1. Introduction

The operation of a hydroelectric power plant at the Madeira River is compromised due to the presence of many logs carried through the water stream that can damage the power plant machinery and reduce its production efficiency. To prevent such damage, nine assemblies of containment structures, called log booms (Figures 1 and 2), were installed near the power plant and across the river to retain and deflect these logs. Nevertheless, as the influence of logs and stream can exert large loads over the assemblies, they are under the risk of structural failure; so a numerical tool based on the finite element method (FEM) was developed to predict the distributions of loads and equilibrium configurations of the lines, therefore assisting their surveillance and maintenance. The numerical tool was part of a research and development (R&D) project developed by the Institute for Technological Research (IPT), and it can be extended to similar problems in which the structures behave likewise, as is the case of fishing nets and cleaning structures [1]. Moreover, while commercial software could be used for the same purpose, the intrinsic drawbacks of large computation time, automation, and customization

mainly composed of steel, making the unit respond linearly until approximately 80% of the yield stress [6], but the displacements are geometrically nonlinear, as the forces, the geometric stiffness, and the assembly equilibrium configuration

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

Consider the three-dimensional truss element composed by six degrees of freedom (DoF) that represent all the possible translation directions its nodes have. Such element is used to model the structural behavior of each log boom (Figure 1). With this numbering, the element constitutive stiffness matrix KEe is

> 100 000 000

�100 000 000

00 0 01 0 00 1

000 0 �1 0 0 0 �1

These matrices represent the stiffness of the truss due to its constitutiveness (as KEe depends on the stiffness module Ee, on the cross-sectional area Aϕ, and on the undeformed length L0) and existence of external influence (counterbalanced by the tension T). Consequently, the tangential element matrix KTe, defined as the overall

An assembly is then modeled by the joint of N trusses, representing its N log booms, as in Figure 2; thus, to account for the assembly overall stiffness KT, each KTe is linearly transformed from the truss local coordinate systems to the global

Now, considering a function g uð Þ that represents the unbalance between internal and external forces (F and R, respectively), for a given geometric configuration

The equilibrium configuration u<sup>∗</sup> is achieved when g uð Þ¼0, and in order to find it, g u<sup>∗</sup> ð Þ can be expanded into a Taylor series about an arbitrary initial

(1)

(2)

100 000 000

000 0 �1 0 0 0 �1

00 0 01 0 00 1

KTe ¼ KEe þ KGe (3)

g uð Þ� R � F ¼ KTΔu (4)

KEe <sup>¼</sup> EeA<sup>ϕ</sup> L0

while its geometric stiffness matrix KGe is

KGe <sup>¼</sup> <sup>T</sup> L0

assembly counterpart, and they are later superimposed.

u, there is a relation between g uð Þ and KT:

:

stiffness, is simply their sum:

configuration u<sup>j</sup>

27

depend on one another.

DOI: http://dx.doi.org/10.5772/intechopen.87034

2.1 Mathematical model

Figure 1.

Log boom unit schematics and its truss element representation with six DoF. The local reference frame is in blue, while the global in red.

difficulties were generally not justified by the potential increase in accuracy, as the tool would eventually be integrated in a larger numerical simulator.

### 2. Numerical model

The mathematical model used in the present formulation is based on the works of Felippa and Gavin [2–4], whereas the numerical implementation was developed within the Matrix Laboratory (MATLAB) environment [5]. The material deformations are assumed elastic only (i.e., no plasticity or yielding) as each log boom is

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

mainly composed of steel, making the unit respond linearly until approximately 80% of the yield stress [6], but the displacements are geometrically nonlinear, as the forces, the geometric stiffness, and the assembly equilibrium configuration depend on one another.

## 2.1 Mathematical model

Consider the three-dimensional truss element composed by six degrees of freedom (DoF) that represent all the possible translation directions its nodes have. Such element is used to model the structural behavior of each log boom (Figure 1).

With this numbering, the element constitutive stiffness matrix KEe is

$$\mathbf{K}\_{\text{Ee}} = \frac{\mathbf{E}\_{\text{e}} \mathbf{A}\_{\Phi}}{\mathbf{L}\_{0}} \begin{bmatrix} 1 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \\ 0 & 0 & 0 & 0 & 0 & 0 \\ -1 & 0 & 0 & 1 & 0 & 0 \\ \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{bmatrix} \tag{1}$$

while its geometric stiffness matrix KGe is

$$\mathbf{K}\_{\mathbf{G}\_{\mathbf{e}}} = \frac{\mathbf{T}}{\mathbf{L}\_{0}} \begin{bmatrix} \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} & \mathbf{0} & -\mathbf{1} \\ \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & -\mathbf{1} & \mathbf{0} & \mathbf{0} & \mathbf{1} \end{bmatrix} \tag{2}$$

These matrices represent the stiffness of the truss due to its constitutiveness (as KEe depends on the stiffness module Ee, on the cross-sectional area Aϕ, and on the undeformed length L0) and existence of external influence (counterbalanced by the tension T). Consequently, the tangential element matrix KTe, defined as the overall stiffness, is simply their sum:

$$\mathbf{K\_{Te}} = \mathbf{K\_{Ee}} + \mathbf{K\_{Ge}} \tag{3}$$

An assembly is then modeled by the joint of N trusses, representing its N log booms, as in Figure 2; thus, to account for the assembly overall stiffness KT, each KTe is linearly transformed from the truss local coordinate systems to the global assembly counterpart, and they are later superimposed.

Now, considering a function g uð Þ that represents the unbalance between internal and external forces (F and R, respectively), for a given geometric configuration u, there is a relation between g uð Þ and KT:

$$\mathbf{g(u)} \equiv \mathbf{R} - \mathbf{F} = \mathbf{K\_T} \Delta \mathbf{u} \tag{4}$$

The equilibrium configuration u<sup>∗</sup> is achieved when g uð Þ¼0, and in order to find it, g u<sup>∗</sup> ð Þ can be expanded into a Taylor series about an arbitrary initial configuration u<sup>j</sup> :

difficulties were generally not justified by the potential increase in accuracy, as the

Log boom unit schematics and its truss element representation with six DoF. The local reference frame is in blue,

The mathematical model used in the present formulation is based on the works of Felippa and Gavin [2–4], whereas the numerical implementation was developed within the Matrix Laboratory (MATLAB) environment [5]. The material deformations are assumed elastic only (i.e., no plasticity or yielding) as each log boom is

tool would eventually be integrated in a larger numerical simulator.

Log boom lines and example of assembly of trusses used to model them.

Truss and Frames - Recent Advances and New Perspectives

2. Numerical model

Figure 2.

26

Figure 1.

while the global in red.

