# Developments Of Numerical Model Biology Essay

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## Introduction

One of the appropriate methods to study the effects of strengthening existing structures is simulation by using computer software. In most of the structural analysis software, Finite Elements Method (FEM) is used. The strengthened steel structures may be simulated in either 2D or 3D case (Colombi, 2009; Deng et al., 2004; Haghani et al., 2009; Linghoff and Al-Emrani, 2009; Linghoff et al., 2009; and Seleem et al., 2010). Normally, to simulate the 2D and 3D models the 2D shell and 3D solid elements are used, respectively. In meshing, application of an appropriate meshing method and meshing size helps to increase the accuracy of the modelling. Also, choosing a suitable boundary condition and loading can increase the simulation modelling accuracy. Generally, structures may be analyzed in linear or nonlinear approaches. In this chapter, the methods of numerical studies for achieving all the above mentioned will be introduced. In addition, the results of some simulated specimens in the 2D/3D and linear/nonlinear are compared to the experimental results to choose the best modelling and analyzing methods having highest accuracy.

## Finite Element Method

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The finite element method is a numerical procedure that can be applied to obtain solutions to a variety of problems in engineering. The origin of the modern finite element method may be traced back to the early 1900s when some researchers approximated and modelled elastic continua using discrete equivalent elastic bars. However, Courant (1943) has been credited with being the first person to develop the finite element method (Moaveni, 2008).

## ANSYS

In this research, the ANSYS Finite Element Program (User Guides for ANSYS, 2007) is used to develop the numerical simulation of the strengthened steel I-beams by using CFRP strips. The software was available in Computer Laboratory of Civil Engineering Department and CADCAM Laboratory of Manufacturing Department. An extensive number of models were developed compared to the experimental works to investigate various parameters thereby making the research more economic. In addition, as some parameters are not measurable by using experimental tests e.g. strain and stress on adhesive, utilizing software to simulate the specimen's behaviour is required.

ANSYS is the advanced FEM software, which is capable of simulating various engineering problems. It is a comprehensive general-purpose finite element computer program that contains more than 100,000 lines of code. ANSYS is capable of performing static, dynamic, heat transfer, fluid flow, and electromagnetism analyses. ANSYS has been a leading FEA program for about 40 years (since 1971).

ANSYS is suitable for simulating CFRP strengthened steel beams (Colombi, 2006). In this research ANSYS is used to investigate different parameters of strengthened steel I-beams.

## Modelling method

Simulation of strengthened steel beams may be modelled in 2D (Colombi, 2006) because of the reduced running time, or in 3D (Deng et al., 2004; Haghani et al., 2009; Linghoff and Al-Emrani, 2009; Linghoff et al., 2009; and Seleem et al., 2010). Also, because of the reduction in the running time, half of a steel beam (Haghani et al., 2009; and Linghoff et al., 2009) or one quarter of the beam (Linghoff and Al-Emrani, 2009) may be simulated. However, full length specimens may also be simulated (Seleem et al., 2010).

In this research, the specimens are simulated in the full-3D modelling case. In order to investigate the accuracies of the 2D and 3D simulation modelling, some specimens are simulated in the 2D modelling case, and the results of different parameters in the 2D and 3D modelling cases are compared to the experimental results.

## 2D modelling

In the 2D case, all elements are modelled by using 2D plane elements. In this case, the structure is simulated in the plane, and the dimensions of the structure in out-plane is defined as thickness. Normally, this method is used because of the reduced running time. Figure 4.1 shows a flexural strengthened steel I-beam, including steel I-beam, steel stiffener, adhesive and CFRP plate, which was simulated in the 2D modelling case.

Figure 4.1 Two dimensional (2D) modelling

## 4.3.2 3D modelling

In the 3D case, all the elements are simulated by using 3D solid elements that have dimensions in all directions. The problems observed in the 3D case are the higher running time because of the larger amount of elements and equations. This problem is more significant when the nonlinear analysis method is used. Designers may prefer to simulate the strengthened structures in the 2D case because of the lower running time. Figure 4.2 indicates a simulated flexural strengthened steel I-beam in the 3D modelling case, which includes steel I-beam, steel stiffener, adhesive and CFRP plate.

## Material properties

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The same material properties for steel I-beam, steel plate, bolt, adhesive, and CFRP plates (which are mentioned in section 3.2) are applied in the numerical simulation.

Figure 4.2 Three dimensional (3D) modelling

## Element selection

One of the most important parameters in the FEM simulation is the selection of the appropriate elements. Several elements are provided in ANSYS that use different cases. For the selection of the elements, modelling case (2D or 3D), material property (e.g. elastic, plastic, linear and nonlinear), structural behaviour (e.g. structural element, mass, pipe, axisymmetric, shell, membrane, plane stress, plane strain, link, joint, spring, and fluid), and loading state (e.g. static, dynamic, heat transfer, fluid flow, and electromagnetism) are considered to choose the proper elements.

