Web viewλr = obtained from AISC manual depending on the beam is compact section or non compact section. λ = Width/ Thickness ratio

CIE 633

FINITE ELEMENT ANALYSIS

PROJECT

BEAM ANALYSIS

BY

Kalpesh Parikh

1

Acknowledgment

My deepest gratitude goes to Dr. Richard Wilson Perkins for his continuous and constructive advice and

follows up. His successive advisories and comments were the pillars in my every step during the Project.

I am thankful to him for the fact that he has inspired and helped me to know about the Finite Element

Analysis.

2

Index

1. Introduction 04

2. Buckling Analysis 05

3. Modal Analysis 24

4. Transient and Dynamic Analysis 29

5. Conclusion 35

-Reference 36

- Appendix 37

3

1 Introduction

The Project analysis is done in the software called ANSYS .Basically Software takes all the Inputs and it has in its memory all the solver which are required to be carried out. So I basically started using an W shape section (I section) and here i took structural components as a Beams. Where I studied the various behavior of the structure in order to understand that when we design some structural component we should be clear with the behavior of the structure. So that we don’t land up to an failure of the structure.

General Information

A wide flange structural steel beam W10 x 26 (10 means deep and 26 is weight of the section lb/ft)

Material properties

• ASTM Designation: A992 Steel Section• Modulus of Elasticity: E =30000 Ksi • Poisson ratio =0.3• Density of Steel= 490 lb/ft3• Yield Strength Fy = 50 ksi • Tensile Strength Fu = 65 ksi

Cross sectional Properties

Flange Thickness (tf) = 0.44 inch Web Thickness (tw) = 0.3 inch Flange width (bf) = 5.75 inch Depth of section (d) = 10.3 inch Moment of Inertia in z-z direction (Izz) = 14.1 inch4

Moment of Inertia in y-y direction (Iyy) = 144 inch4

4

2. Buckling Analysis

Buckling loads are critical loads where certain types of structures become unstable. Each load has an associated buckled mode shape; this is the shape that the structure assumes in a buckled condition. There are two primary means to perform a buckling analysis:

1. Eigenvalue

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.

2. Nonlinear

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.

Study of Buckling Analysis and its comparison with various boundary condition and loading condition:

1 Cross section used W10 x 26 as defined.

2 Elements used is Beam 188, Beam 189, Beam 4, and Shell element 63 and Shell element 93

3 Study of Eigenvalue Analysis.

Model Descriptions

1. Beam 4 Element was model using the cross sectional properties and followed and two key points were used to define the length of the beam.. Boundary condition and loading condition was assigned. Analysis is Buckling so general post processor was defined. As per the required analysis

2. Beam 188 element and Beam 189 was model using standard SECDATA and SECTYPE and then I assigned the sectional properties to the command, four key points were used to define the length and width of the beam.. Boundary condition and loading condition was assigned. Analysis is Buckling so general post processor was defined. As per the required analysis

3. Shell 63 and Shell 93 element was modeled using real constant and modeling I beam is not so proper as it goes into it as shown below (Area overlapped)

5

These is only model imperfection involved in the shell element as it makes shell element more stronger. Apart from that the main think is to define loading at various position in order to get proper result so as it is made up of three part web, flange and Flange I applied loading at all three position and at the tip of the flange at node 1 and node 2 so in order to get buckled the structure.

The Euler buckling load for a cantilever beam length L=120 inch, with the above material and cross sectional properties is given by

When L = 120

Pcr = π 2∗E∗Izz

(2L )2 =72.47 kips

When L = 60


(2L )2 = 289.91 kips

Case 1 Cantilever Beam Subjected Axial Load

When Length is 120 inch Using Various Element Beam 188, Beam 189, Beam 4, Shell Element 63 and Shell 93 Element. Analysis – Eigenvalue Analysis

Buckling Loads obtained using Set list when simulated

Element Theoretical Pcr

in KipsAnsys Pcr in

Kips

% of Comparison of Theoretical with

AnsysBeam 4 72.47 72.48 100.0137988Beam 188 72.47 71.635 98.84779909Beam 189 72.47 71.635 98.84779909Shell 63 72.47 71.994 99.34317649Shell 93 72.47 71.523 98.69325238

Study of Variation in Length


6

Comparision Table showing Theoretical value and Ansys Eulers Buckling Load


in KipsAnsys Pcr in

Kips



Interpretation

The results form ANSYS are almost the same as the theoretical buckling load obtained using Euler Formula.

Below Figures shows the buckled shape of the cantilever beam for the BEAM4, BEAM188, BEAM189,

SHELL63 and SHELL93 models respectively.

The deformed shapes in all the cases (where the beam is modeled using the beam and shell elements), clearly

agree with the mode shape that would result if a first mode Euler buckling is to occur. It is also important to

note from the buckling shapes of BEAM188, BEAM189, SHELL63 and SHELL93 that the buckling occurs

about the weak axis.

Distribution of load helped to identify the correct buckling value and behavior of the buckling which was

required s it is important how we apply loads in the beam to get correct buckling.

As I Tried to study the variation of the length in the beam I reduced my length to 50% and noted the result the

result are not changing as it has nothing to do with geometric propertie s , apart from how you apply loads.

Figure Buckling for Beam 4

7

Figure Showing Buckling for Beam 188 and Beam 189

Figure Showing Buckling for Shell 63 and Shell 93

Case 2 Simply Supported Beam Subjected to axial Load:

The Modeling Procedure remains same I tried all 5 element that are described for cantilever beam to study the

Eigenvalue buckling the model built up procedure remains same only changes would be boundary conditions it was

really painful to understand the behavior when I inserted the boundary condition as a simply supported then I found

in the textbook Structural Stability by Dr. Eric M Lui and Chen in that boundary conditions I need to apply (ROTX)

need to be restrained at one end of the beam in order to give structural stability. So now the behavior of the structure

8

were quiet reasonable. So I applied restrained on the translation (Ux, Uy, Uz in node 1 and Uy in node 2 ) and made

restrained in (Rotx) then simulate as using the post processor for Eigen value buckling.

The Euler buckling load for a simply supported

When L = 120


( L )2 = 289.91 kips

Where E = 30000 ksi

Izz = Moment of Inertia about weak axis =14.1 inch4

Study of Eigenvalue Using simply supported boundary condition




in KipsAnsys Pcr in

Kips



Interpretation:

1. Beam 4 give same result as we obtained by theoretical value reason behind these is there is no local buckling. Now the question comes what is local buckling generally it is defined as λ> λr Local Buckling will occur in the beam λr = obtained from AISC manual depending on the beam is compact section or non compact section

λ = Width/ Thickness ratio (We can determine Flange value as well as web value)

2 Beam 188 gives bit off result reason is that there is a local buckling effect taking place in the flange portion and the web portion of the beam. So we can interprete from that generally the local buckling takes place if the section is assembled and it is general effect is under when we use high yield strength steel so these clear says that it is not adequate using beam188 element and same apply to beam 189 element as these also shows the Local buckling in the web and flange portion.

