Master
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Analysis and design of load bearing structures Aalborg University School of Civil Engineering and Science
MSc Structural and Civil Engineering, 1st semester Group B219 Fall 2013
School of Engineering & Science Sohngårdsholmsvej 57 DK-9000 Aalborg Øst Telephone (+45) 9940 8530 http://www.ses.aau.dk
Title: Analysis of a steel beam with holes Theme: Analysis and design of load-bearing structures Project period: MSc 1st Semester Fall 2013 Project group: Group B219 Members: Liuba Agapii Jonas Sneideris Paulius Bucinskas Nina Korshunova Alexandru-Iulian Radu Nicolas Martinez Almario Supervisor: Christian Frier No. printed copies: 6 No. of pages: 104
Synopsis:
Before a structure can be manufactured engineer has to evaluate the results received from different types of calculations. This report consists of analytical, numerical and experimental analysis of a beam perforated with holes. The main focus of this paper is to assess the behavior of the beam in linearelastic area in two different loading cases. In the analytical part the beam is analyzed in accordance to the following beam theories: Bernoulli-Euler and Timoshenko. These two approaches are compared and the result of it is the basis for calculation of the necessary values. The experimental analysis gives real results of the test object. Firstly the material properties are tested and the obtained results are used in analytical and numerical parts. Secondly the beam is tested and after processesing the received measurements, the results are compared with calculations. The numerical analisys is performed by using our own written code in Matlab R2013b and by commercial software ABAQUS CAE v.6.12. The following models were created: two dimensional (2D) shell and three dimensional (3D) shell and solid. 2D-shell model analisys was performed both in Matlab and Abaqus, also 3D-shell and solid models were analysed in Abaqus. The conclusion is made out of the comparison of three different approaches in analyzing the beam.
Completed: 16 Dec 2013 The contents of this report are freely accessible, however publication (with source references) is only allowed upon agreement with the authors.
Preface The report was prepared by the group B219 at 1st semester of MSc program Structural and Civil Engineering at Aalborg University. The project was completed in the module “Analysis and design of load-bearing structures” under supervision of Christian Frier. The paper was handed in December 2013. This report is based on the following courses: Material Modeling in Civil Engineering and Structural Mechanics and Dynamics. The main aim is to learn how to apply different approaches in solving the problem and comparing the obtained results. The group would like to thank the supervisor of the project Christian Frier and all members of the group for hard work during the semester.
Reading guide The Harvard system of referencing is used in this report. Through the report in brackets the name of the author and year of publication with page is sited (f. x. Lars Damkilde, 2013, p.5). The material that was read and not quoted can be found after the report in the bibliography. Here the references are written in the following order with full information: author (in case of more than one author it is stated the first name and “et al.”), year of publication, title, the edition. For a paper from the Internet it is written as following: author, year of publication, title, designation, name of institution submitted, the source which was the basis for writing the chapter of the project is stated before this chapter. The numeration of tables, figures and equations starts with the number of the chapter and continue through the appropriate part. The appendix is numbered in the same manner. All calculations presented in the appendix are done by our group.
Figure: Project structure.
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Notations Symbol A Av b E F G I Kt M N Q q S t u w z α β γ ε θ κ ν σ τ ϕ
Description Area of cross section Shear area Width of the cross section Elastic modulus Concentrated force Shear modulus Moment of inertia Stress consentration factor Moment Normal force Shear force Distributed load Static moment of cross section Thickness Projection of displacement to x-direction Projection of displacement to y-direction Distance from neutral axis Rotation Shear correction factor Shear strain Normal strain Total cross section rotation Curvature Possion‘s ratio Normal stress Shear stress Shear angle
4
Table Of Contents Table Of Contents Chapter 1 Introduction 1.1 Analitycal Analysis . . . . . . . . 1.2 Experimental Analisys . . . . . 1.3 Numerical Analisys . . . . . . . 1.4 Comparisons and Conclusions
5
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9 10 10 10 10
Chapter 2 Beam characteristics and Model 2.1 Beam description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Beam Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Cross section characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 11 11 12
Chapter 3 Material Properties 3.1 Specimen set-up . . . . . . . . . . . . . 3.2 Strain gauge set-up . . . . . . . . . . . 3.3 Spicemen test . . . . . . . . . . . . . . 3.4 Material properties and data analysis
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17 17 18 19 20
Chapter 4 Analytical part 4.1 Simplifications and boundary conditions . . . . . . . . . . . . . . . . . 4.2 Bernoulli-Euler beam theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Constitutive equation and section integration . . . . . . . . . . 4.2.3 Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Differential Equation . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Timoshenko Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Kinematics conditions . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Constitutive equation . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Equlibrium condition . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Comparison between Bernoulli-Euler and Timoshenko beam theories 4.5 Stresses related to the cross section . . . . . . . . . . . . . . . . . . . . . 4.5.1 Stress concentration factor . . . . . . . . . . . . . . . . . . . . . 4.5.2 Calculation of stresses . . . . . . . . . . . . . . . . . . . . . . . .
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23 23 25 25 25 26 27 30 31 32 32 38 39 39 41
Chapter 5 Experimental Part 5.1 Beam set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Rosette strain gauges set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43 43 46
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5.3
Channels set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
49
5.4
Data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Chapter 6
Numerical Analysis
53
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.2
Model set-up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.2.1
Boundary conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.2.2
Applied load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
6.2.3
Supports of the beam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
6.2.4
Symmetry conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
The meshing of the models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
6.3
6.4
6.5
6.3.1
2D Matlab model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
58
6.3.2
3D Abaqus models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Convergense analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
60
6.4.1
2D Matlab model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
61
6.4.2
3D Abaqus shell model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
6.4.3
3D Abaqus solid model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
66
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
Chapter 7
Comparison-Conclusion Chapter
73
7.1
Displacement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
7.2
Stresses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
75
7.3
Discussions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
80
Bibliography
83
A Appendix
85
A.1
Diagrams for the shear force and the moment for both loading cases . . . . . . .
85
A.2
Calculation of moment of inertia and area . . . . . . . . . . . . . . . . . . . . . . .
86
A.3
Finite element theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
A.3.1
FEM - Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
A.3.2
FEM - Types of elements . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
A.3.3
Finite element calculations . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
A.3.4
FEM - Plane strain and plane stress . . . . . . . . . . . . . . . . . . . . . .
94
Results obtained from different numerical models . . . . . . . . . . . . . . . . . .
94
A.4
B Digital Appendix
103
B.1
Calculation of moment of inertia and shear area . . . . . . . . . . . . . . . . . . . 103
B.2
Calculation of coefficients from material test . . . . . . . . . . . . . . . . . . . . . 103
B.3
Calculation of displacement of the beam using Bernoulli-Euler theory . . . . . . 103
B.4
Calculation of displacement of the beam using Timoshenko theory . . . . . . . . 103
B.5
Comparison of beam theories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
B.6
Calculation of normal stresses and shear stresses . . . . . . . . . . . . . . . . . . 103
B.7
Main test calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.8
2D MatLab . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.9
2D Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
B.10 3D Shell Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 6
B.11 3D Solid Abaqus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
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Introduction
1
In building construction beams with perforated holes are used often. The main advantage of this sort of beams is that ventilation, heating, plumbing, electicity and other communications can be installed through the holes of the beams, saving usefull space (Figure 1.1)
Figure 1.1. Beams perforated with holes in the structure
For this type of beams a simple analytical solution to find deflection does not exist. Beam theories are used for calculations and analysis of the beam behaviour. Later more advanced approach are inplemented in numerical models. Finally, an experiment is performed and obtained results are compared to the calculations. The main aim of the project is to find out how much the holes in the web affect the overall behaviour of the beam. Also to compare different calculation methods with the results from experimental and numerical parts. 9
1.1 Analitycal Analysis Analytical analysis considers the Bernoulli-Euler and Timoshenko beam theories. The descriptions of these theories can be found in Chapter 4.2 and 4.3 respectively. These two theories have different approaches for calculations. The main difference is that Timoshenkotheory considers shear deformations, while Bernoulli-Euler does not. These two theories present different results in displacement. It is expected to obtain a higher value for displacement using Timoshenko theory, because of shear force influence.
1.2 Experimental Analisys The experimental analysis provides the actual deformations, stresses and strains of the test specimen. Two experiments were performed: the first one - testing the material properties. It was performed using the same type of steel as the beam is made of. After the material properties test, the yield stress, elastic modulus, shear modulus and Poisson’s ratio were obtained. These values were later used in numerical and analytical parts. The second experiment is the main test, where displacements, stresses and strains are measured during deformation of the beam. The experiment is performed just in elastic state applying two different loading cases (see Chapter 2.2).
1.3 Numerical Analisys An approximate finite element model is used in numerical analysis to simplify calculations of complex problems. The model consists of elements that combined together form the original geometry. Two dimensional FEM model is created using linear strain triangular (LST) elements and isoparametric six node triangular element calculated using a code written in “Matlab” and “Abaqus” respectively. It was decided to make two different types of three dimensional models using computer software “Abaqus”: shell and solid. Two shell models were made, one using STR3 and STRI65 elements. Also two solid models were created: one using linear four node tetrahedral, another quadratic ten node tetrahedral elements. Different types of elements are compared and the best suited type is chosen for both 2D and 3D models.
1.4 Comparisons and Conclusions The comparison is made in order to consider which approach is more reliable in our case. A conclusion is made out of the comparison where it is stated uncertainties and assumptions.
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Beam characteristics and Model
2
2.1 Beam description The project analyses a HEA-140 steel profile, which belongs to a European steel profiles data base. The analysed case presents a beam with a span of 1940 mm in total. A special feature for this beam is the circular holes cut into the web. These holes have a diameter of 82 mm and are placed into the web precisely. The beam supports are placed at a distance of 1870 mm centred via the beam center and these are simple supports. The right support Figure 2.1, has blocked translation both on x and z axes and free rotation; blocked translation only on z axes is provided by the left support. The Figure 2.1 presents idealized boundary conditions.
Figure 2.1. Analysed beam elevation (HEA-140 steel profile) with the 82mm diameter holes.
2.2 Beam Model A beam model is the computational model of a real beam, assumed to a line which represents the imaginary axis of the real beam. Moreover, the user can load it in any cases of loading, such as concentrated forces, uniform loads and so on, in the idea of obtaining the most stressed areas or the maximum efforts for the specific diagrams. The project considers two different loading cases with two symmetrical concentrated external forces: These loading cases are presented in all parts of the project. Moreover, the same loading cases are also implemented in experimental test in order to compare test results to the results obtained from analytical and numerical calculations. 11
Case 1: The beam is loaded with two forces (F) applied symmetrically and with a span of 1000 mm between them (Figure 2.2). The F forces are both equal to 18 kN ( this value is determined in chapter 4.5.1 ) and represented by two concentrated forces.
Figure 2.2. Case 1- beam model with loads (F-concentrated forces).
Case 2: The beam is loaded with two forces (F) with intensity of 11 kN (this value is determined in chapter 4.5.1) and applied symmetrically and with a span of 240 mm between them Figure 2.3.
Figure 2.3. Case 2-beam model with loads (F-concentrated forces).
2.3 Cross section characteristics Cross section of an HEA is very similar to an H section (Figure 2.4). This type of sections are also called “double wide flange”. Euro profiles always has a curvature between the flanges and the web, concluding that HEA-140 has a curvature with a radius equal to 12 mm. Table 2.1 presents the sectional features and characteristics of an HEA-140 with curvatures. Figure 2.5 represents the cross section of an HEA-140 without curvatures. Section without curvatures is an advantage because it is easier to calculate section properties by hand and to portray the cross section in numerical calculation stage. Table 2.2 shows features and data of mentioned cross section(Figure 2.5)
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Figure 2.4. Cross section of a HEA-140 steel profile.