According to the abovementioned points, in the 2D modelling case, the 2D structural shell element eight nodes 93 (Figure 4.3.a) is selected. All elements are modelled by using this element (Eight nodes 93). This element is not a plane-stress or plane-strain element. One of the most important behaviours of unconstrained steel beams is lateral-torsional-buckling. If the plane-stress or plane-strain element are chosen, then out-plane deformations are not achievable appropriately.

In the 3D modelling case, the 3D solid element ten nodes 187 (Figure 4.3.b) are used.

Figure 4.3 Selected elements (a) 2D-Shell-8 nodes 93, (b) 3D-Solid- 10 nodes 187

## Meshing method

The selection of the appropriate meshing method and mesh size increases the accuracy of the simulations.

To mesh the elements, a combination of auto and map meshing are used. In the critical regions, such as the end of the plates, below the loads, supports, and anchor plates the elements were meshed smaller than the other regions to increase the accuracy of the FEM calculation.

Normally, by increasing the number of elements (finer mesh size) the accuracy of the FEM analysis is increased. However, after using a specific number of elements the variation of results was still steady. Moreover, making the mesh size finer causes an increase in the element and equation number and results in a longer running time. To verify the appropriateness of the mesh size, for the non-strengthened specimen (F1), three different mesh sizes are selected for the specimen - 16781, 22874, and 67183 elements. Then, the results of the numerical simulation and experimental test are compared, as shown in Figure 4.4. It shows that all three mesh sizes produced the same result for vertical deflection at the mid-span, therefore, it shows the suitability of the selected mesh size. Consequently, this meshing method is used for meshing all the specimens.

## Surface interaction

The simulated CFRP strengthened steel I-beam is a composite structure that includes different elements that are connected to each other by using different common surfaces. The full interactions of the common surfaces are completely defined between the steel I-beam, adhesive, CFRP strips, bolts, and steel plates to achieve the failure modes (Linghoff and Al-Emrani, 2009). Debonding, peeling, and splitting occur when the plastic strains exceed the ultimate strain.

## Boundary condition

To simulate the support in the 2D modelling case, a point is constrained vertically at each support. In the 3D case, a line is constrained at the support in the vertical direction.

## Specification of the simulated models

Numerical models for specimens under flexural and shear are developed. Totally, more than one hundred specimens are simulated by ANSYS (Narmashiri and Jumaat, 2009a, b, c, d, e, f; Jumaat and Narmashiri, 2009; Narmashiri et al., 2010a, b), but here, only fifty specimens are investigated.

Figure 4.4 Vertical deflection (F1)

## Flexural specimens

To investigate the effect of in-plane CFRP end cutting shapes in the experimental studies, three end cutting shapes (rectangular, trapezoidal, and semicircular) are chosen (Figure 4.5 a, b, and e). In addition, two more end cutting shapes (trapezoidal with different dimensions and triangular shapes) are selected (Figure 4.5 c and d).

To investigate the effects of mechanical fastening CFRP, the same specimens as the experimental studies are simulated in ANSYS. Figure 4.6 shows the simulated CFRP anchoring system of the steel I-beam.

In order to investigate the effect of CFRP length on the flexural behaviour of steel I-beams, five lengths are chosen (600, 1000, 1500, 1700, and 1800 mm). The last three lengths are tested in the experimental works.

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The specifications of the numerically simulated flexural specimens are shown in Table 4.1.

Figure 4.5 Different in-plane CFRP end cutting shapes

Figure 4.6 Modelled bolts and plate as end anchor

## Shear specimens

For the shear strengthening study, finite element models were developed to achieve these objectives, i.e.: (a) investigation on the effects of applying different CFRP ratio on web (Figure 4.7), (b) effects of single or double sides strengthening of the web, and (c) application of CFRP as local strengthening reinforcement. The same specimens as tested in the experimental work are chosen for the numerical studies. Tables 4.2 and 4.3 show the specifications of the simulated shear specimens.

In order to examine the effect of applying different CFRP ratios on the web, four ratios of 0.00 (S 1, 2, 3, 4), 0.48 (S9, 10), 0.72 (S 7, 8, 12), and 1.00 (S 5, 6, 11) are chosen.

To investigate the effectiveness of applying CFRP on single and double sides of the web, the CFRP strips are pasted on one side of the web (S6, 8, 10) or both sides of the web (S 5, 7, 9, 11, 12).

