Study the Crippling Load in Buckling Analysis- Hani Aziz Ameen

Study of crippling Load Dr. Hani Aziz Ameen

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In Buckling Test

for different materials

Asst. Prof. Dr. Hani Aziz Ameen

Technical College - Baghdad

Dies and Tools Engineering Department

E-mail: [email protected]

mailto:[email protected]


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1- Introduction

Structural members which carry compressive loads may be divided into two broad categories depending on their relative lengths and cross-sectional dimensions. Short, thick members are generally termed columns and these usually fail by crushing-when the yield stress of the material in compression is exceeded. Long, slender columns or struts, however, fail by buckling some time before the yield stress in compression is reached. The buckling occurs owing to one or more of the following reasons:

(a) the strut may not be perfectly straight initially;

(b) the load may not be applied exactly along the axis of the strut;

(c) one part of the material may yield in compression more readily than others owing to some lack of uniformity in the material properties throughout the strut.

At values of load below the buckling load a strut will be in stable equilibrium where the displacement caused by any lateral disturbance will be totally recovered when the disturbance is removed. At the buckling load the strut is said to be in a state of neutral equilibrium, and theoretically it should then be possible to gently deflect the strut into a simple sine wave provided that the amplitude of the wave is kept small. This can be demonstrated quite simply using long thin strips of metal, e.g. a metal rule, and gentle application of compressive loads.

Theoretically, it is possible for struts to achieve a condition of unstable equilibrium with loads exceeding the buckling load, any slight lateral disturbance then causing failure by buckling; this condition is never achieved in practice under static load conditions. Buckling occurs immediately at the point where the buckling load is reached owing to the reasons stated earlier.[7]

1-1 Buckling Analysis

Buckling analysis is a technique used to determine buckling loads (critical loads at which a structure becomes unstable) and buckled mode shapes (the characteristic shape associated with a structure's buckled response). .[7]

1-2 Types of Buckling Analyses

Two techniques are available in the ANSYS Multiphysics, ANSYS Mechanical, ANSYS Structural, and ANSYS Professional programs for predicting the buckling load and buckling mode shape of a structure: eigenvalue (or linear) buckling analysis, and nonlinear buckling analysis. Because the two methods can yield dramatically different results, it is necessary to first understand the differences between them. .[7]


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1- 2 -1 Eigenvalue Buckling Analysis

1- 2 – 2 Nonlinear Buckling Analysis

1- 2 -1 Eigenvalue Buckling Analysis

Eigenvalue buckling analysis predicts the theoretical buckling strength of an ideal

elastic structure. It computes the structural eigenvalues for the given system loading and

constraints. This is known as classical Euler buckling analysis. Buckling loads for

several configurations are readily available from tabulated solutions. However, in real-

life, structural imperfections and nonlinearities prevent most real world structures from

reaching their eigenvalue predicted buckling strength; ie. it over-predicts the

expected buckling loads. This method is not recommended for accurate, real-world

buckling prediction analysis.

Eigenvalue buckling analysis predicts the theoretical buckling strength (the bifurcation point) of an ideal linear elastic structure. (b).) This method corresponds to the textbook approach to elastic buckling analysis: for instance, an eigenvalue buckling analysis of a column will match the classical Euler solution. However, imperfections and nonlinearities prevent most real-world structures from achieving their theoretical elastic buckling strength. Thus, eigenvalue buckling analysis often yields unconservative results, and should generally not be used in actual day-to-day engineering analyses and fig(1) show the Eigenvalue Buckling curve.[1], .[7]

1- 2 – 2 Nonlinear Buckling Analysis

Nonlinear buckling analysis is more accurate than eigenvalue analysis because it

employs non-linear, large-deflection, static analysis to predict buckling loads. Its mode

of operation is very simple: it gradually increases the applied load until a load level is

found whereby the structure becomes unstable (ie. suddenly a very small increase in the

load will cause very large deflections). The true non-linear nature of this analysis thus

permits the modeling of geometric imperfections, load perterbations, material

nonlinearities and gaps. For this type of analysis, note that small off-axis loads are

necessary to initiate the desired buckling mode.