Table 2.1. Data and characteristics of a HEA-140 steel profile.
Table 2.2. Data and characteristics of a HEA-140 steel profile (profile without interior curves).
Figure 2.6 represents the cross section of an HEA-140 without curvatures and entire hole. It is placed on the beam elevation, in the center of each hole and practically could represent a critical section in the calculation model.
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Figure 2.5. The assumed cross section of a HEA-140 steel profile (profile without interior curves).
Figure 2.6. Cross section of a HEA-140 steel profile (profile without interior curves and with the 82 mm diameter hole).
Table 2.3 show data and other features of the above pictured cross section, with the entire hole. The average moment of inertia along the beam is 1, 4639 × 107 mm 4 . This value was calculated using Matlab function which finds value of moment of inertia for every x (xdistance from the start of the beam till the considered cross section) and average all obtained numbers(Figure 2.7). All calculations can be found in Appendix A.2 and Digital Appendix B.1.
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Table 2.3. Data and characteristics of a HEA-140 steel profile (profile without interior curves and with the entire hole).
Figure 2.7. Dependence of the moment of inertia from the distance.
Moreover, the same calculations were made for the whole and shear area of the cross section.
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Material Properties
3
In order to obtain better results from analytical, numerical and experimental model, it is important to use real material properties, determined via an experiment. The purpose of this chapter is to determine Young’s modulus, Shear modulus and Poisson coefficient. Tensile test is required to find these material properties.
3.1 Specimen set-up Before testing the specimen, some preparatory operations have to be made. A steel plate is cut from the beam’s web using a CNC water jet cutting machine; the operation involves cutting steel using a very precise water jet at a very high velocity. To mount the strain gauges on the steel plate, a clean surface is mandatory in order to obtain accurate results. The cleaning is done with an operation called sandblasting and it involves a sand jet propagated with high velocity which takes away a thin layer of steel (the residual layer) until the metallic luster is achieved. After a visual survey over the sandblasted surface, the surface is cleaned again because of the thin layer of metallic dust that was produced in the sandblasting stage. Acetone is used as an cleaning solvent, it removes the residual dust from the sandblasted surface.
Figure 3.2. Photo of how cyanoacrylate glue is used to mount one of the strain gauges
Figure 3.1. Photo of how acetone is used to remove the residual dust
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Strain gauges are now mounted with a cyanoacrylate glue. The contact surface between the strain gauges and steel should be without any air pockets, otherwise the results would be affected.
Figure 3.3. Photo of how the cables are attached to the strain gauges
Figure 3.4. Photo of the groups strain gauges
3.2 Strain gauge set-up Strain gauges are sensors with tiny internal structure, they are machine made, with the capability of measuring strains up to 8, 5 · 10−15 [µm/m] which leads directly to small stress increments. For our specimen test four strain gauges were used, two on each side. Figure 3.5 shows the position of strain gauges, x-x axis represents not only the specimen length axis, but also the applied force direction. Strain gauges are placed perpendicular one from another with a distance of 5[mm] between them. Each sensor is connected to a channel and the table 3.1 shows the sensors and their respective channels. Strain gauges used have these characteristics: 12[mm] in length and 6[mm] in width, a resistance of 120[Ω] ±0.35%, a strain gauge factor of 2.07 ± 1%, transverse sensitivity−0.1% and a thermal coefficient of the gauge factor of 93±10[10−6 / ◦C ], for steel α coefficient is equal to 10.8[10−6 / ◦C ] (Manufacturer technical data).
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Figure 3.5. Strain gauges set-up
Table 3.1. Centralized positions of strain gauges
3.3 Spicemen test Material is tested using Mohr & Federhaff Universal Static Testing Machine. The specimen is positioned in clamps, in a way that gauges are located the middle between the clamps. The aim of this test is to obtain the stress (σx ) and strains (εx , ε y ) that are acting in the test subject .They are found by monitoring strain gauges signals (voltage [V ]) and then converting them into displacements. Tensile load is applied in continuous steps until 90% of the elastic stage is reached, test is performed without creating any permanent deformations. Linear regression is performed (Digital Appendix B.2 ) for the data received from the elastic stage and Young’s modulus is found. Poisson coefficient is calculated by performing linear regression to the strains. Later the experiment continues- the load is increased until steel begins its post-elastic behaviour. Under the same load, permanent deformations appear, steel’s internal particles are reorganizing (also known as strengthening material), a thermal process is generated and heat is 19
released. New internal structure of particles creates some extra strength in the material. Load is further increased. Finally test specimen breaks with a ductile fracture, and the the collapsing force is registered. This type of fracture is very well known for steel structures and is also called the warning breaking due to the visible deformations.
Figure 3.6. Test set-up
3.4 Material properties and data analysis The material used in the project is Steel S355, this information was provided before performing the test. The tested beam has an HEA-140 profile, it is an Europrofile steel product manufactured by rolling steel ingots. These types of steel products are called laminated profiles. Test specimen has a cross section of 50[mm] by 5.5[mm] and a length of 500 [mm], it was cut from HEA-140 profile beams web. Test was performed three times for different maximum load as it is stated in Table 3.2, the data from these tests can be found in the appendix.(Digital Appendix B.2). For each test, specimen was only loaded in the elastic state, to avoid permanent displacements, also the strain gauges register data only for the elastic deformations. The experimental data was processed using linear regression. Linear regression creates a straight line through a set of points, in a way that squared residuals are as small as possible. After that strength of material is calculated by dividing the force (N ) by the cross-section area (A) Eq. 3.4.0.1. Young’s modulus is obtained by the use of Hooke’s law Eq. 3.4.0.2 and the Poisson coefficient by dividing ε y by εx Eq. 3.4.0.3.
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After data interpretation, an average maximum normal stress was determined σx = 321.705[N /mm 2 ] which is close to table value. σx =
N A
(3.4.0.1)
where σx is the normal stress, N is the normal force and A is the area of the cross section.
E=
σx εx
(3.4.0.2)
where E is the Young’s modulus and εx is the normal strain.
ν=
εy
(3.4.0.3)
εx
where ν is the Possion’s ratio and ε y is the strain in y-direction
G=
E 2 · (1 − ν)
(3.4.0.4)
G- is the Shear modulus.
Table 3.2. The average value for Young’s module and Poisson’s ratio for performed tests
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Table 3.3. Young’s modulus values for different tests
Table 3.4. Poisson coefficient values for different tests
22
Analytical part
4
4.1 Simplifications and boundary conditions An analytical approach is used when there is a need to solve a given problem as fast as possible. Only not very detailed models can have analytical solutions, because more complicated models would produce overcomplicated calculations. In this project we use simple analytical approach and make the following simplifications:
• We neglect the weight of the beam in analytical part calculations. • The real cross section has curvatures. We simplify the geometry by changing the curvatures into right angles. • We model the beam with simple supports and two loads in two different loading cases. It is impossible to take into account all possible loading cases which can occur in actual conditions as we have limited laboratory equipment. • In static system the overhanging parts of the beam are neglected. There is no moment and no shear force in these ends. • We also consider only linear elastic behaviour of the beam. It was done to avoid yielding. Dynamic behaviour is not analyzed in this project. • Buckling is not considered. • Temperature and atmospheric conditions are neglected. • Moreover, we simplify the model and calculate strain in x-direction and displacement in z-direction. • In real conditions loads and supports are distributed, in our case it is assume that they are point loads and point supports. • The assumptions applied for each beam theory will be described later.
To find constants in differential equations in beam theories, boundary conditions needs to determine. The static model and equilibrium equations are stated in 4.2 and 4.3 of this report. The defined kinematic boundary conditions are based on continuous deformations. The beam is divided in 3 elements: 1. 0 ≤ x ≤ a mm 2. a ≤ x ≤ b mm 3. b ≤ x ≤ L mm 23
Figure 4.1. To determine boundary conditions.
To the simply supported beam (Figure 4.1) the kinematic boundary conditions are defined as:
w 1 (x) = 0, if x = 0 w 3 (x) = 0, if x = L w 1 (x) = w 2 (x), if x = a w 2 (x) = w 3 (x), if x = L - b θ1 (x) = θ2 (x), if x = a θ2 (x) = θ3 (x), if x = L - b
(4.1.0.1)
where w is deflection in z-direction; θ is rotation of the corresponding cross section. The direction of the displacement and rotation is shown in Figure 4.2.
Figure 4.2. Beam Bending Andersen & Nielsen [2008]
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In order to be able to solve the given problem we have to set the static boundary conditions. We have pinned support at x=0 and roller support at x=L . The considered problem is statically determined. The diagrams for moment and shear force distribution for both loading cases are described in chapter 2.2 can be found in Appendix A.1. The chosen signs direction are presented in Figure 4.3
Figure 4.3. Equilibrium for infinitesimally small beam element with distributed load, q Haukaas [2013a].
4.2 Bernoulli-Euler beam theory The Bernoulli-Euler beam theory allows us to determine the deflection of straight loaded beam. It is assumed that the material is linear elastic (Hooke‘s law). The theory’s main assumption is: during bending the plane sections and cross sections remain plane and perpendicular to the neutral axis (see Figure 4.2). According to this assumption it can be considered that the effects of shear force are neglected. Other assumptions are that: Poisson‘s ratio is equal to zero and the shape of cross section does not change.Andersen & Nielsen [2008], Haukaas [2013a]
4.2.1 Equilibrium The equilibrium equations are based on the following considerations: 1) equilibrium in the xdirection for the infinitesimal beam element; 2) distributed load, q, acts on opposite direction to z-axis Figure 4.3. According to Figure 4.3 the equilibrium equations can be written down. Vertical equilibrium yields: Q − qd x(Q + dQ) = 0
⇒
dQ = −q dx
(4.2.1.1)
Moment equilibrium about the rightmost edge yields: (M + d M ) − M −Qd x + qd x ·
dx =0 2
⇒
dM =0 dx
(4.2.1.2)
4.2.2 Constitutive equation and section integration The material law throughout linear elastic theory is Hooke‘s law: σ = E ·ε
(4.2.2.1) 25
where σ − normal stress; E − Young‘s module; ε − strain. Notice that Eq 4.2.2.1 is based on two-dimensional elasticity‘s theory and is called plane stress material law.Haukaas [2013a]. Plane stress material law in x-direction can be written down:
σx x = E · εx x = −E
µ
¶ d2 w ·z dx 2
(4.2.2.2)
Axial stresses over the cross-section are integrated using formula below: Z M = −σ · z · dA
(4.2.2.3)
A
where A- cross-section area. Haukaas [2013a].
4.2.3 Kinematics Strains in the cross-sections can be computed using Navier‘s hypothesis for beam bending(Figure 4.4).
Figure 4.4. Navier‘s hypothesis for beam bending Haukaas [2013a].