Truss and Frames - Recent Advances and New Perspectives

$$\mathbf{g(u^{\*})} = \mathbf{g(u^{\dagger})} + \frac{\mathbf{dg(u)}}{\mathbf{du}} \Big|\_{\mathbf{u}} = \mathbf{u^{\dagger}} \left(\mathbf{u^{\*} - u^{\dagger}}\right) + \frac{1}{2} \frac{\mathbf{d^{2}} \,\mathbf{g(u)}}{\mathbf{du}^{2}} \Big|\_{\mathbf{u}} = \mathbf{u^{\dagger}} \left(\mathbf{u^{\*} - u^{\dagger}}\right)^{2} + \mathbf{O(u^{3})} = \mathbf{0} \tag{5}$$

Truncating Eq. (5) to the first order, u<sup>∗</sup> can be approximated by the unbalance and its first derivative:

$$\mathbf{u}^\* \approx \mathbf{u}^\circ - \left(\frac{\mathbf{d} \, \mathbf{g}(\mathbf{u})}{\mathbf{d} \mathbf{u}}\Big|\_{\mathbf{u}} - \mathbf{u}^\circ\right)^{-1} \mathbf{g}(\mathbf{u}^\circ) \tag{6}$$

which, however, is a linear approximation of this nonlinear formulation. Given such nonlinearity, the problem must be solved numerically, and all the equations presented insofar are part of an iterative scheme, in which j (j¼0, 1, …) is the j th iteration. Therefore, numerically speaking, successive expansions must be taken about the consecutive values of u until a satisfactory equilibrium u<sup>∗</sup> Num is achieved within a specified tolerance. So, from Eq. (6), at each iteration an improved equilibrium configuration ujþ<sup>1</sup> is obtained. Also, rearranging Eq. (4) in terms of the same improved equilibrium configuration, one obtains:

$$\mathbf{u}^{\dagger+1} = \mathbf{u}^{\dagger} - \left[\mathbf{K}\_{\Gamma}^{\dagger}\right]^{-1} \left(\mathbf{R}^{\dagger} - \mathbf{F}^{\dagger}\right) \tag{7}$$

A series of velocity magnitudes V and incidence angles β were simulated such that normal and tangential coefficients (Cz and Cx, respectively) were calculated [7–9]. From them, fitting curves in the form of Eq. (8) were adjusted, whose values

Example of Cx and Cz coefficients as functions of the velocity magnitude and incidence angle.

Cx <sup>¼</sup> K1x cos Kð Þ 2x<sup>β</sup> K3x

Each part of the log boom that is free to rotate was assigned with a pair of coefficients (i.e., the grids and the chassis) such that the total hydrodynamic

force acting on the trusses is retrieved based on them. So, for the i

2 ρV2

2 ρV<sup>2</sup>

Rxi Vi; <sup>β</sup><sup>i</sup> ð Þ¼ <sup>1</sup>

Rzi Vi; <sup>β</sup><sup>i</sup> ð Þ¼ <sup>1</sup>

Cz <sup>¼</sup> K1zsen Kð Þ 2z<sup>β</sup> K3z (8)

<sup>i</sup>ð Þ CcziACN þ CGziAGN (9)

<sup>i</sup>ð Þ CcxiACN þ CGxiAGN

th log boom

of K depend on V (Figure 4):

(i ¼ 1, …, NÞ,

29

Figure 3.

Figure 4.

CFD domain used to obtain the external force database.

DOI: http://dx.doi.org/10.5772/intechopen.87034

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

As expected, by similarity, K<sup>j</sup> <sup>T</sup><sup>¼</sup> dguð Þ du u¼uj � � , i.e., the tangential matrix is a function of uj . So, the numerical method consists of starting from an initial geometry, calculating the external and internal forces, calculating the assembly tangential matrix, and updating the geometric configuration. This process is repeated until the internal and external forces are sufficiently close.

#### 2.2 Force calculations

The loads acting on the assembly originate from the hydrodynamic interaction and from the log accumulation and must be counterbalanced by the internal forces, which are a function of the mechanical properties of the material. Although the formulation is tridimensional, for the purposes by which the model was made, it is sufficient to obtain the equilibrium configuration on the XZ plane. Hence, the calculations presented next will only be on this plane.

#### 2.2.1 External loads

To estimate the external forces, a database was created with the aid of computational fluid dynamics (CFD) (Figure 3). The software Siemens Star-CCM+, version 12.02.011, was used for these numerical simulations: both water and air were admitted incompressible, and the flow field was modeled using the unsteady Reynolds-averaged Navier-Stokes equations implicitly (at the first iterations, the movements were frozen to ensure fluid stabilization and then gradually released, still seeking stabilization); the interaction between the fluids was represented by the volume of fluid Eulerian multiphase model, and the multiphase conditions were defined by the flat wave model with a numerical damping, added in the momentum equation on the vertical direction, on both the inlet and outlet boundary conditions to minimize the effects of reflection and ensure better convergence; finally, the k-ω SST model was adopted.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

Figure 3. CFD domain used to obtain the external force database.

g u<sup>∗</sup> ð Þ¼ g u<sup>j</sup> � � <sup>þ</sup> dg uð Þ

and its first derivative:

du u <sup>¼</sup> uj <sup>u</sup><sup>∗</sup> � <sup>u</sup><sup>j</sup> � � <sup>þ</sup>

<sup>u</sup><sup>∗</sup> <sup>≈</sup> <sup>u</sup><sup>j</sup> � dg uð Þ

same improved equilibrium configuration, one obtains:

internal and external forces are sufficiently close.

calculations presented next will only be on this plane.

As expected, by similarity, K<sup>j</sup>

tion of uj

2.2 Force calculations

2.2.1 External loads

SST model was adopted.

28

� � � �

Truss and Frames - Recent Advances and New Perspectives

1 2 d2 g uð Þ

Truncating Eq. (5) to the first order, u<sup>∗</sup> can be approximated by the unbalance

du u ¼ uj � � �

which, however, is a linear approximation of this nonlinear formulation. Given such nonlinearity, the problem must be solved numerically, and all the equations presented insofar are part of an iterative scheme, in which j (j¼0, 1, …) is the j

> T h i�<sup>1</sup>

. So, the numerical method consists of starting from an initial geometry,

du u¼uj �

The loads acting on the assembly originate from the hydrodynamic interaction and from the log accumulation and must be counterbalanced by the internal forces, which are a function of the mechanical properties of the material. Although the formulation is tridimensional, for the purposes by which the model was made, it is sufficient to obtain the equilibrium configuration on the XZ plane. Hence, the

To estimate the external forces, a database was created with the aid of compu-

tational fluid dynamics (CFD) (Figure 3). The software Siemens Star-CCM+, version 12.02.011, was used for these numerical simulations: both water and air were admitted incompressible, and the flow field was modeled using the unsteady Reynolds-averaged Navier-Stokes equations implicitly (at the first iterations, the movements were frozen to ensure fluid stabilization and then gradually released, still seeking stabilization); the interaction between the fluids was represented by the volume of fluid Eulerian multiphase model, and the multiphase conditions were defined by the flat wave model with a numerical damping, added in the momentum equation on the vertical direction, on both the inlet and outlet boundary conditions to minimize the effects of reflection and ensure better convergence; finally, the k-ω

calculating the external and internal forces, calculating the assembly tangential matrix, and updating the geometric configuration. This process is repeated until the

iteration. Therefore, numerically speaking, successive expansions must be taken about the consecutive values of u until a satisfactory equilibrium u<sup>∗</sup> Num is achieved within a specified tolerance. So, from Eq. (6), at each iteration an improved equilibrium configuration ujþ<sup>1</sup> is obtained. Also, rearranging Eq. (4) in terms of the

<sup>u</sup><sup>j</sup>þ<sup>1</sup> <sup>¼</sup> <sup>u</sup><sup>j</sup> � <sup>K</sup><sup>j</sup>

<sup>T</sup><sup>¼</sup> dguð Þ

g u<sup>j</sup> � � �

du2 <sup>u</sup> <sup>¼</sup> uj <sup>u</sup><sup>∗</sup> � <sup>u</sup><sup>j</sup> � �<sup>2</sup>

<sup>R</sup><sup>j</sup> � <sup>F</sup><sup>j</sup> � � (7)

� , i.e., the tangential matrix is a func-

� � �

��<sup>1</sup>

<sup>þ</sup> O u<sup>3</sup> � � <sup>¼</sup> <sup>0</sup>

(5)

(6)

th

Figure 4. Example of Cx and Cz coefficients as functions of the velocity magnitude and incidence angle.