Nonlinear buckling analysis is usually the more accurate approach and is

therefore recommended for design or evaluation of actual structures. This technique

employs a nonlinear static analysis with gradually increasing loads to seek the load level

at which your structure becomes unstable


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Using the nonlinear technique, your model can include features such as initial imperfections, plastic behavior, gaps, and large-deflection response. In addition, using deflection-controlled loading, you can even track the post-buckled performance of your structure (which can be useful in cases where the structure buckles into a stable configuration, such as "snap-through" buckling of a shallow dome), fig(1) show the Nonlinear Buckling curve . .[7]

Fig ( 1 ) Buckling Curves.[7]

1-3 Equations of Buckling Analyses

1-3-1 Euler’s Column Theory

It is based in the determination of the buckling load for long column and

struts. He derived an equation, for the buckling load of long columns based on the

bending stress neglecting the effect of direct stress. Direct stress in a long column is

negligible as compared to bending stress, Euler‟s formula can not be used in case of

short columns, because the effect of direct stress is considerable and hence can not be

neglected.

1-3-2 Euler’s Buckling load P for various

End Conditions. .[1]


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a .When both Ends fixed ( equivalent length L=2

l )

2

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l

EIP

…………………..( 1 )

E= Young‟s modulus for the column material.

I= least moment of inertia of the column section.

l Length of the column.

b. When one End fixed and Other Hinged ( equivalent length L= 2

l )

2

22

l

EIP

…………………( 2 )

c - When both Ends hinged ( equivalent length L=l ).fig( 2 ).

2

2

l

EIP

……………………..( 3 )

d. When on End fixed and the other free ( equivalent length L= l2 ) fig( 3 ).

2

2

4l

EIP

Fig( 2 ) Fig(3 )

y L= l

P

P

P F

L= 2 l


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1-3-2 Rankine Gorden Formula.

This formula is applicable for intermediate columns in which both the direct as well as buckling stresses are taken into account. From Rankine Gordon formula , the buckling or crippling load P is given by relation. .[1]

2

1

k

la

AfP c …………………..( 4 )

Where cf Crushing stress of the column material

A= Area of the column cross-section

a= Rankine‟s constant ( less than 0 ).

l Equivalent length of column

K= Least radius of gyration.

Note: „ a „ are generally adopted:

(i): Mild steel = 1/7500

(ii): Cast iron = 1/1600

1-3-3 Johnson’s Formula for safe load on

columns. .[1]

(i): Straight line formula for column

k

LnfAP …………………….( 5 )

(ii) Parabolic formula for columns.

2

k

LrfAP …………………..( 6 )

Where:


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P = Safe load on the column

f Stress in the column material.

A= Area of cross-section of column

n, r = Constant whose value depend upon the material of the column.


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2-Buckling of Compression Members

This experiment will investigate columns that are acted upon by compressive loads

and compare the results with the Euler buckling equation.

2-1 Test Procedure

1. Take all measurements of the bar to be tested and then place the test

specimen in the testing machine. Check to see if the bar is bowed to one side.

2. Connect the displacement measuring transducer and the force transducer to

the testing machine.

3. Some of the ends of the specimen are designed to simulate some fixed

support and free end so be sure that they are clean and well lubricated.

4. The testing machine will slowly apply either an increasing load at free end or

displacement to the bar. Which is it?

5. Slowly apply a compressive load while recording both displacement and

load. Carefully watch the bar to see when it starts to displace to the side.

Continue until the bar buckles. Note the maximum load and record it. Did it

displace in the direction that the bar was initially bowed?

6. Repeat for the other bars.

2-2 Practical Notes :

1. In general follow the specified format for an Abstract or as specified in class.

This means write no more than one half page of text, single spaced. You

should include: what you did; why you did it; what the results were, and a

discussion of those results, all in a half page.