9

3 Mode shape difference are found when I changes the number of element so it signifies the importance of the element size if we want to study the behavior of mode shape.

4 As Shell 63 and Shell 93 value is to high as I signifies that shell element would not be an good idea to study the exact behavior of the buckling as it has model imperfection as web goes into the flange portion so when we study the Eigen value buckling it requires more load to buckle because of its strong action in the model. As always buckling occurs about the weak axis where there is an Inertia value is lesser so in Shell63 and Shell 93 in both Local buckling take place in the web and the flange portion.

5 When we calculate the stiffness of the beam for shell 93 an shell 63 we can find that the stiffness is better of shell 93 and shell 63 apart from that when we study the effect of Stress in both the beam during the buckling Mode we can see There are more stresses in the shell 63 compare to shell 93 element.

6. The behavior using simply supported boundary condition is very unusual as the constraining of model is an big challenge to get the adequate results.

7. For Practical purpose when we do the execution of steel beam the behavior of shell element would more be given focus as in actual high rise building we always use High strength steel where the yield strength and local buckling is always there compare to low yield strength so we can say if you have more yield strength we have more chances of local buckling taking place in the structure.

8. Study of variation in length was made only the percentage of accuracy of the ansys remains same so I did not tabulated the value for it so there is no percentage changes in the ansys.

CASE 3: FIXED – PINED END CONDITIONS


Eigen value buckling the model built up procedure remains same only changes would be boundary conditions

become Translation in x,y and z were restrained and other end I made restrained to all the degree of freedom. And

following same repetitive steps for general post processors.

The Euler buckling load when boundary condition is fixed and Pinned end


(0.7∗L )2 = 591.67 kips

Izz =Weak axis of moment of inertia (Why I can say that only z-z axis is weak in moment of inertia it is very much clear from the concept of mechanics of materials that bending takes place in the z-z axis ).

Here study is made on the all the elements used to compare theoretical eigenvalue buckling with Ansys so that we can understand that when boundary condition changes what are the changes.

Study of Eigenvalue Using Fixed and Pinned boundary condition


10


Element Theoretical Pcr in Kips

Ansys Pcr in Kips

% of Comparison of Theoretical with Ansys

Beam 4 591.67 594 100.39Beam 188 591.67 574.51157 97.1Beam 189 591.67 574.51157 97.1Shell 63 591.67 829.048004 140.12Shell 93 591.67 1192.451718 201.54

Interpretation

1 When Beam 4 Element used the results are perfect as per the theoretical value. I did not notice indication of

local buckling in the flange and web portion of the beam.

2 Results are same as obtained in the simply supported beam there is a local buckling effect in the shell element

and beam element local buckling taking place in the flange an d web portion .same case as it makes sense

because I have modified boundary conditions and obtained the result.

3 Shell element showing higher value as compared to theoretical reason is same as mentioned in the simply

supported beam.

4 Variation of Length is being studied the percentage of the accuracy for ansys does not changes.

Study of Lateral Torsional Buckling of beam

Lateral torsional buckling occurs when the distance between the lateral brace points is large enough that the beam

fails by lateral , outward movement in combination with a twisting action (∆ and Ѳ respectively. Generally beam

with wider flanges provide more resistance to lateral displacement . In general adequate restraint against lateral

torsional buckling is accomplished by the addition of the brace or similar restraint somewhere between the centroid

of the member and the compression flange.

The Euler Buckling Load formula obtained from Timeshenko and Gere

When Length = 120 inch

Pcr = 4.013∗E

L2 ∗√ Izz∗J2 (1+V )❑

=17.45 kips

Izz = Weak axis of moment of Inertia.

J= torsional value for the given section

11

When Length =60 inch

Pcr = 4.013∗E

L2 ∗√ Izz∗J2 (1+V )❑

=69.82 kips

Case 4 : Study of Lateral torsional Buckling of Cantilever beam

Modeling Procedure


Eigen value buckling the model built up procedure remains same only changes would be boundary Now the

loading was applied laterally to beam and the Post Processor remains same and eigen value analysis is being

performed.

When Length is 120 inch Using Various Elements Beam 188, Beam 189, Beam 4, Shell Element 63 and Shell 93 Element. Analysis – Eigen value Analysis. Load Applied Laterally to the beam.



in KipsAnsys Pcr in

Kips


AnsysBeam 188 17.45 17.101 98Beam 189 17.45 17.101 98Shell 63 17.45 25.4072 145.6Shell 93 17.45 32.9805 189

Study of Variation in Length




in KipsAnsys Pcr in

Kips


AnsysBeam 188 69.82 68.4236 98Beam 189 69.82 68.4236 98

12

Shell 63 69.82 108.221 155Shell 93 69.82 131.9598 189

Interpretation

1. Result obtained for Beam 188 and Beam 189 are near to the theoretical value which makes sense. And the

behavior shown in the ansys matches the theoretical behavior of the beam means lateral displacement in

combination with twisting show exact behavior of the lateral torsional buckling.

2. Shell 63 and Shell 93 elements are of concern because of the Model imperfection that is the Area

overlapped changes the ansys behavior as the K matrix changes means when we calculate the stiffness of

the beam using rotation and torsion the overlapping of the elements add to an more result hence the results

are higher as we can get the same result as theoretical value required.

3. Study of length variation is being carried out and result are same as expected by theoretical nothing

changes so effect of length variation does not changes the behavior of the structure.

4. Below figure shows the behavior of the lateral torisonal buckling we can notice the lateral displacement in

combination with twisting.

Figure Showing Buckling Using Lateral Load –Beam 188

13

Figure Showing Buckling Using Lateral Load –Beam 189

Figure Showing Buckling Using Lateral Load –Shell 63

Figure Showing Buckling Using Lateral Load –Shell 93

Study of effect of holes in the cantilever beam

Modeling Procedure

14

The beam is modeled using shell 63 and shell 93 element and the six holes where made into the beam and then the basic post processor would be Eigenvalue Analysis. Here two variation were made . a case of when subjected to axial load and when subjected to lateral load.

