A REPORT ON STRUCTURAL ANALYSIS OF BEAMS USING ANSYS

PREPARED BY

Vishnu Prasad K 2009A4PS072G

Based on Mechanics of Solids

At

BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE, PILANI, GOA CAMPUS Semester-I 2010-11

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ABSTRACT

Background

Beams continue to be one of the most used methods of providing structural integrity for any

kind of construction. The project aims at generating a code that can be used with the ANSYS

structural solver for calculating the deflection as well as shear and bending stresses generated

in a beam of general cross section under generally applicable loading conditions.

Results

Two sets of codes were generated, giving two possible approaches at completely modeling and

solving the beam – loads problem. The accuracy of both the codes was tested by comparing the

data generated by ANSYS solver with theoretical values.

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Table of contents

Introduction Page 3

Beam under action of a point load page 5

Procedure Page 5

Results Page 7

Theoretical Solution Page 8

Code Page 11

Beam under uniform loading page 13

Procedure Page 13

Results Page 16

Theoretical Solution Page 17

Code Page 20

Conclusion Page 23

References Page 24

APPENDIX A Page 25

APPENDIX B Page 30

APPENDIX C Page 36

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Introduction

Structural analysis comprises the set of physical laws and mathematics required to study and

predicts the behavior of structures. The subjects of structural analysis are engineering artifacts

whose integrity is judged largely based upon their ability to withstand loads. From a theoretical

perspective the primary goal of structural analysis is the computation of deformations, internal

forces, and stresses. In practice, structural analysis can be viewed more abstractly as a method

to drive the engineering design process or prove the soundness of a design without a

dependence on directly testing it.

This project mainly concerns with the static structural analysis of the behavior of beams

subjected to various loading conditions. The main objective of the project is to generate a set of

codes that can be used to accurately model the behavior of a beam under various loading

conditions and thus predict various factors that are of prime importance from the point of view

of structural integrity of the beam like deflection of the beam under the applied load as well as

the stresses developed in the beam due to the applied load. These factors are of particular

concern as it directly represents the safe limits to which the beam can be used.

Beams continue to be one of the most used methods of providing structural integrity for any

kind of construction. A wide variety of cross sections are used for beams taking into

considerations the kind of loads and stresses the beams are expected to bear as well as the

type of working conditions that the beam might be subjected to. I-beams are beams with an I-

or H-shaped cross-section. I-beams are used as major support trusses in building, to ensure that

a structure will be physically sound. Steel is one of the most common materials used to make I-

beams, since it can withstand very heavy loads and is relatively cheap. Composite I-beams are

also available, with layers of other materials encasing the outside of the steel I-beam.

Over the years, beams have been well studied and a set of theories have been put forward that

accurately describes behavior of beams under loads. The Timoshenko beam theory, one of the

widely used theories used for modeling beams, was developed by Ukrainian-born scientist

Stephen Timoshenko. The model takes into account shear deformation and rotational inertia

effects, making it suitable for describing the behavior of short beams as well as sandwich

composite beams. The resulting equation is of 4th order partial differential equation with a

second order spatial derivative. The finite element method provides an approximate numerical

solution to the said partial differential equation.

The ANSYS Mechanical solver, which is one of the most acclaimed FEA softwares available, has

been used to solve the beam load – deflection problem and obtain detailed data on the

deflection of the beam as well as the shear and bending stresses developed in the beam under

load. The HE120A standard beam was chosen for analysis and two cases were analyzed and the

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results obtained were compared against theoretical results obtained using the Euler – Bernoulli

equation.

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BEAM UNDER ACTION OF A POINT LOAD

Consider the HE 120A beam, very commonly used in construction purposes , of length 3 m

clamped at both the ends and under the action of a 30 kN load as shown in the figure.

Calculate the vertical deflection in the beam due to the loading.

Calculate the maximum and minimum shear stress developed in the beam.

Calculate the maximum and minimum bending stress developed in the beam.

PROCEDURE

The very first step to obtaining the solution of any system using ansys is the selection of

a proper element type that best represents the problem at hand and at the same time

solves for the parameters that you require.

The problem at hand is a typical beam load – deflection problem. Ideally, BEAM4 should

be able to solve for deflections, however, there are more advanced element types in

ansys that offer a more comprehensive solution of the same problem. One such

element is the BEAM1882 element which is based on the Timoshenko beam theory1.