2. For each bar compare the measured buckling load with that predicted by

Euler's equation:

3. Use E=as type of column material.

4. When the ends are designed to provide no moment, they should represent

simple supports and k is assumed to be equal to one. Is this a valid

assumption? Is there any moment at the ends?

5. Look up the values of k for the boundary conditions which you identify for

the other columns. Based upon the experimental data, calculate what might

be better values for k.

6. Put a title on the text page and on the graph, your name, and the school

name.

7. Explain any differences between experimental results and theoretical

predictions.

http://gaia.ecs.csus.edu/~ce113/ce113-report-format.html#abstract


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8. Plot load vs. displacement, with load on the y axis, and size this graph to fit

below the text on the same page.

9. Show the theoretical and experimental values for Pcr on the graphs. If there

are a lot of data points, do not plot symbols.

2-3 Device Use

Many types are used for Buckling test .

1- columns test type. 2- Buckling test of a thin-walled CFRP cylinder

fig( 4 ) show the column test type


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Fig ( 5 ): Buckling test of a thin-walled CFRP cylinder


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3- Introduction

This chapter includes the theoretical considerations of buckling phenomenon under

axial compression load with some values must be obtained , and also the analysis by the

finite element method.

Columns are a long vertical member or slender bar subjected to an axial compressive

load, and when the column is loaded and the load is continuously increased , a certain

value of load shall be reached when the column will just be slightly deflected or a little

lateral displacement will take place in it.

At this position the internal forces which tend to straighten the column are just equal

to the applied load. The minimum limiting load at which the column tends to have

lateral displacement or tends to buckle is known as buckling or crippling load.

Buckling takes place about the axis having minimum radius of gyration, or least

moment of inertia[1].

Safe load : It is the load to which a column is actually subjected and is well below the

buckling load. It is obtained by dividing critical load by factor of safety .

Long and short columns:

Sometime the columns, whose slenderness ratio is more then 80, are known as

long columns, and those whose slenderness ratio is less than 80 are known as short

columns.

Those columns whose slenderness ratio lies between 80 and 120 are known as medium

size or intermediate columns. And the long column will more than this ratio.[1]

Types End Conditions

A loaded column can have any one of the following four end conditions:

1. Both end hinged.

2. Both end fixed.

3. One end fixed and the other hinged.

4. One end fixed and the other free.


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Slenderness ratio: The ratio of the length of column to the minimum radius of gyration of the cross-

section area of the column is known as slenderness ratio as table ( 1 )

.

r

l

r

klRS e. ……………………( 7 )

Where:

S.R= Slenderness ratio.

Le: effective length, taking into account the manner of attaching the ends

(note that Le=KL).

r: smallest radius of gyration the cross section of gyration K= The value of "K" depends on how the ends of the column are secured, as shown in Fig.( 6 ). It should be noted that the values listed for "K" are based on the expected shape of the deflected column when buckling takes place.

Fig( 6 ) Values of K for effective length, Le=KL for different connections [ 8 ].

The estimation of slenderness for the slenderness ratio which eliminates the column

types such as (long, intermediate and short), table ( 1 ). These ratios are taken into

consideration in the calculation throughout this work .


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Table ( 1 ); slenderness ratio for columns for different materials (S.R= Le/r) [8 ].

Material

Short column

(strength limit )

Intermediate column

(Inelastic stability limit)

Long column (Elastic

stability limit)

Structural steel

S .R<40

40<S.R<150.

S.R>150

Aluminum Alloy

AA6061 -T6

S.R<9.5

9.5<S.R<66

S.R>66

Aluminum Ally

AA2014-T6

S.R<12

12<S.R<55

S.R>55

Wood

S .R< 11

11 <S.R< (18-30)

(18-30)<S.R<50

Assumption in Column Test: According to the theory of column formula is based on the following assumption

must taken during Column test (Buckling Test ):

1. The material of the column is perfectly homogeneous and isotropic.

2. The applied load is perfectly axial.

3. The column is initially straight and of uniform lateral dimensions.

4. the length of column is very large as compared to its cross-sectional dimensions.

5. limit of proportionality is not exceeded.

