Beams under Bending Loads: Comparison of Theoretical and Numerical Solutions

Beams under Bending Loads:Comparison of theoretical and numerical solutions
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Abstract
This report aims at an examination of the importance of the Finite Element Analysis in obtaining the
properties of I-beam cross-section member in determining values such as stresses at certain points,
second moment of area of the component and the position of the centroid in an I cross-section beam
subjected to a moment of bending my about the centroidal y axis and a moment of bending mz about
the centroidal z axis. The bending, lateral twisting and displacement are the main modes of failure that
determine any beam design. Structural analysis and simulation using ANYSYS are completed on the

I-beam cross-section and the results compared with the theoretical values obtained from calculations
in order to ascertain and approve results from the simulation. Finite Element Analysis of any
component is the most appropriate method for carrying out analysis and simulation of the section.
Using the study it is observed that both numerical and theoretical values are almost similar and thus
the FEA method can be applied in determining the suitability and use of any section under any
application.

Table of Contents
1. Aims & Objectives..........................................................................................................................4
2. FEA Literature Review....................................................................................................................4
3. Theoretical Equations and Calculations.........................................................................................7
4. Numerical Screenshots..................................................................................................................9
5. Results Summary - Tabulated Theory and Numerical Values......................................................10
6. Discussion....................................................................................................................................10
7. Conclusion...................................................................................................................................10
8. References...................................................................................................................................11

Table of Figures
Figure 1: Warping......................................................................................................................................... 4
Figure 2: I-beam cross-section....................................................................................................................... 6
Figure 3: Test Data........................................................................................................................................ 6

1. Aims & Objectives
To compare theoretical and numerical solutions for an I-beam cross-section under bending loads.
2. FEA Literature Review
In many scenarios, the sections of a structure are required to conform to the standards of being able to
withstand the various types of loadings that may be applied on it. The combination of this forces
acting on a structural member causes the deformation of the section. The loads are assumed to act
independent of each other due to the stress produced by each kind of applied force. The cumulative
force is obtained from the addition of the individual stress components of the loaded cross-section.
Force of shear and the bending moment are the two types of resultants that occur when loading of
transverse loads takes place in the beam planes. The beam’s boundary conditions, distance and the
force are the functions that well describe the bending moment. The deformation of the beam is closely
related to the sign conventions of the moments and the internal forces.
The action of applied torsional forces and moments or eccentric forces on the cross section of the
member results in what is known as torsion. Hence, twisting of the member about the longitudinal
axes and passing via the shear centre of the beam occurs when a member is subjected to torsion.
As illustrated in Figure 1; Shear centre is the point at which bending occurs without twisting in the I
cross-section as a result of the passing of the lateral (or transverse) loads through the cross section. It
is also the rotation centre, when only the applied force is pure torque. An assumption is made that a
cross section that is plane remains the same way (plane) during and after application of torsion on
circular bars where twisting occurs. However, warping takes place when the component with solid I
cross-section channel are under effects of torsion. In order to have no stresses of longitudinal in a
component then warping should be allowed at both ends.
Figure 1: Warping
Torsion is eliminated in any case where the resultant force obtained from a loaded section is made to
pass via the shear centre of the I-cross section. In many instances, the forces applied passes through
the centre of gravity. In addition, the centre of shear matches with the centre of gravity for a double
symmetric section; hence the torsion is eliminated. Effects of torsion may be influenced by many