Strain in the cross section is defined with the formula below:
ε=
du dx
(4.2.3.1)
where u is displacement. Eq. 4.2.3.1 means that displacement u (elongation or shorten of upper or lower layers of crosssection) is divided by original length x. According Figure 4.4 it is obvious that displacement u is related with the rotation of cross-section dθ. Under the assumption of Bernoulli-Euler beam theory (during the bending the plane sections and cross sections remain plane and perpendicular to the neutral axis) the axial displacement of each layout in the cross-section is expressed in formula: d u = −z · d θ
(4.2.3.2) 26
Also according Figure 4.4 the rotation of the cross-section is defined as:
θ=
dw dx
(4.2.3.3)
Combining Eq. 4.2.3.2 and Eq. 4.2.3.3 the axial displacement is obtained: u = −z ·
dw dx
(4.2.3.4)
Finally, using equations mentioned above in sub-article “ 4.2.3 Kinematics” the kinematic equation for beam members is obtained:
ε=
du dθ d2 w = −z · = −z · dx dx dx 2
(4.2.3.5)
Infinitesimal beam part in the Figure 4.4 is curved because of the bending moment effect. The curvature in Bernoulli-Euler beam theory is defined as:
κ≈
dθ d2 w ≈ dx dx 2
(4.2.3.6)
Relation between bending moment and curvature is defined:
M = κ·E ·I
(4.2.3.7)
where E- elasticity modulus and I- moment of inertia.
4.2.4 Differential Equation Combining equations for equilibrium, material law, section integration and kinematics, differential equation is obtained:
d2 M dQ d2 =− q =− = − dx dx 2 dx 2
Z
d2 σ·zdA = − 2 dx A
Z
d2 E ·ε·zdA = − 2 dx A
Z A
E·
d2 w 2 d4 w ·z dA = −E ·I · dx 2 dx 4 (4.2.4.1)
where E- elasticity modulus is assumed constant over the cross-section; Moment of inertia is defined:
Z
Iy =
z 2 dA
(4.2.4.2)
A
27
Since the boundary conditions are specified it is more convenient to use the following formulas. Shear force: Q =E ·I ·
d3 w dx 3
(4.2.4.3)
Bending moment:
M =E ·I ·
d2 w = κ·E ·I dx 2
(4.2.4.4)
The cross-section rotation:
θ=
dw dx
(4.2.4.5)
Eq. 4.2.4.4 can be expressed as:
d2 w M = 2 dx E·I
(4.2.4.6)
Integrating Eq. 4.2.4.6 yields to Eq. 4.2.4.7, Eq. 4.2.4.8 and Eq. 4.2.4.9:
dw 1 = dx
Z
dw 2 = dx
Z
dw 3 = dx
Z
M1 dx +C 1 E·I
(4.2.4.7)
M2 dx +C 2 E·I
(4.2.4.8)
M3 dx +C 3 E·I
(4.2.4.9)
The displacements of the neutral axis of the beam are defined using Eq. 4.2.4.7, Eq. 4.2.4.8 and Eq. 4.2.4.9:
Z Z
w 1 (x) =
Z Z
w 2 (x) =
M1 dxdx +C 1 · x +C 4 E·I
(4.2.4.10)
M2 dxdx +C 2 · x +C 5 E·I
(4.2.4.11) 28
Z Z
w 3 (x) =
M3 dxdx +C 3 · x +C 6 E·I
(4.2.4.12)
Constants C 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 are determined by using boundary conditions described in Eq.( 4.1) and using computer program “Matlab”. The calculations and results are presented in digital appendix B.3. On the following charts (Figure 4.5, Figure 4.6) the resulting curve of the displacement is shown for different loading cases.
Figure 4.5. Displacement of the beam in the first load case.
In Table 4.1 it is shown the maximum value of the displacement using different values of moment of inertia It is clear that the difference between the received results is relatively small. Between the average moment of inertia and the minimum value it is 1.96% . So, it can be concluded that there is no need to use the moment of inertia like a function, because it makes calculations too complicated. Also, in comparison with the first loading case, the second way of loading decreases the value of the moment and also leads to 7,84% lower value for the displacement.
29
Figure 4.6. Displacement of the beam in the second load case.
Table 4.1. The comparison of the values of displacement
4.3 Timoshenko Beam Theory The Timoshenko beam theory retains the assumption that the cross section remains plane during bending. However, the assumption that it must remain perpendicular to the neutral axis is changed. In other words, the Timoshenko beam theory also consider shear deformations.Haukaas [2013b] To derive the Timoshenko differential, kinematic and the constitutive conditions has to be fulfilled. 30
4.3.1 Kinematics conditions
Figure 4.7. Assumptions for Timoshenko Beam Theory.
According Figure 4.7, the relation between displacement and rotation is defined in Eq.( 4.3.1.1).
α=
dw dx
(4.3.1.1)
Also, based on Figure 4.7 the relation between the shear angle ϕ and the rotation α can be written as:
θ = α+ϕ =
dw +ϕ dx
(4.3.1.2)
Normal strain:
εx x =
du dx
(4.3.1.3)
Shear strain: γx z =
du dw + dz dx
(4.3.1.4)
The displacement in the x direction: u = −θ · z
(4.3.1.5) 31
Substitution of Eq.( 4.3.1.5) in Eq. ( 4.3.1.3) and Eq. ( 4.3.1.3) yields:
d2 w dϕ + ·z dx dx
εx x = −
(4.3.1.6)
γx z = ϕ
(4.3.1.7)
4.3.2 Constitutive equation The material law throughout linear elastic theory is Hooke‘s law: σ = E ·ε
(4.3.2.1)
Substitution of the normal strain and shear strain equations in Hooke‘s law yields stress σx x and shear stress τx z equations.
σx x = E · εx x = −E
µ
¶ d2 w dϕ + ·z dx 2 dx
(4.3.2.2)
τx z = G · γx z = G · ϕ,
(4.3.2.3)
where G is the shear modulus defined by: E 2(1 + v)
G=
(4.3.2.4)
4.3.3 Equlibrium condition Z µ
Z
M=
A
Z
Q=
A
−σx x · z · d A =
τx z · d A =
Z A
A
¶ d2 w dϕ + E · z2 · d A dx 2 dx
G ·ϕ·d A
(4.3.3.1)
(4.3.3.2)
Shear is defined area as Av =βv · A , where A is the actual area and βv is an shear correction factor.
Z A
τ · d w · d A = d wv · Q
(4.3.3.3)
If kinematic relationship is d w = γd x then:
Z A
(γ · d w) · t τ · d A = (γ · d x) · Q
(4.3.3.4) 32
Substitution of the material law γ =
τ G
in Eq. 4.3.3.4 yields to:
Z ³ ´ τv τ ·dx ·τ·d A = ·Q · dx G A G
(4.3.3.5)
Furthermore, on the right-hand side the average shear stress is written in terms of the shear Q force on the cross-section, i.e., τv = Av , where A v is an auxiliary shear area that is defined as:
Z ³ ´ 1 Q τ ·dx ·τ·d A = · ·Q · dx G G Av A
τ=
(4.3.3.6)
Q·S , I·b
(4.3.3.7)
where b is the width of the cross section.
Z A
¶ µ 1 Q·S 2 1 Q ·dx · ·d A = · ·Q · dx G I·b G βv · A
(4.3.3.8)
Resolving Eq. 4.3.3.8 βv can be written as following:
βv =
A
I2 R ¡ S ¢2 A b
(4.3.3.9) dA
The equation can be simplified for the I- beam:
βv ≈
Aw e b A
(4.3.3.10)
33
Figure 4.8. Shear area.
According to the Figure 4.8 it is assumed that A w e b ≈ is shear area for the I-beam profile. The bending moment M(x) and the shear force Q(x) can be written following:
µ
M=
Q=
¶ d2 w dϕ + EI dx 2 dx
(4.3.3.11)
µ 3 ¶ dM d w d2 ϕ = + EI dx dx 3 dx 2
(4.3.3.12)
Q = G · Av · ϕ
(4.3.3.13)
Knowing the shear force in the beam Q(x), the distributed load can be found q(x):
dϕ dϕ 1 dQ = −G · A v · → =− ·q dx dx dx G · Av
(4.3.3.14)
¶ d4 w d3 ϕ EI + dx 4 dx 3
(4.3.3.15)
q =−
µ
q=
The equation for the quasi static bending of a linear elastic, isotropic, homogeneous beam of constant cross-section beam is:
d4 w 1 d2 q q = · + dx 4 G · A v dx 2 E · I
(4.3.3.16)
34
The bending moment, M and the shear force, Q can be identified through equilibrium conditions:
M d2 w dϕ = + EI dx 2 dx
(4.3.3.17)
Q = G · Av · ϕ
(4.3.3.18)
dQ 1 dϕ · = dx G · A v dx
(4.3.3.19)
Replacing Eq. 4.3.3.19 in Eq. 4.3.3.17 yields:
d2 w M dQ 1 = − · 2 dx EI dx G · A v
(4.3.3.20)
Integrating Eq. 4.3.3.20 yields to Eq. 4.3.3.21, Eq. 4.3.3.22 and Eq. 4.3.3.23:
1 dw 1 =− · Q1 · d x + dx G · Av
Z
dw 2 1 =− · Q2 · d x + dx G · Av
Z
dw 3 1 =− · Q3 · d x + dx G · Av
Z
M1 d x +C 1 EI
(4.3.3.21)
M2 d x +C 2 EI
(4.3.3.22)
M3 d x +C 3 EI
(4.3.3.23)
The displacements of the neutral axis of the beam, w(x) are defined as following:
Z
w 1 (x) = −
Z
w 2 (x) = −
Z
w 3 (x) = −
Q1 d xd x + G · Av
Z Z
Q2 d xd x + G · Av
Z Z
Q3 d xd x + G · Av
Z Z
M1 d xd x +C 1 · x +C 4 EI
(4.3.3.24)
M2 d xd x +C 2 · x +C 5 EI
(4.3.3.25)
M3 d xd x +C 3 · x +C 6 EI
(4.3.3.26)
35
θ1 (x) =
Z
θ2 (x) =
Z
θ3 (x) =
Z
M1 d x +C 1 EI
(4.3.3.27)
M2 d x +C 2 EI
(4.3.3.28)
M3 d x +C 3 EI
(4.3.3.29)
The constants C 1 ,C 2 ,C 3 ,C 4 ,C 5 ,C 6 can be determined using boundary conditions (Chapter 4.1, Eq. 4.1, Eq. 4.3.3.27, Eq. 4.3.3.28 and Eq. 4.3.3.29). It was done using “Matlab”. The calculations and results are presented in the digital Appendix B.4. On the following chart (Figure 4.9 and Figure ??) the resulting curve of the displacement is shown.
Figure 4.9. Displacement of the beam in the first loading case.
36
Figure 4.10. Displacement of the beam in the second loading case.
In the Table 4.2 it is shown the maximum value of the displacement using different values of moment of inertia.
Table 4.2. The comparison of the values of displacement.
It can be concluded that the difference between minimum and average value becomes significant in this theory (16.82%). Therefore it was decided to use minimum values for final results to be on the safe side. Also, the displacement is lower in the second case of 2.37% than in the first one.
37
4.4 Comparison between Bernoulli-Euler and Timoshenko beam theories In this chapter Bernoulli-Euler beam theory is compared with Timoshenko beam theory. The comparison is based on the displacements which are defined by these two theories. The equation of displacement for Timoshenko beam theory are shown in the Eq. 4.3.3.27, Eq 4.3.3.28and Eq. 4.3.3.29. The equation of displacement for Bernoulli-Euler beam theory are Eq. 4.2.4.10, Eq. 4.2.4.11 and Eq. 4.2.4.12
Figure 4.11. The displacement of the beam calculated by Timoshenko and Bernoulli-Euler theory for both of the loading cases(minimum value of cross section properties is used).
In the Figure 4.11. it is ploted charts by which the results can be compared. In the Table 4.3 it is shown the value of the displacement for minimum value of moment of inertia (the worst case) and for different cases of loading.
Table 4.3. The comparison of the values of displacement.