The One end fixed, the other free was taken in this project.

The fundamental mode of buckling in this case therefore is given when

2

nL

And

EI

PL

2

Or

2

2

4L

EIP

for theoretical Solution

3-1 Finite Elements

The finite element methods has a powerful tool of numerical solution of a wide range of engineering problems. It is used to calculate stress and temperature distribution in solid structures of complex geometries subjected to complicated loadings.[2] The basic ideas of the finite elements methods originated from advances in structural analysis in 1941, Hrenikoof [3] presented a solution of elasticity problems using the “ frame work method “ . Courant’s paper[4], which used it to solve model torsion problems appeared in 1943. turner..et al [5].derived


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stiffness matrices for plate ,truss, beam and other elements and presented their finding in 1956. In the late 1960 and early 1970 , finite element analysis was applied to nonlinear problems and large deformations. Oden’s Book [6]on nonlinear continuer appeared in 1972. Today , the developments in mainframe computers and availability of powerful microcomputer has brought this method within reach of engineers working.[2].

The finite element method is a numerical technique for solving problems which are described by partial differential equations or can be formulated as functional minimization. A domain of interest is represented as an assembly of finite elements. Approximating functions in finite elements are determined in terms of nodal values of a physical field which is sought. A continuous physical problem is transformed into a discretized finite element problem with unknown nodal values. For a linear problem a system of linear algebraic equations should be solved.

3-1-1 Element use [7].

BEAM3 fig ( 7 ) is a uniaxial element with tension, compression, and bending capabilities. The element has three degrees of freedom at each node: translations in the nodal x and y directions and rotation about the nodal z-axis. See BEAM3 in the ANSYS, Inc. Theory Reference for more details about this element. Other 2-D beam elements are the plastic beam (BEAM23) and the tapered unsymmetric beam (BEAM54).

Fig( 7 ) element Beam3 [7]

mk:@MSITStore:C:/Program%20Files/Ansys%20Inc/v100/CommonFiles/HELP/en-us/ansyshelp.chm::/thy_el3.html

mk:@MSITStore:C:/Program%20Files/Ansys%20Inc/v100/CommonFiles/HELP/en-us/ansyshelp.chm::/theory_toc.html

mk:@MSITStore:C:/Program%20Files/Ansys%20Inc/v100/CommonFiles/HELP/en-us/ansyshelp.chm::/Hlp_E_BEAM23.html



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BEAM3 Input Summary

Nodes

I, J

Degrees of Freedom

UX, UY, ROTZ

Real Constants

AREA - Cross-sectional area

IZZ - Area moment of inertia

HEIGHT - Total beam height

SHEARZ - Shear deflection constant

ISTRN - Initial strain

ADDMAS - Added mass per unit length

Note

SHEARZ goes with the IZZ. If SHEARZ = 0, there is no shear deflection in the element Y direction.

Material Properties

EX, ALPX (or CTEX or THSX), DENS, GXY, DAMP

Surface Loads

Pressure --

Body Loads

Temperatures -- T1, T2, T3, T4

3-2 Procedure for Eigenvalue Buckling Analysis

Again, remember that eigenvalue buckling analysis generally yields unconservative results, and should

usually not be used for design of actual structures. If you decide that eigenvalue buckling analysis is

appropriate for your application, follow this procedure:

1. Build the model.

2. Obtain the static solution.

3. Obtain the eigenvalue buckling solution.

4. Expand the solution.

5. Review the results.

3-2-1 Obtain the Static Solution

The procedure to obtain a static solution is the same as described in [ 7 ], with the following

exceptions:

Prestress effects must be activated. Eigenvalue buckling analysis requires the stress stiffness

matrix to be calculated.


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Unit loads are usually sufficient (that is, actual load values need not be specified). The

eigenvalues calculated by the buckling analysis represent buckling load factors. Therefore, if a

unit load is specified, the load factors represent the buckling loads. All loads are scaled. (Also,

the maximum permissible eigenvalue is 1,000,000 - you must use larger applied loads if your

eigenvalue exceeds this limit.)

