

R a t

i o =

( C / C c )

Fig. 4.1. Damping Ratio Versus Natural Frequency.

As can be seen in fig. 4.1, the first portion (frequency range: 0.15-2.5 Hz) the curveshows marked non-linearity, and beyond thereafter the variation is practically linear.

Thus, a set of values 1 , 2 , 3... n and 1 , 2 , 3... n , have been assumed as thecorresponding damping ratio for ith mode considering a linear relationship and thedamping ratio thus obtained is given by:

( ) 111

1

+

= i

m

mi

(4.3)

where: i = damping ratio for the ith mode (for all i m); 1 = damping ratio for the first mode;

m = damping ratio for the mth

significant mode considered in the analysis; i = natural frequency for the ith mode; 1 = natural frequency for the first mode; m = natural frequency for the mth significant mode considered in the analysis;

For structures having large degrees of freedom, it is only the first few modes whichcontribute to the significant dynamic behavior.


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For most of the engineering structures, the number of significant modes by which almost95% of the mass has participated is usually around 3 at minimum and about 25 atmaximum [9] .Base on an eigenvalue solutionand modal mass participationresult, one can identify thesignificant modes ( m) and follow the procedure outlined at the end of this chapter step by

step to determine Rayleigh coefficients and .To calculate 1 , 2 , 3... m a modal analysis with damping neglected can be performed. It can be demonstrated that the difference between damped circular frequency D and undamped circular frequency is minimal.Thus, solving equations (1.22) and (1.25) at equilibrium and comparing the solutionseasily can be concluded that:

21 = D (4.4)

The difference D and depends on the value of which for the majority of realstructures ranges between 0.01 and 0.1 [10] . For the extreme value of =0.1 equation

(4.4) yields: 99.0= D (4.5)Equation (4.5) indicates that the frequency of the damped system can be taken as equal tothe natural frequency of the corresponding undamped system. This observation can beextended to multi-degree-of-freedom systems and is of paramount importance in theevaluation of the natural frequencies and mode shapes of real structures.

Damping Ratios for Various Systems.

System Damping Ratio Metals (in elastic range)


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= M M T ~ (4.6)

Considering I the influence vector which represents the displacements of the massesresulting from static application for a unit ground displacement, the modal participation

factorsmatrix can be calculated as:

M MI

M MI T

T

T P ~

== (4.7)

or individually:

iT

i

T i

M MI

i P =

(4.7a)

and modal mass participation ratios:

= iii

m P M

i M 2

(4.7b)


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Analytical Experiments

Further on, some analytical experiments will be considered to determine applicability of the method for three different types of tubular free standing structures, these being: highmast, antenna and transmission towers.

The structures considered are twelve sided, tapered multi-sections, interconnected usingslip joints or flanged connections.

Transmission Tower Antenna Tower High Mast Tower A variety of heights and weights were chosen to best represent the large array of applications for these structures.All these structures were designed and built by West Coast Engineering Group Ltd., thelargest Canadian Manufacturer of Poles.For these structures the following parameters were calculated; the first eight naturalfrequencies, the modal mass participation factors, as well as the cumulative mass

participation and the damping ratios with a realistic value of 0.05 for and consideredfor this type of structure.The software used to calculate the above parameters was Algor.


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High Mast Towers

Table. 4.1 High Mast Towers - Natural Frequencies.Tubular High Mast Poles Natural Frequencies

Height Bottom O.D.TopO.D. Weight (Hz)

(m) (m) (m) (kg) 1 2 3 4 5 6 7 8 18.30 0.60 0.20 860 2.03 8.27 20.45 37.25 59.69 85.21 115.28 147.4634.50 0.85 0.40 5630 0.93 3.13 7.65 13.87 22.67 33.04 45.51 59.7938.06 0.85 0.20 4000 0.75 2.57 6.28 11.42 18.66 27.25 37.59 49.44

Table. 4.2 High Mast Towers - Modal Participation Factors.