factors such as the arrangement of the load, the restrains on the beam against warping, the beam’s
boundary conditions and the type of cross-section (close or open cross section).
Generally in steel structures, torsion being an inappropriate method of resisting loads should be
avoided as much as possible. In cases where torsion is unavoidable, the use of box girders or closed
sections is recommended due to the increase of torsional resistance as compared to the torsional
resistance of the sections that are open. Briefly; when subjecting sections to torsion, the most
preferred type of section is the closed section. However, torsional loading in the cross-section occurs
when the resultant force do not pass via the shear center. Moreover, the section might be an open
cross-section which has reduced torsional resistance. Therefore, this study focuses on the comparison
of the theoretical and numerical solutions for an I-beam cross-section under bending loads and is thus
subjected to stresses due to shear, torsion and due to bending.
Nowadays, the finite element analysis method is a great tool used in most fields of engineering such
as mechanical and civil engineers. The FEA (finite element analysis) has many merits in that there are
no boundary conditions, geometric, material and loading properties restrictions hence it can be used
for solving various types of problems.
Moreover, members that have different mathematical descriptions and different behaviours can be
joined together for analysis. The best known existing experimental method to compare the theoretical
and the numerical solutions that have been conducted in this study is simulation of the component
using a computer software and in this case the ANSYS. Specific hand calculations of the area moment
of inertia of the section and stresses at two specific points A and B have been performed in order to
verify and compare with the results obtained from the FEA. Furthermore, the obtained results from
the FEA are only approximations. Hence, there is a slight difference between the FEA solution and
the theoretical solution. In this study the simulation of the I-beam cross section has been executed in
ANYSYS Finite Element Analysis software in order to study its different behaviours under loads at
specific points on the member.
The following are some of the reviewed literature on similar topic:
i) Torsion and bending of beams with a hollow flange channel (Hong-Xia & Mahen, 2015)
This paper illustrated detailed items concerning an investigative anaysis of the combined torsional and
bending behaviour of the Lite Steel beam (hollow channel of a flanged beam) using Finite Element
Analysis and through experimental work. The study under Experimental work involved three Lite
Steel beam sections under a mid-span eccentric load tested to failure. Simulation was accurately
carried out under certain boundary conditions of the beam to simulate different loading eccentricities
by test supports that are suitable in a test rig that is special. The tested Lite Steel beams FE models
were established by the use of ANSYS, and their modes of failure, applied load, ultimate strength,
curves of displacement were obtained and a similarity made with the obtained experimental results.
The developed FE models were validated after the results from the FE analysis and the test results
agreed well. An investigation into the eccentricity effects and location effects of the load applied, and
spans were also conducted from parametric studies of the validated finite element models of Lite Steel
beam. The results showed that as the eccentricity loading increases then there is a significant
reduction of the bending moment

The paper presented the tests details, FEA, and LSBs study of parametric subjected to a combination
of torsion, bending and the results obtained.
ii) Lateral twisting and displacement of I-girder with webs of corrugated profile under uniform
bending (Jiho et al, 2009)
Presentation of the paper was on FEA and under uniform bending of the I-girder lateral twisting and
displacement with webs having a corrugated profile was done and compared with theoretical results.
This paper also discussed on similar studies that have been done on the I-girder’s torsional rigidity
and bending with corrugated webs. Moreover, shear centre location and warping constant calculation
approximate methods are proposed. Under uniform bending and with webs of a corrugated profile, the
I-girder’s lateral displacement and twisting strength can be calculated easily using the methods
proposed. Comparison is made on the results obtained from the proposed methods with those obtained
from the FEA. Thus successfully the proposed methods are verified based on these comparisons.
Lastly an investigation and a discussion were made on the effect of the corrugated web profiles on the
I-girder’s lateral displacement and twisting strength. It was concluded that with flat webs on the I-
girder, the warping constant is smaller than with corrugated web profiles on the I-girder, while the
modulus of shear of the flat plates is larger than that of the plates with a corrugated profile.
iii) Study of lateral displacement and twisting behaviour of beam with a steel section having a
web of trapezoidal profile by Finite Element Analysis and Experimental investigation (Denan
et al, 2010)
Presentation of the paper was on numerical study and experimental work on lateral displacement and
twisting behaviour of beam with a steel section having a web of trapezoidal profile. A flat webbed
conventional beam was used as the basis for comparison. In the experiment, sections with a length of
5m and 200 by 80 mm nominal dimension were vertically loaded while the lateral displacement were
unrestricted to allow for lateral displacement and twisting. In the analysis, eigen-value displacement
and twisting analysis in the FE method was applied in determining the critical displacement and
twisting load. FE can be applied in determining lateral displacement and twisting elasticity moment of
the section. The results indicate that thickness of the corrugation influences resistance to lateral
displacement and twisting. From the study of lateral displacement and twisting behaviour of beam
with a steel section having a web of trapezoidal profile by Finite Element Analysis and Experimental
investigation, it can be agreed that the corrugated web section with the trapezoidal shape have higher
to lateral displacement and twisting resistance as compared to flat web section. Resistance to lateral
displacement and twisting is seen in corrugation sections which are thicker. The contributing factor to
higher resistance to lateral displacement and twisting is the higher value of inertia moment on the
section about minor axis.
iv) Castellated Steel Beam Finite Element Analysis (Wakchaure & Sagade, 2012)
The paper presents a castellated Steel Beam of I section which web openings having an increase in
depth. ANSYS14 software package was used for modelling of the section for behaviour analysis of
the steel beam having a cross-section of the beam being I-shaped. The beam was simply supported
and two point loads applied and then analysis carried out. The beam deflection at its centre and the
several failure patterns are investigated. The increased depth of beams is then compared with one
another and with original section for criteria of serviceability and some parameters. From the FEA
results, conclusion is made that, the CBM behaves accordingly with respect to the requirements of
serviceability up to a maximum depth of 0.6h of the web opening. Local effects in the castellated
beams occur due to the presence of holes in the web. Different modes of failure were captured

effectively from the FEA. The outcome also illustrated that as the opening depth increases, the
castellated beam flexural stiffness decreases and hence these types of beams are highly regarded for
industrial buildings and structures that are multi-storeyed.
3. Theoretical Equations and Calculations
Figure 2: I-beam cross-section
Figure 3: Test Data
The I – beam cross-section is divided into three rectangles i.e, Top Flange, Web and Bottom Flange.
Area of Top Flange ( A1)
¿ 180 ×37=6660 mm2
Area of Web ( A2)
¿ 210 ×38=7980 mm2
Area of Bottom Flange ( A3)
¿ 306 ×36.5=11169 mm2

