Dynamic Analysis with Damping for Free-Standing Structures Using Mechanical Event Simulation

Abstract

This paper presents a new engineering approach to the design of free-standing structures subject to dynamically induced loads. This paper outlines procedures for Mechanical Event Simulation, the most advanced engineering simulation concept available—and also accounts for damping—to investigate, analyze and optimize very complex 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 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 apply them statically. The simulated interaction will be a realistic one, triggering the fatigue sources, which are the cyclic loads.

1. Introduction

One of the most amazing achievements in engineering is the practical application of numerical simulation. Leibnitz’s dream (Erikson et al., 1996), almost three centuries ago, of developing a generally applicable method 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 computing domains. Practically, numerical simulation has been applied successfully for more than fifty (50) years in the main engineering disciplines. Today, the market offers a wide variety of general-purpose, numerical simulation software, 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 the world around us as it really is—in a nonlinear fashion.

The Mechanical Event Simulation (MES) (González and Bussler, 2000) concept introduced and developed by ALGOR, Inc. represents a paradigm shift in engineering design. It allows engineers and designers 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 its surroundings 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 most critical parameters the analyst must account for is the damping coefficient. Characterization of damping forces in a vibrating structure has long been an active area of research in structural dynamics. Despite significant research, however, a thorough understanding of damping mechanism has not been attained. A major reason for this is that the state variables that govern damping forces are generally not as clear as they are for inertia and stiffness forces. There are advanced research results to identify a general model of damping (Adhikari, 2000) or the estimation of damping in a random vibrating system (Rudinger, 2002). The most common approach is to use viscous damping or Rayleigh damping, in which it is assumed that the damping matrix is proportional to the mass [M] and stiffness matrices [K], or:

[1]          

For large systems, identification of valid damping coefficients and β for all significant modes is a very complicated task. This paper presents the theoretical basis of event simulation, together with a methodology to incorporate damping for free-standing structures with many degrees of freedom, including transmission line towers, windmills, antennas or highmast towers.

2. Mechanical Event Simulation Concept

Event simulation as an engineering methodology (González and Bussler, 2000) is vastly different from the techniques that have been taught to engineers since the known origin of formal engineering training beginning with the Greek Mathematician Archimedes around 200 BC. Event simulation is the process of simulating a physical event in a virtual laboratory. To perform an engineering analysis using event simulation, a different viewpoint from that of 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 the displacement vector, {U}, from the force vector, {F} and the stiffness matrix, [K]. Subsequently, the stresses are calculated from the equation {σ}= E{ε}, where {ε} is the strain vector, which is a normalized displacement vector. E is Young's modulus that corresponds to Hooke's constant, k. This method works well if the analyzed system is always at rest. However, in practical mechanical or structural engineering, the static case would never dictate the design. The design must consider the "worst case scenario," which generally occurs when the system is 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 the entire event, not simply to obtain a static solution. A useful by-product of simulating the event is the forces generated by motion. The theory behind MES is based on the general finite element equilibrium equations clearly presented by Bathe (Bathe, 1982). Weaver and Timoshenko (Weaver et. al, 1990), Inman (Inman, 2001) and Hutton (Hutton, 2004) later made major contributions in modeling and incorporating damping into the free vibration mathematical models. The derivation of MES equations is presented in detail at http://www.wceng-fea.com/dil.pdf and is presented briefly, below.

For a general three-dimensional body, discretized in m finite elements, under external surface forces f ^S, body forces f ^B and concentrated forces F ⁱ, the equilibrium equations of the element assemblage are:

[2]         

Where:
- C is the damping matrix of the element assemblage,

[3]         

- M is the mass matrix of the element assemblage,

[4]         

- K is the stiffness matrix of the element assemblage,

[5]         

- R is the matrix of the resultant of the applied loads on the three-dimensional body,
and:
- k^(m) is the elemental damping property;
- H^(m) is the elemental displacement interpolation matrix;
- ρ^(m) is the elemental mass density;
- B^(m) is the strain-displacement matrix;
- C^(m) is the elemental damping matrix.

Figure 1. General three-dimensional body.

Equation 2 is the basic equation of virtual engineering; note how it models the combination 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 the formula {σ} = E{ε}, where {ε} (the strain vector) is easily obtained from the displacement vector {U}.

3. How to Incorporate Damping into Mechanical Event Simulation?

In practice, it is difficult (if not impossible) to determine the element damping parameters for general finite element assemblages, in particular because the damping properties are frequency-dependent. For this reason, the matrix C is generally not assembled from element damping matrices. Instead, it is constructed using the mass matrix and stiffness matrix of the complete element assemblage, together with experimental results on the amount of damping.