The difference in two theories is presented in both loading cases. It can be explained as following. In Timoshenko beam theory the shear force is taken in to consideration. Timoshenko beam theory will provide bigger displacement for short and thick beams, where the shear force will have an important influence. In our case, the Timoshenko beam theory has significant contribution in first loading case on the displacement. To prove the conclusion that 38
difference between these two theories becomes bigger because of the decreasing of the length the following chart was done. (Digital Appendix B.5)
Figure 4.12. The dependence of the difference between theories from the length of the beam.
For convenience it was considered to make the comparison between two theories with the loading case when the span between the forces is 1 mm. All calculations were made with the minimum value of moment of inertia for the beam. The length of the beam was varied from 0 to 1870 mm.
4.5 Stresses related to the cross section 4.5.1 Stress concentration factor Stresses in the cross section weakened by the hole will have a different stress distribution in comparison with the stress distribution of a whole cross section. The dependence is that the maximum value of the stress is right on the edge of the hole, and it decreases rapidly in the direction away from the hole. This is called “stress concentration effect” (Figure 4.13).
Figure 4.13. Anticipated stress distribution.
39
Calculations of the stress concentration factor are very complicated. But if some assumptions are made an appropriate formula for calculating the stress concentration can be found. The choice of the formula depends on the loading case, the type and dimensions of the hole. However, it is important to note that K t is an ideal value based on linear elastic behaviour and it only depends on the proportions of the dimensions of the stress raiser and the notched part. To calculate the stress concentration factor in our project, the formula from the table is used. Pilkey & Pilkey [2008]. Two types of calculation is done. The first one is for a single circular hole in an infinite plate (Figure 4.14)
Figure 4.14. Single circular hole in a infinite plate in bending, m 1 and m 2 - bending moments Pilkey & Pilkey [2008].
In our case the element is under the simple bending, so the following formula can be used:
s
K t = 3.000 − 0.947
d d d + 0.129 , 0 ≤ ≤ 7.0 t t t
(4.5.1.1)
where d - the diameter of the holes, t - thickness of the plate. Calculating with the dimensions the result for stress concentration factor is 1.46. The second case is for single row for circular holes in infinite plate (Figure 4.15)
µ ¶2 µ ¶3 d d d d K t = 1.787 − 0.060 − 0.785 + 0.217 , 0 ≤ ≤ 1.0 L L L L
(4.5.1.2)
where L - the distance between the center of the holes. So, K t = 1.746 The different approaches to calculate the stress concentration factor was used in order to compare these results with the results from the numerical part. After determining the stress concentration factor the maximum value of the force without having the yielding in the cross section was defined. The maximum allowed stress for the weakened cross section:
σ=
σy i e l d Kt
=
335M P a = 191.87M P a 1.746
(4.5.1.3) 40
Figure 4.15. Single row for circular holes in infinite plate in bending Pilkey & Pilkey [2008]
The maximum allowed moment:
M m a x = σm a x · Ww e a k e n e d = 191.87 · 145859 = 27.99kN · m
(4.5.1.4)
The maximum value of allowed force can be defined from the following formula:
1 27.99 1 a2 Mm a x M m a x = F (l −a)+ F = = 55.77kN (4.5.1.5) → Fm a x = a2 12 2 8 l 0.5(l − a) + 8l 0.5(1.87 − 1) + 8·1.87 For the second loading case: Fm a x =
Mm a x 0.5(l − a) +
a2 l
=
27.99 2
0.24 0.5(1.87 − 0.24) + 8·1.87
= 34.18kN
(4.5.1.6)
Both values were divided by the safe factor of 3. The force for first loading case in our project was determined 18kN , for the second one - 11kN .
4.5.2 Calculation of stresses By combining of Eq. 4.2.2.3 and Eq. 4.2.4.2 Navier’s equations for calculations of stresses were obtained:
σ=−
M ·z I
(4.5.2.1)
minus sign appears because it is compressive (negative) stresses in the positive z-axis domain that gives a positive bending moment, i.e., bending moment with tension at the bottom. Shear stresses were calculated using Juravski formula (Eq. 4.3.3.7). All the calculations can be found in Digital Appendix B.6. 41
Experimental Part
5
In order to compare results with the analytical and numerical models, an experimental test has been performed. Displacements and strains are subjects ware measured at specific and precisely positioned points. This chapter refers to effects induced by loads applied to a HEA-140 profile beam. Holes are a subject to a special stress distribution case; that’s why, unidirectional and rosette strain gauges measure the strains in the influenced area, around a hole.
5.1 Beam set-up It is necessary to set-up the beam model in an adequate way to obtain results that are as close as possible to the results obtained from analytical and numerical calculations.
Figure 5.1. Beam set-up - position of loading cases (it was considered one case at a time)
The Figure 5.1 shows both loading cases, loading case number 1 is dotted in the drawing. The tested beam has a cross section HEA-140, an original length of L o r g = 1940[mm] and a length between the supports of L = 1870[mm]. Holes are cut using the same technology of CNC water-jet cutting machine. Supports are made of a steel roller with a diameter of 15[mm]and a length of 140[mm], beneath this roller a steel plate of 30 × 25[mm] by 140[mm] is placed. Supports base is a steel plate, with the following dimensions: 80 × 25 − 140[mm]. All three components are clamped together to form the roller support. The supports are placed 35[mm] from the each end of the beam. In terms of structural engineering, a roller support means a free rotation around its axis and restrained displacement on x-direction (x-x direction represents 43
the axis which is orthogonal to the deformed beam axis). Both ends of the model have the same supports as illustrated in the picture below (see Figure 5.3)
Figure 5.2. Beam set-up - roller support (Segment A-A)
Two applied loads, which basically represent two concentrated forces for each loading case, are the only external factors for which model is analysed. Two loading cases are analysed and two tests have been conducted for each of loading cases (Figure 5.1). In both loading cases, vertical forces are applied via manual hydraulic jacks. In the analytical calculations, forces are considered to be concentrated. But in reality a force can not be applied in a point, it always is distributed to a surface, which can only simulate a concentrated force. Each force is applied by the help of a “sandwich” made of two steel plates, two plastic plates and a grease core. The resulting surface between steel plate and the top flange is 50[mm] by 140[mm].
44
Figure 5.3. Beam set-up - applied force (Segment B-B)
Figure 5.4. Photo of loading case 1
Figure 5.5. Photo of loading case 2
The entire sub assembly can be seen in the picture above (Figure 5.3). Each loading case contains two forces [F ] witch are equal to each other, first case has F equal to 18[kN ], while in the second- F is equal to 11[kN ]. Both cases have the forces applied symmetrical from the beam center as follows: 500[mm] between the forces in first the situation and 120[mm] in the second.
45
Figure 5.6. Beam set-up - B219 group strain gauges and rosettes placed around 4th hole (beam segment is positioned up side down for a better illustration of Ch 14 and Ch 15 - Segment C-C)
5.2 Rosette strain gauges set-up A rosette strain gauge can provide a more complicated stress state of a point, because it measures data in three directions. Having not only two orthogonal directions [x-x; y-y], but also a third one - between previous two [x-y], lets us record the strains in all directions. Line x-y has a 45 ◦ orientation, its strains are directly related with the shear stress.
Figure 5.7. Geometric strain
46
Computational method for finding the strains: u1 =
∂u y y
d y; u 2 =
0 0
Leng t h(a b ) =
∂u y ∂u x ∂u x d y; u 3 = d x; u 4 = d x; ∂y ∂x ∂x
q
Leng t h(a 0 b 0 ) ≈ d x + u 3
d x + 2 · u 3 + u 32 + u 42
(5.2.0.1)
(5.2.0.2)
The normal strain in the x direction of the rectangular element is defined as: εx =
εy =
ext ensi on l eng ht (a 0 b 0 ) − l eng ht (ab) ∂u x = = or i g i nal l eng ht l eng ht (ab) ∂x
∂u y
(5.2.0.3)
(5.2.0.4)
∂y
The shear strain is defined as the change of the angle between and :
γx y = α + β
(5.2.0.5)
tan α =
u4 d x + u3
(5.2.0.6)
tan β =
u2 d x + u1
(5.2.0.7)
γx y =
∂u y ∂x
+
∂u x ∂y
(5.2.0.8)
Wikipedia [2013]
47
Figure 5.8. Rosette gauge
The strain formula in x-y direction is: εx y = ε45 −
ε90 + ε0 2
(5.2.0.9)
The shear strain formula is given by: γx y = 2 · εx y
(5.2.0.10)
Normal strains are defined in the material properties part, stresses in x-x, y-y and x-y directions, are evaluated using Hooke’s law (for plane stress).
E σx 2 1−ν σy = ν τx y 0
ν E 1−ν2
0
0 εx 0 · εy G γx y
(5.2.0.11)
Where E is the stiffness matrix: E ν 0 2 1−ν E E= ν 0 1−ν2 0 0 G
(5.2.0.12)
σx = [(εx + ν · ε y ) · E ]/(1 − ν2 )
(5.2.0.13) 48
- normal stress on x-x direction
σx = [(ε y + ν · εx ) · E ]/(1 − ν2 )
(5.2.0.14)
- normal stress on y-y direction
τx y = G · γx y
(5.2.0.15)
- shear stress on x-y direction
5.3 Channels set-up Rosette strain gauges have the ability to measure three-directional strains. All the preparatory operations as in the material testing stage were performed: sand blasting, cleaning of the surface, gluing the strain gauges, testing them for errors, preparing the model to be tested, these steps are important and mandatory in order to obtain accurate results. Table 5.1 shows the connection between each strain gauge/rosette direction and its channel. The experimental model operates with a total of twenty-four channels. As shown in the list, the first nine strain gauges are used by the entire semester, they are assigned in a succeeding order and have the following purposes: • Ch1 and Ch2 are placed along the supports axis to demonstrate the existence of shear force. • Ch3 and Ch4 have the same role and are positioned at the same distance from the flanges like the previous two and are near the 5t h hole. • Ch5 and Ch6 aims to show the normal stresses on the top and the bottom flanges and are placed on the beam symmetry axis. • Ch7 and Ch8 shows the forces applied to the model. • Ch9 is the displacement transducer and it measures the displacement at middle of the beam Each project group had six unidirectional strain gauges and three rosette strain gauges, the placement of these gauges was chosen by each group individually. We have chosen to place the sensors around the 4th hole to measure stress distribution around it. In the second loading case this hole is affected by a full shear force and from 85% to 95% of the total bending moment. Because of that the hole becomes more oval, as the stress increases. Picture above (Figure 5.6 shows unidirectional and rosette strain gauges positioning. Ch14 refers to Section 1-1 and Ch15 to Section 3-3, they are placed along the model symmetry axis and register extreme strains in accordance to extreme stresses for these two sections. Section 1-1 is characterized not only by the Ch14 but also by Ch10 and Ch11 they have the same role- x-x direction strain measurements. Section 3-3 has its cross-section influenced by the entire hole diameter, 82[mm], Ch12 and Ch13 are placed with the same x-x orientation, though they are place exactly inside the hole. This section aims to illustrate the effects of the hole to the normal stress distribution. It is a very important factor not only in the experimental part but also in the 49
Table 5.1. Centrelized positions of strain gauges and rosettes
previous chapters. Section 2-2 also shows the effects of the hole on the stress distribution as it is located 10[mm] from the edge of the hole as illustrated in the picture ( Figure 5.6). Rosettes are placed along Section 2-2, they measure strains in three directions. Section 22 has the most influential position and by the help of rosettes and close experimental data analysis interpretations can be easly performed. Moreover, this section is different from the other sections because, it is the only section in which measurements are made by the help of rosette strain gauges, thus allowing us to determine the full stress states. Channels 17, 20 and 23 are orientated in the x-x direction in the section 2-2. For the same section, strains on y-y axis are captured by Ch18, Ch21 and Ch24. Previous table (Table 5.1) shows the channels that correspond to each strain gauge plus strain gauge orientation. Conventional axis for all three directions have been chosen showed in the illustrations and those are:
50
• x-x is the axis parallel to the symmetry axis it goes along the models length and is perpendicular to the loading orientation; • y-y axis is orthogonal to x-x direction and parallel to the loading orientation; • x-y or y-x axis is the 45 ◦ vector resulting from the composition of the first two principal directions.