Case 5 Effect of Holes in cantilever beam using shell 63 element and shell 93 element when subjected to axial load

When Length is 120 inch Using Various Element Shell element 63 and Shell 93 element. Analysis – Eigen value Analysis. Six hexagonal holes ( h= 4 inch)

Comparison of Shell Element when subjected to axial load with and without effect of holes of ansys value

Element Ansys Pcr With Hole

Ansys Pcrwithout Hole Comments

Shell 63 65.55 71.994 Results are Better when used Shell 93Shell 93 69.051 71.523

Interpretation

1. In cantilever beam when holes are made then its euler’s buckling load decreases for both the Shell element.2. Shell 93 elements results are better because we can see that effect of holes has not taken much effect in the

buckling load.3. We can notice the behavior of the buckling for the shell 93 element as shown below

Figure showing buckling behavior of shell 93 element with and without hole when subjected to axial load

Case 6 Effect of Holes in cantilever beam using shell 63 element and shell 93 element when subjected to Lateral Load

When Length is 120 inch Using Various Element Shell element 63 and Shell 93 element.

15

Analysis – Eigen value Analysis. Six hexagonal holes ( h= 4 inch) Loading is lateral load

Comparison of Shell Element when subjected to lateral load and study the effect of with and without effect of holes of ansys value

Element Ansys Pcr With Hole

Ansys Pcrwithout Hole Comments

Shell 63 25.41 25.4072 Results were suprising i got better result for shell 63 but cannot interpreteShell 93 31.98 32.98

Interpretation

1. In cantilever beam when holes are made then its euler’s buckling load Shell 63 elements I was not able to interprete it

2. results are better because we can see that effect of holes has not taken much effect in the buckling load.3. We can notice the behavior of the buckling for the shell 63 element as shown below

Figure showing buckling behavior of shell 63 element with and without hole when subjected to lateral load

Case 7 Effect of Number of Modes

Model Description

Here i tried element Beam 189 and made a model of cantilever beam subjected to axial loading model was made in same fashion as previously describe for the cantilever beam but now i made variation in the method as subspace method and Block lancoz method and changing number of modes to 3,4 and 5 and try to study the behavior due to change in the modes. Reason of trying with one element is to notice the difference is coming for one element then it should come with other element also

16

When Length is 120 inch Using Element Beam 189 Analysis – Eigen value Analysis. Loading is Axial load.

Comparison of number modes and two different method for eigenvalue analysis

Element No. of ModesAnsys Pcr using Subspace

methodAnsys Pcr using Block

Lancoz method

Beam 189

3 71.635 71.6354 71.635 71.6355 71.635 71.635

Interpretation

1 Compared and obtained result that there is no changes in the eigenvalue due to change in number of modes. Behavior for pldisp,1 remains same if I simulate with 3 modes or 4 modes or 5 modes.

2 Even I tried to change the method to notice some difference but both the method gave same result so we can say these the method which we solve will play importance in the modal analysis and that to we can add some equation solver to see its effect to change in mode shape. So buckling there is no significance changing methods for analysis.

Figure showing buckling behavior of Beam 189 element when subjected to Axial load using both method

Case 8 Effect of Shift Variables

Model Description

Here i tried element Beam 188 and made a model of cantilever beam subjected to axial loading model was made in same fashion as previously describe for the cantilever beam but now i made variation in the method as subspace method and Block lancoz method and changing number of Shifts in the command to 0,0.5 and 1 and try to study the behavior due to change in the modes. Reason of trying with one element is to notice the difference is coming for one element then it should come with other element also

When Length is 120 inch Using Element Beam 188 Analysis – Eigen value Analysis. And notice the difference in behavior due to shift variable Loading is axial load.

Comparison using Shift variable and two different method for eigenvalue analysis

17

Element Shift

VariableAnsys Pcr using Subspace

methodAnsys Pcr using Block

Lancoz method Comments

Beam 188

0 71.635 71.635Effect of Shift

Variable is notice in Subspace

Method

0.5 218.34 71.635

1 218.34 71.635

Interpretation

1. When I command I tried with changing in the shift variables using block lancoz method I did not notice any changes in the buckling value.

2. But when I tried with subspace method I notice the difference when I put the shift variable as 0.5 the eigenvalue changes and it takes more load to buckle.

3. What is shift variable when we plot a graph of load vs displacement shift variable is assigned using symbol (λ) we can see below how it shifts the value.

4. Genrally effect of shift variable is due to complexity in the models.

Non Linear Analysis

Model description

Cantilever beam using all 5 element type was used to compute the eigenvalue using Non linear analysis for that model remains same only the post processor changes. The difference is that we put the Euler’s Buckling Load and we factored it by two times and then we put that into the command and monitor the graph and notice the displacement.

The main modeling is how we apply lateral load

Element Apply Load Weak axis Plot DisplacementBeam 4 Y direction small lateral

LoadZ-Z axis is weak axis Uy

Beam188 Z direction small lateral load

Y-Y axis is weak axis Uz

Beam 189 Z direction small lateral load


Shell 63 Z direction small lateral load


Shell 93 Z direction small lateral load


18

Modeling of the model all different elements was challenge it all depends on the orientation how we place the beam but from mechanics of material we know that the bending axis is z-z it is the weakest axis.

Method used here is Newton Raphson Method

Comparison of Eigenvalue obtained from Set list using non linear approach

Element

Theoretical Pcr

in KipsAnsys Pcr in

Kips



Figure Showing Plot of Load vs deflection for Beam 4

19

Figure Showing Plot of Load vs deflection for beam 189

Figure Showing Plot of Load vs deflection for beam 188

Figure Showing Plot of Load vs deflection for Shell 63 Element

20

Figure Showing Plot of Load vs deflection for Shell 93 Element (Uz)

Interpretation

1. 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.

2. As we can see from the graph of shell element and beam element the value which I obtained theoretically from euler’s buckling load as the same load we can see that there is an buckling indication (means displacement is noticed in the graph).

3. The value which I obtained from eigenvalue for shell element where to high but it is figured out in the non linear analysis as it does not account much for model imperfection it gives better result compare to eigenvalue. Almost wecan read from the graph the value at which the buckling starts and afterwards it gives high displacement.

Displacement figure during buckling

21

Figure showing Displacement at time Buckling Figure showing Displacement at time Takes Place – For Beam 188 (Uz) Buckling takes place- For Beam189 (Uz)

Figure showing Displacement at time Buckling Figure showing Displacement at time Takes Place – For Shell 63 (Uz) Buckling takes place- For Shell 93 (Uz)

Stresses Study

22

It was interesting to study stresses when the Buckling takes place for that I used the same model cantilever beam using all the 5 element type when we do nonlinear analysis model the point where the buckling starts I noted the stresses at that point where I went through all the stresses all 14 type of stresses I found the maximum among which gave the stresses for beam was Von Mises Stress.