The BEAM188 is represented using a line element in ansys and the cross – section of the

beam (defined for the purpose of calculating the shear stress distribution) is manually

defined using the SECTYPE and SECDATA commands. It is essential to change the

KEYOPT values allotted to the solver to ensure that the solver solves for shear stress

distributions also. This done by changing the automatically allotted KEYOPT 4 value of 0

to 2.

A key element size of 0.1 meter was chosen as appropriate. It divides the whole length

of 3 meters into 31 nodes.

The material for the beam is defined with a modulus of elasticity 200 GPa and Poisson’s

ratio of 0.3. The Poisson's ratio affects the shear-correction factor and shear-stress

distribution slightly but the element defined ignores these effects given the fact that

their contribution is minor in nature.

1 | Refer Appendix C 2| Refer Appendix A for details regarding usage.

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The degrees of freedom of the nodes at both the ends are completely restricted to

model the clamping of the ends of the beam and the load of 30 kN is applied at a

distance of 1m from one of the ends.

The solve command was executed and the details of vertical displacement as well as

shear stress distributions were obtained using suitable commands.

Figure 1. Vertical deflection of the beam (Deformed shape + undeformed shape)

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Results

Vertical deflection of the beam

x UZ(x) Solution

3.00 0.00000

0.10 -0.00011

0.30 -0.00060

0.60 -0.00170

0.90 -0.00281

1.20 -0.00327

1.50 -0.00308

1.80 -0.00251

2.10 -0.00173

2.40 -0.00093

2.70 -0.00029

MAXIMUM ABSOLUTE VALUES

DISTANCE X =1.3M

VALUE -0.32655E-02

The complete solution of the deflection problem is given in appendix B.

Shear and bending stresses developed in the beam

Due to the extensive data generated in the solution process, the complete solution is given in

appendix B.

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Theoretical solution

Consider the beam load – deflection problem:

Calculate the vertical deflection in the beam due to the loading.

Calculate the maximum and minimum shear stress developed in the beam.

Calculate the maximum and minimum bending stress developed in the beam.

Method of solution: Euler – Bernoulli beam theory

Analysis:

Modulus of elasticity E = 200 GPa

Moment of inertia I = 5.8017e-6 m4

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x UY(x) Theoretical Solution Difference

3 0.00000 0.00000 0.00000

0.1 -0.00005 -0.00011 0.00006

0.3 -0.00043 -0.00060 0.00017

0.6 -0.00138 -0.00170 0.00033

0.9 -0.00233 -0.00281 0.00048

1.2 -0.00279 -0.00327 0.00047

1.5 -0.00269 -0.00308 0.00039

1.8 -0.00221 -0.00251 0.00030

2.1 -0.00151 -0.00173 0.00022

2.4 -0.00079 -0.00093 0.00014

2.7 -0.00023 -0.00029 0.00007

-- Theoretical solution

-- ANSYS Solution

+ Error

Comparison of results

The error that’s shown in the figure can be justified by the fact that Timoshenko theory was

used by the FEM formulation by ansys where as Euler’s theory was used in obtaining the

theoretical solution. However , it must be noted that the error is very low in the order of

0.1mm.

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Shear and bending stresses

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Code

/PREP7

MP,EX,1,2e11

MP,NUXY,1,0.3

ET,1,188,,,,2

SECTYPE,2,BEAM,I,IBEAM

SECDATA,120e-3,120e-3,114e-3,8e-3,8e-3,5e-3

SECNUM,2

K,,0,0

K,,3,0

L,1,2

KESIZE,ALL,0.1

LMESH,ALL

FINISH

/SOLU

NSEL,S,LOC,X,1

F,12,FZ,30e3

NSEL,S,LOC,X,0

D,1,ALL,0

NSEL,S,LOC,X,3

D,2,ALL,0

NSEL,ALL

ANTYPE,STATIC,NEW

SOLVE

/POST1

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NSEL,ALL

NLIST

PLNSOL,U,Z

PRNSOL,U,Z

PRESOL,S

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BEAM UNDER UNIFORM LOADING

Consider the HE 120A beam, very commonly used in construction purposes, of length 3 m

clamped at both the ends under the action of a uniform load of intensity 10 kN/m as shown in

the figure:

Calculate the vertical deflection in the beam due to the loading.