Pole Modal Participation Factor Cumulative

MassHeight (%) Participation

(m) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 (%)

18.30 49.75 20.50 9.55 5.63 3.48 2.67 1.75 1.53 94.934.50 42.08 21.63 10.56 6.73 3.85 3.13 1.97 1.76 91.738.06 43.06 21.10 10.43 6.54 3.80 3.04 1.94 1.70 91.6

P i /3 44.96 21.08 10.18 6.30 3.71 2.95 1.89 1.66 92.73

Table. 4.3 High Mast Towers - Damping Ratios.Pole Modal Damping Ratios

Height (%)(m) 1 2 3 4 5 6 7 8

18.30 0.0631 0.2098 0.5125 0.9319 1.4927 2.1305 2.8822 3.686734.50 0.0501 0.0862 0.1945 0.3486 0.5679 0.8268 1.1383 1.4952

38.06 0.0521 0.0740 0.1610 0.2877 0.4678 0.6822 0.9404 1.2365

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

P1 P2 P3 P4 P5 P6 P7 P8

Fig. 4.2 Highmast Towers -Modal Mass Participations

Modal Mass Partic ipations


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Fig. 4.3 Highmast Towers - Modal Damping Ratio vs. Modal Frequency

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50

Modal Frequency (Hz)

M o

d a l

D a m p

i n g

R a t

i o ( % ) 18.3m

34.5m

38.06m

Fig. 4.3A Highmast Poles - Modal Damping Ratio vs. Modal Frequency

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 1 2 3 4 5 6


M o

d a l

D a m p

i n g

R a t

i o ( % ) 18.3m

34.5m

38.06m


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Antenna Towers

Table. 4.4 Antenna Towers - Natural Frequencies.Tubular Antenna Poles Natural Frequencies

Height BottomO.D. TopO.D. Weight (Hz)

(m) (m) (m) (kg) 1 2 3 4 5 6 7 8 10.00 0.50 0.35 1734 5.16 27.10 72.09 133.27 213.48 299.84 403.54 502.0419.50 0.65 0.35 3854 1.81 8.73 22.86 42.79 69.77 100.79 138.48 178.1020.50 0.65 0.35 3865 1.64 7.90 20.71 38.83 63.38 91.74 126.22 162.7231.70 0.98 0.38 3925 1.07 4.60 11.68 21.71 35.38 51.39 70.74 91.9034.50 0.85 0.40 7058 0.87 3.57 9.27 17.19 28.39 41.12 57.26 74.2136.00 1.05 0.60 12188 0.96 4.22 11.13 20.77 34.19 49.35 68.34 87.9538.00 1.25 0.61 6900 1.04 4.34 11.26 20.81 34.17 49.12 67.88 87.1040.80 0.61 0.61 8825 0.28 1.65 4.47 8.65 14.19 20.94 28.84 37.8640.80 0.76 0.76 8076 0.37 2.11 5.62 10.81 17.66 25.91 35.41 46.2048.30 1.53 0.75 20818 0.37 2.42 6.69 12.48 19.62 29.11 40.82 55.26

Table. 4.5 Antenna Towers - Modal Participation Factors.

Pole Modal Participation Factor Cumulative

MassHeight (%) Participation

(m) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 (%)10.00 58.51 19.93 7.75 4.17 2.66 1.76 1.31 0.89 97.019.50 55.19 20.11 8.30 4.57 2.86 2.03 1.43 1.14 95.620.50 55.19 20.10 8.28 4.56 2.85 2.02 1.43 1.13 95.631.70 50.26 20.36 9.09 5.20 3.22 2.37 1.62 1.35 93.534.50 49.21 21.52 9.56 5.31 3.30 2.35 1.66 1.32 94.236.00 51.63 21.35 9.14 4.91 3.12 2.13 1.57 1.19 95.038.00 49.60 21.57 9.55 5.30 3.32 2.36 1.67 1.33 94.740.80 51.95 20.33 7.92 4.14 2.54 1.73 1.23 0.91 90.840.80 54.71 20.88 8.59 4.47 2.76 1.93 1.36 0.98 95.748.30 67.70 17.03 8.24 4.38 2.00 0.46 0.13 0.03 100.0

P i /10 54.40 20.32 8.64 4.70 2.86 1.91 1.34 1.03 95.20


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Table. 4.6 Antenna Towers - Damping Ratios.Pole Modal Damping Ratios