5.4 Data analysis The obtained data provides us with information about the strains, using the above mentioned formulas (Eq. 5.2.0.11 and the following equations) they are converted in to stresses. In following table (Table 5.2) the computed results are presented with connection to each channel and the type of strain gauge. In sections 1-1 and 3-3 unidirectional strain gauges were used, because using rosettes everywhere would be too expensive. Because unidirectional strain gauges provide information about strains in only one direction, stresses were calculated using Hooke’s law (Eq. 3.4.0.2). This assumption can lead to some errors in the obtained results. Graphs with the experimental results (Digital Appendix B.7 can be found in the chapter 7 (Comparisons-Conclusions)
51
Table 5.2. Centralized experimental model data- stresses of all Channels
Table 5.3. Centralized experimental model data- Displacements
52
Numerical Analysis
6
6.1 Introduction This chapter covers numerical analysis of the beam. Four different models were created using a code written in “MatLAB” and commercial finite element program Abaqus CAE. The main purpose of the numerical part is to investigate the behavior of the beam in two different loading cases using Finite Element method. The created models are compared with each other. The results from the most appropriate model are then compared with the values obtained from analytical and experimental parts. The following models are presented in this chapter: a 2D model which is created using an assumption of plane stress and plain strain; a 3D solid model and a 3D shell model. The magnitudes of forces were calculated to avoid yielding and exceeding linear elastic state in the beam (see Chapter 4.5.1). The theory about finite element method can be found in Appendix A.3
6.2 Model set-up All models presented in the numerical analysis are created using elastic parameters: Young’s modulus E = 2, 32 × 105 N /mm 2 and Poisson’s ratio ν = 0, 27. These values were obtained during the main beam test (see Chapter 5). 2D Matlab, 2D plane stress and plane strain Abaqus and 3D shell Abaqus models were created using cross section without curvatures while 3D solid Abaqus model based on cross section with curvatures. Different cross sections and their parameters can be found in Chapter 2.3 “Cross section characteristics”.
6.2.1 Boundary conditions Boundary conditions include applied forces, supports of the beam and symmetry planes. Those conditions are used in both 3D solid and 3D shell models. 2D plane stress and plane strain model is not divided in parts by symmetry planes and the whole beam is modelled. For this reason 2D model uses boundary conditions just for applied loads and beam supports.
6.2.2 Applied load Two loading cases (see Chapter 2.2 “Beam model”) are implemented in numerical models. To avoid infinite stresses, forces are applied as pressure loads on surfaces instead of concentrated forces in points. They are distributed in two 50 × 140mm size areas as in the laboratory during 53
the main test (see Chapter 5.1). Forces in both loading cases are applied in the negative ydirection (see Figure 6.1 and Figure 6.2).
Figure 6.1. Full beam model with applied forces in the 1st loading case.
Figure 6.2. Full beam model with applied forces in the 1st loading case.
6.2.3 Supports of the beam The supports in Abaqus are modelled as close as possible to the real experimental model supports (see Chapter 5.1). They are located 35mm from beam ends on the bottom flange (see Figure 6.3). Both supports are applied as lines over the width of the beam. A roller support (see Figure 6.3) is used at one beam end which restricts model movement on y-direction (see Figure 6.5).
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Figure 6.3. Roller support.
A pinned support (see Figure 6.4) is designed at the other end.
Figure 6.4. Pinned support.
It constrains displacement in y-direction in the applied section and rigid body motion in zdirection (see Figure 6.5). In both loading cases, the beam is equally supported. In contrast, point supports are used in two dimensional models with the same movement restriction. Point and line supports causes stress concentration in the areas where they are applied. To avoid this phenomenon plate supports could be modelled. It would help to distribute stresses in the wider area. In this project it was decided not to use plates like supports because the stress concentration in supports areas does not have big influence on the final results obtained from the models. Also, point/line supports were chosen to create similar support conditions like in the laboratory-roller supports (see Chapter 5.1).
Figure 6.5. Full beam model supported by roller at one end and pinned support at another.
55
6.2.4 Symmetry conditions In order to obtain more accurate results while saving computational power and time creating a finer mesh in more interesting areas, only a quarter of the beam is designed in both 3D solid and 3D shell models. This is done using two symmetry planes. The 1s t is a vertical plane which splits the cross section in two parts along the beam. The 2n d is a vertical cross section cut which is applied exactly in the middle of the beam (see Figure 6.6).
Figure 6.6. Symmetry planes in full beam model.
To obtain the same results from the quarter and the full beam models the following boundary conditions have to be specified. A roller support is applied on the cross section in the middle of the beam to keep the model from rigid body motion on z-axis (see Figure 6.7). It also restricts rotations around x and y-axes. Another roller support is designed on the second symmetry plane which splits the cross section in to two parts (see Figure 6.8). In this case displacement on x-direction and rotations around y and z-axes are constrained. Just a quarter of the total force in both loading cases is taken into consideration because the full model is split in to four parts. For the same reason load distribution area is four times smaller than in the full model.
56
Figure 6.7. The support applied on the cross section in the middle of the beam illustrates 1st symmetry plane.
Figure 6.8. Roller assigned along beam model interprets 2nd symmetry plane
Symmetry conditions specified above are used in both loading cases for all 3D models. It was decided not to use the symmetry conditions in the Abaqus 2D plane stress and plane strain model and in 2D model created in MatLAB because a fine mesh was generated for the full model and accurate results were obtained. Also the main purpose of the creation of 2D model in Abaqus is to compare it with the 2D MatLAB model.
6.3 The meshing of the models One of the most important steps of creating the model is to generate a fine mesh in order to get an accurate results. Higher density meshes produce more accurate results but at the same time require a lot of computational power and time. Increasing the number of elements in the whole model causes a lack of operational memory and calculations stop. That’s why it is important to find a mesh with an optimal number of elements. 57
6.3.1 2D Matlab model To be able to model the beam in MatLAB, a script for generating the mesh was created. Linear strain triangular elements (LST) were used (see Appendix A.3). Using the created script it is possible to change the mesh density (it is explained further). After performing convergence analysis, the mesh density which does not take too much time to compute and still gives accurate results, was determined. By using a mesh with approximately 50 000 nodes the final results were obtained. The nodes around the holes are generated using a circle equation, and with increasing distance from the holes edge- ellipse equations are used. The distances between the holes are meshed by means of line equation (see Figure 6.9).
Figure 6.9. Nodes created around hole.
The mesh density can be increased or decreased by changing variables cc, gg, nn in the MatLAB script input. The variable cc describes the mesh density around the holes, while gg and nn describe the mesh density between the holes and at the beam ends respectively (see Figure 6.10 and Figure 6.11).
Figure 6.10. Generated mesh on the part of the beam. In this case cc = 4, gg = 4 and nn = 4.
58
Figure 6.11. Generated mesh on the part of the beam. In this case cc = 8, gg = 8 and nn = 8.
6.3.2 3D Abaqus models In order to obtain reliable results and avoid calculation process interuption mesh is set up with varying density through the model (see Figure 6.12 and Figure 6.13 ). Finer mesh is created:
• around the holes; • in the top flange where the force is applied; • in the web section where the strain gauges are installed; • in the bottom flange at the one end where the beam is supported; • in the bottom flange at half the beams length (see Figure 6.14).
Figure 6.12. Side one of varying mesh density through the model. In this case mesh consists of 86 490 elements and 158 810 nodes.
59
Figure 6.13. Side two of varying mesh density through the model. In this case mesh consists of 86 490 elements and 158 810 nodes.
6.4 Convergense analysis After modelling the beam, a convergence analysis is made for deflection and stress. The size and the number of the total used finite elements influence the analysis results. Increasing the number of the elements will lead to more accurate results for the stated problem. A finer mesh was made around the holes, on the surfaces where the loads are applied and on the monitor points on the bottom flange. The mesh is created with various meshing techniques based on uniform seeding. In convergence analysis two different points are observed (see Figure 6.14):
• For von Mises stress, the point located on the top flange, 850 mm from the left-beam end in Abaqus and the middle point in the bottom flange in MatLAB are chosen (see Figure 6.15 ). • For deflection, the point is established in the middle of the beam on the bottom flange.
Because of the lack of time the different point for von Mises stresses convergence test in MatLAB was chosen to avoid rewriting script for convergence analysis. Von Mises stresses were determined in both MatLAB and Abaqus because they represent real stress state more general (the formula for calculations includes all stress components). The results for other stress components are presented in the Appendix A.4.
60
Figure 6.14. Marked points for convergence test for 3D Abaqus models
Figure 6.15. Marked point for both von Mises stress and displacement convergence tests for 2D Matlab model
6.4.1 2D Matlab model Convergence analysis was performed by increasing values of variables cc, gg, nn, which describe the mesh coarseness, as it is presented in the Table 6.1.
61
Table 6.1. Steps for increasing the mesh density in convergence analysis.
The following figures (see Figure 6.16 and Figure 6.17) show the results from performed convergence analysis for LST elements (see Appendix A.3). It can be noticed that the convergence of stress is slower than that of displacement (see Table 6.2).
Figure 6.16. Convergence analysis in terms of displacement for MatLAB 2D model
62
Figure 6.17. Convergence analysis in terms of von Mises stresses for MatLAB 2D model.
Table 6.2. The results from convergence test for von Mises stresses and displacement for 2D MatLAB model.
6.4.2 3D Abaqus shell model Rough mesh is increased with the step of 20[mm] and global seed in mesh around the holes is half of the rough mesh. Two types of elements are tested: 3-node triangular thin shell and 6-node triangular thin shell. Elements description can be found in Appendix A.3
63
Table 6.3. Created meshes for 3D shell Abaqus model.
The following figure (see Figure 6.18) and table (see Table 6.4) show the results from performed convergence analysis.
Figure 6.18. Convergence analysis in terms of displacement for Abaqus 3D shell model.
An approximation of using a shell in modelling of a structure is based on the advantage of the dimensions of the shell (thickness is small compared to the other dimensions) Simulia [2010]. Three-dimensional shell elements are 4- to 8-node isoparametric quadrilaterals or 3- to 6-node triangular elements in any 3-D orientation. Each shell element node has 5 degrees of freedom (DOF) - three translations and two rotations. The translational DOF are in the global Cartesian coordinate system. Mirza & Smell [2011] There are two types of shell elements in Abaqus: conventional and continuum shell elements. In conventional shell elements the thickness of material is defined through section properties. On the other hand, continuum shell elements resemble three-dimensional solid elements. Simulia [2010] Shell formulation assumes thin shell problems and thick shell problems. Thick shell problem takes in consideration transverse shear deformation and thin shell problem neglects transverse 64
Table 6.4. The results from convergence test for displacement for 3D Abaqus shell model.