Calculate the maximum and minimum shear stress developed in the beam.

Calculate the maximum and minimum bending stress developed in the beam.

PROCEDURE

It is possible to proceed just as in the first problem and obtain the solution for the

required parameters. But here I like to present an option which gives much more

control over the number of computations as well as the overall accuracy of the results.

It can be seen from the output of the first problem that a huge and unnecessary amount

of data was calculated during the shear stresses and bending moment calculations. The

computational time can be optimized and cost can be reduced to a large extent by

manually performing the beam cross section definition and meshing process.

A csv (comma separated value) sheet of the coordinates of the I section of the HE120A

beam can be obtained from the manufacturer’s database.

A PLANE82 element, ideally suited for 2D meshing is used for defining the cross section

as well as creating the mesh. The cross section can be of any arbitrary shape, offset by

any value from the origin. The ansys solver automatically accounts for these factors and

calculates the necessary values for the beam. Moreover, PLANE82 is one of the few

elements supported by SECWRITE command, which is used to save the created mesh.

For a detailed and accurate result of the shear and bending stresses at cross sections, a

finer area mesh is to be used. However this project uses a coarser mesh due to certain

constrains.

It is recommended that the mesh be refined at the flange-web intersection points, as

shown in the following figure, to improve accuracy of the result.

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Figure. Ideal mesh for detailed shear and bending stress analysis

Figure. Ideal mesh for faster computation with preference to deflection under load only

The meshed cross section is imported and assigned to the BEAM188 element definition

by using a SECREAD command.

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The key element size is taken to be 0.1 meters just as in the above problem. It divides

the whole length of 3 meters into 31 nodes.

The material for the beam is defined with a modulus of elasticity 200 GPa and Poisson’s

ratio of 0.3. The Poisson's ratio affects the shear-correction factor and shear-stress

distribution slightly but the element defined ignores these effects given the fact that

their contribution is minor in nature.

The degrees of freedom of the nodes at both the ends are completely restricted to

model the clamping of the ends of the beam and the load intensity of 10 kN/m is

applied throughout the distance of 3m.

The solve command was executed and the details of vertical displacement as well as

shear stress distributions were obtained using suitable commands.

Figure. Vertical deflection of the beam (Deformed + undeformed shape)

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Results

The vertical deflection of the beam obtained using the ansys solver is given below.

Distance X UZ (x) Meters

0 0

3 0

0.3 0.00034

0.6 0.00093

0.9 0.00153

1.2 0.00195

1.5 0.00211

1.8 0.00195

2.1 0.00153

2.4 0.00093

2.7 0.00034

MAXIMUM ABSOLUTE VALUES

DISTANCE X=1.5

VALUE 0.21075E-02

The complete solution for the deflection problem is given in appendix B.

Shear and bending stresses developed in the beam

Due to the extensive data generated in the solution process, the complete solution is given in

appendix B.

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Theoretical Solution

Consider the beam load- deflection problem given below:

Calculate the vertical deflection in the beam due to the loading.

Calculate the maximum and minimum shear stress developed in the beam.

Calculate the maximum and minimum bending stress developed in the beam.

Method of solution: Euler – Bernoulli beam theory

Analysis:

Modulus of elasticity E = 200 GPa

Moment of inertia I = 5.8017e-6 m4

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On evaluating the of the deflection of the beam, the following data was obtained:

Distance Theoretical Solution Error

0.00 0.00000 0.00000 0.00000

3.00 0.00000 0.00000 0.00000

0.30 0.00024 0.00034 -0.00010

0.60 0.00074 0.00093 -0.00018

0.90 0.00128 0.00153 -0.00024

1.20 0.00168 0.00195 -0.00028

1.50 0.00182 0.00211 -0.00029

1.80 0.00168 0.00195 -0.00028

2.10 0.00128 0.00153 -0.00024

2.40 0.00074 0.00093 -0.00018

2.70 0.00024 0.00034 -0.00010

All dimensions given are in meters.