Figure showing Stresses at time Buckling takes Figure showing Stresses at time Buckling takes Place – For Shell 93 (von mises Stress) Place- For Shell 63 (Von mises Stress)

Figure showing Stresses at time Buckling takes Figure showing Stresses at time Buckling takes Place – For Beam 188 (von mises Stress) Place- For Beam 189 (Von mises Stress)

Interpretation

23

When Stresses study are made it is important to know which one gives maximum stresses in the case where I find

the stresses for the Beam element an d shell element what I observe that maximum stresses we can see from figure .

The Comparison between element we do then the Shell element gave more stresses in von mises and where as stress

by beam element was less on the comparison.

3 Modal analysis24

Modal Analysis is the study of dynamic properties of structures under vibrational excitation . The main purpose to study modal analysis in the beam is to understand it behaviors.

We can determine Natural Frequency, Mode Shapes and Mode participation factors means how much a given mode participates in a given direction.

Basically I followed Method and try to obtained result for both the method

1. Subspace Method2. Reduced Method

Basic Information about model which I used

Material properties

• Modulus of Elasticity: E =30000 Ksi • Poisson ratio =0.3• Density of Steel= 490 lb/ft3


Thickness in Y direction (ty) = 0.44 inch Thickness in Z direction (tz) = 0.3 inch Moment of Inertia in z-z direction (Izz) = 14.1 inch4


Area of Cross section = 7.61 inch2

Study of Modal Analysis and its comparison with various Method

1 Cross section used as defined.

2 Elements used is Beam 4 and Mass 21

3 Study of Natural Frequency.

Model Descriptions

1. Beam 4 Element using Subspace method was model using the cross sectional properties and followed and two key points were used to define the length of the beam..which is L=120 inch.

2. I use a case of Beam as a Cantilever beam subjected with two concentrated masses in order to carry out the modal analysis it is important to find the weight of the beam and Moment of inertia for the masses. Below show the calculation for the beam carried out which was given in the command

E= 30000 ksi

nu =0.3

Gfactor=1

Gmod= Gfactor*E/(2*(1+nu)) = 1*30000/(2*(1+0.33)) =11278.196

25

g =386

A = 7.61 inch2, Iyy =144 inch2 Izz=14.1 inch2

Specific Gravity = Weight/ Force = Iron density/ Water Density = 7850/1000 =7.85

Wt density = 7.85 *62.4/1728 =0.28167

ρ= wt.density/g = 0.28167/386 =7.29 *10-4

Wt. Beam = ρ*A*g =7.29 * 10-4 *7.61*386 = 2.143

Mass m1 = 0.25/g……………… Assumed

Mass m2 = 0.25/g………………. Assumed

Calculation of Moment of inertia

Z1= m1 (x12 + y1

2)/12, Z2= m2 (x22 + y2

2)/12

Im1= Z1 +m1*(x1/2)2 Im2= Z2 +m2*(x2/2)2

X1 = 2, Y1 = 1, X2 = 2, Y2= 1………….Assumed

3. Element use to define mass was Mass 21 element and I assign the location at the tip and the mid node of beam 4 element. Mass 21 element was created using real constant

4. Now its time to add the post processor to obtained the desired result so I used subspace method to simulate my first result to determine the natural frequency and obtain the mode shape.

Theoretical Calculation of Natural Frequency obtained from the Structural dynamics by Roy R craig.

ώ n = αn2 √(EI/mL4)

αn = 1.875 ………..To obtain mode 1

αn = 4.694…………To obtain mode 2

αn = 7.844…………To obtain mode 3

(Dependent on *Mass,*Length and *Moment Inertia)

Total Mass = (Mass of Beam + Mass of m1 and Mass of m2) = (2.643/g)

Length =120 inch

Izz= Weak axis of moment of inertia from mechanics of materials = 144.1 inch4.

Ansys Model

26

Figure Showing Beam 4 element

used and obtained-Mode 1 and Mode 2 subspace method

Table Showing Comparision of Natural Frequency of Theory with Ansys when use subspace method

ElementMode

Theory Natural Frequency

Ansys Natural Frequency % Ansys Accuracy

Beam 4

1 5.2683 5.1 96.812 35.08068 33.96 96.813 245.5623 238.41 97.09

Interpretation

1. Result obtained from the ansys for natural frequency is not so accurate may be I am missing some concept.

2. But it is closer by from here onwards we can know add forces and notices the behavior and notice the transient dynamic analysis for the same model.

Case 2 Using Reduced Method

1 Cross section used as defined.

2 Elements used is Beam 4 and Mass 21

3 Study of Natural Frequency

Model Descriptions

27

1. Beam 4 Element using Reduce method was model using the cross sectional properties and followed and two key points were used to define the length of the beam. which is L=120 inch.

2. Only difference is that now we add Master degree of freedom to the beam and use the post processor for reduced method. . These are degrees of freedom that govern the dynamic characteristics of a structure.

3. Rest everything remain same as we have defined earlier. And let us find the result

Figure Showing Beam 4 element

used and obtained-Mode 1 and Mode 2 by Reduced method

Table Showing Comparision of Natural Frequency of Theory with Ansys when use Reduced method

ElementMode

Theory Natural Frequency

Ansys Natural Frequency % Ansys Accuracy

Beam 4

1 5.2683 5.1 96.812 35.08068 33.96 96.813 245.5623 238.41 97.09

Interpretation

28

1. Result obtained from the ansys for natural frequency is not so accurate may be I am missing some concept.

2. But it is closer by from here onwards we can know add forces and notices the behavior and notice the transient dynamic analysis for the same model.

3. Result are dot same for both the method reason there is not so complexity in the model the model is not so complex if model becomes then we can notice the difference in the reduced method.

4. It is better idea if in the execution we have complex structure it is better from the Ansys to calculate using reduced method obtained the analysis.

5. Subspace method and reduced method are frontal solver. But Subspace method is much slower then reduced method.

4 Transient Dynamic Analysis

29

Transient dynamic analysis is a technique used to determine the dynamic response of a structure under a time-varying load. The time frame for this type of analysis is such that inertia or damping effects of the structure are considered to be important. Cases where such effects play a major role are under step or impulse loading conditions. For my case, we will impact the end of the beam with an impulse force and view the response at the location of impact.

Figure Showing Example how I will apply impulse force at certain time and release after certain time

Since an ideal impulse force excites all modes of a structure, the response of the beam should contain all mode frequencies. However, we cannot produce an ideal impulse force numerically. We have to apply a load over a discrete amount of time dt

After the application of the load, we track the response of the beam at discrete time points for as long . The smaller the time step, the higher the mode frequency we will capture.