Figure 6.19. Difference between continuum and conventional shell models
shear deformation. Shell element has both displacement and rotation degrees of freedom. In our case, the beam was modelled like a conventional thin shell model. The displacement is assumed to vary linearly through the thickness. Wemper & Talaslidis [2003] The convergence for Abaqus 3D shell model was performed only in terms of displacement because it is complicated to interpret stresses. Stress components in the shell are interpreted in the local directions, which are dependent on the orientation of each element and not on the orientation of the global axes. Stress components S11 and S22 are stresses in local 1 and 2 directions. Stress component S33 is normal to the surface therefore it is always equal to zero. Local 1- and 2directions lie in plane of the shell. Default local 1-direction is the projection of global 1-axis onto shell surface. If global 1-axis is normal to shell surface, local 1-direction is the projection of the global 3-axis on shell surface. Local 2-direction is perpendicular to local 1-direction on surface of the shell, so that local 1-direction, local 2-direction, and the positive normal to the surface form a right-handed set (see Figure 6.20),Morley [1995]
65
Figure 6.20. Shell element sign convention Morley [1995].
6.4.3 3D Abaqus solid model A test is performed by doubling the global seeds on the edges with every new mesh until it reaches 4[mm] in fine mesh areas and 8[mm] in rest of the beam. Later, a more accurate mesh is set up just in fine mesh parts (see Table 6.5).
Table 6.5. Created meshes for 3D solid Abaqus model.
Since the 3D solid Abaqus model consists of more than 87 000 quadratic elements it is not possible to run the model due to the limited virtual memory of the computer. But it is not necessary to continue the test since the convergence is reached (see Figure 6.21, Figure 6.22, Table 6.7 and Table 6.8). Displacement grows slightly with 10−3 [mm] increment after 87 000 linear elements bound. The same result is obtained using just 9200 quadratic elements. The results (see Figure 6.21 and Figure 6.22) prove that both types of elements can be converged, but ten-node tetrahedrons provide more accurate results. For the above mentioned reasons C3D10 elements are used in the final mesh. Distorted elements have a deformed geometry, i.e. one point (nodal or mid-side) is displaced far away from the rest of element points. Data 66
regarding this type of elements was collected during the test as well (see Table 6.22). It is very important to avoid distorted elements because when force is applied, strain value in such element and around it becomes unreliable. It can be noticed in Table 6.22 that number of distorted elements decreases with a higher density mesh (there are some inconsistencies because some individual elements get distorted in areas where mesh is rough).
Table 6.6. Distorted elements number in different meshes.
In this case it is not important that number of distorted elements increases at the end of the test (see Table 6.21) because quadratic elements (mesh number 8) are decided to use in the final 3D solid Abaqus model.
Figure 6.21. Convergence analysis in terms of von Misses stresses for Abaqus 3D solid model.
67
Figure 6.22. Convergence analysis in terms of displacement for Abaqus 3D solid model.
Table 6.7. 4-node linear tetrahedral elements.
68
Table 6.8. 10-node quadratic tetrahedrons.
Finally, 3D solid model mesh consisting of 87855 ten-node tetrahedron elements is decided to be used for the final model. Based on the convergence test, mesh with a lower element number could be used to obtain accurate results but since some distorted elements are provided it is better to use finer mesh.
6.5 Results In this sub-chapter the results obtained from different numerical models for displacement and von Mises stresses for the second loading case are presented. A more detailed discussion is developed in the following chapter (see Chapter 7). All obtained results from numerical part can be found in Appendix ref{ap:A.4 The biggest deflection (1,517 mm) was determined in the point located on the bottom flange in the middle of the beam (see Figure 6.23).
Figure 6.23. Deflection of 2D MatLAB model (deflection is not in scale).
As shown in the Figure 6.24 the biggest displacement is in the middle point of the area where the load is applied. The monitor point is in the middle of the beam and the deflection obtained from this point is equal to 1,526 mm. The difference between 2D MatLAB and 2D plane stress and plane strain models is very small because the same assumptions were made both for MatLAB model and Abaqus model. The only one difference between these two models is the used element type. The obtained displacements in the 3D models are shown in Figure 6.25 and Figure 3.6.4. More reliable results are determined in the 3D solid model (deflection is equal to 1,581 mm at monitor point) than in the 3D shell model (deflection is equal to 1,424 mm). That happens because in the 3D solid model the cross section of the beam better resembles the real cross 69
Figure 6.24. Deflection of 2D plane stress and plane strain Abaqus model.
section (e.g., curvatures are modelled). Since elements used are different in both models, the resulting deflection can also be affected.
Figure 6.25. Deflection of 3D Abaqus solid model.
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Figure 6.26. Deflection of 3D Abaqus shell model.
In the following figure (see Figure 6.27) the distribution of von Mises stresses in 2D MatLAB model is presented. It was decided to show von Mises stresses because it is common used failure criterion which represents the obtained results well. The maximum stresses appear around the hole because of the influence of stress concentration factor. This phenomenon was also discussed in analytical calculations (see Chapter 4.5.1)
Figure 6.27. Von Mises stresses for 2D Matlab model.
The von Mises stresses of 2D plane stress and plane strain model in Abaqus (see Figure 6.28), (maximum value is equal to 149,1MPa) are very close to results from 2D model in MatLAB (maximal value is at the edge of the fourth hole at 135 ◦ angle and the magnitude is 145, 9M P a). The small difference in results might appear because of different types of elements.
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Figure 6.28. von Mises stresses of 2D Abaqus model.
The maximum value of von Misses stress of Abaqus 3D solid model (269,4MPa) appears in the area where line support is applied (see Figure 6.29 upper left corner). Distribution of von Mises stresses in pressure load area varies approximately from 67MPa to 110MPa. It is visualized in Figure 6.29 lower right corner. All models can be found in Digital Appendix B.8, B.9, B.10 and B.11.
Figure 6.29. von Mises stresses of 3D Abaqus solid model.
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Comparison-Conclusion Chapter
7
This chapter aims to compare and discuss results gained from the analytical, numerical and experimental analysis. The following part will be divided into three main sub chapters: discussion of displacement, stresses and conclusion. All models uses parameters which were obtained via experimental workshop see Chapter 3.4.
7.1 Displacement Bar-chart and table below show the obtained displacement for seven types of performed analysis. Under the first loading case, displacement is bigger in accordance to second loading case. Hence, displacement is directly affected by the increasing of moment and applied force.
Table 7.1. Displacement obtained from different models.
In the analytical model, the displacement calculated with Timoshenko beam theory is bigger than the displacement calculated with Euler - Bernoulli because the first theory considers the 73
shear deformation. The difference between analytical and numerical results was expected because of assumptions made in beam theories regarding cross section characteristics. The 2D Abaqus and 2D MatLAB models provide almost exact results because as mentioned before models were created using plane stress and plane strain (see Table 7.2). Difference can be explained by different element types used. Abaqus 3D solid model compared to 3D shell model gives more realistic results because 3D solid model represents the beam cross section better.
Table 7.2. Displacement comparison.
Figure 7.1. Displacement of the bottom flange along the beam from different models for the first loading case
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Figure 7.2. Displacement of the bottom flange along the beam from different models for the second loading case.
Experimental results give the biggest displacement due to some error sources. Firstly, the real beam cross section’s dimensions vary along the beam which can cause a difference in the measured displacement. In order to check, the dimensions of the real cross section were measured in three different positions, e.g. flanges thickness varies up to 0.9[mm], the diameter of the holes is not constant throughout the entire beam. On the other hand materials proprieties that were obtained during material test can include some uncertainties. Inaccurate strain gauges installation, processing the material properties data using linear regression, the test specimen which might not represent the material of the beam precisely can all be a cause of the difference between calculations and test results. Also, the displacement transducer can cause an error if it is not properly zero-calibrated and/or vertically placed. Furthermore, modelling of loads and boundary conditions can cause some uncertainties because of made assumptions
7.2 Stresses Besides displacement, stresses were also a point of interest in this project. Three cross-sections (named: 1-1, 2-2, 3-3) were considered for further stress-distribution analysis. Results for stresses from analytical and numerical analysis were obtained in the same positions where the strain gauges were placed during the experimental set up. Stress distribution obtained from all the parts is presented in the following graphs and the values for the interesting points are in the following tables .
75
-60,00
30,00
0,00
-1,00
1,50
1,00
3D Abaqus
2D MatLAB
0,00
3D Abaqus
5,00
2D Abaqus
2D MatLAB
2D Abaqus
Experimental
3D Abaqus
Analytical
-66,50
-56,50
-46,50
-36,50
-26,50
-16,50
-6,50
3,50
h [mm]
3D Abaqus 2D MatLAB
0,00
Experimental
-66,50
-66,50
-49,88
-33,25
-16,63
V [M Pa]
13,50
23,50
33,50
43,50
53,50
63,50
0,20
60
-5,00
0
-60
2,00
0,50
0,00
-1,50
-2,00
60,00
Stress distribution for 1st loading case (18 kN)
Analytical 3D Abaqus
0,00
h [mm]
W [M Pa]
2D Abaqus
0,60
2D MatLAB
2D Abaqus
W [M Pa]
16,63
33,25
49,88
66,50
Shear stress distribution Section 3-3
0,80
2D MatLAB
3D Abaqus
-30
2D Abaqus
30
Experimental 2D Abaqus 2D MatLAB
-49,88
-33,25
-16,63
0,00
-60,00
-66,50
-56,50
-46,50
V [M Pa]
h [mm]
40,00
Analytical
-80,00
-60,00
-36,50
-26,50
-16,50
-6,50
3,50
16,63
33,25
49,88
66,50
Normal stress distribution Section 3-3
-40,00
-66,50
-56,50
-46,50
-40,00
W [M Pa]
13,50
23,50
33,50
43,50
53,50
63,50 h [mm]
Shear stress distribution Section 2-2
60,00
Experimental
-30,00
-20,00
-0,50
-36,50
0,00
h [mm]
Normal stress distribution Section 2-2
-20,00
-26,50
V [MPa]
20,00
40,00
60,00
80,00
Shear stress distribution Section 1-1
-0,20
-16,50
-6,50
3,50
13,50
23,50
33,50
43,50
53,50
63,50 h [mm]
Normal stress distribution Section 1-1
1,00
0,40
80,00
20,00
0,00
-80,00
10,00
0,00
16,63
33,25
49,88
66,50
30
20
10
0
-10
-20
-30
-40
-50
-60
-18,00
-21,00
-24,00
0,00
-6,00 -15,00
-20,00
-25,00
0,00
20
-40
2D MatLAB
3D Abaqus
ʍ [MPa]
2D Abaqus
-40
Analytical
0
Experimental
-66,50
-49,88
-33,25
-16,63
20
Analytical 3D Abaqus
-66,50
-49,88
-33,25
-16,63
0,00 40
-10,00
-30,00
60
0
-60
3,00 -80
-9,00
-12,00
-15,00
-27,00
-30,00
Stress distribution for 2nd loading case (11 kN)
Experimental 2D Abaqus 2D MatLAB
0,00
h [mm]
60
2DMatLAB
2D Abaqus
40
Analytical 3D Abaqus
IJ [M Pa]
16,63
33,25
49,88
66,50
Normal stress distribution in Section 3-3
IJ [M Pa]
3D Abaqus
Analytical
0,00
16,63
33,25
49,88
Shear stress distribution Section 3-3 66,50
-50,00
Experimental 2D Abaqus 2D MatLAB
-66,50
-49,88
-33,25
ʍ [MPa]
16,63
33,25
49,88
66,50
h [mm]
-40,00
3D Abaqus
Analytical
-66,50
-49,88
-33,25
-16,63
0,00
h [mm]
Shear stress distribution in Section 2-2
2D MatLAB
2D Abaqus
-66,50
-49,88
-33,25
-16,63
h [mm]
0,00
2D MatLAB
2D Abaqus
40
3D Abaqus
50
Analytical
60
Experimental
-3,00
-16,63