Height (%)(m) 1 2 3 4 5 6 7 8

10.00 0.1338 0.6784 1.8026 3.3319 5.3371 7.4961 10.0886 12.5510

19.50 0.0591 0.2211 0.5726 1.0703 1.7446 2.5200 3.4622 4.452620.50 0.0562 0.2007 0.5190 0.9714 1.5849 2.2938 3.1557 4.068231.70 0.0501 0.1204 0.2941 0.5439 0.8852 1.2852 1.7689 2.297834.50 0.0505 0.0963 0.2344 0.4312 0.7106 1.0286 1.4319 1.855636.00 0.0500 0.1114 0.2805 0.5205 0.8555 1.2343 1.7089 2.199038.00 0.0500 0.1143 0.2837 0.5215 0.8550 1.2285 1.6974 2.177840.80 0.0963 0.0564 0.1173 0.2191 0.3565 0.5247 0.7219 0.947240.80 0.0768 0.0646 0.1449 0.2726 0.4429 0.6487 0.8860 1.155548.30 0.0768 0.0708 0.1710 0.3140 0.4918 0.7286 1.0211 1.3820

High nonlinearity zone.

0.00

10.00

20.00

30.00

40.00

50.00

P1 P2 P3 P4 P5 P6 P7 P8

Fig. 4.4 Antenna Towers - Modal Mass Participations

Modal Mass Participations


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Fig. 4.5 Antenna Towers - Modal Damping Ratio vs. Modal Frequency

0.00

0.20

0.40

0.60

0.80

1.00

1.20

0 10 20 30 40 50


M o

d a

l D a m p

i n g

R a

t i o

( % )

Tapered 19.5m

Tapered 31.7m

Tapered 36m

Tapered 38m

Round 40m

Tapered 48.3m

Fig. 4.5A Antenna Towe rs - Modal Damping Ratio vs. Modal Frequency

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 1 2 3 4 5 6


M o

d a l

D a m p

i n g

R a t

i o ( % )

Tapered 19.5m

Tapered 31.7m

Tapered 36m

Tapered 38m

Round 40m

Tapered 48.3m


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Transmission Towers

Table. 4.7 Transmission Towers - Natural Frequencies.Tubular Transmission Poles Natural Frequencies

Height BottomO.D. TopO.D. Weight (Hz)

(m) (m) (m) (kg) 1 2 3 4 5 6 7 8 21.60 0.81 0.43 2085 1.84 8.84 23.06 43.00 69.86 100.40 127.41 175.6624.40 0.97 0.25 2140 1.86 7.15 17.49 31.99 51.67 74.77 101.91 132.2924.70 0.89 0.25 2215 1.66 6.53 16.09 29.55 47.88 69.43 94.98 123.4525.30 0.97 0.25 2420 1.73 6.66 16.29 29.81 48.19 69.82 95.26 123.7627.40 1.35 0.51 7965 1.97 8.41 21.11 38.69 62.21 88.67 120.19 152.7428.30 1.17 0.31 3300 1.67 6.42 15.68 28.65 46.21 66.80 90.91 177.78

Average Values 1.79 7.34 18.29 33.62 54.34 78.32 105.11 147.61

Table. 4.8 Transmission Towers - Modal Participation Factors.

PoleModal ParticipationFactor

CumulativeMass

Height (%) Participation

(m) P 1 P 2 P 3 P 4 P 5 P 6 P 7 P 8 (%)21.60 55.00 20.17 8.38 4.64 2.91 2.07 1.45 1.16 95.824.40 47.49 20.49 9.88 5.95 3.64 2.82 1.87 1.60 93.724.70 48.31 20.47 9.71 5.77 3.54 2.71 1.82 1.54 93.925.30 47.43 20.48 9.87 5.93 3.63 2.81 1.87 1.59 93.627.40 51.17 20.55 9.32 5.40 3.37 2.52 1.66 1.42 95.428.30 47.32 20.51 9.91 5.98 3.66 2.84 1.88 1.61 93.7

P i /6 49.45 20.45 9.51 5.61 3.46 2.63 1.76 1.49 94.4

Table. 4.9 Transmission Towers - Damping Ratios.Pole Modal Damping Ratios

Height (%)(m) 1 2 3 4 5 6 7 8

21.60 0.0596 0.2238 0.5776 1.0756 1.7469 2.5102 3.1854 4.391624.40 0.0599 0.1822 0.4387 0.8005 1.2922 1.8696 2.5480 3.307424.70 0.0566 0.1671 0.4038 0.7396 1.1975 1.7361 2.3748 3.086525.30 0.0577 0.1703 0.4088 0.7461 1.2053 1.7459 2.3818 3.094227.40 0.0619 0.2132 0.5289 0.9679 1.5557 2.2170 3.0050 3.818728.30 0.0567 0.1644 0.3936 0.7171 1.1558 1.6704 2.2730 4.4446