The main Fundamental is

Time = (1/No of discrete point * Highest mode of frequency)

Material properties

• Modulus of Elasticity: E =30000 Ksi • Poisson ratio =0.3• Density of Steel= 490 lb/ft3


30

Thickness in Y direction (ty) = 0.44 inch Thickness in Z direction (tz) = 0.3 inch Moment of Inertia in z-z direction (Izz) = 14.1 inch4


Area of Cross section = 7.61 inch2

Method used

The Full Method

All types of non-linearity are allowed. It is dependent on spacing in the disk to use full system matrices. This is the easiest method to use.

The Mode Superposition Method

This method requires a preliminary modal analysis, as factored mode shapes are summed to calculate the structure's response. It is the quickest of all the methods.

Case1 Cantilever with two concentrated masses

Modeling Procedure

1 Same Model is used which was modeled for modal analysis as I am performing mode of superposition method so I need the factored mode shape from the modal analysis.

2 Damping Method #2 is added zeta , alpha and beta was added as Beam 4 supports most of the application . Like just for discussion it has gyroscopic effect which can be added as a real constant.

3 Now here I applied Impulse force at the tip of beam at time =0.01 and widraw that force at time 0.02

4 Now in /post26 we can note the dynamic response at the time 0.005, we can note the displacement in y direction at the tip and middle to effect of impulse force.

Figure Showing at the Substep 1 and 20

31

Table Showing Load step and Sub step and Time

Set TimeLoad Step Sub Step

1 4.42E-03 1 120 0.1001 1 20

Figure Showing Time vs Displacement at the tip and Middle

Interpretation

1 We can see the plot of Uy displacement at the position of the tip and Middle. As we can see that tip displacement is more at time 0.1 sec and mid displacement is less at same time.

2 Like if we imagine the beam when force is applied at the tip at one time we can see that the beam displaces more at the tip and less at the middle portion

3 The Behavior of beam can be known clearly how beam behave under loading condition.

Case 2 A cantilever beam has attached masses. One mass is free to hop or Leave the surface

Case Description –

Two concentrated masses are affixed to a cantilever beam. A third mass is resting on the top of the beam. A hammer Strike is applied at the beam free end. The load is of the form of a triangle. The object is to investigate the possibility of a hop of the third mass which is connected to the beam using contact12 elements. Springs elements, Combin14, are used to simulate the attachment to beam of hopping mass.

Modeling Procedure

32

1. Element chosen to built the Beam is Beam4 then i assign the geometric properties of the Cylinder which are going to be attached to the beam.

2. Then I assigned the various elements like Mass21 for Masses, Contac12 to give contact surface to the cylinder and the Combin 14 in Uy dof and Ux dof that is Spring element

3. These all element were generate by the real constant as shown below.

Real Constant Assign Properties Purposer1 Area, Izz, Tz,Tx,h Beam Model Creationr2 mass1,Izz1 Accelerometer#1-Cylinderr3 mass2,Izz2 Accelerometer#2-Cylinderr4 mass3,Izz3 Free Cylinderr5 Uy Dof Contact elementr6 Ux Dof Spring attached to accelerometer

to the Beamr7 thk Spring for fastening free cylinder.

4 Accelerometer #1 & #2, Free cylinder, springs, Contact element were assigned with respect to the keypoint.5 After these is done assigning a boundary condition to its6 Now hammer Load is applied at the tip of the beam at time t=0.01 sec and removed at time t=0.02 and

effect are noted till time t=0.5 sec.7 In the post processor Command was given to plot y displacement at 2 location.

Figure showing Front view of the model

Figure Showing Eplot of the Beam Figure showing at last sub step where can see Hop of Free mass

33

Figure Showing Uy Plot for beam displacement at the tip and mid accel

Interpretation

1 The Graph of Y displacement looks reasonable was lumpm,on ( Lumped mass option). The displacement shows between time 0.01 sec to 0.075 sec after that there is a constant line of Uy these signifies that as a hammer load was released at time 0.02 The load impact remains till 0.075 sec only after that it stabilize.

2 These result may be very important design firm to note the response based on the loading which was given and look at the behavior of Beam.

3 As we can see that the mass which was kept free on the third cylinder it hop out of the cylinder as shown above in one of the figure.

Case 3 A Block falls on the Cantilever Beam

Case description: A Blocks Falls on the cantilever beam and study of displacement is noted at each position of the beam.

Modeling Procedure

1 Model was Prepared using Beam 4 element for Beam structure, Plane 42 element for the Block, Contac171 was given between Beam and Block as Block strikes the beam so targe169 element was taken in to consideration.

2 With Real constant Block and Beam were made. Block was made at the centre of beam . At the centre of the Beam in order to give stability to the structure the structure was Constrained at centre.

3 Method used for Transient Analysis is full method4 Modeling was very simple inafe here Block Falls after some time and effects is studied.

34

Figure showing eplot Figure Showing Block Position Load Step3 and Sub Step10

Figure Showing Y displacement at Node at Various Nodes

Interpretation

1 The Behavior of the Beam can notice at various time when Block Falls on the beam we can see at time 0.28 sec where all the nodes have almost same displacement even it is a peak point of the graph where n maximum displacement take place.

2 At different time we noted different displacement of the beam . from all of the study main purpose would be that we full fill our design criteria..

35

Conclusion

1 I study the behavior of the beam using various analysis. The main aim of doing Buckling Analysis, Modal analysis and Dynamic Analysis was to understand the behavior of the Beam under each analysis.

2 From The Buckling Analysis, I carried out Eigenvalue analysis we have seen the buckling in beams takes place at the same load one which is given by Euler’s Buckling Load.

3 Spotted that if complexity in the model is involved ansys solve easily using the shift variables.

4 Non Linear Study was made and it gave better result than the Eigenvalue analysis as in that geometric imperfection is not much of effect.

5 Modal analysis was carried out where I tried and spotted that the natural frequency given by ansys where slightly lower than I calculate by the theoretical formula.

6 In Modal Analysis we came to know the natural frequency of the beam. I came to know even while comparing between two methods if the complexity involved in the model ansys can solve complex structure using reduced method.

7 In Transient Dynamic Analysis, I several cases were tried applying an impulse force for certain time and know the response of structure due to that impulse force.

“Collapse Structure of WTC- we don’t want to see these again”

Being a Civil Engineers it is my duty to Analyze the structure Correctly and design it correctly based on that analysis

36

References

1. “Concepts and applications of Finite Element Analysis” – 4th Edition by Robert D Cook,

David S. Malkus, Michael E. Plesha, Robert J. Witt.

2. “ Structural Analysis- 3rd Edition by Aslam Kassimali

3. “ Structural Steel Design”- 1st Edition by Abi Aghayere and Jason Vigil

4. “ Design of Steel Structure- 3rd Edition by Edwin H Gaylord

5. “ Stability of Structure”- 2nd edition by Dr. Eric m Lui and Chen

6. “Theory of Structures” –By Timoshenko, Stephen.