-20
-66,50
0,00
16,63
33,25
49,88
66,50
Normal stress distribution in Section 2-2
-5,00
-49,88
ʏ [Mpa]
h [mm]
-20
-33,25
ı [M pa]
16,63
33,25
49,88
66,50
Shear stress distribution in Section 1-1
-10,00
-16,63
h [mm]
Normal stress distribution in Section 1-1
10,00
-20,00
-30,00
-60,00
80
-60
-80
5,00
RG 1 RG 2 RG 3
Strain gauges
SG 10 SG 11 SG 12 SG 13 SG 14 SG 15 RG 1 RG 2 RG 3
Strain gauges
-‐3,33 17,90 -‐64,80 65,00 56,85 53,33 -‐18,61 0,15 9,30
0,00 0,00 0,00
-‐3,61 6,96 -‐1,46
Analytical Experimental
2D MatLAB -‐2,42 5,35 -‐0,75
2D Abaqus -‐2,64 5,80 -‐0,75
3D Abaqus -‐2,93 5,97 -‐0,73
-‐2,40 16,35 -‐68,52 63,73 49,62 57,15 -‐54,11 -‐3,75 19,50
-‐16,89 -‐17,52 -‐17,18
-‐3,56 -‐25,69 -‐27,45
2D MatLAB -‐25,63 -‐18,69 -‐26,13
2nd loading case 11 kN Analytical Experimental
-‐3,45 17,97 -‐34,76 34,76 45,96 55,03 -‐25,88 -‐3,75 19,50
2D MatLAB -‐3,89 15,09 -‐64,54 60,13 45,35 50,70 -‐58,78 -‐7,33 47,96
3D Abaqus -‐3,43 16,37 -‐62,07 61,20 50,05 45,85 -‐19,32 0,65 8,45
2D MatLAB -‐4,76 17,50 -‐62,28 65,93 51,82 49,98 -‐12,68 1,20 5,82
2D Abaqus -‐3,09 15,79 -‐60,18 60,03 53,04 49,98 -‐17,62 0,68 8,16
11 kN
1st loading case 18 kN
Analytical Experimental
-‐3,97 20,64 -‐34,20 34,20 52,78 54,16 -‐27,39 -‐3,96 20,63
Analytical Experimental
2nd loading case
18 kN
1st loading case
2D Abaqus -‐19,16 -‐21,75 -‐23,43
2D Abaqus -‐2,59 13,94 -‐65,37 59,70 46,37 51,17 -‐47,91 -‐6,82 38,85
3D Abaqus -‐19,44 -‐22,35 -‐23,68
3D Abaqus -‐2,77 14,45 -‐65,85 62,24 43,75 47,05 -‐48,56 -‐7,25 39,19
Cross-section 1-1 is located in the middle between two holes of the beam ( see Graphs 7.2 and 7.2). During the experiment unidirectional strain gauges were used in this cross-section because of the limitation of available equipment. Therefore, the full real stress state in this points can’t be obtained. Nevertheless normal stresses from the experiment for both loading cases in cross-section 1-1 (see Graphs 7.2 and 7.2) do not contradict assumptions made in analytical and numerical parts of the project. Normal stresses are consistent throughout all the performed calculations and the experimental results. On the other hand shear stresses were not expected to appear in the first loading case (see Graphs 7.2 and 7.2), but the experimental data showed that they exist. This can be explained by the complicated stress state existing in the real beam perforated with holes. Moreover, the shear stress in the analytical part was calculated using the Juravski’s formula (or also known as Grasshof’s formula),Chapter 4.5.2. However, when beam cross section varies throughout the beam the extended formula provide more accurate results. In this case, shear stress distribution depends both on the position of neutral axis with respect to longitudinal edges of the beam and the ratio between the shear force and the bending moment. The following formula is used for symmetric beams with varying cross section:
τ=
QS M d S M S d I + − bI bI d x bI 2 d x
(7.2.0.1)
where: Q- Shear force, S - Static moment of cross section M - Moment b - Width of cross section I - Moment of inertia dS d I d x , d x - First deriative of static moment and moment of inertia. Kirilenko. & Pinchuk [2010] Besides, shear stress for the second loading case is also hard to predict because of abovementioned reasons and bigger influence of the existing the shear force. It is easy to notice that when the cross-section along the beam is constant, the terms S and I are also constant and the derivative of them cancels the last two terms, resulting in Juravski’s formula as previously defined in analytical chapter 4.5.2:
τ=
QS bI
(7.2.0.2)
Cross section 2-2 is located very close to a hole (see Graphs 7.2 and 7.2). In the experimental part multi-axial strain gauges, also known as rosette strain gauges were used. This type of strain gauges represents the real stress state of point better then unidirectional ones. For the 1st loading case the normal stresses obtained in the cross-section are lower than those in Section 1-1 (see Graphs 7.2 and 7.2). In contrast, for the 2nd loading case they become bigger than in the Section 1-1 (see Graphs 7.2 and 7.2). It can be explained by the presence of the influence of the stress concentration factor. Nevertheless, unexpected behaviour in the first loading case can appear because of more complicated stress state in the real beam than it was modelled analytically and numerically. Also, in the experiment applied forces were not exactly equal to 79
each other. That can cause some decrease or increase of the value of the moment and lead to different results for stresses. This can be checked by integrating the stress curve over the distance to obtain the real value of the moment, but that was not done because of lack of time. Cross section 3-3 goes through the center of the hole (see Graphs 7.2 and 7.2). The type of strain gauges used for this cross section is the unidirectional ones. Thus the same limitations for obtained experimental results as mentioned in the cross sections 1-1 description apply here as well. Although, these limitations do not cause big uncertainties in results when the unidirectional strain gauges that are placed on the flanges of beam. Normal stresses obtained in the experiment prove the influence of the stress concentration factor (see Graphs 7.2 and 7.2). In the previous chapters 4.5.1 it was calculated that the stress concentration factor is 1.646. The experiment value was determined by division of the maximum obtained stress value at the edge of the hole by the nominal one. The result is 1,9. It was explained earlier, that all the existing tables for the stress concentration factor do not give exact prediction of this value because of the lack of existing experimental research and the complicated behaviour of the stresses in reality due to the presence of the stress concentrator (e.g. holes, cracks etc). Furthermore, dimensions used for analytical calculations of the stress concentration factor can differ from the real beam cross section’s dimensions due to irregularity of the real cross section which can cause difference between theoretical and experimental values. Moreover, the distribution of stress concentration factor further from the hole’s edge is too hard to predict. But we can conclude according to the numerical analysis the dependence will have a secondorder polynomial distribution (see Graphs 7.2 and 7.2). The following sources of errors can cause some uncertainties in the obtained results. As it was mentioned above the use of unidirectional strain gauges in the experiment instead of rosette ones can lead to errors in calculations of normal stresses. Also inaccurate strain gauge installation can result in errors in measured strains and consequently in determined stresses. As well the presence of stress concentrators in the structure is not well-investigated yet and can cause unexpected behavior of stresses and strains. Furthermore, there are irregularities along the real beam and it is hard to determine the influence they have to the results.
7.3 Discussions The main aim of the project is to investigate the influence of holes to the behaviour of a structure. The analysis was performed with analytical, numerical and experimental approaches. A comparison between different calculations was made in order to evaluate the accuracy of the applied methods. It can be concluded that obtained displacements in all of the models are similar to each other. There is the difference between analytical and numerical approach because the holes were considered in simplified way by taken the minimum values of cross section properties. It was found that real deflection will be between the solution for the beam without any holes and the solution with the minimum values of cross section properties determined from the real beam model. However, an unexpected behaviour of the beam during the experimental analysis was observed. The obtained maximum displacement is bigger than in calculations and models, which can be explained by the above-discussed reasons(Chapter 7.2). As well as estimation of deformations of the beam perforated with holes the investigation of the stress distribution was performed. Due to the holes, no simple analytical expressions exist for calculating stresses. With some assumptions the stresses were obtained in three different cross sections. The influence of stress concentrator is hard to estimate analytically 80
because of uncertainties regarding the stress concentration. Moreover, it causes concentration of stresses not only around the edge of the hole but also at the surface of the flange. According to the results the stress concentration factor going further from the hole will follow a secondorder polynomial distribution. In addition, two loading cases in the project were chosen in order to investigate the influence of the shear force to the behaviour of the beam. The intensity of the forces was determined so that it would not cause plastic deformation in the beam (see chapter 4.5.1). Installed strain gauges in the experiment set up were situated in such a way that in the 2nd loading case the influence of the shear force to the determined stresses could be evaluated. It can be seen (see Graphs 7.2 and 7.2) that as expected the presence of the shear force will cause bigger shear stresses in the analysed cross sections compared to the case 1. It has to be mentioned that the assumptions of analytical calculations of stresses should be made carefully, because a presence of a shear force causes an increase of shear deformations in the structure and at the same time a more complicated stress state. The presence of shear force and stress concentrators can cause yielding earlier because shear stress will exceed a point of shear yield limit. It was concluded that the use of Juravski formula can cause errors in the obtained results due to the varying cross section of the analysed beam. Further investigation is needed to approve this conclusion. Furthermore, a more complicated stress state appears in the real beam than it is assumed in the calculations. Hence, it is difficult to quantify errors and inaccuracies due to complicated analysed phenomena.
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Bibliography (2006). ‘Module for plane stress and plane strain analysis’. http://www.engineering.ucsb. edu/~hpscicom/projects/stress/introge.pdf. L. Andersen & S. R. Nielsen (2008). Elastic Beams in Three Dimensions, dce lecture notes no. 23 edn. J. Clausen (2013). ‘Analysis and Design of Load-Bearing Structures’. Aalborg University . T. Court & S. Gloucestershire (2011). ‘Simula UK RUM 2011’. download/rum11/UK/Feedback-Presentation-1.pdf.
http://www.simulia.com/
R. D.Cook, et al. (2002). Concepts and Applications of Finite Element Analysis. John Wiley Sons, Inc., fourth edition edn. T. Haukaas (2013a). ‘Euler-Bernoulli Beams’. http://www.inrisk.ubc.ca. T. Haukaas (2013b). ‘Timoshenko Beams’. http://www.inrisk.ubc.ca. V. Kirilenko. & E. Pinchuk (2010). ‘Concerning the distribution of shear stress in beam with variable cross-section’ . S. Mirza & M. Smell (2011). ‘Simulation Keep it SIMPLE SMART’. Autodesk University . D. C. Morley (1995). ‘ABAQUS-SHELL Element Sign Convention’. http://www-h.eng.cam.ac. uk/help/programs/fe/abaqus/faq68/abaqus.shell.conv.html. W. D. Pilkey & D. F. Pilkey (2008). Stress Concentration Factors. John Wiley Sons, Inc., Hobroken, New Jersey. D. Simulia (2010). ‘Abaqus 6.10 documentation’. G. Wemper & D. Talaslidis (2003). Approximations. CRC Press LLC.
Mechanics of Solids and Shells:
Theories and
T. F. E. Wikipedia (2013). ‘Deformation (mechanics)’. http://en.wikipedia.org/w/index.php? title=Deformation_(mechanics)&oldid=583208647.
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Appendix
A
A.1 Diagrams for the shear force and the moment for both loading cases
Figure A.1. Moment and shear force diagrams for the 1s t loading case
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Figure A.2. Moment and shear force diagrams for the 2n d loading case
A.2 Calculation of moment of inertia and area In the parts of beam where the cross section is without the holes the moment of inertia was determined like:
Iw hol e =
b · h 3 − (b − t w ) · h 13
(A.2.0.1)
12
where: b- width of the flange h- height of the cross section t w - thickness of the web h 1 - height of the web
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Figure A.3. To the determining the moment of inertia in the cross section weakened by the hole.