-- Theoretical solution

-- ANSYS solution

+ Error

Comparison of results

It is clear from the data generated that both the solutions are in excellent agreement of each

other. The error in data generated is much less as compared to the first case.

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Shear and bending stress

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CODE

PART ONE – CROSS SECTION MODELING

/PREP7

et,1,82

k,, -0.0600 , 0.0570

k,, 0.0600 , 0.0570

k,, 0.0600 , 0.0490

k,, 0.0025 , 0.0490

k,, 0.0025 , -0.0490

k,, 0.0600 , -0.0490

k,, 0.0600 , -0.0570

k,, -0.0600 , -0.0570

k,, -0.0600 , -0.0490

k,, -0.0025 , -0.0490

k,, -0.0025 , 0.0490

k,, -0.0600 , 0.0490

A,1,2,3,4,5,6,7,8,9,10,11,12

KESIZE,ALL,0.007

AMESH,ALL

SECWRITE,c:\mesh,,,1

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PART TWO – BEAM ANALYSIS

/PREP7

MP,EX,1,2e11

MP,NUXY,1,0.3

ET,1,188,,,,2

SECTYPE,2,BEAM,mesh,IBEAM

SECREAD,c:\mesh,,,mesh

SECNUM,2

K,,0,0

K,,3,0

L,1,2

KESIZE,ALL,0.1

LMESH,ALL

FINISH

/SOLU

NSEL,ALL

SFBEAM,ALL,ALL,PRES,-10E3

NSEL,S,LOC,X,0

D,1,all,0

NSEL,S,LOC,X,3

D,2,all,0

NSEL,ALL

ANTYPE,STATIC,NEW

SOLVE

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/POST1

NSEL,ALL

NLIST

PLNSOL,U,Z

PRNSOL,U,Z

PRESOL,S

Notes on code:

Refine the mesh as required by modifying part one code.

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Conclusion

The aim of the project was to put forward a set of codes that can be used to completely and

accurately model the behavior of beams under load.

The data obtained using ansys was within agreeable limits of the theoretical solution, taking

into consideration the fact that the beam element uses Timoshenko solution, whereas Euler –

Bernoulli equation was used for finding out the theoretical values.

Given these facts, the code put forward in this project can be used for analysis of beams under

any given loading and support conditions.

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References

"An Introduction to the Mechanics of Solids" - Crandall and Katt

Ansys documentation

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APPENDIX A

BEAM188 Element Description

BEAM188 is suitable for analyzing slender to moderately stubby/thick beam structures. The element is based on Timoshenko beam theory which includes shear-deformation effects. The element provides options for unrestrained warping and restrained warping of cross-sections.

The element is a linear, quadratic, or cubic two-node beam element in 3-D. BEAM188 has six (or seven) degrees of freedom at each node. These include translations in the x, y, and z directions and rotations about the x, y, and z directions. A seventh degree of freedom (warping magnitude) is optional. This element is well-suited for linear, large rotation, and/or large strain nonlinear applications.

The element includes stress stiffness terms, by default, in any analysis with large deflection. The provided stress-stiffness terms enable the elements to analyze flexural, lateral, and torsional stability problems (using Eigen value buckling, or collapse studies with arc length methods or nonlinear stabilization).

Elasticity, plasticity, creep and other nonlinear material models are supported. A cross-section associated with this element type can be a built-up section referencing more than one material.

BEAM188 Element Technology and Usage Recommendations

BEAM188 is based on Timoshenko beam theory, which is a first-order shear-deformation theory: transverse-shear strain is constant through the cross-section (that is, cross-sections remain plane and undistorted after deformation). The element supports an elastic relationship between transverse-shear forces and transverse-shear strains.

Use the slenderness ratio of a beam structure (GAL2 / (EI) ) to judge the applicability of the element, where:

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G Shear modulus A Area of the cross-section L Length of the member (not the element length) EI Flexural rigidity

A slenderness ratio greater than 30 is recommended.

BEAM188 Assumptions and Restrictions

The beam must not have zero length. Cross-section failure or folding is not accounted for. Rotational degrees of freedom are not included in the lumped mass matrix if offsets are

present. Only moderately "thick" beams can be analyzed.

BEAM188 Input Data

The geometry, node locations, coordinate system, and pressure directions for this element are shown in Figure 188.1: BEAM188 Geometry.