Note that eigenvalues represent scaling factors for all loads. If certain loads are constant (for

example, self-weight gravity loads) while other loads are variable (for example, externally

applied loads), you need to ensure that the stress stiffness matrix from the constant loads is not

factored by the eigenvalue solution.

One strategy that you can use to achieve this end is to iterate on the eigensolution, adjusting the

variable loads until the eigenvalue becomes 1.0 (or nearly 1.0, within some convergence

tolerance). Design optimization could be useful in driving this iterative procedure to a final

answer.

Consider, for example, a pole having a self-weight W0, which supports an externally-applied

load, A. To determine the limiting value of A in an eigenvalue buckling solution, you could

solve repetitively, using different values of A, until by iteration you find an eigenvalue

acceptably close to 1.0.

3-3 Problem Description

Four type of materials were taken ( Steel ,Wood, Silicon, Aluminum ) under compressive load and the Buckling Phenomenon happened during testing. To Determine the critical buckling load with maximum and minimum displacement coordinate points of a column one fixed end and free other . The bar has a cross-sectional height h, and area A. Only the upper half of the bar is modeled because of symmetry. The boundary conditions become free-fixed for the half-symmetry model. The moment of inertia of the bar is calculated as I = Ah

2/12

problem sketch


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Introducing ANSYS:

ANSYS V10 ( Analysis System Software ) is a finite element analysis

software enables engineers to perform the following tasks :

• Build computer models or transfer CAD models of structures, products,

components or systems.

• Apply operating loads or other design performance conditions.

• Study physical responses, such as stress levels, temperature distributions, or

the impact of electromagnetic fields.

• Optimize a design early in the development process to reduce production costs.

• Do prototype testing in environments where it otherwise would be undesirable or

impossible ( for example biomedical applications ) .

Using The ANSYS V10 Program :

The ANSYS program is designed to run by three methods which are :

1- Interactive method, which is also called Graphical User Interface (GUI).

2- ANSYS Parametric Design Language (APDL).the advance methods in

programmable engineering design.

ANSYS Structural Analyses :

Structural analysis is probably the most application of the finite element method.

The term structural implies not only civil engineering structures such as bridges,

building and foundations, but also mechanical structures such as ship hulls, aircraft,

bodies and machine housings as well as mechanical components such as pistons,

machine parts and tools…etc.

• The procedure for a static structural analysis consists of three main steps:

1- Build the model and its meshing .

2- Apply loads and obtain the solution .

3- Review the results .

the type of problem involved, an ANSYS analysis consists of the same steps: modeling,

meshing, solution, and post processing.


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The modeling is drawing in a 2D or 3D representation of the problem.

During the meshing phase you will define material properties and choose an element

suitable for the problem.

The last step of the meshing phase is to discredited the model i.e. create the mesh.

In the solution phase, boundary conditions and loads need to be defined.

The types of loads and boundary conditions you select depend on the simplifications

being made.

then attempt to solve the system of equations defined by the mesh and boundary

conditions.

Finally, when the solution is complete, you will need to review the results using the post

processor. These results may be color contour plots, line plots, or simply a list of DOF

results for each node.

The easiest way to do and learn my problems is by using the pictures doing step – by –

step in ANSYS output :

Say we would like to analyze a column with different materials under axial load for

Buckling Phenomenon .


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4- Resultant View

Four test cases shown in this chapter for the same column made from deferent

materials ( Aluminum, Silicon, Wood, Steel ), column dimensions are :

Length= 101.6 along the Y-axis

Area= 6.35 X 6.35 square shape cross-section

Case 1 : 2024 Aluminum Alloy , E= 72 GPa, ν = 0.33

Theoretical Critical load = 2329 N Ansys Program = 2332 N

Fig( 8 ) : show the mode shape

Fig( 9 ) : show the Max , Min Deflection value


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Case 2 : Silicon , E= 2.5 N/ mm2, ν = 0.41

Theoretical Critical load = 0.080884 N Ansys Program = 0.080967 N




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Case 3 : Wood , E= 200 GPa , ν = 0.3