A series of velocity magnitudes V and incidence angles β were simulated such that normal and tangential coefficients (Cz and Cx, respectively) were calculated [7–9]. From them, fitting curves in the form of Eq. (8) were adjusted, whose values of K depend on V (Figure 4):

$$\mathbf{C}\_{\mathbf{x}} = \mathbf{K}\_{\mathbf{1}\mathbf{x}} \cos \left(\mathbf{K}\_{\mathbf{2}\mathbf{x}} \mathfrak{f}\right)^{\mathbf{K}\_{\mathbf{x}}}$$

$$\mathbf{C}\_{\mathbf{z}} = \mathbf{K}\_{\mathbf{1}\mathbf{z}} \mathrm{sen}(\mathbf{K}\_{\mathbf{2}\mathbf{z}} \mathfrak{f})^{\mathbf{K}\_{\mathbf{3}\mathbf{z}}} \tag{8}$$

Each part of the log boom that is free to rotate was assigned with a pair of coefficients (i.e., the grids and the chassis) such that the total hydrodynamic force acting on the trusses is retrieved based on them. So, for the i th log boom (i ¼ 1, …, NÞ,

$$\mathbf{R}\_{\rm xi}(\mathbf{V}\_{\rm i}, \boldsymbol{\beta}\_{\rm i}) = \frac{1}{2} \rho \mathbf{V}\_{\rm i}^{2} (\mathbf{C}\_{\rm cxi} \mathbf{A}\_{\rm CN} + \mathbf{C}\_{\rm Gxi} \mathbf{A}\_{\rm GN})$$

$$\mathbf{R}\_{\rm xi}(\mathbf{V}\_{\rm i}, \boldsymbol{\beta}\_{\rm i}) = \frac{1}{2} \rho \mathbf{V}\_{\rm i}^{2} (\mathbf{C}\_{\rm cxi} \mathbf{A}\_{\rm CN} + \mathbf{C}\_{\rm Gzi} \mathbf{A}\_{\rm GN}) \tag{9}$$

Therefore, for the same i

DOI: http://dx.doi.org/10.5772/intechopen.87034

2.3 Initial condition

convergence (Figure 7).

in which Ri

Figure 7.

31

!

there is no "element 0," but T!

start at its correct location. Then, at each Δ

increments Δ

th log boom, stretched to a new length Li (thus having new

(11)

!<sup>j</sup> the iterative scheme was run until

are not known a priori (note that

1); thus, a recursive scheme starting from one of

!<sup>j</sup> has to be carefully

(12)

(13)

coordinates ui), the internal tension acting upon it is calculated as in Eq. (11):

Li � L0 L0

This tension must be transformed to internal forces Fi that act on the nodes and are represented on the global coordinate system. Considering the same angle θi, the

> FXi ¼ Ti cosð Þ θ<sup>i</sup> FZi ¼ Ti sin ð Þ θ<sup>i</sup>

Originally, the solution strategy was to adopt the line completely stretched along

However, such quasi-static approach is computationally expensive because of

chosen to avoid numerical divergence. Thus, an improved initial condition was proposed in which the assembly was initially approximated by a catenary whose "weight vector" was obtained from the average of the hydrodynamic forces along s (Figure 8). Then, with the initial geometry determined, the external and internal forces could be calculated based on node equilibrium (Figure 9), as in Eq. (13):

> <sup>i</sup>�<sup>1</sup> and Ti !

!<sup>j</sup> toward its final location, while the rightmost mooring point would

the X direction and then gradually move the leftmost mooring point by small

Ti ¼ EA<sup>ϕ</sup>

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

internal forces are obtained through a similar transformation:

the repetitiveness of the iterative procedure. Additionally, Δ

T ! <sup>i</sup>�<sup>1</sup> þ T ! <sup>i</sup> þ R ! <sup>i</sup>�<sup>1</sup> þ R ! i <sup>2</sup> <sup>¼</sup><sup>0</sup> !

<sup>0</sup> ¼ T !

<sup>¼</sup> RiXX^ <sup>þ</sup> RiZZ and T ^ !

Example of initial condition and its evolution until final convergence.

#### Figure 5.

Example of the variation of the velocity magnitude and incidence angle along a projected line that connects the mooring points. The angles are measured taking the north coordinate of reference; thus, it is not β itself, but a function of it.

ACN and AGN are the chassis and grid front areas, and ρ is the water density. The V and β combinations used in the simulations were arithmetically averaged from velocimetry data measured in situ along a line s that connects the assemblies' mooring points (Figure 5).

Once the hydrodynamic coefficients were obtained, the contribution from the logs that accumulate along the lines was assumed to be increments on the normal coefficients [10] and estimated with the aid of data from experiments at the towing tank of the IPT (Figure 6). Thus, considering already such increments, Eq. (9) accounts for the total external forces acting on the center of the element, which are equally distributed to its nodes and further transformed from local to global coordinates as well, according to the angle θ<sup>i</sup> between the two reference frames:

$$\begin{aligned} \mathbf{R\_{Xi}} &= -\mathbf{R\_{zi}} \sin \left( \theta\_{\mathbf{i}} \right) + \mathbf{R\_{zi}} \cos \left( \theta\_{\mathbf{i}} \right) \\ \mathbf{R\_{Zi}} &= \mathbf{R\_{zi}} \cos \left( \theta\_{\mathbf{i}} \right) + \mathbf{R\_{zi}} \sin \left( \theta\_{\mathbf{i}} \right) \end{aligned} \tag{10}$$

So, RXi=2 and RZi=2 act on the X and Z directions of each node element.

#### 2.2.2 Internal loads

The tension distribution throughout the trusses is calculated based on classic linear relations of solid mechanics, and it acts on the axial direction of the element.