Table 4.1 Specifications of the simulated flexural strengthened beams

## Anchor plate at ends

1

F1-FEM (F1-3D-NL)

(F1-Elem:16781)

N/A

N/A

N/A

N/A

N/A

N/A

3D

Nonlinear

2

(F1-Elem:22874)

N/A

N/A

N/A

N/A

N/A

N/A

3D

Nonlinear

3

(F1-Elem:67183)

N/A

N/A

N/A

N/A

N/A

N/A

3D

Nonlinear

4

F1-3D-L

N/A

N/A

N/A

N/A

N/A

N/A

3D

Linear

5

F1-2D-NL

N/A

N/A

N/A

N/A

N/A

N/A

2D

Nonlinear

6

F1-2D-L

N/A

N/A

N/A

N/A

N/A

N/A

2D

Linear

7

F3-FEM

(F3-3D-NL)

S

1500

1.2

Rectangular

N/A

N/A

3D

Nonlinear

8

F3-3D-L

S

1500

1.2

Rectangular

N/A

N/A

3D

Linear

9

F3-2D-NL

S

1500

1.2

Rectangular

N/A

N/A

2D

Nonlinear

10

F3-2D-L

S

1500

1.2

Rectangular

N/A

N/A

2D

Linear

11

F5-FEM

S

1530

1.2

Rectangular

N/A

N/A

3D

Nonlinear

12

F6-FEM

S

1570

1.2

Trapezoidal-1

N/A

N/A

3D

Nonlinear

13

F7-FEM

S

1560

1.2

Semicircular

N/A

N/A

3D

Nonlinear

14

F8-FEM

S

1700

1.2

Rectangular

N/A

N/A

3D

Nonlinear

15

F9-FEM

S

1800

1.2

Rectangular

N/A

N/A

3D

Nonlinear

16

F10-FEM

(F10-3D-NL)

CF1

1500

1.4

Rectangular

N/A

N/A

3D

Nonlinear

17

F10-3D-L

CF1

1500

1.4

Rectangular

N/A

N/A

3D

Linear

18

F10-2D-NL

CF1

1500

1.4

Rectangular

N/A

N/A

2D

Nonlinear

19

F10-2D-L

CF1

1500

1.4

Rectangular

N/A

N/A

2D

Linear

## Anchor plate at ends

20

F11-FEM

(F11-3D-NL)

CF2

1500

1.4

Rectangular

N/A

N/A

3D

Nonlinear

21

F11-3D-L

CF2

1500

1.4

Rectangular

N/A

N/A

3D

Linear

22

F11-2D-NL

CF2

1500

1.4

Rectangular

N/A

N/A

2D

Nonlinear

23

F11-2D-L

CF2

1500

1.4

Rectangular

N/A

N/A

2D

Linear

24

F12-FEM

S

1500

1.2

Rectangular

PL. A

N/A

3D

Nonlinear

25

F13-FEM

S

1500

1.2

Rectangular

PL. B

N/A

3D

Nonlinear

26

F14-FEM

S

1500

1.2

Rectangular

PL. B

PL. B

3D

Nonlinear

27

F15-FEM

CF2

1500

1.4

Rectangular

N/A

PL. B

3D

Nonlinear

28

F16-FEM

CF2

1500

1.4

Rectangular

PL. B

PL. B

3D

Nonlinear

29

F17-FEM

CF2

1500

1.4

Rectangular

N/A

N/A

3D

Nonlinear

30

F18-FEM

S

1550

1.2

Trapezoidal-2

N/A

N/A

3D

Nonlinear

31

F19-FEM

S

1610

1.2

Triangular

N/A

N/A

3D

Nonlinear

32

F20-FEM

S

600

1.2

Rectangular

N/A

N/A

3D

Nonlinear

33

F21-FEM

S

1000

1.2

Rectangular

N/A

N/A

3D

Nonlinear

34

F22-FEM

S

1500

2

Rectangular

N/A

N/A

3D

Nonlinear

35

F23-FEM

S

1500

4

Rectangular

N/A

N/A

3D

Nonlinear

36

F24-FEM

CF2

1500

2

Rectangular

N/A

N/A

3D

Nonlinear

37

F25-FEM

CF2

1500

4

Rectangular

N/A

N/A

3D

Nonlinear

Figure 4.7 Shear strengthening modelling with the CFRP ratio of 0.48

To study the effects of applying CFRP strips on the compressive flange to increase the local stiffness, specimens S11 and S12 are selected.

## Analysis method

In this research static analysis is used. The static analysis of structures may be utilized for linear or nonlinear analysis.

The total load (P) is divided into several point loads, all of which are applied to the beam on the top flange the same as in the experimental test.