7. “ University Alberta” – Ansys help

http://www.mece.ualberta.ca/tutorials/ansys/IT/Transient/Transient.html

37

Appendix

Beam 4 Eigenvalue buckling of a an I-shaped solid beam.!!Eigenvalue buckling of a an I-shaped solid beam.!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid cantilever beam- BEAM4/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam4!keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyyk,1,k,2,120

38

l,1,2LESIZE,1,,,4lmesh,1! Boundary Conditionsd,1,all!f,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fx,-1 !Initial small loadsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3 !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode

39

/eshape,1pldisp,1 !Plot mode shape

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!!Eigenvalue buckling of a an I-shaped solid beam.!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid Cantilever beam - BEAM188/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam188keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120k,1,k,2,120k,3,0,4k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26

40

SECDATA,bf,bf,dg,tf,tf,twSECNUM,1LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,all!f,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fx,-1 !Initial small loadsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,lanb,3,0.5 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3,,,YES !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

41

/title,Buckling of solid cantilever beam- BEAM189!!Eigenvalue buckling of a an I-shaped solid beam.!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid cantilever beam- BEAM189/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam189keyopt,1,8,2keyopt,1,9,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5! GEOMETRY INFORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120k,1,k,2,120k,3,0,4k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26SECDATA,bf,bf,dg,tf,tf,twSECNUM,1

42

LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,all!f,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fx,-1 !Initial small loadsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3,0.5 !Use subspace method, extract 3 modesoutres,all,allsavesolvefinish/soluexpass,onmxpand,3,,,YES !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode

43


Buckling of a cantilever beam using shell63/filnam,cantileverbucklingshell63/title,Buckling of solid cantilever beam - shell63/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,Shell63mp,ex,1,exxmp,nuxy,1,nu! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26r,1,twr,2,tflb=120k,1,,dg/2k,2,lb,dg/2k,3,lb,-dg/2k,4,0,-dg/2k,5,0,dg/2,-bf/2k,6,0,dg/2,bf/2k,7,lb,dg/2,bf/2k,8,lb,dg/2,-bf/2k,9,0,-dg/2,-bf/2k,10,0,-dg/2,bf/2k,11,lb,-dg/2,bf/2k,12,lb,-dg/2,-bf/2a,1,2,3,4a,1,2,7,6

44

a,1,2,8,5a,3,4,10,11a,3,4,9,12k=12f=4w=8lsel,s,Length,,lblesize,all,,,klsel,s,Length,,bf/2lesize,all,,,flsel,s,Length,,dglesize,all,,,wtype,1real,1amesh,1type,1real,2amesh,2,3type,1real,2amesh,4,5! Boundary Conditionsnsel,s,loc,x,0d,all,all/pbc,f,1/pbc,u,1nsell,allsavefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,all!nsel,s,loc,x,lb!nsel,r,loc,z,0!f,all,fx,-1/(w+1) !Equal Initial small load at center line only!f,all,fx,-1/(4*f+w+3) !Equal Initial small load at all tip nodes!**LOADING APPLIED TO THE FLANGE AND THE WEB PROPORTIONAL TO CORRESPONDING AREAnsel,s,loc,x,lb

45

nsel,r,loc,z,0f,all,fx,-.3/(w+1) !WEBnsel,s,loc,x,lbnsel,r,loc,Y,DG/2f,all,fx,-.35/(2*F+1) !UPPER FLANGEnsel,s,loc,x,lbnsel,r,loc,Y,-DG/2f,all,fx,-.35/(2*F+1) !LOWER FLANGEnsell,allNODE1=NODE(LB,-DG/2,0)f,NODE1,fx,(-.35/(2*F+1))+(-.3/(w+1))NODE2=NODE(LB,DG/2,0)f,NODE2,fx,(-.35/(2*F+1))+(-.3/(w+1))nsell,allsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3 !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

!Eigenvalue BUCKLING PROBLEM/filnam,SOLID BEAM/title,Buckling of solid Cantilever Shell 93/prep7

46

/plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,Shell93mp,ex,1,exxmp,nuxy,1,nu! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26r,1,twr,2,tf!keyopt,1,3,2!e=156.25!b=112.5!ho=375!dt=(dg-ho)/2lb=120k,1,,dg/2k,2,lb,dg/2k,3,lb,-dg/2k,4,0,-dg/2k,5,0,dg/2,-bf/2k,6,0,dg/2,bf/2k,7,lb,dg/2,bf/2k,8,lb,dg/2,-bf/2k,9,0,-dg/2,-bf/2k,10,0,-dg/2,bf/2k,11,lb,-dg/2,bf/2k,12,lb,-dg/2,-bf/2a,1,2,3,4a,1,2,7,6a,1,2,8,5a,3,4,10,11a,3,4,9,12k=12f=4

47

w=8lsel,s,Length,,lblesize,all,,,klsel,s,Length,,bf/2lesize,all,,,flsel,s,Length,,dglesize,all,,,wtype,1real,1amesh,1type,1real,2amesh,2,3type,1real,2amesh,4,5! Boundary Conditionsnsel,s,loc,x,0d,all,all/pbc,f,1/pbc,u,1nsell,allsavefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,all!nsel,s,loc,x,lb!nsel,r,loc,z,0!f,all,fx,-1/(w+1) !Equal Initial small load at center line only!f,all,fx,-1/(4*f+w+3) !Equal Initial small load at all tip nodes!**LOADING APPLIED TO THE FLANGE AND THE WEB PROPORTIONAL TO CORRESPONDING AREAnsel,s,loc,x,lbnsel,r,loc,z,0f,all,fx,-.3/(2*w+1) !WEBnsel,s,loc,x,lbnsel,r,loc,Y,DG/2f,all,fx,-.35/(4*F+1) !UPPER FLANGE

48

nsel,s,loc,x,lbnsel,r,loc,Y,-DG/2f,all,fx,-.35/(4*F+1) !LOWER FLANGEnsell,allNODE1=NODE(LB,-DG/2,0)f,NODE1,fx,(-.35/(4*F+1))+(-.3/(2*w+1))NODE2=NODE(LB,DG/2,0)f,NODE2,fx,(-.35/(4*F+1))+(-.3/(2*w+1))nsell,allsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3 !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

/filnam,catellatedbeam/title,Buckling of Castellated Beam/prep7 /plopts,logo,0/plopts,date,0antype,staticantype,staticexx=30000 nu=0.3