Theoretical Critical load = 6470.76 N Ansys Program = 6477 N




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Case 3 : Wood , E= 10000 N/ mm2, ν = 0.29

Theoretical Critical load = 323.53 N Ansys Program = 323.86 N




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5- Discussion & Conclusion :

Buckling has become more of a problem because collapse is unexpected and

there are few warning signs. When a structure have dimension ratio more than

1:3 ( D / L ) or ( compressive area to length of volume of spacemen ) and

subjected to a compression load undergoes visibly large displacements transverse

to the load, then it is said to buckle. Buckling may be demonstrated by pressing the

opposite edges of a flat sheet of cardboard towards one another. For small loads

the process is elastic since buckling displacement disappear when the load is

removed.

By using Euler’s Formula find no account of direct stress. as already stated,

direct stress in long columns or struts is small, which is of considerable importance

in the case of short or medium struts. This means that Euler’s formula may give a

buckling load for such struts far in excess of load which they can withstand under

direct compression. This formula is applicable to an ideal strut only and the loads

applied may not be exactly axial.

If the actual effective slenderness ratio "Le/r" is greater than "C", then the

column is long, and the Euler formula, defined in the next section should be used to

analyze the column. If the actual ratio "Lc/r" is less than "C" then the column is

short. In these cases, either tangent modulus theory, reduced modulus theory,

Johnson formula, or the direct compressive stress formula should be used, as

discussed in a later section.

Whenever a given column is being analyzed to determine the load it will carry, the

value of "C" and the actual ratio "Le/r" should be computed first to determine

which method of analysis should be used. Notice that "C" depends on the material

properties yield strength "ay" and modulus of elasticity "E". In working with

steel, "E" is usually taken to be (200-210 Gpa). Using this value and assuming a

range of values for yield strength, the values . for "C" shown in

A long column fails by buckling, whereas a short column fails by crushing, and

The strength of a column to resist buckling depends upon its:

1. Slenderness ratio


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2. End conditions of column.

In this chapter comparison between four cases( Aluminum , Silicon , Steel, Wood)

is presented , the comparison is based on analysis performed in Chapter four on

the specimen having the properties given in Table ( 2 )

Material Max Deflection Coordinates Min Deflection Coordinates

X-axis ( mm ) Y-axis ( mm ) X-axis ( mm ) Y-axis ( mm )

Aluminum

2024 Alloy

0.97 40.64 4.28 91.44

Silicon 1.48 50.8 2.09 60.96

Steel 4.28 91.44 2.09 60.96

Wood 4.28 91.44 0.97 40.64

Table (2 ) show the Max, Min Deflection Coordinates in column 101.6 mm

And if the test compared with each other we can find the following notes:

The max. point in Aluminum & Silicon is far from applied Force but in Steel and Wood

the max near to the applied load, this will given us the indicator about the Failure Point

and Broken location place.

5-1 Recommendations :

The test can be modified to study a columns under high speed rotation.

Study of lateral Buckling under combined loading.

Study Buckling for piping for deference in-out diameters.

Study Buckling for slender with holes.

Study Buckling for a columns of composite materials.

Measure the Buckling Strain rate using strain gauges.

Study Creep - Buckling combination Phenomenon.


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5- 2 References :

[1]: Mathur S.B “ Mechanical Engineering “ Dhanpat Rat & Sons, 1985.

[2]: Tim Langlais , “ ANSYS Short Course “, [email protected]

August 16, 1999. from internet .

[3]: Hrenikoff,A.,” Solution of problems in elasticity by the frame work method”. J. of

Appl. Mech.,8: 169-175 (1941).

[4]: Courant, R. “ Variation methods for the solution of problems of equilibrium and

vibrations “, Bulletin of American Mathematical Society, 49:1-23 (1943) .

[5]: Turner ,M.J.,R.W.clough,H.C.Martin and L.J. Topp “ stiffness and deflection

analysis of complex structures” J. of Aero. Science 23(9):805-824(1956).