Figure 6. Example of experiment to measure the influence of the logs on the assembly.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

Therefore, for the same i th log boom, stretched to a new length Li (thus having new coordinates ui), the internal tension acting upon it is calculated as in Eq. (11):

$$\mathbf{T}\_{\mathrm{i}} = \mathbf{E} \mathbf{A}\_{\phi} \left( \frac{\mathbf{L}\_{\mathrm{i}} - \mathbf{L}\_{0}}{\mathbf{L}\_{0}} \right) \tag{11}$$

This tension must be transformed to internal forces Fi that act on the nodes and are represented on the global coordinate system. Considering the same angle θi, the internal forces are obtained through a similar transformation:

$$\begin{aligned} \mathbf{F\_{Xi}} &= \mathbf{T\_i} \cos \left(\theta\_i\right) \\ \mathbf{F\_{Zi}} &= \mathbf{T\_i} \sin \left(\theta\_i\right) \end{aligned} \tag{12}$$

#### 2.3 Initial condition

ACN and AGN are the chassis and grid front areas, and ρ is the water density. The V and β combinations used in the simulations were arithmetically averaged from velocimetry data measured in situ along a line s that connects the assemblies'

Example of the variation of the velocity magnitude and incidence angle along a projected line that connects the mooring points. The angles are measured taking the north coordinate of reference; thus, it is not β itself, but a

Once the hydrodynamic coefficients were obtained, the contribution from the logs that accumulate along the lines was assumed to be increments on the normal coefficients [10] and estimated with the aid of data from experiments at the towing tank of the IPT (Figure 6). Thus, considering already such increments, Eq. (9) accounts for the total external forces acting on the center of the element, which are equally distributed to its nodes and further transformed from local to global coordinates as well, according to the angle θ<sup>i</sup> between the two reference frames:

> RXi ¼ �Rzi sin ð Þþ θ<sup>i</sup> Rxi cosð Þ θ<sup>i</sup> RZi ¼ Rzi cosð Þþ θ<sup>i</sup> Rxi sin ð Þ θ<sup>i</sup>

So, RXi=2 and RZi=2 act on the X and Z directions of each node element.

The tension distribution throughout the trusses is calculated based on classic linear relations of solid mechanics, and it acts on the axial direction of the element.

Example of experiment to measure the influence of the logs on the assembly.

(10)

mooring points (Figure 5).

Truss and Frames - Recent Advances and New Perspectives

Figure 5.

function of it.

2.2.2 Internal loads

Figure 6.

30

Originally, the solution strategy was to adopt the line completely stretched along the X direction and then gradually move the leftmost mooring point by small increments Δ !<sup>j</sup> toward its final location, while the rightmost mooring point would start at its correct location. Then, at each Δ !<sup>j</sup> the iterative scheme was run until convergence (Figure 7).

However, such quasi-static approach is computationally expensive because of the repetitiveness of the iterative procedure. Additionally, Δ !<sup>j</sup> has to be carefully chosen to avoid numerical divergence. Thus, an improved initial condition was proposed in which the assembly was initially approximated by a catenary whose "weight vector" was obtained from the average of the hydrodynamic forces along s (Figure 8). Then, with the initial geometry determined, the external and internal forces could be calculated based on node equilibrium (Figure 9), as in Eq. (13):

$$
\overrightarrow{\mathbf{T}}\_{i-1} + \overrightarrow{\mathbf{T}}\_{i} + \frac{\overrightarrow{\mathbf{R}}\_{i-1} + \overrightarrow{\mathbf{R}}\_{i}}{2} = \overrightarrow{\mathbf{0}} \tag{13}
$$

in which Ri ! <sup>¼</sup> RiXX^ <sup>þ</sup> RiZZ and T ^ ! <sup>i</sup>�<sup>1</sup> and Ti ! are not known a priori (note that there is no "element 0," but T! <sup>0</sup> ¼ T ! 1); thus, a recursive scheme starting from one of

Figure 7. Example of initial condition and its evolution until final convergence.

Truss and Frames - Recent Advances and New Perspectives

For the log boom assembly, the fixed DoF are related to the mooring points, so ΔuF are their known displacements, and g uð Þ<sup>F</sup> are the unknown reaction forces, whereas the free DoF are related to the unknown displacement of the rest of the line; ΔuL and g uð Þ<sup>L</sup> are the known difference between the external hydrodynamic

and internal loads. Thus, the linear system (Eq. (4)) can be rewritten as:

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

DOI: http://dx.doi.org/10.5772/intechopen.87034

¼ � <sup>K</sup><sup>j</sup> T h i�<sup>1</sup> g uð Þ<sup>L</sup>

KLL KLF

<sup>L</sup> <sup>þ</sup> KLFΔuj

FFΔu<sup>j</sup>

These sub-systems are solved recursively: the first of Eq. (18) is solved for Δuj

Several times, the initial condition itself is not sufficient to ensure convergence because the average velocity used to calculate the catenary "weight" might not create a representative starting geometry. So, whenever the direct method fails, a sub-relaxation parameter is applied to the force unbalance as a way to prevent large deflections through the line, which is especially interesting during the first iterations; the restrained deflections are generally sufficient to adequately converge the simulation. The sub-relaxation Ω, 0 < Ω ≤ 1 is applied in the updating process of

Ω is usually a function of the iteration, but it must necessarily achieve unity prior to final convergence. An example of how this parameter varies with the

After numerical convergence, a series of variables can be outputted, such as the final geometry, the velocity magnitude and incidence angle acting over each log boom, the reaction forces, and the tension distribution along the length of the line LA. These variables, especially the last one, are of great importance, once they help predict whether or not the simulated condition threatens the integrity of the line. Examples of converged line and tension distribution are presented on Figures 11

and, sequentially, the reaction forces as calculated from the second one. Moreover,

<sup>L</sup> <sup>þ</sup> <sup>K</sup><sup>j</sup>

g uð Þ<sup>F</sup> � �<sup>j</sup>

<sup>F</sup> <sup>¼</sup> g u<sup>j</sup> � �

<sup>F</sup> <sup>¼</sup> g u<sup>j</sup> � �

L

F

g u<sup>j</sup> � � <sup>¼</sup> g u<sup>j</sup>�<sup>1</sup> � � <sup>þ</sup> <sup>Ω</sup><sup>j</sup> Rj � <sup>F</sup><sup>j</sup> � � (19)

KFL KFF " # (17)

(16)

(18)

L

ΔuL ΔuF � �<sup>j</sup>

Additionally, the matrix KT can be interpreted as follows:

Kj LLΔu<sup>j</sup>

8 ><

>:

Kj FLΔu<sup>j</sup>

static condensation is needed regardless of the initial condition used.

Thus, from Eq. (4),

2.5 Sub-relaxation

g uð Þ such that

iteration is shown in Figure 10.

2.6 Post-processing

and 12.

33

KT ¼

Figure 8.

Schematics of the new initial condition to obtain the tension distribution throughout the elements.

Figure 9. Example of improved initial condition (catenary shape).

the mooring points is performed, as (13) has only one unknown at any of these locations. The internal forces are then calculated using Eq. (12), and the iterative

scheme starts by either stretching the geometry by a small amount δ ! from the catenary geometry or, equivalently, by creating a small unbalance g.