Equation 2 can be transformed in terms of generalized displacements X_(t) using the transformation:

[6]         

where P is a square matrix and X_(t) is a time-dependent vector of order n. Theoretically, there can be many different transformation matrices P, which would reduce the bandwidth of the system matrices. However, in practice, an effective transformation matrix is established using the displacement solutions of the free vibration equilibrium equations with damping neglected,

[7]         

with solution having the form

[8]         

where Φ is a vector of order n, t the time variable, t₀ a time constant, and ω a constant identified to represent the frequency of vibration (rad/sec) of the vector Φ.

Substituting Equation 8 into Equation 7 provides the generalized eigenproblem, from which Φ and ω must be determined.

[9]         

The eigenproblem in Equation 9 yields n eigensolutions (ω₁², Φ₁), (ω₂², Φ₂), ... (ω_n², Φ_n), where the eigenvectors are M-orthonormalized; i.e.,

[10]         

and

[11]         

The vector Φ_i is called the i^th-mode shape vector and ω_i is the corresponding frequency of vibration (rad/sec). It should be emphasized that Equation 7 is satisfied using any of the n displacement solutions Φ_isin ω_i(t-t₀), i=1,2…,n. Defining a matrix Φ whose columns are the eigenvectors, a diagonal matrix which stores the eigenvalues ω_i² on its diagonal and because the eigenvectors are M-orthonormal, Equation 1 becomes:

[12]         

which is valid only when the damping matrix is proportional to the mass and stiffness matrix [M] and [K].

[13]         

For large systems, identification of valid damping coefficients, and β, for all significant modes is a very complicated task. As shown by Chowdhury and Dasgupta (Chowdhury and Dasgupta, 2003), the relationship between the damping ratio and natural frequency for a free-standing structure will appear as shown in Figure 2. Based on the fact that above frequencies close to one Hertz, the relationship is practically linear, one could set up a methodology (Chowdhury and Dasgupta, 2003) to estimate the Rayleigh damping coefficients that will approximate the damping for all frequency modes in this linear range. The applicability of this methodology was investigated by the author for a large array of transmission, antenna and highmast towers, which covers practically all potential applications for these structures. A methodology was set up that was applicable to tubular, multisided, tapered, free-standing structures with a height less than ~50 m.

Figure 2. Damping ratio versus natural frequency.

The use of this methodology, in conjunction with Mechanical Event Simulation, is a completely new approach in structural dynamic analysis (Giosan, 2005), giving the design engineer more degrees of freedom in investigating the response of the critical structural connections to cyclic-induced loads, which are the fatigue sources.

3.1 Methodology to Calculate Rayleigh Damping Coefficients and β (Chowdhury and Dasgupta, 2003)

• Perform a modal frequency analysis¹ to calculate the natural frequencies; determine the m value for which the cumulative modal mass participation is close to 95% or higher.
• Consider the 2.5m vibration modes;
• Select ζ₁, the damping ratio for the system’s first vibration mode;
• Select ζ_m, the damping ratio for the system’s m^th vibration mode;
• For intermediate modes i, where 1<i<m, obtain ζ_i from Equation 14 based on linear interpolation.

[14]         

• For modes greater than m (m<i≤2.5m), extrapolate the values based on Equation 15:

[15]         

• Select first set of data: ω₁, ω_m and ζ₁, ζ_m.
• Based on the first set of data, calculate β with Equation 16:

[16]         

Back-substituting the values of β in Equation 17:

[17]         

obtain .

• Next, select a second set of data consisting of: ω₁, ω_2.5m and ζ₁, ζ_2.5m.
• Recalculate β and based on Equation 16 and Equation 17, respectively.

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

• Plot the four sets of data based on Equation 18 and determine what data fits best with the linear interpolation curve for the first m significant modes.

[18]         

• Select the corresponding values for > β as the desired values that will give the incremental damping ratio based on Rayleigh damping.

4. Numerical Application – 400 kV Transmission Tower

Let’s consider the transmission tower shown in Figure 3. This tower supports three 400 kV phases, has a height of 40 m, 7 m span between two phases, 1.2 m diameter at the base and total mass of 15000 kg. The tower is anchored using 24 anchor bolts with a diameter of 38 mm. To level the tower, leveling nuts are used, as shown in Figure 4. Above the base plate there is a 300 mm by 750 mm inspection access opening with a 12.7 mm thick reinforcing ring. We will try to investigate the response of the structure considering damping, under dynamically induced forces of wind on the electrical cables and of an unexpected rupture of one of the electrical conductors.

Figure 3. 400 kV transmission tower.		Figure 4. Pole bottom detail.