i /6 0.0587 0.1868 0.4586 0.8411 1.3589 1.9582 2.6280 3.6905


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23

0.00

5.00

10.00

15.00

20.00

25.00

30.00

35.00

40.00

45.00

50.00

P1 P2 P3 P4 P5 P6 P7 P8

Fig. 4.6 Transmission Towers - Modal Mass Participations

Modal Mass Participations

Fig. 4.7 Transmission Towers - Modal Damping Ratio vs. Modal Freque ncy

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0 20 40 60 80 100 120 140 160


M o

d a l

D a m p

i n g

R a t

i o ( % )


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24

Analyzing the curves presented in figs. 4.1, 4.3, 4.5 and 4.7 it can be concluded that for an equation of form:

bx y xa += ,

When x is small, the first term a/x dominates and as x increases this term diminishesapproaching zero and the term bxstarts to dominate the equation.In other words if the analyzed structure is very flexible and has a very low fundamentalfrequency it will display non-linear damping behavior in the beginning with respect tofrequency, and will converge to a linear proportionality with frequency as the eigenvaluesincrease with each subsequent mode.In figs. 4.3A and 4.5A it can be seen that the 38.06m high mast tower and respectivelythe 40m and 48.3m antenna towers display some nonlinearities at the beginning of theanalyzed domain. However, extreme applications were considered for these structures.

In most cases, the towers are designed to have a reasonable rigidity and would have ahigher fundamental frequency value (see transmission towers), and the term bx willusually dominate. Moreover, considering the fact that the non-linear range is very smallfor most of the investigated free standing structures, it is not unrealistic to assume thedamping ratio for each mode is linearly proportional to the frequency of the system.Furthermore, the methodology outlined below will assist to identify if the investigatedstructure shows a linear damping variation in respect with natural frequencies.

Methodology to calculate Rayleigh Damping Coefficients and 2

Perform a modal frequency analysis to calculate the natural frequencies; tabulatethe results (see tables: 4.1, 4.2 and 4.3) determine the m value for which thecumulative modal mass participation is close to 95% or higher.

Consider the 2.5mvibration modes; Select 1, the damping ratio for the systems first vibration mode; Select m , the damping ratio for the systems mth vibration mode; For intermediate modes i, where 1


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25

Based on the first set of data calculate with equation (4.9):

22

1

11 22

m

mm

= (4.9)

Back-substituting the values of in expression (4.10):

21112 += (4.10)

Obtain .

Next select a second set of data consisting of: 1 , 2.5mand 1 , 2.5m. Recalculate and based on equations (4.9) respectively (4.10).

Now one has the three sets of data a, b, and c,below:

a) Based on linear interpolation;b) Based on data set: 1 , m and 1 , m.. c) Based on data set: 1 , 2.5m and 1 , 2.5m. d) Obtain a fourth set of data based on the averages of b) and c).

Plot the four sets of data based on equation (4.2) and check which data fits best with the linear interpolation curve for the first m significant modes.

Select the corresponding values for and as the desired values which

will give the incremental damping ratio based on Rayleigh damping.In some cases it may happen that values will show variation in higher modes, beyond m significant modes, but this is irrelevant as long as the values matchclosely for the first m modes, since the contribution of (higher) modes greater than m as can be seen above are deemed insignificant for the system.


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26

5. Numerical Application 400 kV Transmission Tower

Lets consider the transmission tower shown in fig. 5.1.

This tower supports three 400 kV phases, has a height of 40m, 7m span between two phases, 1.2m diameter at the base and total weight of 15 tons.The tower is anchored using 24 anchor bolts witha diameter of 38mm. To level the tower, levellingnuts are used as shown in fig. 5.2.Above the base plate there is a 300mm by750mm inspection access opening with a12.7mm thick reinforcing ring.

Fig. 5.1 400 kV Transmission Tower Fig. 5.2 Pole Bottom Detail

We will try to investigate the response of the structure considering damping, under dynamic induced forces by wind on the electrical cables and by an unexpected rupture of one of the electrical conductors.Using the methodologies presented in the design standards ( [17] to [26] ):

- all forces will be approximated and applied in a static fashion;- the damping coefficients will be the same for all natural frequency modes;

This approach distorts the system response by eliminating the most dangerous loads, thecyclic ones, which potentially generate fatigue in the systems critical connections.