The beam element is a one-dimensional line element in space. The cross-section details are provided separately via the SECTYPE and SECDATA commands.

BEAM188 Cross-Sections

BEAM188 can be associated with these cross-section types:

Standard library section types (SECTYPE,,BEAM) Generalized beam cross-sections (SECTYPE,,GENB) Tapered beam cross-sections (SECTYPE,,TAPER)

SECTYPE COMMAND

SECTYPE SECTYPE, SECID, Type, Subtype, Name Type

BEAM — Defines a beam section.

Subtype When Type = BEAM, the possible beam sections that can be defined for Subtype are:

RECT Rectangle

QUAD Quadrilateral

CSOLID Circular solid

CTUBE Circular tube

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CHAN Channel

I I-shaped section

Z Z-shaped section

L L-shaped section

T T-shaped section

HATS Hat-shaped section

HREC Hollow rectangle or box

It’s required to supplement the definition of section type with a definition of the section. This is done using the SECDATA command or the SECREAD command, if already meshed cross section is available. In such cases, the solver automatically calculates the required parameters for the analysis like moment of inertia, shear center etc.

SECDATA

SECDATA,VAL1,VAL2,VAL3,VAL4,VAL5,VAL6,VAL7,VAL8,VAL9,VAL10 Describes the geometry of a section. VAL1, VAL2, VAL3. . . VAL10 Values, such as thickness or the length of a side or the numbers of cells along the width that describes the geometry of a section.

Type: BEAM

Data to provide in the value fields:

W1, W2, W3, t1, t2, t3

where

W1, W2 = Width of the top and bottom flanges

W3 = Overall depth

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t1, t2 = Flange thicknesses

t3 = Web thicknesses

BEAM188 Output Data

The solution output that can be obtained from BEAM188 is in two forms:

Nodal displacements and reactions included in the overall nodal solution Linearized Stress

SDIR is the stress component due to axial load.

SDIR = Fx/A, where Fx is the axial load and A is the area of the cross-section.

SByT and SByB are bending-stress components.

SByT = -Mz * ymax / Izz

SByB = -Mz * ymin / Izz

SBzT = Mz * zmax / Iyy

SBzB = Mz * zmin / Iyy

The Element Output Definitions table uses the following notation:

A colon (:) in the Name column indicates that the item can be accessed by the Component Name method (ETABLE, ESOL). The O column indicates the availability of the items in the file Jobname.OUT. The R column indicates the availability of the items in the results file.

Table 188.1 BEAM188 Element Output Definitions

Name Definition O R

SF:y, z Section shear forces 2 Y

SE:y, z Section shear strains 2 Y

Fx Axial force Y Y

Transverse-Shear Stress Output

The BEAM188 formulation is based on three stress components:

one axial

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two shear stress

The shear stresses are caused by torsional and transverse loads. BEAM188 is based Timoshenko beam theory. The transverse-shear strain is constant for the cross-section; therefore, the shear energy is based on a transverse-shear force. The shear force is redistributed by predetermined shear-stress distribution coefficients across the beam cross-section, and made available for output purposes.

The accuracy of transverse-shear distribution is directly proportional to the mesh density of cross-section modeling.

The Poisson's ratio affects the shear-correction factor and shear-stress distribution slightly, shear distribution calculation ignores the effects of Poisson's and this effect is ignored.

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APPENDIX B

BEAM UNDER ACTION OF A POINT LOAD

Deflections under load (Ansys values and theoretical solution)

x UY(x) Theoretical ( mm ) Solution (mm) Difference(mm)