Total Area A = A1 + A2 + A3 = 25809 mm2
Due to symmetry, centroid lies on axis z-z. The bottom of the cross-section is chosen as the reference
axis to locate the centroid.
The distance of the centroid from the bottom of the cross-section:
¿ ∑ of the momentsof the areas the rectangles about the bottom of the cross−section
Total areaof the sections
¿
6660× [ ( 210+36.5 ) + 37
2 ]+7980 × [ 210
2 +36.5 ]+11169 × 36.5
2
25809
¿ 120.0319 mm
Therefore the centroid position from the bottom of the section ( z) = 120.03 mm.
With reference to the axes of the centroid y-y and z-z, the centroid of the rectangles A1 is (0.0,
144.97), and that of A2 is (0.0, 21.47) and that of A3 is (0.0, 101.78)
I yy= 180 ×373
12 +6660 ×144.972 + 38× 2103
12 +7980 × 21.472 + 306 × 36.53
12 +11169 ×101.782
¿ 290674870.5 mm4
I zz= 37 ×1803
12 + 210× 383
12 + 36.5 ×3063
12
¿ 106093967 mm4
Polar moment of inertia
¿ I yy + Izz
¿ 290674870.5+106093967
¿ 396768837.5 mm4

4. Numerical Screenshots

5. Results Summary - Tabulated Theory and Numerical Values
Theoretical
I zz 106093967 mm4
I yy 290674870.5 mm4
I yz 0
Stress at A 203.56
Stress at B -255.97
Height of centroid 120.0319 mm
Angle of neutral axis 65.03 °
Table 1: Summary results for Theoretical and Numerical Values
6. Discussion
Table 1 shows the comparison of the area moment of inertia, of the section, the stresses at A and B
between Theoretical solutions obtained from calculations and FEA numerical solution. It can be
deduced from the table that, the second area moments of the section in both the theoretical solution
and the numerical solution are almost the same. However, the values of the stresses in the theoretical
solution differ with those from the numerical solution. The tested Finite element models were
developed and modelled by the use of ANSYS, and their modes of failure, applied load, ultimate
strength, the second moments of area were obtained and a comparison made with the results obtained
from the experimental test. Both the results obtained from analysis and from the experiment agreed
well and thus FE models development were validated.
7. Conclusion
This study has reported the theoretical and numerical solutions for calculating the second area of
moment of the cross-section, stresses at A and B deflection of I-beam cross-section under a moment
of bending of magnitude my about the centroid of y axis and a bending moment of mz about the
centroid of z axis. The finite element analysis based on the above parameters yielded good results.
Both the results obtained from analysis and from the experiment agreed well and thus FE models
development were validated.
Numerical
I zz 106 ×106 mm4
I yy 291 ×106 mm4
I yz 0.792 ×10−8 mm4
Stress at A 283.5
Stress at B 153

8. References
Divahar R., P. S. Joanna,“ Lateral Buckling Of Cold Formed Steel Beam with Trapezoidal Corrugated
Web”, International Journal Of Civil Engineering And Technology (IJCIET) Volume 5, Issue
3, March (2014), pp. 217-225
Denan Fatimah, Mohd Hanim Osman & Sariffuddin Saad , “The Study of Lateral Torsional Buckling
Behaviour of Beam With Trapezoid Web Steel Section By Experimental And Finite Element
Analysis”, IJRRAS 2 (3) March 2010
Hong-Xia Wan, Mahen Mahendran , “Bending and torsion of hollow flange channel beams”
Engineering Structures 84 (2015) 300–312, 2015
Jiho Moon a, Jong-WonYi b, ByungH.Choi c, HakEunLee, “Lateral– torsional buckling of I-girder
with corrugated webs under uniform bending”, Thin-Walled Structures 47 (2009) 21–
30,2009
Wakchaure M.R. & Sagade A.V., “Finite Element Analysis of Castellated Steel Beam”, International
Journal of Engineering and Innovative Technology (IJEIT) 2, Issue 1, July 2012