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27

To avoid this inconvenience we will use a dynamic, fully nonlinear approach applyingthe Mechanical Event Simulation concept 3.

We will consider a 60s mechanical event defined in table 5.1 and fig. 5.3.In order to not generate excessive perturbation, the gravitational acceleration will be

applied gradually from 0m/s2

to 9.81m/s2

on the system (during the first 10s). When thesystem is completely stabilized (at time 35s) wind pressure forces on the electricalconductors will gradually be applied.

Table 5.1 Mechanical Event DescriptionMechanical Even Duration = 60 s

GravitationalWind Pressure

ForceWind Pressure

ForceTime Acceleration on Cables 2-6 on Cable 1

(s) (m/s2) (N/m2) (N/m2)0 0 0 0

10 9.81 0 030 9.81 0 035 9.81 400 1200

35.1 9.81 408 040 9.81 1200 045 9.81 1200 0

45.1 9.81 0 060 9.81 0 0

Structure Loading Diagram

0

200

400

600

800

1000

1200

0 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

W i n d P r e s s u r e

F o r c e

( N / m 2 ) o r

G r a v i

t a t i o n a l

A c c e l e r a t

i o n

( m / s 2 ) Gravitational AccelerationWind Pressure Force on Cables 2-5

Wind Pressure Force on Cable 1

Fig. 5.3 Mechanical Event Simulation Loading Diagram

Some instability in the system will be generated by applying various wind pressure forceson conductor 1 and simulating a rupture of this conductor (at time 40s).

3 The methodology presented here is been used with success by author for many structures designed and built by West Coast Engineering Group Ltd.


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28

The problem requires plotting the axial stress in bolt #1 and to checking the stressdistribution around the (hand) hole reinforcing ring, and in the weld between the base

plate and shaft during the mechanical event.

Algor software will be used to simulate this mechanical event.

For these types of complicated nonlinear applications, the author proposes the following procedure which has been incorporated as a standard design procedure into the WestCoast Engineering Group Ltd. Design Process.

1) Develop a simple beam finite element model to simulate accurately the geometricaland structural details of the investigated system (Fig. 5.4).

This beam FE model will be used to determine thesystems natural frequency modes (Table 5.2).

2) Using the methodology presented in chapter 4calculate the Rayleigh damping coefficients which will

approximate the damping for all important naturalfrequency modes (Table 5.2).

3) Using truss (or beam) finite elements simulate theelectrical conductors.

4) Apply the loads (shown in Table 5.1) and boundaryconditions to simulate the rupture of conductor #1.

5) Choose the start Time Step for the nonlinear solution: 1/10.

6) Graph the results in all critical areas (above the base plate connection and all flanged connections).

Fig. 5.4 Pole beam FE Model

In this example, focus is on the pole bottom portion so the dynamic induced loads weregraphed at 2m above the base plate and are presented in figs. 5.5 to 5.7.


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Table 5.2 Natural Frequency Modes and Rayleigh Damping Coefficients.

A B C D

Frequency Frequency Linear Up to 6th modeUp to 18th

modeAverage((B+C)/2)

Mode Dampingapprox.

Dampingapprox.

Damping DampingNr. (Hz) (%) (%) (%) (%)1 0.407 0.0200 0.0200 0.0200 0.02002 2.064 0.0241 0.0100 0.0091 0.00953 7.477 0.0375 0.0238 0.0205 0.02214 12.339 0.0495 0.0382 0.0328 0.03555 15.937 0.0584 0.0490 0.0420 0.04556 25.391 0.0817 0.0775 0.0664 0.07207 32.793 0.1000 0.1000 0.0857 0.09288 47.443 0.1362 0.1445 0.1237 0.13419 55.903 0.1571 0.1702 0.1457 0.1580

10 61.211 0.1702 0.1863 0.1596 0.173011 74.818 0.2038 0.2277 0.1950 0.211412 81.973 0.2215 0.2495 0.2136 0.231513 86.647 0.2330 0.2637 0.2258 0.244714 95.333 0.2545 0.2901 0.2484 0.269315 111.235 0.2938 0.3385 0.2898 0.314116 115.225 0.3036 0.3506 0.3002 0.325417 127.509 0.3340 0.3880 0.3322 0.360118 140.831 0.3669 0.4285 0.3669 0.3977