Using the methodologies presented in the design standards (AASHTO, 2001; CAN/CSA-S6-00; NBC, 2005, etc.):

- all forces will be approximated and applied in a static fashion;
- the damping coefficients will not fit properly for all natural frequency modes.

4.1 How the Mechanical Event Simulation Approach Works?

Let’s consider a 60 s mechanical event defined in Table 1 and Figure 5. In order to avoid generating excessive perturbation, the gravitational acceleration will be applied gradually from 0 m/s² to 9.81 m/s² on the system (during the first 10 s). When the system is completely stabilized (at time 35 s), wind pressure forces on the electrical conductors will gradually be applied. Some instability in the system will be generated by applying various wind pressure forces on conductor 1 and simulating a rupture of this conductor (at time 40 s). The problem requires plotting the axial stress in bolt #1 and checking the stress distribution around the hand-hole reinforcing ring and in the weld between the base plate and shaft during the mechanical event. ALGOR software was used to simulate this mechanical event. For these types of complicated nonlinear applications, the author proposes a procedure that has been incorporated as a standard design procedure into the West Coast Engineering Group Ltd. design process, as briefly outlined below:

1) Develop a beam finite element model that accurately simulates the geometrical and structural details of the investigated system, which will be used to determine the system’s natural frequency modes.
2) Using the methodology presented in Section 3.1, calculate the Rayleigh damping coefficients.
3) Using truss (or beam) finite elements, simulate the electrical conductors.
4) Apply the loads (shown in Table 1) and boundary conditions to simulate the rupture of conductor #1.

Table 1. Mechanical event description

Figure 5. Mechanical event simulation – loading diagram.

5) Choose the start time step for the nonlinear solution: ω₁/10.
6) Caption and graph the results in all critical areas (above the base plate connection and all flanged connections).

In this example, focus is on the pole bottom portion so the dynamically induced loads are graphed at 2 m above the base plate as presented in Figures 6 to 8.

Figure 6. Fz dynamically induced load.		Figure 7. Fx dynamically induced load.

7) Develop a detailed finite element model for the area of interest (in our case, the bottom of the pole as shown in Figure 9) and check the stability of this model prior to applying the dynamically induced loads captured from the beam finite element model (Step 6).

Figure 8. Fy dynamically induced load.		Figure 9. Pole bottom – detailed finite element model.

The correct application of the boundary conditions is critical for the model’s response to the dynamically induced loads. In our specific case, we were interested in tracking the stress distribution around the hand-hole reinforcing ring and in the welded connection between the base plate and the shaft. We also want to graph the axial stress in anchor bolt #1.

The finite element model was built using plate finite elements to simulate the shaft and hand-hole reinforcing ring, brick finite elements to simulate the weld between the shaft and the base plate (two layers), three layers of brick finite elements to simulate the base plate, and brick finite elements to simulate the washers and nuts. The anchor bolts were simulated using beam finite elements. The model also simulates pre-tensioning of the anchor bolts, which will induce a high pressure at the contact between the washers and the base plate (see Figure 10). The model was checked for different mesh sizes and the results were convergent. The model was run for the above-described load case (see Figure 5) and the results are presented in Figures 10 and 11.

In Figure 10, it can be clearly seen how the areas of interest respond to the applied loads. The highest stress is at contact between the washers and the base plate, above the shaft/base plate welded connection and around the welded hand-hole reinforcing ring.

Figure 10. Pole bottom – stress distribution.		Figure 11.  Axial stress variation in bolt #1.

It is very important to plan the work before building a finite element model. In this case, our model is built to permit even further and easy investigation of the pole bottom. For example, using plate finite elements to simulate the hand-hole reinforcing ring, it is easy to investigate the effect of the thickness of the hand-hole reinforcing ring on the stress distribution around this opening.

In Figure 11, the axial stress variation in bolt #1 between t=30 s and t=48 s during the analyzed mechanical event is presented. It is very important to observe that the dynamically induced loads applied to the model triggered the first two natural frequency modes (ω₁=0.407 Hz and ω₂=2.064 Hz). The model presents behavior of the areas of interest with a very high degree of detail.

5. Conclusions

This paper presents a new approach in designing free-standing structures that are subject to dynamically induced loads, the sources of structural fatigue. The procedures outlined in this paper involve the most advanced engineering simulation concept available – Mechanical Event Simulation – to investigate, analyze, and optimize very complex geometries (see Figure 12) that are usually the most critical areas for free-standing structures.

These complicated connections cannot generally be investigated using closed-form solutions. The high degree of detail makes it practically impossible for even a very experienced design engineer to predict the system’s response to dynamically induced loads.

