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Finite Element Analysis of Steel Cantilever I beam
COURSE PROJECT REPORT
(of CE-620)
by
Bharti Banshiwal (173040018)
Kalyani Ambhore (183040053
Radhika Pajgade (184040006)
Under the guidance of
Prof. Yogesh.M. Desai
Department of Civil Engineering
Centre for Computational Engineering & Science (CCES)
Indian Institute of Technology Bombay
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Contents List of Tables Error! Bookmark not defined.
Abstract 5
Introduction 6
I Problem Formulation 7
1. Form Geometry 7
II Assigning Material Properties 10
2. Create Assembly 13
3. Define the Type of Element 14
4. Define Step 15
III Application of Loads 16
5. Apply Boundary conditions (Fixed end) 16
6. Apply displacements as boundary conditions 18
7. Create a Job 21
8. Getting the output values 21
(a). Reaction Forces 21
(b). Stresses and Deflections 21
IV Results and Discussion 22
1. Mesh Optimisation using Abaqus 23
Effect of reduced Integration 23
Comparison between 1-D and 3-D analyses 24
Displacement and Stresses along the beam length 24
References 26
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List of figures
Figure 1 Deflection of cantilever beam subjected to point load ‘P’at free end ............................. 6
Figure 2 Deflection of cantilever beam with uniformly distributed load of ‘w’ kN/m .................. 6
Figure 3 2 noded (linear) and 3 noded (quadratic element) [2] ...................................................... 6
Figure 4 8 noded and 20 noded brick element [2] .......................................................................... 6
Figure 5 Cantilever beam with I cross section ................................................................................ 7
Figure 6 –Window for creating part (1-D) ...................................................................................... 8
Figure 7 Window for creating the part in 3D .................................................................................. 8
Figure 8 Geometry of the I section ................................................................................................. 9
Figure 9 Window for assigning length to the section .................................................................... 9
Figure 10 Isometric view of the cantilever beam with I cross section.......................................... 10
Figure 11 Window for assigning geometric properties in 1-D ..................................................... 11
Figure 12 Window for assigning material properties in 1-D and 3-D .......................................... 12
Figure 13 Window for creating a section ...................................................................................... 12
Figure 14 Section Assignment to the cantilever beam.................................................................. 13
Figure 15 Window for creating an instance .................................................................................. 13
Figure 16 Window for creating mesh controls ............................................................................. 14
Figure 17 Window for defining element type ............................................................................... 15
Figure 18 Window for creating the loading step .......................................................................... 15
Figure 19 Window for editing the loading step ............................................................................ 16
Figure 20 Window for defining restraints ..................................................................................... 17
Figure 21 Selection of region to apply boundary conditions ........................................................ 17
Figure 22 Window for editing the boundary condition ................................................................ 18
Figure 23 Window for creating displacement boundary condition ............................................. 18
Figure 24 Window for selecting a region for applying load ......................................................... 19
Figure 25 Window for editing the boundary conditions ............................................................... 19
Figure 26 Isometric view of the cantilever beam with applied displacement............................... 20
Figure 27 Isometric view of the cantilever beam with an applied uniform displacement over the
section ........................................................................................................................................... 20
Figure 28 Cantilever beam in 1-D with applied displacement at the free end .............................. 20
Figure 29 Window for editing job ................................................................................................ 21
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Figure 30 Window showing results .............................................................................................. 22
Figure 31 Displacements along length of beam ............................................................................ 24
Figure 32 Bending stresses along the length of beam ................................................................... 24
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List of Tables
Table 1 Mesh Optimization in Abaqus ......................................................................................... 23
Table 2 Comparison of reaction forces for linear and quadratic element ..................................... 23
Table 3 Effect of Reduced integration .......................................................................................... 23
Table 4 Comparison of 1-D and 3-D results ................................................................................. 24
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Abstract
A steel cantilever beam is analyzed for various loading conditions The cross section chosen for the beam
is a hot-rolled steel beam adopted from IS:808-1989 [1]. 1-D and 3-D model of the cantilever beam is
created in ABAQUS software. For 1-D, 2 noded linear line element and 3 noded quadratic line element is
chosen. and results from both are compared with each other. For 3-D modelling of beam 8 noded
hexagonal brick element as well as 20 noded hexagonal brick element are chosen. the results obtained
from both are compared. Results from 1-D and 3-D analysis in FEM are compared with the analytical
results. A displacement controlled approach is used and the corresponding loads are evaluated at the free
end. From the results, it is seen that the 1-D model for the beam gives much better results than the 3-D
model. There is an error of about 6% in the value of loads (Reaction forces) as obtained from the 3-D
analysis. Some analyses are performed with a reduction in element size for convergence study. Secondary
unknowns like bending stresses are also obtained and are compared with analytical results which match
well with each other.