0.00 0.00 0.00 0.00

3.00 0.00 0.00 0.00

0.10 -0.05 -0.11 0.06

0.20 -0.20 -0.32 0.11

0.30 -0.43 -0.60 0.17

0.40 -0.71 -0.94 0.22

0.50 -1.04 -1.31 0.27

0.60 -1.38 -1.70 0.33

0.70 -1.72 -2.10 0.38

0.80 -2.04 -2.47 0.43

0.90 -2.33 -2.81 0.48

1.00 -2.55 -3.08 0.53

1.10 -2.71 -3.21 0.50

1.20 -2.79 -3.27 0.47

1.30 -2.81 -3.26 0.45

1.40 -2.78 -3.19 0.42

1.50 -2.69 -3.08 0.39

1.60 -2.57 -2.92 0.36

1.70 -2.40 -2.73 0.33

1.80 -2.21 -2.51 0.30

1.90 -1.99 -2.26 0.27

2.00 -1.76 -2.00 0.25

2.10 -1.51 -1.73 0.22

2.20 -1.27 -1.46 0.19

2.30 -1.02 -1.19 0.17

2.40 -0.79 -0.93 0.14

2.50 -0.58 -0.69 0.11

2.60 -0.39 -0.48 0.09

2.70 -0.23 -0.29 0.07

2.80 -0.11 -0.15 0.04

2.90 -0.03 -0.05 0.02

Average 0.25

Std. Deviation 0.16

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Shear stress distribution

There are over 31 nodes because of the key element size of 0.1 m and at each node, and at

each node, the shear and bending stress distributions are evaluated at 51 points. Due to this

fact, only the solution at x=1 meter is given here. The complete solution can be obtained by

running the code and using PRESOL, S.

ELEMENT NODE 3 (X= 1m)

SEC NODE SXX SXZ SXY

1 2.1579E+07 -1.8023E+04 8.9260E+05

3 2.1579E+07 -3.0667E+05 5.6764E+06

17 1.8551E+07 1.8872E+07 8.6903E+06

15 1.8551E+07 -2.2639E+06 -8.9260E+05

5 2.1579E+07 -3.0667E+05 -5.6764E+06

19 1.8551E+07 1.8872E+07 -8.6903E+06

7 2.1579E+07 -1.8023E+04 -8.9260E+05

21 1.8551E+07 -2.2639E+06 8.9260E+05

27 -4.7730E-09 4.4332E+07 -1.0966E-07

25 4.6566E-10 4.4332E+07 2.1653E-07

35 -1.8551E+07 1.8872E+07 8.6903E+06

33 -1.8551E+07 1.8872E+07 -8.6903E+06

31 -1.8551E+07 -2.2639E+06 8.9260E+05

47 -2.1579E+07 -3.0667E+05 -5.6764E+06

45 -2.1579E+07 -1.8023E+04 -8.9260E+05

49 -2.1579E+07 -3.0667E+05 5.6764E+06

37 -1.8551E+07 -2.2639E+06 -8.9260E+05

51 -2.1579E+07 -1.8023E+04 8.9260E+05

Max= 2.1579E+07 8.6903E+06 4.4332E+07

Min= -2.1579E+07 -8.6903E+06 -2.2639E+06

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BEAM UNDER UNIFORM LOADING

Deflections under load (Ansys values and theoretical solution)

Distance Theoretical(mm) Solution(mm) Error(mm)

0 0.000 0.000 0.000

3 0.000 0.000 0.000

0.1 0.030 0.067 -0.037

0.2 0.113 0.184 -0.072

0.3 0.236 0.339 -0.104

0.4 0.388 0.522 -0.133

0.5 0.561 0.722 -0.160

0.6 0.745 0.930 -0.185

0.7 0.931 1.138 -0.207

0.8 1.112 1.339 -0.226

0.9 1.283 1.526 -0.243

1 1.436 1.694 -0.257

1.1 1.569 1.837 -0.269

1.2 1.675 1.953 -0.278

1.3 1.754 2.038 -0.284

1.4 1.802 2.090 -0.288

1.5 1.818 2.108 -0.290

1.6 1.802 2.090 -0.288

1.7 1.754 2.038 -0.284

1.8 1.675 1.953 -0.278

1.9 1.569 1.837 -0.269

2 1.436 1.694 -0.257

2.1 1.283 1.526 -0.243

2.2 1.112 1.339 -0.226

2.3 0.931 1.138 -0.207

2.4 0.745 0.930 -0.185

2.5 0.561 0.722 -0.160

2.6 0.388 0.522 -0.133

2.7 0.236 0.339 -0.104

2.8 0.113 0.184 -0.072

2.9 0.030 0.067 -0.037

Average -0.1864

Deviation 0.0935

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Shear stress distribution

There are over 31 nodes because of the key element size of 0.1 m. Due to the fact that the

cross sectional area has been manually meshed , the solution is evaluated at almost 200 points

at each node. Due to this, only the solution at x=1 meter is given here. The complete solution

can be obtained by running the code and using PRESOL, S.