## Nonlinear analysis

Normally, much of nature includes nonlinear systems, therefore, most actual models behave nonlinearly. The sources of nonlinear behaviour of structures include: (a) change in system condition compared to primary condition, (b) nonlinear geometrical behaviour, which includes structures with large deformation, and (c) nonlinear material properties. According to the abovementioned points, steel structures behave nonlinearly.

In this case, the load is applied to the structure incrementally in a few steps. The ultimate bearing load is achievable by using the nonlinear analysis. The Newton-Raphson method is used.

For the nonlinear analysis of the steel beams, nonlinear properties of the materials are defined for the steel beam, steel plates, bolts, and adhesive as isotropic materials, however, for CFRP, linear and orthotropic properties are defined (Linghoff et al., 2009). Also, the load is applied incrementally in several steps.

## Linear analysis

In the linear case, the ultimate bearing load is not measurable directly. According to the nonlinear analysis results, the maximum load bearing capacity of the specimens was around 200kN, therefore, in the linear analysis the total load of 200kN is applied to the beams. The load is applied to the structures in one single step.

In the linear analysis of structures, the load is applied to the structures in one single step, and the linear properties of the materials are defined. In this case, the failure modes are not achievable directly, and the ultimate strain and stress must be checked manually.

Table 4.2 Specifications of the simulated shear strengthened beams (Full length)

## Steel stiffener

Modelling type

Analysing type

Running time (Sec)

Supports

1

S1-FEM

N/A

0

0

0

0

N/A

N/A

N/A

3D

Nonlinear

2

S2-FEM

S1-FEM

0

0

0

0

N/A

N/A

N/A

3D

Nonlinear

3

S3-FEM

N/A

0

0

0

0

N/A

3D

Nonlinear

4

S4-FEM

S3-FEM

0

0

0

0

N/A

3D

Nonlinear

5

S5-FEM

N/A

1

1

1

1

N/A

3D

Nonlinear

6

S6-FEM

N/A

1

0

0

1

N/A

3D

Nonlinear

7

S7-FEM

N/A

0.72

0.72

0.72

0.72

N/A

3D

Nonlinear

8

S8-FEM

N/A

0.72

0

0

0.72

N/A

3D

Nonlinear

9

S9-FEM

N/A

0.48

0.48

0.48

0.48

N/A

3D

Nonlinear

10

S10-FEM

N/A

0.48

0

0

0.48

N/A

3D

Nonlinear

11

S11-FEM

N/A

1

1

1

1

(1.2*45*1000)*2

N/A

N/A

3D

Nonlinear

12

S12-FEM

N/A

0.72

0.72

0.72

0.72

(1.2*45*1000)*2

N/A

N/A

3D

Nonlinear

## Selection of modelling and analysis methods

To simulate the CFRP strengthened steel I-beams, 2D, 3D, linear, and nonlinear methods can be used. In this section, the proper methods to model and analyse the CFRP strengthened steel beams are chosen based on the accuracies of FE results compared to the experimental results.

To investigate the results of different modelling and analysing approaches, three flexural strengthened beams (F3, F10, and F11) are selected (see Table 4.1 for the beams' specifications). These specimens are modelled and analyzed using the 2D/3D and linear/non-linear methods to verify the accuracy of FE results and selecting the proper method.

To investigate the accuracy, the tensile strain on the bottom flange at the mid-span for F10 (Figure 4.8), compressive strain on the top flange at the mid-span for F3 (Figure 4.9), tensile strain on the CFRP plate at the mid-span for F11 (Figure 4.10), vertical deflection of the beam at the mid-span for F11 (Figure 4.11), and horizontal deflection of the beam at the mid-span for F10 (Figure 4.12) are chosen for investigation. In these figures the horizontal axis are strain in Microns or deflection in mm.

As these figures show, the best accuracy to the experimental results is achieved by the 3D modelling case and nonlinear analysis. Therefore, these approaches are chosen to simulate and analyse all the specimens.

Figure 4.8 Tensile strain on bottom flange (F10)

Figure 4.9 Compressive strain on top flange (F3)

Figure 4.10 Tensile strain on CFRP plate (F11)

Figure 4.11 Vertical deflection (F11)

Figure 4.12 Horizontal deflection (F10)

## Concluding remark

In this chapter, the numerical simulation programs were explained in detail. First, the FEM was described in general. Second, ANSYS software was introduced. Then, two modelling methods, 2D and 3D, were described. In order to introduce the simulation method, element selection, meshing method, surface interaction, and boundary conditions were investigated. Subsequently, specifications of the flexural and shear specimens were indicated as tables. Then, linear and nonlinear analysis methods were introduced. Finally, the appropriate modelling and analysis methods were chosen based on the results of simulation and analyses of different specimens compared to the experimental results.