The moment of inertia in the cross section weakened by the hole is determined by the following formula (Figure A.3):
Iw i t hhol e = Iw hol e
Where D = 2 · y = 2 ·
tw · D 3 12
(A.2.0.2)
p R 2 − (x − f − n · e)2
x- distance from the start of the beam f - distance from the start of the beam till the first hole e- distance between two holes n = 0 : (v − 1), v- the number of the holes
87
Figure A.4. Dependence of the area of the cross section from the distance.
A.3 Finite element theory A.3.1 FEM - Introduction Finite element analysis (FEA), also called the finite element method (FEM), is a method for numerical solution of the field problems D.Cook et al. [2002]. An analysis domain is divided in smaller sub domains, of variable shapes and dimensions. Field variables are approximated as continuous functions with the ability to be integrated. The accuracy of the result obtained from FEM analysis is directly depended on numerical process, the major factor being the utilized algorithms. FEM characteristics: 1. Dimension - elements can have one (1D), two (2D) or three (3D) dimensions. There are also special elements without geometrical dimensions (elastic supports and point masses). 2. Nodal points - every element has a finite number of nodal points. They are defining the geometry and localized degrees of freedom. In general, for simple elements (linear elements) nodal points are positioned at the corners or at the ends of elements. Higher order elements can have vertex and mid-side nodes. 3. Element geometry - elements can have curved (parabolic and cubic elements) or straight (linear elements) sides. 4. Degrees of freedom -can include both deflections and rotations in all directions in this project. In more general problems such parameters like temperature, pressure, composition, vibration and etc. can be described by DOFs. 5. Nodal forces -are in correspondence with the degrees of freedom (displacement is linked with a force and rotation is assigned to a moment). 88
6. Material property -constitutive laws define material’s behaviour. Hooke’s law is the most simple. It corresponds to material behaviour in linear elastic stage. In our case, the material is characterized by Young’s Module, Poisson coefficient and linear thermal expansion.
Figure A.5. Division process.
A.3.2 FEM - Types of elements In FEM, we can use different types of elements: Melosh (linear and quadratic), isoparametric four-node and eight-node, triangular three-node and six-node (see Figure A.6 and Figure A.7) and six-node isoparametric triangular element (see Figure A.6 and Figure A.7). It is not appropriate to use Melosh elements because they must be rectangular and positioned along the coordinate axis. Convergence analysis in our project was performed with both three – node and six - node triangular elements. Better results were obtained with six - node elements. Therefore, it was decided to use LST in final models. Three-node triangular elements (CST) produce a constant strain field because of linear displacement field. While six-node triangular elements (LST) produce a linear strain field. Triangular elements can be rotated arbitrary. Both CST and LST elements have straight sides. They have two degrees of freedom for each node. For 2D “MatLAB” model it was decided that LST elements are sufficient enough. In 3D shell model instead of LST, six-node isoparametric elements are used. Isoparametric elements use the same set of shape functions to represent both the uniform changes on the initial and secondary conditions and also on local coordinates of elements. The shape functions are defined by natural coordinates, such as triangle coordinates for triangles and square coordinates for any quadrilateral. The advantages of isoparametric elements include the ability to map more complex shapes and have more compatible geometries.
89
Figure A.6. Nodes order on triangular CST element (on the left) and isoparametric element (on the right).
Figure A.7. Nodes order on triangular CST element (on the left) and isoparametric element (on the right).
90
For 3D solid model mesh two different types of elements are tested: • linear 4-node tetrahedral (C3D4) (see Figure A.8); • quadratic 10-node tetrahedral (C3D10) (see Figure A.9). Both element types belong to the same 3D strees element family in computer software Abaqus. Tetrahedral elements are used in solid model in purpose of applying free structured fully automatic mesh. Other elements, like hexahedrons (bricks), do not provide this option because of complex model geometry. In accordance to apply mesh using hex elements, beam model has to be separated in to simple geometry segments which make whole model even more complicated. C3D4 linear elements are convenient for irregular meshes. These elements are defined by four nodes and three active degrees of freedom in each node: displacements in x, y and z-directions. Linear tetrahedral is a first order element. That means it only has one integration point (see Figure A.8). For this reason it takes less time for computer calculations but final results are not very accurate. Simulia [2010]
Figure A.8. Four-node tetrahedral element (C3D4)Simulia [2010].
C3D10 element has guadratic shape because of the mid-points on every edge. This tetrahedral element has ten nodes in total and three active degrees of freedom per node (u,v and w). Quadratic element is second order and has four integration points (see Figure A.9). This property provides a more accurate results but at the computational time increases Finally, 10-node elements (ten-node tetrahedral elements for solid model) are superior to 6node ones because of these reasons:
• They represent complicated geometry better and include second order terms; • Elements with four nodes are usually over stiff and for this reason very fine mesh is required to get accurate results (see Chapter 6.4 “Convergence analysis”). • Ten – node elements provide accurate results in small-displacement problems, as long as analysis is performed for an integral model ; • For computer software Abaqus it is better to use quadratic elements in order to avoid distorted elements and inaccurate results in the area of interest.Court & Gloucestershire [2011] 91
Figure A.9. Ten-node tetrahedral element (C3D10)Simulia [2010].
A.3.3 Finite element calculations Finite element method is based on the following governing equation:
~ =~ K ·U f
(A.3.3.1)
~ is a vector of displacement and ~ Where, K is the stiffness matrix, U f is a force vector. All calculations performed in MatLAB are based in the following theory where formulas for calculations using LST elements are presented. Local element stiffness matrix is defined by integration over the element area. The integration can be only solved numerically:
Z
Ke =
A
BT · D · B · t · d A
≈
Ke =
n X i =1
B iT · D · B i · t · Wi · d et (J i )
(A.3.3.2)
Where B - the strain interpolation matrix containing the derivatives of the shape functions; Dthe elastic constitutive matrix; A- element area; t - thickness of the element; J i - Jacobian. Jacobian for an LST element can be determined by the following formula:
³ ³ ´´ 1 J i = d et J ξi1 , ξi2 , ξi3 = A 2
(A.3.3.3)
Where ξi1 , ξi2 , ξi3 - are the natural coordinates of the triangular element.
εx ε(x, y) = = B(x, y)U εy 2εx y
(A.3.3.4)
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Where U- is displacement. Stresses are determined by the following formula: σx σ(x, y) = σ y Dε(x, y) = DB(x, y)U σx y
(A.3.3.5)
Strain interpolation matrix in the Eq. A.3.3.2 can be determined:
∂ ∂x
˜ ∇ ˜ = B = ∇N, 0
∂ ∂y
0
∂ ∂y ∂ ∂x
"
N1 N= 0
0 N1
··· ···
N6 0
0 N6
#
(A.3.3.6)
˜ - is the differential operator;N - shape functions. To perform integration in Eq. A.3.3.2 Where ∇ triangular Gauss quadrature should be used. The following table presents area coordinates and weight functions for that.
Table A.1. Gauss point for triangles Clausen [2013]
To perform Gauss integration element is divided into small areas where every small area has Gauss point in the center. Weight factor means relation between the size of small area and the entire area. B (strain interpolation matrix) is assumed the same for the small area as for respective Gauss point. 93
A.3.4 FEM - Plane strain and plane stress Loads, supports and material characteristics are assumed to be independent of z-axis. Therefore, the beam can be modelled in 2D domain with a given thickness. When the thickness of the flanges is more considerable than the thickness of the web, the flanges are subject to a plane strain analysis and web - to plane stress.
Figure A.10. Assumption of plane stress and plane strain calculation.
Plane stress case describes that normal component in z-direction and shear components, which are perpendicular to x-y plane, of Cauchy Stress Tensor are equal to zero. In contrast, plane strain is defined “to be a state of strain, in which the strain normal to the x-y plane and the shear strain γ x-z and γ y-z are assumed to be zero”. com [2006]
A.4 Results obtained from different numerical models
Figure A.11. 2D Matlab model:Deformed mesh for the second loading case.
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Figure A.12. 2D Matlab model: Normal stress (σx x ) distribution for the second loading case.
Figure A.13. 2D Matlab model: Normal stress (σ y y ) distribution for the second loading case.
Figure A.14. 2D Matlab model: Shear stress (τx y ) distribution for the second loading case.
Figure A.15. 2D Matlab model: Von Mises stress distribution for the second loading case.
Figure A.16. 2D Matlab model: Normal stress (σx x ) distribution for the first loading case.
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Figure A.17. 2D Matlab model: Normal stress (σ y y ) distribution for the first loading case.
Figure A.18. 2D Matlab model: Shear stress (τx y ) distribution for the first loading case.
Figure A.19. 2D Abaqus Model:Deformed beam for the second loading case.
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Figure A.20. 2D Abaqus Model: Normal stress (σ22 ) distribution for the second loading case
Figure A.21. 2D Abaqus Model: Shear stress (σ12 ) distribution for the second loading case
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Figure A.22. 2D Abaqus Model: Von Mises stress for the second loading case
Figure A.23. 2D Abaqus Model: Normal stress (σ22 ) distribution for the first loading case
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Figure A.24. 2D Abaqus Model: Shear stress (σ12 ) distribution for the first loading case
Figure A.25. 3D Abaqus Shell Model: Deformed beam for the second loading case
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Figure A.26. 3D Abaqus Solid Model: Deformed beam for the second loading case
Figure A.27. 3D Abaqus Solid Model: Normal stress (σ33 ) distribution for the second loading case
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Figure A.28. 3D Abaqus Solid Model: Shear stress (τ13 ) distribution for the second loading case
Figure A.29. 3D Abaqus Solid Model: Von Mises stress distribution for the second loading case
Figure A.30. 3D Abaqus Solid Model: Normal stress (σ33 ) distribution for the first loading case
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Figure A.31. 3D Abaqus Solid Model: Normal stress (σ33 ) distribution for the first loading case
Figure A.32. 3D Abaqus Solid Model: Shear stress (σ13 ) distribution for the first loading case
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Digital Appendix
B
B.1 Calculation of moment of inertia and shear area It was done with the help of Matlab. File Inertia.m contains the calculation of moment of inertia for every point. File sheararea.m contains calculation of shear area.
B.2 Calculation of coefficients from material test Files test 1.xls, test 2.xls, test 3.xls contain the data from material test and the calculation of Young’s modulus and Poisson’s Ratio.
B.3 Calculation of displacement of the beam using Bernoulli-Euler theory File Euler_final.m contains the calculation of displacement of the beam and also displays the necessary plot.
B.4 Calculation of displacement of the beam using Timoshenko theory File Timoshenko_final.m contains the calculation of displacement of the beam and also displays the necessary plot.
B.5 Comparison of beam theories Folder Difference between Timoshenko and Euler contains the comparison between the displacements of the beam calculated with the decreasing of the length.
B.6 Calculation of normal stresses and shear stresses File Sigma-Tau analytical.xls contains tables with calculation of normal stresses and shear stresses in cross sections. 103
B.7 Main test calculations File Main test.xls contains the data obtained from the strain gauges and displacement for maximum applied load in both cases. This file contains also the calculation of stresses.
B.8 2D MatLab Folder 2D MatLAB contains finite element calculations.
B.9 2D Abaqus File 2D1000.cae contain the model of the beam with mesh, boundary condition and both loading cases.
B.10 3D Shell Abaqus Files 3D Shell 11kN.cae and 3D Shell 18kN.cae contain the quarter of the beam with mesh, with boundary condition and with both loading cases.
B.11 3D Solid Abaqus File 3D solid model.cae contains the quarter of the beam with mesh, with boundary condition and with both loading cases.
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