49

et,1,Shell93mp,ex,1,exxmp,nuxy,1,nu! GEOMETRY FORMATION!###########################e=8b=2h=10d=e!########################tf=0.44tw=0.26r,1,twr,2,tfbf=5.75dg=10.3+h/2L=E+2*B+Dk,1,k,2,Lk,3,L,dgk,4,,dgk,7,0,0,-bf/2k,8,0,0,bf/2k,9,L,0,bf/2k,10,L,0,-bf/2k,11,0,dg,-bf/2k,12,0,dg,bf/2k,13,L,dg,bf/2k,14,L,dg,-bf/2a,1,2,3,4 ! WEB AREAa,1,2,10,7a,1,2,9,8a,4,3,13,12a,4,3,14,11

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@! CASTELLATION OF BEAM WEBdt=(dg-h)/2K,15,d/2+b,dt,k,16,d/2,dg/2,k,17,d/2+b,dg-dt,

50

k,18,d/2+e+b,dg-dt,k,19,d/2+e+b+b,dg/2k,20,d/2+e+b,dta,15,16,17,18,19,20ASBA,1,6,,AGEN,6,all,,,d+e+2*bNUMMRG,KPREAL,1AMESH,ALLASEL,S,LOC,Y,DGREAL,2aCLEAR,ALLREAL,2AMESH,ALLASEL,S,LOC,Y,0REAL,2aCLEAR,ALLREAL,2AMESH,ALLASEL,ALL! Boundary Conditionsnsel,s,loc,x,0d,all,all/pbc,f,1/pbc,u,1nsell,allsavefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,all

!**LOADING APPLIED TO THE FLANGE AND THE WEB PROPORTIONAL TO CORRESPONDING AREAnsel,s,loc,x,6*Lnsel,r,loc,z,0f,all,fx,-.3/23 !WEBnsel,allf,3485,fy,0

51

f,3486,fy,0f,3537,fy,0nsel,s,loc,x,6*Lnsel,r,loc,Y,DGf,all,fx,-.35/9 !UPPER FLANGEnsel,s,loc,x,6*Lnsel,r,loc,Y,0f,all,fx,-.35/9 !LOWER FLANGEnsell,allNODE1=NODE(6*L,0,0)f,NODE1,fx,(-.35/9-.3/23)NODE2=NODE(6*L,DG,0)f,NODE2,fx,(-.35/9-.3/23)nsell,allsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3 !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

Non Linear Analysis of the I beam- Beam 4

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!!Eigenvalue buckling of a an I-shaped solid beam.

52

!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid cantilever beam- BEAM4/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam4!keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5Pcr=72.47Fmax=2*PcrL=120! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyyk,1,k,2,120l,1,2LESIZE,1,,,4lmesh,1! Boundary Conditionsd,1,allf,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1

53

savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/solusolcontrol,onmonitor,1,2,uymonitor,2,2,fymonitor,3,2,fx/gst,onnropt,auto !Solution method is selectednlgeom,onsstif,onautots,onpred,on !Activates a predictor in nonlinear analysisoutres,all,all*do,i,1,10time,i*Fmax/10f,2,fx,-(i*Fmax/10)lswrite,i*enddosavelssolve,1,10,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!Postprocessing:!!In post1 use set,list to list loads sets.!Use /dscale,window,multiplier to magnifiy displacement plot as !desired. Eg., /dscale,1,100 will magnify the plot by 100.!Use pldisp,1 to observe deflection shape for sequential load steps.!!Use post26 to get the time history plot. See below.!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/post26nsol,2,2,u,y,uy2prod,3,1,,,LOAD,,,Fmaxxvar,3

54

/axlab,x,Total Load, lbs/axlab,y,Displacement, inchplvar,2


!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Non Linear Buckling of solid Cantilever beam - BEAM188/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam188keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5Pcr=72.47Fmax=2*PcrL=120! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyyk,1,k,2,120k,3,0,4

55

k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26SECDATA,bf,bf,dg,tf,tf,twSECNUM,1LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,allf,2,fz,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/solusolcontrol,onmonitor,1,2,uzmonitor,2,2,fzmonitor,3,2,fx/gst,onnropt,auto !Solution method is selectednlgeom,onsstif,onautots,onpred,on !Activates a predictor in nonlinear analysisoutres,all,all*do,i,1,10time,i*Fmax/10f,2,fx,-(i*Fmax/10)lswrite,i*enddosavelssolve,1,10,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@!Postprocessing:!

56

!In post1 use set,list to list loads sets.!Use /dscale,window,multiplier to magnifiy displacement plot as !desired. Eg., /dscale,1,100 will magnify the plot by 100.!Use pldisp,1 to observe deflection shape for sequential load steps.!!Use post26 to get the time history plot. See below.!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/post26nsol,2,2,u,z,uz2prod,3,1,,,LOAD,,,Fmaxxvar,3/axlab,x,Total Load, lbs/axlab,y,Displacement, inchplvar,2


!/filnam,Solid Ibeam/title,Non Linear Buckling of solid cantilever beam- BEAM189/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam189keyopt,1,8,2keyopt,1,9,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5Pcr=72.47Fmax=2*PcrL=120

! GEOMETRY INFORMATION

57

! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyy

k,1,k,2,120k,3,0,4k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26SECDATA,bf,bf,dg,tf,tf,twSECNUM,1LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,allf,2,fz,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fx,-1 !Initial small loadsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

58

/solusolcontrol,onmonitor,1,2,uzmonitor,2,2,fzmonitor,3,2,fx/gst,onnropt,auto !Solution method is selectednlgeom,onsstif,onautots,onpred,on !Activates a predictor in nonlinear analysisoutres,all,all*do,i,1,10time,i*Fmax/10f,2,fx,-(i*Fmax/10)lswrite,i*enddosavelssolve,1,10,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@!Postprocessing:!!In post1 use set,list to list loads sets.!Use /dscale,window,multiplier to magnifiy displacement plot as !desired. Eg., /dscale,1,100 will magnify the plot by 100.!Use pldisp,1 to observe deflection shape for sequential load steps.!!Use post26 to get the time history plot. See below.!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/post26nsol,2,2,u,z,uy2prod,3,1,,,LOAD,,,Fmaxxvar,3/axlab,x,Total Load, lbs/axlab,y,Displacement, inchplvar,2

59

!Non Linear Buckling of a cantilever beam using shell63/filnam,cantileverbucklingshell63/title,Buckling of solid cantilever beam - shell63/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,Shell63mp,ex,1,exxmp,nuxy,1,nuPcr=72.47Fmax=2*PcrL=120! GEOMETRY FORMATION! A W10x26bf=5.75

dg=10.3tf=0.44tw=0.26a=7.61Izz=14.1Iyy=144lb=120r,1,twr,2,tf

k,1,,dg/2k,2,lb,dg/2k,3,lb,-dg/2k,4,0,-dg/2k,5,0,dg/2,-bf/2k,6,0,dg/2,bf/2k,7,lb,dg/2,bf/2