Rayleigh Damping Coefficients

B B= 0.0061

B= 0.0153C C= 0.0052

C= 0.0154

D D= 0.0056

D= 0.0153


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400 kV Transmission Y Tower Damping Ratio vs. Natural Frequency

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0 20 40 60 80 100 120 140

Natural Frequency - (Hz)

D a m p

i n g

R a

t i o

=

( C / C c

)

A

B

C

D

Fig. 5.4 400 kV Transmission Tower Averaged Damping Ratio.

Fz vs. Time

-350000

-300000

-250000

-200000

-150000

-100000

-50000

0

0 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

F z

( N )

Fz vs. Time

Fig. 5.5 400 kV Transmission Tower Fz Dynamic Induced Load.


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Fx vs. Time

-40000

-30000

-20000

-10000

0

10000

20000

30000

40000

0 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

F x

( N )

Fx vs. Time

Fig. 5.6 400 kV Transmission Tower Fx Dynamic Induced Load.

Fy vs. Time

-20000

-15000-10000

-5000

0

5000

10000

15000

20000

250000 5 10 15 20 25 30 35 40 45 50 55 60

Time (s)

F y

( N )

Fy vs. Time

Fig. 5.7 400 kV Transmission Tower Fy Dynamic Induced Load.


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For example: using plate finite elements to simulate the hand hole reinforcing ring, it iseasy to investigate the effect of thickness of the hand hole reinforcing ring on the stressdistribution around this opening.

Fig. 5.9 400 kV Pole Bottom Stress Distribution

400 kV Transmission Tower Anchor Bolt #1 - Axial Stress vs. Time

-12000000

-10000000

-8000000

-6000000

-4000000

-2000000

0

2000000

4000000

30 32 34 36 38 40 42 44 46 48

Time (s)

A x i a l S t r e s s

( N / m

2 )

Fig. 5.10 Axial Stress Variation in Bolt #1.


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34

In fig.5.10 is presented the axial stress variation in bolt#1 between t=30s and t=48sduring the analyzed mechanical event.It is very important to observe that the dynamic induced loads applied to the modeltriggered the first two natural frequency modes ( 1=0.407 Hz and 2=02.064 Hz).

The model presents behaviour of the areas of interest with a very high degree of detail.

6. Conclusions

How important is durability?In 1982 [27] , the Battelle Laboratories were commissioned by the United Statesgovernment to estimate the annual cost of fatigue and fracture to the countrys economy.Battelle concluded that the cost was 4.4 percent of the gross national product, or in other words, billions of dollars, and that this cost could be reduced by 29 percent theapplication of current technology.

This study says nothing about the amount of money wasted by over-designed products toavoid fatigue and fracture problems. So, how important is durability? Very. It is alsowidely recognized that approximately 80 to 90 percent or more of mechanical failuresarise from fracture and fatigue problems.This Paper presents a modern, new approach in designing free standing structures that aresubject to dynamic induced loads, the sources of structural fatigue.The procedures outlined in this paper involve the most advanced engineering simulationconcept Mechanical Event Simulation to investigate, analyze, and optimize verycomplex geometries (see fig.6.1) that are usually the most critical areas for free standingstructures.

Fig.6.1 Critical Connection for a Transmission Tower.

These complicated connections, can not usually be investigated using closed formsolutions. The high degree of detail makes it practically impossible for even a veryexperienced design engineer, to predict the systems response to dynamic induced loads.


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Most of the design standards ( [17] to [26] ) present methodologies to design thestructures for fatigue, but lack in providing methods to realistically account for thefatigue sources being the interactions between the analyzed system and the(surrounding) real world elements, which in most cases are of nonlinear nature.The new design approach proposed in this paper places the system in direct contact with

the surrounding environment and calculates the loads based on this interaction.Using this method the engineer no longer has to approximate the loads and to apply themstatically. The simulated interaction will be a realistic one triggering the fatigue sourceswhich are the cyclic loads.When analyzing the effect of cyclic loads on systems, it is very important to account for the damping effect. Some design standards and technical literature do offer values for damping ratios determined for the systems first natural frequency mode.