#### 2.4 Static condensation

An intrinsic characteristic of structural problems is the presence of fixed degrees of freedom (F) in addition to the free ones (L). Thus, a common practice in FEM, known as static condensation, is to renumber the DoF seeking to reorder and split the linear system into two sub-systems. So, the displacement array can be written as:

$$
\Delta \mathbf{u} = \begin{Bmatrix} \Delta \mathbf{u}\_{\rm L} \\ \Delta \mathbf{u}\_{\rm F} \end{Bmatrix} \tag{14}
$$

and the unbalance array can be similarly split:

$$\mathbf{g(u)} = \begin{Bmatrix} \mathbf{g(u\_L)} \\ \mathbf{g(u\_F)} \end{Bmatrix} \tag{15}$$

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

For the log boom assembly, the fixed DoF are related to the mooring points, so ΔuF are their known displacements, and g uð Þ<sup>F</sup> are the unknown reaction forces, whereas the free DoF are related to the unknown displacement of the rest of the line; ΔuL and g uð Þ<sup>L</sup> are the known difference between the external hydrodynamic and internal loads. Thus, the linear system (Eq. (4)) can be rewritten as:

$$\left\{ \begin{matrix} \Delta \mathbf{u}\_{\rm L} \\ \Delta \mathbf{u}\_{\rm F} \end{matrix} \right\}^{\dagger} = - \left[ \mathbf{K}\_{\rm T}^{\dagger} \right]^{-1} \left\{ \begin{matrix} \mathbf{g}(\mathbf{u}\_{\rm L}) \\ \mathbf{g}(\mathbf{u}\_{\rm F}) \end{matrix} \right\}^{\dagger} \tag{16}$$

Additionally, the matrix KT can be interpreted as follows:

$$\mathbf{K}\_{\rm T} = \begin{bmatrix} \mathbf{K}\_{\rm LL} & \mathbf{K}\_{\rm LF} \\\\ \mathbf{K}\_{\rm FL} & \mathbf{K}\_{\rm FF} \end{bmatrix} \tag{17}$$

Thus, from Eq. (4),

$$\begin{cases} \mathbf{K}\_{\rm LL}^{\rm j} \Delta \mathbf{u}\_{\rm L}^{\rm j} + \mathbf{K}\_{\rm LF} \Delta \mathbf{u}\_{\rm F}^{\rm j} = \mathbf{g} (\mathbf{u}^{\rm j})\_{\rm L} \\\\ \mathbf{K}\_{\rm FL}^{\rm j} \Delta \mathbf{u}\_{\rm L}^{\rm j} + \mathbf{K}\_{\rm FF}^{\rm j} \Delta \mathbf{u}\_{\rm F}^{\rm j} = \mathbf{g} (\mathbf{u}^{\rm j})\_{\rm F} \end{cases} \tag{18}$$

These sub-systems are solved recursively: the first of Eq. (18) is solved for Δuj L and, sequentially, the reaction forces as calculated from the second one. Moreover, static condensation is needed regardless of the initial condition used.

### 2.5 Sub-relaxation

the mooring points is performed, as (13) has only one unknown at any of these locations. The internal forces are then calculated using Eq. (12), and the iterative

Schematics of the new initial condition to obtain the tension distribution throughout the elements.

Truss and Frames - Recent Advances and New Perspectives

An intrinsic characteristic of structural problems is the presence of fixed degrees of freedom (F) in addition to the free ones (L). Thus, a common practice in FEM, known as static condensation, is to renumber the DoF seeking to reorder and split the linear system into two sub-systems. So, the displacement array can be

> <sup>Δ</sup><sup>u</sup> <sup>¼</sup> <sup>Δ</sup>uL ΔuF

g uð Þ¼ g uð Þ<sup>L</sup>

g uð Þ<sup>F</sup>  !

from the

(14)

(15)

scheme starts by either stretching the geometry by a small amount δ

catenary geometry or, equivalently, by creating a small unbalance g.

and the unbalance array can be similarly split:

2.4 Static condensation

Example of improved initial condition (catenary shape).

written as:

32

Figure 9.

Figure 8.

Several times, the initial condition itself is not sufficient to ensure convergence because the average velocity used to calculate the catenary "weight" might not create a representative starting geometry. So, whenever the direct method fails, a sub-relaxation parameter is applied to the force unbalance as a way to prevent large deflections through the line, which is especially interesting during the first iterations; the restrained deflections are generally sufficient to adequately converge the simulation. The sub-relaxation Ω, 0 < Ω ≤ 1 is applied in the updating process of g uð Þ such that

$$\mathbf{g(u^{j})} = \mathbf{g(u^{j-1})} + \mathfrak{Q}^{j}(\mathbb{R}^{j} - \mathbf{F}^{j}) \tag{19}$$

Ω is usually a function of the iteration, but it must necessarily achieve unity prior to final convergence. An example of how this parameter varies with the iteration is shown in Figure 10.

#### 2.6 Post-processing

After numerical convergence, a series of variables can be outputted, such as the final geometry, the velocity magnitude and incidence angle acting over each log boom, the reaction forces, and the tension distribution along the length of the line LA. These variables, especially the last one, are of great importance, once they help predict whether or not the simulated condition threatens the integrity of the line. Examples of converged line and tension distribution are presented on Figures 11 and 12.

3. Results

without logs) were compared.

form of Eq. (21):

behavior:

35

theoretical and numerical values:

3.1 Verification from a theoretical catenary model

was consecutively increased by a value of approximately ffiffiffi

form of representing the number of elements was defined:

In order to verify and validate the tool, thus assessing the numerical method developed, many simulations were compared to theoretical, experimental, and numerical data. As the ASME Verification & Validation Committee defines [11], validation determines the degree of accuracy that a numerical model has in representing the real world from its perspective, whereas verification evaluates its intrinsic errors and uncertainties. So, the committee states that verification precedes validation, as they must be performed to ensure the numerical reliability of the solution. In this work code and solution verifications were done by systematically refining the discretization and comparing the outputs with theoretical data from the problem of a catenary [12]. Following, the validation was done for two major cases: by comparing the solution with experimental results from a 1:10 scale model tested in the towing tank at the Naval and Ocean Engineering Laboratory of the IPT [13] and by comparing the solution with numerical simulations from SIMPACK® Multi-Body Simulation software in prototype scale [12, 14]; the later comparison had to be done against numerical data because it was unfeasible to instrument the in situ assemblies. Additionally, only hydrodynamic cases (i.e.,

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

DOI: http://dx.doi.org/10.5772/intechopen.87034

While evaluation of the error was performed through comparison with the catenary model, estimation of the error was done through classical Richardson extrapolation [12]. Code and solution verifications were performed by systematic grid refinement, with the coarsest mesh having 100 elements, the finest 800, and in between these two, successive meshes were created such that the number of trusses

the committee [11]. For the analysis, a refinement ratio r that is a dimensionless

<sup>r</sup> <sup>¼</sup> hN hNMax

hN is a representative grid size that, for the present formulation, assumes the

P<sup>N</sup> <sup>i</sup>¼<sup>1</sup> L0

Thus, according to Eq. (20), r¼1 represents the finest grid and r¼0 an extrapolation in which the number of elements would hypothetically tend to infinity, as their length would decrease accordingly. In the convergence region, the behavior of the solution takes the form of Eq. (23), in which fext is the extrapolated value, p the order of convergence, and C a constant to be determined. From this expression, a fitting curve is plotted against r for the assessment of the convergence

f hð Þ¼ <sup>N</sup> <sup>f</sup> ext <sup>þ</sup> Ch<sup>p</sup>

The relative error is simply evaluated by calculating the difference between the

hN ¼

2

<sup>N</sup> <sup>¼</sup> L0 (21)

<sup>N</sup> (22)

<sup>p</sup> , as recommended by

(20)

#### Figure 10.

Example of Ω behavior as a function of the iteration.