[6]: Oden,J.T. “ Finite elements of Nonlinear Continuer “ ,New York, McGraw-Hill,

1972.

[7] ANSYS, Inc. “Ansys Manuals Ver.10” Elements Reference, .

www.ANSYS.com,2006.

[8] Al-jubori H. Kifah “ columns Lateral Buckling Under Combined Dynamic Loading”

Ph.D thesis, University of Technology,Baghdad,2005.

http://www.ansys.com/


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Appendix (A)

7.5. Sample Buckling Analysis (GUI Method)

In this sample problem, you will analyze the buckling of a bar with hinged ends.

7.5.1. Problem Description

Determine the critical buckling load of an axially loaded long slender bar of length

with hinged ends. The bar has a cross-sectional height h, and area A. Only the upper

half of the bar is modeled because of symmetry. The boundary conditions become free-

fixed for the half-symmetry model. The moment of inertia of the bar is calculated as I =

Ah2/12

7.5.2. Problem Specifications

The following material properties are used for this problem:

E = 72 GPa Poisson Ratio= 0.33

The following geometric properties are used for this problem:

L = 101.6mm

A = 6.35 x 6.35 mm2

h = 6.35 mm

I = Ah2/12

Loading for this problem is:

F = 1 lb. Unite Force

7.5.3. Problem Sketch

7.5.3.1. Set the Analysis Title

After you enter the ANSYS program, follow these steps to set the title.

1. Choose menu path Utility Menu> File> Change Title.

2. Enter the text "Buckling of a Bar with Hinged Ends" and click on OK.


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7.5.3.2. Define the Element Type

In this step, you define BEAM3 as the element type.

1. Choose menu path Main Menu> Preprocessor> Element Type>

Add/Edit/Delete. The Element Types dialog box appears.

2. Click on Add. The Library of Element Types dialog box appears.

3. In the scroll box on the left, click on "Structural Beam" to select it.

4. In the scroll box on the right, click on "2D elastic 3" to select it.

5. Click on OK, and then click on Close in the Element Types dialog box.

7.5.3.3. Define the Real Constants and Material Properties

1. Choose menu path Main Menu> Preprocessor> Real Constants>

Add/Edit/Delete. The Real Constants dialog box appears.

2. Click on Add. The Element Type for Real Constants dialog box appears.

3. Click on OK. The Real Constants for BEAM3 dialog box appears.

4. Enter values for area, for IZZ, and for height.

5. Click on OK.

6. Click on Close in the Real Constants dialog box.

7. Choose menu path Main Menu> Preprocessor> Material Props> Material

Models. The Define Material Model Behavior dialog box appears.

8. In the Material Models Available window, double-click on the following options:

Structural, Linear, Elastic, Isotropic. A dialog box appears.

9. Enter value for EX (Young's modulus), and click on OK. Material Model

Number 1 appears in the Material Models Defined window on the left.

10. Choose menu path Material> Exit to remove the Define Material Model

Behavior dialog box.

7.5.3.4. Define Nodes and Elements

1. Choose menu path Main Menu> Preprocessor> Modeling> Create> Nodes>

In Active CS. The Create Nodes in Active Coordinate System dialog box

appears.

2. Enter 1 for node number.

3. Click on Apply. Node location defaults to 0,0,0.

4. Enter 11 for node number.

5. Enter 0,L,0 for the X, Y, Z coordinates.

6. Click on OK. The two nodes appear in the ANSYS Graphics window.




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Note

The triad, by default, hides the node number for node 1. To turn the triad off,

choose menu path Utility Menu> PlotCtrls> Window Controls> Window

Options and select the "Not Shown" option for Location of triad. Then click OK

to close the dialog box.

7. Choose menu path Main Menu> Preprocessor> Modeling> Create> Nodes>

Fill between Nds. The Fill between Nds picking menu appears.

8. Click on node 1, then 11, and click on OK. The Create Nodes Between 2 Nodes

dialog box appears.

9. Click on OK to accept the settings (fill between nodes 1 and 11, and number of

nodes to fill 9).