ELEMENT NODE 12 DISTANCE X=1M

SEC NODE SXX SXZ SXY

1 -2.20E+07 5085.3 -3.55E+05

2 -1.89E+07 5030.8 -3.62E+05

3 -1.89E+07 5036.6 -7.23E+05

4 -2.20E+07 5092.8 -7.10E+05

10 -2.20E+07 5088.9 -0.34727

11 -1.89E+07 5033.7 0.34929

16 -1.89E+07 5025.9 -1.08E+06

17 -2.20E+07 5095.2 -1.07E+06

22 -1.89E+07 4917.6 -1.45E+06

23 -2.20E+07 5173.7 -1.42E+06

28 -1.89E+07 3420.5 -1.81E+06

29 -2.20E+07 6349.4 -1.78E+06

34 -1.89E+07 -18411 -2.16E+06

35 -2.20E+07 23333 -2.14E+06

40 -1.89E+07 -3.40E+05 -2.47E+06

41 -2.20E+07 2.77E+05 -2.72E+06

46 -1.89E+07 6.32E+06 -4.27E+06

47 -2.20E+07 -2.76E+05 -6.05E+05

52 -1.89E+07 6.32E+06 4.27E+06

53 -2.20E+07 -2.76E+05 6.05E+05

58 -1.89E+07 -3.40E+05 2.47E+06

59 -2.20E+07 2.77E+05 2.72E+06

64 -1.89E+07 -18411 2.16E+06

65 -2.20E+07 23333 2.14E+06

70 -1.89E+07 3420.5 1.81E+06

71 -2.20E+07 6349.4 1.78E+06

76 -1.89E+07 4917.6 1.45E+06

77 -2.20E+07 5173.7 1.42E+06

82 -1.89E+07 5025.9 1.08E+06

83 -2.20E+07 5095.2 1.07E+06

88 -1.89E+07 5036.6 7.23E+05

89 -2.20E+07 5092.8 7.10E+05

94 -1.89E+07 5030.8 3.62E+05

P a g e | 34

95 -2.20E+07 5085.3 3.55E+05

100 -1.89E+07 5033.7 -0.34886

101 -2.20E+07 5088.9 0.34684

106 -1.62E+07 9.96E+06 6.33E+05

107 -1.62E+07 9.96E+06 -6.33E+05

112 -1.35E+07 1.02E+07 72423

113 -1.35E+07 1.02E+07 -72423

118 -1.08E+07 1.04E+07 8286.5

119 -1.08E+07 1.04E+07 -8286.5

124 -8.09E+06 1.06E+07 948.08

125 -8.09E+06 1.06E+07 -948.08

130 -5.39E+06 1.07E+07 108.42

131 -5.39E+06 1.07E+07 -108.42

136 -2.70E+06 1.08E+07 12.245

137 -2.70E+06 1.08E+07 -12.245

142 9.31E-10 1.08E+07 -5.23E-08

143 9.31E-10 1.08E+07 9.60E-08

148 2.70E+06 1.08E+07 -12.245

149 2.70E+06 1.08E+07 12.245

154 5.39E+06 1.07E+07 -108.42

155 5.39E+06 1.07E+07 108.42

160 8.09E+06 1.06E+07 -948.08

161 8.09E+06 1.06E+07 948.08

166 1.08E+07 1.04E+07 -8286.5

167 1.08E+07 1.04E+07 8286.5

172 1.35E+07 1.02E+07 -72423

173 1.35E+07 1.02E+07 72423

178 1.62E+07 9.96E+06 -6.33E+05

179 1.62E+07 9.96E+06 6.33E+05

184 1.89E+07 6.32E+06 4.27E+06

185 1.89E+07 6.32E+06 -4.27E+06

190 2.20E+07 -2.76E+05 6.05E+05

191 2.20E+07 -2.76E+05 -6.05E+05

196 2.20E+07 2.77E+05 -2.72E+06

197 1.89E+07 -3.40E+05 -2.47E+06

202 2.20E+07 23333 -2.14E+06

203 1.89E+07 -18411 -2.16E+06

208 2.20E+07 6349.4 -1.78E+06

209 1.89E+07 3420.5 -1.81E+06

214 2.20E+07 5173.7 -1.42E+06

215 1.89E+07 4917.6 -1.45E+06

220 2.20E+07 5095.2 -1.07E+06

221 1.89E+07 5025.9 -1.08E+06

P a g e | 35

226 2.20E+07 5092.8 -7.10E+05

227 1.89E+07 5036.6 -7.23E+05

232 2.20E+07 5085.3 -3.55E+05

233 1.89E+07 5030.8 -3.62E+05

238 2.20E+07 5088.9 -0.34684

239 1.89E+07 5033.7 0.34886

244 1.89E+07 -3.40E+05 2.47E+06

245 2.20E+07 2.77E+05 2.72E+06

250 1.89E+07 -18411 2.16E+06

251 2.20E+07 23333 2.14E+06

256 1.89E+07 3420.5 1.81E+06

257 2.20E+07 6349.4 1.78E+06

262 1.89E+07 4917.6 1.45E+06

263 2.20E+07 5173.7 1.42E+06

268 1.89E+07 5025.9 1.08E+06

269 2.20E+07 5095.2 1.07E+06

274 1.89E+07 5036.6 7.23E+05

275 2.20E+07 5092.8 7.10E+05

280 1.89E+07 5030.8 3.62E+05

281 2.20E+07 5085.3 3.55E+05