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Introduction
Cantilever beam can be subjected to different kinds of loading conditions like point load (e.g due to crane
uniformly distributed load (e.g due to self-weight) or impact load. Cantilever beams are mostly used in
overhangs like balconies in buildings, chajjas over windows and parts of the jib crane etc. In the literature,
analytical solutions are available for different types of loading and end conditions. The analytical results
of cantilever beam deflection are shown in figure 1 and 3. But, in complex building models, finite element
studies are used frequently for the design and analysis of these beams. Different types of models (1-D, 3-
D, 2-D) are used in a variety of available commercial software.
When 2 noded linear beam element is used for the analysis, it gives a linear variation of shape function
whereas in quadratic element its variation is quadratic. Line element as per intrinsic coordinate system is
shown in figure 3. In 3-D, 8-noded linear brick element (hexagonal element) and 20 noded quadratic brick
element are most commonly used mesh types. Hexagonal element in intrinsic coordinate system is shown
in figure 4.
As per plane stress theory as the width of beam is much lesser as compared to its length, we can represent
actual 3-D beam as a 1-D beam. Only σxx is dominating while other stress components negligible
compared to σxx .Variation of primary unknowns is negligible along the cross section.
Figure 1 Deflection of cantilever beam subjected to point load ‘P’at free end
Figure 2 Deflection of cantilever beam with uniformly distributed load of ‘w’ kN/m
Figure 3 2 noded (linear) and 3 noded (quadratic element) [2]
Figure 4 8 noded and 20 noded brick element [2]
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I Problem Formulation
Comparison of analysis of Cantilever beam in 1D and 3D with analytical
Example:
The problem to be modelled in this example is a steel cantilever having I cross-section as shown in the
following figure 5. The geometric properties of I section are - Width of top and bottom flange is 75mm,
the thickness of the flange is 8mm, the thickness of the web is 5mm and the height of the section is
150mm. Young modulus of the section and Moment of inertia of the section are is 20000N/𝑚𝑚2 and
7058143 𝑚𝑚4respectively.
Figure 5 Cantilever beam with I cross section
1. Form Geometry
1-D - Module > Part > Create Part >
Give a name to the part.Choose the modelling space as 2D planar. Select the type of part as ‘deformable’
and select the base feature as ‘wire’.Select approximate mesh size and press continue. The window for
creating a part in 1D is shown in figure 6.
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Figure 6 –Window for creating part (1-D)
3-D - Module > Part > Create Part >
Give a name to the part. Choose the required modelling space. Select the type of part as ‘deformable’.
Select the base feature as solid with ‘Extrusion’ type and press continue. The window for creating a part
in 3D is as shown in figure 7.
Figure 7 Window for creating the part in 3D
Draw the geometry using key points with the following coordinates
{(0,0), (75,0), (75,8), (40,8), (40,142), (75,142), (75,150), (0,150), (0,142), (35,142), (35,8), (0,8), (0,0)}
The I section thus created using the coordinates is shown in figure 8.
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Figure 8 Geometry of the I section
To provide the length to the beam we use extrusion option available in the module. Provide depth (length
of the beam) as 1200 and press ok. The window for defining extrusion depth is shown in figure 9.
Figure 9 Window for assigning length to the section
After extrusion, following the 3D model of the beam as shown in figure 10 is created.
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Figure 10 Isometric view of the cantilever beam with I cross section
Sketch the section for solid extrusion
II Assigning Material Properties
i) Assigning cross-sectional properties for 1-D
Module > Property > Create profile > Name the profile > Choose I > Continue > Edit Profile
> Input I section properties.
The window for creating a I profile is shown in figure 11. Give a name to the profile and choose
the shape of the profile and input various parameters of the I section.
h=150(depth of the section), b1=75 (width of the top flange), b2=75(width of the bottom flange),
t1=8(thickness of top flange), t2=8(thickness of bottom flange), t3=5 (thickness of web)
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Figure 11 Window for assigning geometric properties in 1-D
Module > Property > Assign Beam Orientation > Select the region > Select an appropriate n1
direction (0,0,-1) > OK
ii) Assigning material properties for 1-D and 3-D
Module > Property > Create material
Name the material and choose the material behaviour as Elastic and define Mechanical
properties. Type of material is chosen as ‘isotropic’.The input value of Young’s
Modulus as 210000 and that of Poisson’s ratio as 0.3 and press ok. The window for
editing the material properties is as shown in figure 12.