As presented above, interaction between the system and its surrounding real worldelements will likely trigger more than one frequency mode. Thus it is important to useRayleigh coefficients for a good approximation of damping, at higher frequency modes

as well.A method to account for damping is also outlined in this paper and its applicability to themost common free standing structures is checked.

The methodology presented in this paper was checked using Algor simulation softwarethat incorporates the necessary provisions to implement it.

All simulation procedures were developed and tested by the author as a part of theresearch and development program Stability of Free Standing Structures Under Dynamic Induced Loads initiated by the author, at West Coast Engineering Group Ltd.

Some results and animated files can be seen on the West Coast Engineering Group Ltd.web site: www.wceng-fea.com .


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References

1. Erikson, K (Ed), Estep, D, Hansbo, P and Johnson, C ComputationalDifferential Equations, 1996, Cambridge University Press. ISBN 0521-567-386.

2. Ulises F. Gonzlez, Michael L. Bussler, Algor Inc., Using Mechanical Event

Simulation in the Design Process.3. Sandipon Adhikari, Damping Models for Structural Vibration, Ph. D. Thesis,

Cambridge University, Engineering Department, September 2000.4. Finn Rudinger, Modeling and Estimation of Damping in Non-Linear Random

Vibration, Ph. D. Thesis, Technical University of Denmark, MechanicalEngineering Department, October, 2002.

5. Klaus Jurgen Bathe, Finite Element Procedures in Engineering Analysis.Prentice-Hall Inc, Englewood Cliffs, New Jersey, 1982.

6. W. Weaver, Jr., S.P. Timoshenko, D. H. Young, Vibration Problems inEngineering, Fifth Edition, John Wiley & Sons, 1990.

7. Daniel J. Inman, Engineering Vibration, Second Edition, Prentice Hall, 2001.

8. David V Hutton, Fundamentals of Finite Element Analysis, Mc Graw-Hill, 2004.9. Indrajit Chowdhury, Shambhu P. Dasgupta, Computation of Rayleigh DampingCoefficients for Large Systems, The Electronic Journal of GeotechnicalEngineering, Volume 8/2003, Bundle 8C

10. Constantine C. Spyrakos, Finite Element Modeling in Engineering Practice, Algor Publishing Division, Pittsburg, PA, 1994.

11. Constantine C. Spyrakos & John Raftoyiannis, Finite Element Analysis inEngineering Practice, Algor Publishing Division, Pittsburg, PA, 1997.

12. Vince Adams and Abraham Askenazi, Building Better Product with FEA - 1999.13. Chandrakant S. Desai, Elementary Finite Element Method, Prentice-Hall, Inc.,

1979.

14. Kenneth H. Huebner, The Finite Element Method for Engineers, John Wiley &Sons, 1975.15. D.H. Norrie, G. Devries, An Introduction to Finite Element Analysis, Academic

Press, 1978.16. Eric B. Becker, Graham F. Carey, J. Tinsley Oden, Finite Elements An

Introduction, Volumes 1, 3 & 5 Prentice Hall, 1984.17. Standard Specifications for Structural Supports for Highway Signs, Luminaires

and Traffic Signals, American Association of State Highway and TransportationOfficials, 2001.

18. National Building Code of Canada, National Research Council of Canada, 1995.19. Ontario Highway Bridge Design Code - Ministry of Highways and Communications

Ontario 1983.20. Canadian Highway Bridge Design Code CAN/CSA-S6-00.21. National Research Council, Transportation Research Board NCHRP-Report 469

Fatigue-Resistant Design of Cantilevered Signal, Sign, and Light Supports. 2002.

22. Design of Steel Transmission Pole Structures, Second Edition, 1990, AmericanSociety of Civil Engineering Manuals and Reports on Engineering Practice

N0.72.


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23. Tubular Steel Structures Theory and Design, M. S. Troitsky, The James F.Lincoln Arc Welding Foundation, Cleveland, Ohio, 1982.

24. NTE 003/04/00 Design Standards for Electrical Transmission Lines over 100kV (Replaced Standard PE 104).

25. PE 105 Design Methodology for Electrical Transmission Lines Steel Towers.

26. STAS 10101/20-90 Structural Loads Induced by Wind RomanianStandardization Institute 1990.27. MSC Software How and Why do Structures Fail 2002 MSC Software

Corporation.

Documents

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