286 1.89E+07 5033.7 -0.34929

287 2.20E+07 5088.9 0.34727

Max= 2.20E+07 4.27E+06 1.08E+07

Min= -2.20E+07 -4.27E+06 -3.40E+05

P a g e | 36

APPENDIX C

Timoshenko beam theory

The Timoshenko beam theory was developed by Ukrainian-born scientist Stephen Timoshenko

in the beginning of the 20th century. The model takes into account shear deformation and

rotational inertia effects, making it suitable for describing the behavior of short beams,

sandwich composite beams. The resulting equation is of 4th order, but unlike ordinary beam

theory - i.e. Bernoulli-Euler theory - there is also a second order spatial derivative present.

If the shear modulus of the beam material approaches infinity - and thus the beam becomes

rigid in shear - and if rotational inertia effects are neglected, Timoshenko beam theory

converges towards ordinary beam theory.

In static Timoshenko beam theory without axial effects, the displacements of the beam are

assumed to be given by

where (x ,y ,z) are the coordinates of a point in the beam, ux ,uy, uz are the components of the

displacement vector in the three coordinate directions, is the angle of rotation of the normal to

the mid-surface of the beam, and w is the displacement of the mid-surface in the z-direction.

The governing equations are the following uncoupled system of ordinary differential equations:

The Timoshenko beam theory for the static case is equivalent to the Euler-Bernoulli theory

when the last term above is neglected, an approximation that is valid when

Where L is the length of the beam.

Combining the two equations gives, for a homogeneous beam of constant cross-section,

P a g e | 37

Euler–Bernoulli beam theory

Euler–Bernoulli beam theory (also known as engineer's beam theory, classical beam theory) is a

simplification of the linear theory of elasticity which provides a means of calculating the load-

carrying and deflection characteristics of beams. It covers the case for small deflections of a

beam which is subjected to lateral loads only. It is thus a special case of Timoshenko beam

theory which accounts for shear deformation and is applicable for thick beams.

The Euler-Bernoulli equation describes the relationship between the beam's deflection and the

applied load:

The curve w(x) describes the deflection w of the beam at some position. q is a distributed load;

it may be a function of x, w, or other variables.

Note that E is the elastic modulus and that I is the second moment of area. For an Euler-

Bernoulli beam not under any axial loading this axis is called the neutral axis.

Often, w = w(x), q = q(x), and EI is a constant, so that:

This equation, describing the deflection of a uniform, static beam, is used widely in engineering

practice.

P a g e | 38

HE120A BEAM

The dimensions and configuration of the HE120A beam used in the project is as follows

Type

Beam

height

(mm)

Flange

width

(mm)

Web

thickness

(mm)

Flange

thickness

(mm)

Weight

(kg/m)

Cross-

section

area

Moment of

inertia

in torsion (J)

HE 120 A 114 120 5 8 19.9 25.3 5.81