10. Choose menu path Main Menu> Preprocessor> Modeling> Create>

Elements> Auto Numbered> Thru Nodes. The Elements from Nodes picking

menu appears.

11. Click on nodes 1 and 2, then click on OK.

12. Choose menu path Main Menu> Preprocessor> Modeling> Copy> Elements>

Auto Numbered. The Copy Elems Auto-Num picking menu appears.

13. Click on Pick All. The Copy Elements (Automatically-Numbered) dialog box

appears.

14. Enter 10 for total number of copies and enter 1 for node number increment.

15. Click on OK. The remaining elements appear in the ANSYS Graphics window.

7.5.3.5. Define the Boundary Conditions

1. Choose menu path Main Menu> Solution> Unabridged Menu> Analysis

Type> New Analysis. The New Analysis dialog box appears.

2. Click OK to accept the default of "Static."

3. Choose menu path Main Menu> Solution> Analysis Type> Analysis Options.

The Static or Steady-State Analysis dialog box appears.

4. In the scroll box for stress stiffness or prestress, scroll to "Prestress ON" to select

it.

5. Click on OK.

6. Choose menu path Main Menu> Solution> Define Loads> Apply> Structural>

Displacement> On Nodes. The Apply U,ROT on Nodes picking menu appears.

7. Click on node 1 in the ANSYS Graphics window, then click on OK in the picking

menu. The Apply U,ROT on Nodes dialog box appears.

8. Click on "All DOF" to select it, and click on OK.

9. Choose menu path Main Menu> Solution> Define Loads> Apply> Structural>

Force/Moment> On Nodes. The Apply F/M on Nodes picking menu appears.


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10. Click on node 11, then click OK. The Apply F/M on Nodes dialog box appears.

11. In the scroll box for Direction of force/mom, scroll to "FY" to select it.

12. Enter -1 for the force/moment value, and click on OK. The force symbol

appears in the ANSYS Graphics window.

7.5.3.6. Solve the Static Analysis

1. Choose menu path Main Menu> Solution> Solve> Current LS.

2. Carefully review the information in the status window, and click on Close.

3. Click on OK in the Solve Current Load Step dialog box to begin the solution.

4. Click on Close in the Information window when the solution is finished.

7.5.3.7. Solve the Buckling Analysis

1. Choose menu path Main Menu> Solution> Analysis Type> New Analysis.

Note

Click on Close in the Warning window if the following warning appears:

Changing the analysis type is only valid within the first load step. Pressing OK

will cause you to exit and reenter SOLUTION. This will reset the load step count

to 1.

2. In the New Analysis dialog box, click the "Eigen Buckling" option on, then click

on OK.

3. Choose menu path Main Menu> Solution> Analysis Type> Analysis Options.

The Eigenvalue Buckling Options dialog box appears.

4. Click on the "Block Lanczos" option, and enter 1 for number of modes to extract.

5. Click on OK.

6. Choose menu path Main Menu> Solution> Load Step Opts> ExpansionPass>

Expand Modes.

7. Enter 1 for number of modes to expand, and click on OK.

8. Choose menu path Main Menu> Solution> Solve> Current LS.

9. Carefully review the information in the status window, and click on Close.

10. Click on OK in the Solve Current Load Step dialog box to begin the solution.

11. Click on Close in the Information window when the solution is finished.

7.5.3.8. Review the Results

1. Choose menu path Main Menu> General Postproc> Read Results> First Set.

2. Choose menu path Main Menu> General Postproc> Plot Results> Deformed

Shape. The Plot Deformed Shape dialog box appears.


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3. Click the "Def + undeformed" option on. Click on OK. The deformed and

undeformed shapes appear in the ANSYS graphics window.

7.5.3.9. Exit ANSYS

1. In the ANSYS Toolbar, click on Quit.

2. Choose the save option you want and click on OK.

Study the Crippling Load in Buckling Analysis- Hani Aziz Ameen

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Transcript of Study the Crippling Load in Buckling Analysis- Hani Aziz Ameen