Figure 11.

Example of configuration after convergence, evidencing the final geometry, the velocity profile, and the reaction forces.

Figure 12. Example of tension distribution along the assembly length LA.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

## 3. Results

In order to verify and validate the tool, thus assessing the numerical method developed, many simulations were compared to theoretical, experimental, and numerical data. As the ASME Verification & Validation Committee defines [11], validation determines the degree of accuracy that a numerical model has in representing the real world from its perspective, whereas verification evaluates its intrinsic errors and uncertainties. So, the committee states that verification precedes validation, as they must be performed to ensure the numerical reliability of the solution. In this work code and solution verifications were done by systematically refining the discretization and comparing the outputs with theoretical data from the problem of a catenary [12]. Following, the validation was done for two major cases: by comparing the solution with experimental results from a 1:10 scale model tested in the towing tank at the Naval and Ocean Engineering Laboratory of the IPT [13] and by comparing the solution with numerical simulations from SIMPACK® Multi-Body Simulation software in prototype scale [12, 14]; the later comparison had to be done against numerical data because it was unfeasible to instrument the in situ assemblies. Additionally, only hydrodynamic cases (i.e., without logs) were compared.

#### 3.1 Verification from a theoretical catenary model

While evaluation of the error was performed through comparison with the catenary model, estimation of the error was done through classical Richardson extrapolation [12]. Code and solution verifications were performed by systematic grid refinement, with the coarsest mesh having 100 elements, the finest 800, and in between these two, successive meshes were created such that the number of trusses was consecutively increased by a value of approximately ffiffiffi 2 <sup>p</sup> , as recommended by the committee [11]. For the analysis, a refinement ratio r that is a dimensionless form of representing the number of elements was defined:

$$\mathbf{r} = \frac{\mathbf{h}\_{\text{N}}}{\mathbf{h}\_{\text{N}\_{\text{Max}}}} \tag{20}$$

hN is a representative grid size that, for the present formulation, assumes the form of Eq. (21):

$$\mathbf{h}\_{\mathbf{N}} = \frac{\sum\_{i=1}^{N} \mathbf{L}\_0}{\mathbf{N}} = \mathbf{L}\_0 \tag{21}$$

Thus, according to Eq. (20), r¼1 represents the finest grid and r¼0 an extrapolation in which the number of elements would hypothetically tend to infinity, as their length would decrease accordingly. In the convergence region, the behavior of the solution takes the form of Eq. (23), in which fext is the extrapolated value, p the order of convergence, and C a constant to be determined. From this expression, a fitting curve is plotted against r for the assessment of the convergence behavior:

$$\mathbf{f}(\mathbf{h}\_{\rm N}) = \mathbf{f}\_{\rm ext} + \mathbf{C}\mathbf{h}\_{\rm N}^{\rm P} \tag{22}$$

The relative error is simply evaluated by calculating the difference between the theoretical and numerical values:

Figure 11.

Figure 12.

34

Example of tension distribution along the assembly length LA.

Figure 10.

Example of Ω behavior as a function of the iteration.

Truss and Frames - Recent Advances and New Perspectives

forces.

Example of configuration after convergence, evidencing the final geometry, the velocity profile, and the reaction

Truss and Frames - Recent Advances and New Perspectives

$$\mathbf{E\_r} = \frac{|\mathbf{f\_{ext}} - \mathbf{f(h\_N)}|}{|\mathbf{f\_{ext}}|} \tag{23}$$

and the uncertainty σ estimated through a grid convergence index (GCI):

$$\text{GCI} = \frac{\mathbf{F\_S} \left| \mathbf{f(h\_{N\_{\text{Max}}})} - \mathbf{f(h\_{(N\_{\text{Max}}-1)})} \right|}{\left[ \frac{h\_{N\_{\text{Max}}}}{h\_{(N\_{\text{Max}}-1)}} \right]^{\text{P}} - \mathbf{1}} \tag{24}$$
 
$$\text{GCI}$$

$$\boldsymbol{\sigma} = \frac{\mathbf{GCI}}{\boldsymbol{\Psi} \times \mathbf{f}\_{\text{ext}}} \tag{25}$$

FS and ψ are assigned, 1.25 and 1.1, respectively, so a safe value for σ is estimated. The catenary used for verification has a total length of LC ¼ 200 m, and the mooring points are at ð Þ 0; 200 and ð Þ 10; 20 . A vertical force per unit length F<sup>0</sup> ¼ 617:32 N=m acts throughout, and the minimum and maximum forces over the catenary are used as variables of verification, f hð Þ <sup>N</sup> ; they are both functions of the curve parameter a, which in turn is obtained by solving a transcendental equation that depends on the geometric parameters and on the force distribution:

$$\mathbf{F\_{min}} = \mathbf{a} \times \mathbf{F'} \tag{26}$$

catenary geometry, the verification was conducted with the line completely

Geometry comparison for the theoretical formulation and numerical method.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

DOI: http://dx.doi.org/10.5772/intechopen.87034

Reduced model experiments were conducted at the facilities of the IPT in a 1:10 scale with the purpose of studying the hydrodynamic behavior of such structures [13]. To do so, the model, developed mainly in polycarbonate and polymer, was constructed with the aid of a laser cutting machine and a 3D printer; the assembly was ballasted by adding lead stripes along the structure but without changing its front area as to minimize drag interference; to test both symmetric and asymmetric setups, log booms were added from a five-unit symmetric configuration, while the left mooring point was offset by distances ΔZ along the flow direction, since the model width was limited by the basin; flat plates were attached on both extremities to reduce the interference of the structures and to better control the flow incidence. Four S-Type S9M uniaxial load cells, produced by HBM, were used at each extremity, each of which with a nominal measure limit of 500 N, so the tensile forces acting on the leftmost and rightmost log booms could be acquired. Additionally, to understand how the stain behaves on each log boom and also how it is transferred throughout the structure, water strain gages, manufactured by Kyowa Electronic Instruments, were placed on one of the modules, as shown schematically in Figure 15. These locations were appropriately chosen with the aid of numerical simulations from finite element method: the criterion was to locate the stress concentration area that increased the measurement sensitivity while avoided the high strain gradients, since the latter can contribute to unreliable measurements.

Figures 16–18 compare experimental and numerical results for three different configurations as the function of the velocity. "LE" indicates "left extremity" and "RE" "right extremity," while "Exp" and "Num" refer to experimental and numerical values. No numerical uncertainty was provided since the number of trusses was

The results agree satisfactorily: the differences are more evident as the free stream velocity increases; the numerical method reproduces the experimental observation that the tensile force is larger at the leftmost log boom, an expected trend since tensile loads are higher for elements that are parallel to the stream; the

stretched along the X direction.

maintained equal the number of log booms.