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Figure 12 Window for assigning material properties in 1-D and 3-D
Module > Property > Create section > Name (I beam) > Category(Solid) > Type
(Homogeneous) > Continue > Edit section > Set steel(I beam) > ok
Following window of creating section as indicated in figure 13 pops up then name the section. Assign the
category of the section as ‘Solid’ and choose the type as ‘Homogeneous’ and press continue.
Figure 13 Window for creating a section
Module > Property > Assign section > Select the section > Colour changes to green
implies section is assigned
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The green colour of the section in figure 14 indicates that the properties are assigned to the
section.
Figure 14 Section Assignment to the cantilever beam
2. Create Assembly
Module > Assembly > Create Instance > Parts > Select section > Ok
Figure 15 Window for creating an instance
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3. Define the Type of Element
It is now necessary to define the type of element to use for our problem:
Module > Mesh > Mesh Controls > Hexagonal element shape > Technique > Sweep >
ok
Here ‘hexagonal’ element shape is chosen and the technique of meshing is ‘sweep’.
Figure 16 Window for creating mesh controls
(a). 1-D
Module > Mesh > Assign element types >Family > Beam > Geometric order > Linear > B21 (
2-node linear beam element in a plane is chosen)>ok
(b). 3-D
Module > Mesh > Assign element types >Family > 3D stress > Geometric order > Linear >
Hex > Reduced integration > Second order accuracy >ok
Types of elements used are for the different analyses-
1. C3D8R -Cubic 3D 8 noded linear element with reduced integration
2. C3D8 -Cubic 3D 8 noded linear element without reduced integration
3. C3D20R - Cubic 3D 20 noded quadratic element with reduced integration
4. C3D20 - Cubic 3D 20 noded quadratic element without reduced integration
A standard element from the library is chosen linear geometric order and belonging to the 3D
stress family. ‘Reduced integration’ is used to reduce computation effort and various element
controls are specified and thus an 8 noded linear brick element.The window for defining element
is as shown in figure 17.
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Figure 17 Window for defining element type
Module > Mesh > Seed edges > Region selection > OK
Module > Mesh > Mesh part > Ok to mesh part > Yes Mesh is created.
4. Define Step
Module > Step > Create step > Static, general > Continue > Other > Matrix storage >
Use solver default > Ok
Window for creating step is shown in figure 18.
Figure 18 Window for creating the loading step
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The window for editing the step properties like the method for solving equations and solution
technique is as shown in figure 19. ‘Full Newton’ solution technique is chosen.
Figure 19 Window for editing the loading step
III Application of Loads
For both 1-D and 3-D, the following steps are followed-
Module > Load > Create load > Category > Mechanical > Type of selected steps > Pressure >
Continue > Ok > Select surface for the load > Edit load > Magnitude (= 66.67) > Ok
5. Apply Boundary conditions (Fixed end)
Module > Load > Boundary conditions > Step > Initial > Type of selected selection > Step> Initial>
Type of selected selection > Step > Symmetry/Antisymmetry/Encastre > Continue > Region selection>
Encastre > Ok
Name the boundary condition and determine the category of the step as ‘Mechanical’ and select the step
type. Following figure 20 shows window for creating boundary conditions.
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Figure 20 Window for defining restraints
Select the section of the beam where the boundary condition is to be assigned. Name the
boundary condition and press continue. The boundary condition is applied to the highlighted
section in figure 21. The same procedure is adopted in the 1-D model.
Figure 21 Selection of region to apply boundary conditions
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Define the boundary condition as Encastre to constrain all six degrees of freedom (3 translational
degrees of freedom and 3 rotation degrees of freedom) as shown in figure 22.
Figure 22 Window for editing the boundary condition
6. Apply displacements as boundary conditions
Module > Load > Boundary condition
Name the boundary condition. Select the step as ‘load’ and define the category of boundary condition as
‘Mechanical’ and select the type for selected step as ‘displacement/rotation’ as shown in figure 23.
Figure 23 Window for creating displacement boundary condition
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Select the point where the displacement is to be applied and continue. The window for region
selection is shown in figure 24.
Figure 24 Window for selecting a region for applying load
Edit the boundary condition and give vertical deflection as 20 units downward (i.e U2= -20) as
indicated in the following figure 25.
Figure 25 Window for editing the boundary conditions
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The applied deflection at the node of the farther end of the cantilever is shown in the following
figure 26.
Figure 26 Isometric view of the cantilever beam with applied displacement
Figure 27 Isometric view of the cantilever beam with an applied uniform displacement over the section
Figure 28 Cantilever beam in 1-D with applied displacement at the free end
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7. Create a Job
Module > Job > Create job > Continue > Submit job
Window for editing job is shown in figure 29.