Most of the design standards present methodologies to design the structures for fatigue, but lack methods for realistically accounting for the fatigue sources – these 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.

Figure 12. Critical connections for a transmission tower.

Using this method, the engineer no longer has to approximate the loads and apply them statically. The simulated interaction will be a realistic one, triggering the fatigue sources, which are the cyclic loads.

When analyzing the effect of cyclic loads on systems, it is very important to account for the damping effect. Design standards and the technical literature do offer values for damping ratios determined for the system’s first natural frequency mode. As presented above, interaction between the system and its surrounding real-world elements will likely trigger more than one frequency mode. Thus, it is important to use Rayleigh coefficients for a good approximation of damping, even at higher frequency modes. A method to account for damping is also outlined in this paper and its applicability to the most common free-standing structures checked.

The methodology presented in this paper was checked using ALGOR simulation software, which incorporates the necessary provisions to implement it. All simulation procedures were developed and tested by the author as a part of the research and development program, ”Stability of Free-Standing Structures Under Dynamic Induced Loads,” initiated by the author at West Coast Engineering Group Ltd.

6. Footnotes

1. To estimate the natural frequencies and the modal participation factors, a general-purpose FEA software can be used.
2. The methodology presented here has been used with success by the author for many structures designed and built by West Coast Engineering Group Ltd.

7. References

Adams V., Askenazi A., 1999. Building Better Products with FEA. OnWord Press.
Adhikari S., 2000. Damping Models for Structural Vibration, Ph. D. Thesis, Cambridge University, Engineering Department.
American Association of State Highway and Transportation Officials (AASHTO), 2001. Standard Specifications for Structural Supports for Highway Signs, Luminaires and Traffic Signals.     
American Society of Civil Engineering, 1990. Design of Steel Transmission Pole Structures, Second Edition, Manuals and Reports on Engineering Practice No.72.
Bathe K., 1982. Finite Element Procedures in Engineering Analysis. Prentice Hall, Englewood Cliffs, New Jersey.
Becker E., Carey G., Oden J., 1984. Finite Elements – An Introduction, Volumes 1, 3 & 5, Prentice Hall.
Canadian Standard Association, 2005. Canadian Highway Bridge Design Code – CAN/CSA-S6-00.
Chowdhury I., Dasgupta S., 2003. Computation of Rayleigh Damping Coefficients for Large Systems, The Electronic Journal of Geotechnical Engineering, Volume 8, Bundle 8C.
Desai C., 1979. Elementary Finite Element Method, Prentice Hall.
Erikson, K. (Ed), Estep, D., Hansbo, P. and Johnson, C., 1996. Computational Differential Equations, 1996, Cambridge University Press.
Giosan I., 2005. Stability of Free-Standing Structures Under Dynamic Induced Loads – Research & Development Project – West Coast Engineering Group Ltd.
Giosan I., 2005. Building Better Free Standing Structures, Electrical Line Magazine, Volume 12, No. 1.
Giosan I., 2000. Streamlining Metal Poles, Mechanical Engineering Magazine, April 2000.
González U., Bussler M., 2000. Using Mechanical Event Simulation in the Design Process.
Huebner K., 1975. The Finite Element Method for Engineers, John Wiley & Sons.
Hutton D., 2004. Fundamentals of Finite Element Analysis, McGraw-Hill.
Inman D., 2001. Engineering Vibration, Second Edition, Prentice Hall.
Ministry of Highways and Communications, 1983. Ontario Highway Bridge Design Code – Ontario.
MSC Software Corporation, 2002. MSC Software – How and Why do Structures Fail.
National Research Council, 2002. Transportation Research Board NCHRP-Report 469 Fatigue-Resistant Design of Cantilevered Signal, Sign, and Light Supports.
National Research Council of Canada, 2005. National Building Code of Canada.
Norrie D., Devries G., 1978. An Introduction to Finite Element Analysis, Academic Press.
Rudinger F., 2002. Modeling and Estimation of Damping in Non-Linear Random Vibration, Ph. D. Thesis, Technical University of Denmark, Mechanical Engineering Department.
Spyrakos C., 1994. Finite Element Modeling in Engineering Practice, ALGOR Publishing Division, Pittsburgh, PA.
Spyrakos C. & John Raftoyiannis, 1997. Finite Element Analysis in Engineering Practice, ALGOR Publishing Division, Pittsburgh, PA.
Troitsky M., 1982. Tubular Steel Structures – Theory and Design, The James F. Lincoln Arc Welding Foundation, Cleveland, Ohio, 1982.
Weaver W. Jr., Timoshenko S., Young D., 1990. Vibration Problems in Engineering, Fifth Edition, John Wiley & Sons.