37

3.2 Model scale validation

Figure 14.

$$\mathbf{F\_{max}} = \max\left\{ \frac{\mathbf{l} \times \mathbf{F'}}{\sin\left[\tan^{-1}\left(\frac{l}{\mathbf{a}}\right)\right]} \right\} \tag{27}$$

l is the length of a segment measured from one of the catenary vertices. Figure 13 shows the trends of the numerical forces as a function of the refinement ratio along with the theoretical values, which are Fmin ≈110:793 kN and Fmin ≈133:492 kN. On both cases p is close to 1, i.e., a convergence accuracy of first order. Furthermore, the extrapolated values differ from the theoretical ones by no more than 0:2%, and for the finest grid, the error is within the numerical uncertainty. Finally, all the values are close to the fitting curve, indicating that even for the poorest grids they seem to be within the convergence region: in fact, if the uncertainties remain of the same order regardless of the level of discretization, the disparity between the poorest grid and theoretical value is acceptable [12].

Finally, Figure 14 shows a comparison between numerical and theoretical geometries; as they practically overlap, the verification corroborates the precision of the proposed model. As a note, in order not to be much close to the correct

#### Figure 13.

Convergence of the numerical catenary maximum and minimum forces versus refinement ratio. Values are normalized by the theoretical ones.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

Figure 14. Geometry comparison for the theoretical formulation and numerical method.

catenary geometry, the verification was conducted with the line completely stretched along the X direction.

### 3.2 Model scale validation

Er <sup>¼</sup> j j <sup>f</sup> ext � f hð Þ <sup>N</sup>

and the uncertainty σ estimated through a grid convergence index (GCI):

�

Truss and Frames - Recent Advances and New Perspectives

GCI <sup>¼</sup> FS f hNMax ð Þ� <sup>f</sup>ðhð Þ NMax�<sup>1</sup>

<sup>σ</sup> <sup>¼</sup> GCI ψ � f ext

FS and ψ are assigned, 1.25 and 1.1, respectively, so a safe value for σ is estimated. The catenary used for verification has a total length of LC ¼ 200 m, and the

F<sup>0</sup> ¼ 617:32 N=m acts throughout, and the minimum and maximum forces over the catenary are used as variables of verification, f hð Þ <sup>N</sup> ; they are both functions of the curve parameter a, which in turn is obtained by solving a transcendental equation

> l � F<sup>0</sup> sin tan �<sup>1</sup> <sup>l</sup>

a � � � � ( )

mooring points are at ð Þ 0; 200 and ð Þ 10; 20 . A vertical force per unit length

that depends on the geometric parameters and on the force distribution:

l is the length of a segment measured from one of the catenary vertices. Figure 13 shows the trends of the numerical forces as a function of the refinement

Fmin ≈133:492 kN. On both cases p is close to 1, i.e., a convergence accuracy of first order. Furthermore, the extrapolated values differ from the theoretical ones by no more than 0:2%, and for the finest grid, the error is within the numerical uncertainty. Finally, all the values are close to the fitting curve, indicating that even for the poorest grids they seem to be within the convergence region: in fact, if the uncertainties remain of the same order regardless of the level of discretization, the

ratio along with the theoretical values, which are Fmin ≈110:793 kN and

disparity between the poorest grid and theoretical value is acceptable [12]. Finally, Figure 14 shows a comparison between numerical and theoretical geometries; as they practically overlap, the verification corroborates the precision of the proposed model. As a note, in order not to be much close to the correct

Convergence of the numerical catenary maximum and minimum forces versus refinement ratio. Values are

Figure 13.

36

normalized by the theoretical ones.

Fmax ¼ max

hNMax <sup>h</sup>ð Þ NMax�<sup>1</sup> � �<sup>p</sup>

� �

� 1

<sup>f</sup> ext j j (23)

(24)

(25)

(27)

�

Fmin ¼ a � F<sup>0</sup> (26)

Reduced model experiments were conducted at the facilities of the IPT in a 1:10 scale with the purpose of studying the hydrodynamic behavior of such structures [13]. To do so, the model, developed mainly in polycarbonate and polymer, was constructed with the aid of a laser cutting machine and a 3D printer; the assembly was ballasted by adding lead stripes along the structure but without changing its front area as to minimize drag interference; to test both symmetric and asymmetric setups, log booms were added from a five-unit symmetric configuration, while the left mooring point was offset by distances ΔZ along the flow direction, since the model width was limited by the basin; flat plates were attached on both extremities to reduce the interference of the structures and to better control the flow incidence.

Four S-Type S9M uniaxial load cells, produced by HBM, were used at each extremity, each of which with a nominal measure limit of 500 N, so the tensile forces acting on the leftmost and rightmost log booms could be acquired. Additionally, to understand how the stain behaves on each log boom and also how it is transferred throughout the structure, water strain gages, manufactured by Kyowa Electronic Instruments, were placed on one of the modules, as shown schematically in Figure 15. These locations were appropriately chosen with the aid of numerical simulations from finite element method: the criterion was to locate the stress concentration area that increased the measurement sensitivity while avoided the high strain gradients, since the latter can contribute to unreliable measurements.

Figures 16–18 compare experimental and numerical results for three different configurations as the function of the velocity. "LE" indicates "left extremity" and "RE" "right extremity," while "Exp" and "Num" refer to experimental and numerical values. No numerical uncertainty was provided since the number of trusses was maintained equal the number of log booms.

The results agree satisfactorily: the differences are more evident as the free stream velocity increases; the numerical method reproduces the experimental observation that the tensile force is larger at the leftmost log boom, an expected trend since tensile loads are higher for elements that are parallel to the stream; the

Figure 15.

Schematics of the experimental setup.

later reason, better agreement could result from finer grids, although the one-to-

A prototype-scale validation is presented, and the results were compared to simulations from SIMPACK® Multi-Body Simulation software. The variables of comparison are again the reaction forces at the leftmost and rightmost anchor points, and the relative differences uLE and uRE use the values of SIMPACK® for normalization. In the following tables, SP refers to the results from SIMPACK®, while FEM to the results from the current method. Three validation cases were

The first prototype validation was performed for line 02 with a variable velocity profile, similar to the one presented in Figure 5. An example of converged solution for line 02 is presented in Figure 19, in which the equilibrium configuration,

The results from SIMPACK® and the current method are presented in [14]: the first simulation considers a 20% increase in the normal external force (Cz), while

Table 1 shows that the maximum relative difference for the simulations that happen at the RE (approximately 7.6%), while for the LE the maximum difference is about 5.3%. Such values are satisfactory given the simplicity of the tool, in

SP (kN) FEM (kN) u (%) LE RE LE RE LE RE 1025 814 1076 876 5.0 7.6 1239 1029 1305 1104 5.3 7.3

one representation would be lost.

3.3 Prototype-scale validation

3.3.1 Line 02

Figure 18.

Table 1.

39

the second a 50% increase.

comparison to SIMPACK®.

performed: line 02, line 12, and the set of line 13.

Force comparison as a function of free stream velocity. Eight log boom lines.

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

DOI: http://dx.doi.org/10.5772/intechopen.87034

projected line, and velocity distribution are depicted.