Figure 29 Window for editing job
8. Getting the output values
For extracting the output from the post- processor, the following steps are followed.
(a). Reaction Forces
Module > Visualization > XY DATA > odb field output > Continue > Position >
Unique nodal > Choose reaction force > RF2 > U2> Element/nodes > Node sets >
Reference point > Plot > XY data > Select plot
(b). Stresses and Deflections
First, a node path has to be defined along the beam length to extract the values at those nodes. The
procedue is as follows;
Module > Visualization > Tools (on the Menu Bar) > Path > Create
Now, the output file is generated for these selected nodes.
Module > Visualization > Create XY Data > Path > Continue > XY Data from Path > Choose Path >
Select Field Output (Stress, Displacements etc. ) > Plot
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Figure 30 Window showing results
IV Results and Discussion
Now, since the purpose of this exercise was to verify the results - we need to calculate what we should
find.
Analytical Solution:
From the classical solutions available for a cantilever beam, deflections at the free end are given by-
(1)
We have adopted a displacement controlled analysis approach for the current study. For that, a
certain amount of displacements are applied to the model as a predefined boundary condition and
corresponding resistance provided by the beam at the free end is calculated which represents P in
equation (1).
Here; E= 20000 N/mm2
, I=7058143 𝑚𝑚4, L=1200mm, 𝛿 = 20𝑚𝑚(predefined)
As we have applied vertical displacement of 20mm. We must obtain a load = 49014.88 N
At any section of the beam, bending stress (at the top fibre) are calculated with the flexure formula.
=𝑀∗𝑦
𝐼 (2)
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1. Mesh Optimisation using Abaqus
For the 1-D model; by changing the element size, we get the following results
Table 1 Mesh Optimization in Abaqus
No of Elements Displacement(mm) Reaction Forces (N) (Linear)
1200 20 49063.6
240 20 49063.5
100 20 49063.0
80 20 49062.7
Analytical 20 49014.88
Here, we can clearly see that with the increase in the number of elements, there is no significant effect on
the reactions provided by the beam at the free end. Thus, for the optimum computational efforts, we will
consider the mesh with 240 elements for further study.
Maintaining the aspect ratio of the elements as unity, if we change the element sizes in the 3-D model for
linear and quadratic element types, we get the resistance.,
Table 2 Comparison of reaction forces for linear and quadratic element
No of Elements Displacement(mm) Reaction Forces (N) (Linear) Reaction Forces (N) (Quad)
2598000 20 37996 38750
33728 20 44097.6 45375.1
8640 20 44906.4 45941.6
6000 20 45115.8 45870.4
Analytical 20 49014.88
From the 3-D model, results are converging to the analytical results with the decrease in element numbers
which is not matching with the normal trend of convergence of results with an increase in the number of
elements. For the optimum computational effort and convergence of results, the model with 6000 number
of elements has been used for further analysis.
Effect of reduced Integration
Reduced integration is a method used in the finite element analysis to reduce the computational time and
efforts by using a single point in the element for the calculations. Here in this study, analyses were done
on linear and quadratic elements with and without reduced integration respectively. It was observed from
the results that linear elements are having a significant variation in the reaction forces for both the
approaches whereas, it doesn’t have much effect on the elements with a quadratic variation.
Table 3 Effect of Reduced integration
Type of Element Equation
Reaction Forces (N)
with reduced integration without reduced integration
Linear 44906.4 46325.7
Quadratic 45941.6 46054.3
Analytical results
49014.88
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Comparison between 1-D and 3-D analyses
Table 4 Comparison of 1-D and 3-D results
Type of Model Displacement(mm) Reaction Forces (N)
1-D (Linear Elements) 20 49063
1-D (Quadratic Elements) 20 49063.6
3-D (Linear Elements) 20 45115.8
3-D (Quadratic Elements) 20 45870.4
Analytical 20 49014.88
Displacement and Stresses along the beam length
Figure 31 Displacements along length of beam
Figure 32 Bending stresses along the length of beam
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As seen in the figure31 and 32 , a very close match is obtained among the results of analytical solution, 1-
D and 3-D finite element model. A sudden rise of stresses is seen at the end of the beam in case of 3-D
FE model due to stress concentration at the point of load application.
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References
1. BIS (Indian Standard Code) (1989) IS 808:Dimensions for hot rolled steel beam, column,
channel and angle sections, 3rd revision. Bureau of Indian Standards, New Delhi, India.
2. Dassault Systèmes. Abaqus user’s manual, version 6.14; 2014.