Comparison of results between SIMPACK® and the current method—Line 02.

Figure 16. Force comparison as a function of free stream velocity. Six log boom lines.

number of log booms seems not to affect the accuracy of the results, as the error remains practically the same regardless of the configuration; the larger differences are within 6:0% and are possibly due to reasons such as the model material properties, the modeling of the external forces, and the few number of elements. For the

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

Figure 18. Force comparison as a function of free stream velocity. Eight log boom lines.

later reason, better agreement could result from finer grids, although the one-toone representation would be lost.

### 3.3 Prototype-scale validation

A prototype-scale validation is presented, and the results were compared to simulations from SIMPACK® Multi-Body Simulation software. The variables of comparison are again the reaction forces at the leftmost and rightmost anchor points, and the relative differences uLE and uRE use the values of SIMPACK® for normalization. In the following tables, SP refers to the results from SIMPACK®, while FEM to the results from the current method. Three validation cases were performed: line 02, line 12, and the set of line 13.

#### 3.3.1 Line 02

The first prototype validation was performed for line 02 with a variable velocity profile, similar to the one presented in Figure 5. An example of converged solution for line 02 is presented in Figure 19, in which the equilibrium configuration, projected line, and velocity distribution are depicted.

The results from SIMPACK® and the current method are presented in [14]: the first simulation considers a 20% increase in the normal external force (Cz), while the second a 50% increase.

Table 1 shows that the maximum relative difference for the simulations that happen at the RE (approximately 7.6%), while for the LE the maximum difference is about 5.3%. Such values are satisfactory given the simplicity of the tool, in comparison to SIMPACK®.


Table 1.

Comparison of results between SIMPACK® and the current method—Line 02.

number of log booms seems not to affect the accuracy of the results, as the error remains practically the same regardless of the configuration; the larger differences are within 6:0% and are possibly due to reasons such as the model material properties, the modeling of the external forces, and the few number of elements. For the

Force comparison as a function of free stream velocity. Six log boom lines.

Force comparison as a function of free stream velocity. Seven log boom lines.

Figure 15.

Figure 16.

Figure 17.

38

Schematics of the experimental setup.

Truss and Frames - Recent Advances and New Perspectives

#### 3.3.2 Line 12

The second validation was performed for line 12 with several velocity profiles, also similar to the one presented in Figure 5. Analogously, Figures 19 and 20 depict an example of a converged solution.

Table 2 presents the comparison between SIMPACK® and the developed formulation. For this set of simulations, the maximum percentage difference happens at the LE, and its value is approximately 6.6%, while at the RE this difference is about 5.2%; the values are comparable to those for line 02, except that the extremities at which the maximum occurs are swapped.

3.3.3 Lines 13

The last validation was conducted for the group of lines 13, i.e., for lines 13A, 13B, 13C, and 13D. Figure 21 is an example of a converged simulation for line 13C/13D.

Figure 19. Example of converged simulation for line 02.

Particularly for line 13, the magnitude and incidence angle remained constant throughout them, varying from simulation to simulation. In Table 3 these parameters are summarized: the first columns indicate the velocity range (VR) used in each simulation for the choice of the coefficients of Eq. (8), even though the actual

SP (kN) FEM (kN) u (%) LE RE LE RE LE RE 1234 1666 1295 6.4 4.9 1204 1641 1267 6.6 5.2 1479 1983 1543 6.3 4.3 1578 2158 1659 6.4 5.2 1124 1559 1181 6.6 5.1 1387 1910 1459 6.5 5.2 1487 2071 1564 6.4 5.2 1382 1905 1454 6.5 5.2 1359 1877 1424 6.2 4.8 1306 1821 1374 6.5 5.2 1748 2405 1840 6.6 5.3 1314 1811 1381 6.4 5.1 1414 1888 1425 3.7 0.8

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

DOI: http://dx.doi.org/10.5772/intechopen.87034

Table 4 presents the comparison between SIMPACK® and the developed

the difference is about 5.5% for line 13A/13B and 6.3% for line 13C/13D.

The maximum percentage difference of the simulations happens again at the LE and is approximately 7.0% for line 13A/13B and 7.1% for line 13C/13D. At the RE,

velocity magnitude was sometimes not within that range.

Comparison of results between SIMPACK® and the current method—Line 12.

Example of converged solution for line 13C/13D.

formulation.

41

Figure 21.

Table 2.

Figure 20. Example of converged solution for line 12.


Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures DOI: http://dx.doi.org/10.5772/intechopen.87034

#### Table 2.

3.3.2 Line 12

3.3.3 Lines 13

Figure 20.

40

Figure 19.

Example of converged solution for line 12.

Example of converged simulation for line 02.

an example of a converged solution.

ities at which the maximum occurs are swapped.

Truss and Frames - Recent Advances and New Perspectives

The second validation was performed for line 12 with several velocity profiles, also similar to the one presented in Figure 5. Analogously, Figures 19 and 20 depict

Table 2 presents the comparison between SIMPACK® and the developed formulation. For this set of simulations, the maximum percentage difference happens at the LE, and its value is approximately 6.6%, while at the RE this difference is about 5.2%; the values are comparable to those for line 02, except that the extrem-

The last validation was conducted for the group of lines 13, i.e., for lines 13A, 13B, 13C, and 13D. Figure 21 is an example of a converged simulation for line 13C/13D.

Comparison of results between SIMPACK® and the current method—Line 12.

Figure 21. Example of converged solution for line 13C/13D.

Particularly for line 13, the magnitude and incidence angle remained constant throughout them, varying from simulation to simulation. In Table 3 these parameters are summarized: the first columns indicate the velocity range (VR) used in each simulation for the choice of the coefficients of Eq. (8), even though the actual velocity magnitude was sometimes not within that range.

Table 4 presents the comparison between SIMPACK® and the developed formulation.

The maximum percentage difference of the simulations happens again at the LE and is approximately 7.0% for line 13A/13B and 7.1% for line 13C/13D. At the RE, the difference is about 5.5% for line 13A/13B and 6.3% for line 13C/13D.

#### Truss and Frames - Recent Advances and New Perspectives


tool, to forecast the tension distributions throughout the lines such that it can be used as a simulator for the assessment of the structural safety of log boom

Nonlinear Truss-Based Finite Element Methods for Catenary-Like Structures

The authors would like to express their gratitude to the Santo Antonio Energy for funding the project (PD-06683-0116/2016), through the Research and Development Fund of Brazilian Electricity Regulatory Agency, and the Institute for Technological Research Foundation. They also would like to acknowledge the financial support and scholarship granted by the Coordination of Superior Level Staff Improvement, under projects 1784366 and 133846, and the Foundation for the

assemblies.

Acknowledgements

Author details

43

Jose Rodolfo Chreim\* and Joao Lucas Dozzi Dantas Institute for Technological Research, Sao Paulo, Brazil

provided the original work is properly cited.

\*Address all correspondence to: jrchreim@outlook.com

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium,

institute for Technological Research.

DOI: http://dx.doi.org/10.5772/intechopen.87034

#### Table 3.

Parameters for simulations of lines 13.


Table 4.

Comparison of results between SIMPACK® and the current method—Lines 13.