60

k,8,lb,dg/2,-bf/2k,9,0,-dg/2,-bf/2k,10,0,-dg/2,bf/2k,11,lb,-dg/2,bf/2k,12,lb,-dg/2,-bf/2a,1,2,3,4a,1,2,7,6a,1,2,8,5a,3,4,10,11a,3,4,9,12k=12f=4w=8lsel,s,Length,,lblesize,all,,,klsel,s,Length,,bf/2lesize,all,,,flsel,s,Length,,dglesize,all,,,wtype,1real,1amesh,1type,1real,2amesh,2,3type,1real,2amesh,4,5! Boundary Conditionsnsel,s,loc,x,0d,all,all

/pbc,f,1/pbc,u,1nsell,allsavefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effects

61

outres,all,all!nsel,s,loc,x,lb!nsel,r,loc,z,0!f,all,fz,-1/(w+1) !Equal Initial small load at center line only!f,all,fz-1/(4*f+w+3) !Equal Initial small load at all tip nodes!**LOADING APPLIED TO THE FLANGE AND THE WEB PROPORTIONAL TO CORRESPONDING AREAnsel,s,loc,x,lbnsel,r,loc,z,0f,all,fx,-.3/(w+1) !WEBnsel,s,loc,x,lbnsel,r,loc,Y,DG/2f,all,fx,-.35/(2*F+1) !UPPER FLANGEnsel,s,loc,x,lbnsel,r,loc,Y,-DG/2f,all,fx,-.35/(2*F+1) !LOWER FLANGEnsell,allNODE1=NODE(LB,-DG/2,0)f,NODE1,fz,(.35/(2*F+1))+(.3/(w+1))NODE2=NODE(LB,DG/2,0)f,NODE2,fz,(.35/(2*F+1))+(.3/(w+1))nsell,allsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/solusolcontrol,onmonitor,1,2,uzmonitor,2,2,fzmonitor,3,2,fx

/gst,onnropt,auto !Solution method is selectednlgeom,onsstif,onautots,onpred,on !Activates a predictor in nonlinear analysisoutres,all,all*do,i,1,10time,i*Fmax/10

62

f,2,fx,-(i*Fmax/10)lswrite,i*enddosavelssolve,1,10,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@!Postprocessing:!!In post1 use set,list to list loads sets.!Use /dscale,window,multiplier to magnifiy displacement plot as !desired. Eg., /dscale,1,100 will magnify the plot by 100.!Use pldisp,1 to observe deflection shape for sequential load steps.!!Use post26 to get the time history plot. See below.!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/post26nsol,2,2,u,z,uy2prod,3,1,,,LOAD,,,Fmaxxvar,3/axlab,x,Total Load, lbs/axlab,y,Displacement, inchplvar,2

Lateral Buckling

!Lateral Buckling!Eigenvalue buckling of a an I-shaped solid beam.!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid Cantilever beam - BEAM189/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000

63

nu=0.3 et,1,beam188keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120k,1,k,2,120k,3,0,4k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26SECDATA,bf,bf,dg,tf,tf,twSECNUM,1LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,all!f,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fy,-1 !Initial small loadsave

64

solvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,lanb,3,0.5 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3,,,YES !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@Effect of Shift Variables using Subspace Method!Eigenvalue buckling of a an I-shaped solid beam.!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Buckling of solid Cantilever beam - BEAM188/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,beam188keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3!mp,gxy,1,1e5!mp,gxz,1,1e5! GEOMETRY FORMATION

65

! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120k,1,k,2,120k,3,0,4k,4,0,-4l,1,2LESIZE,1,,,10SECTYPE,1,BEAM,I,W10x26SECDATA,bf,bf,dg,tf,tf,twSECNUM,1LATT,1,,1,,3,4,1lmesh,1! Boundary Conditionsd,1,all!f,2,fy,1 !Apply a small lateral load! Display Commands/pbc,f,1/pbc,u,1savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effectsoutres,all,alld,1,allf,2,fx,-1 !Initial small loadsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,lanb,3,0.5 !Use subspace method, extract 3 modessavesolve

66

finish/soluexpass,onmxpand,3,,,YES !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

Lateral Buckling Loading!Buckling of a cantilever beam using shell63/filnam,cantileverbucklingshell63/title,Buckling of solid cantilever beam - shell63/prep7 /plopts,logo,0/plopts,date,0antype,staticexx=30000 nu=0.3 et,1,Shell63mp,ex,1,exxmp,nuxy,1,nu! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26r,1,twr,2,tflb=120k,1,,dg/2k,2,lb,dg/2k,3,lb,-dg/2k,4,0,-dg/2k,5,0,dg/2,-bf/2k,6,0,dg/2,bf/2

67

k,7,lb,dg/2,bf/2k,8,lb,dg/2,-bf/2k,9,0,-dg/2,-bf/2k,10,0,-dg/2,bf/2k,11,lb,-dg/2,bf/2k,12,lb,-dg/2,-bf/2a,1,2,3,4a,1,2,7,6a,1,2,8,5a,3,4,10,11a,3,4,9,12k=12f=4w=8lsel,s,Length,,lblesize,all,,,klsel,s,Length,,bf/2lesize,all,,,flsel,s,Length,,dglesize,all,,,wtype,1real,1amesh,1type,1real,2amesh,2,3type,1real,2amesh,4,5! Boundary Conditionsnsel,s,loc,x,0d,all,all/pbc,f,1/pbc,u,1nsell,allsavefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,staticpstres,on !Calculate prestress effects

68

outres,all,all!nsel,s,loc,x,lb!nsel,r,loc,z,0f,all,fx,-1/(w+1) !Equal Initial small load at center line onlyf,all,fx,-1/(4*f+w+3) !Equal Initial small load at all tip nodes!**LOADING APPLIED TO THE FLANGE AND THE WEB PROPORTIONAL TO CORRESPONDING AREAnsel,s,loc,x,lbnsel,r,loc,z,0f,all,fy,-.3/(w+1) !WEBnsel,s,loc,x,lbnsel,r,loc,Y,DG/2f,all,fy,-.35/(2*F+1) !UPPER FLANGEnsel,s,loc,x,lbnsel,r,loc,Y,-DG/2f,all,fy,-.35/(2*F+1) !LOWER FLANGEnsell,allNODE1=NODE(LB,-DG/2,0)f,NODE1,fy,(-.35/(2*F+1))+(-.3/(w+1))NODE2=NODE(LB,DG/2,0)f,NODE2,fy,(-.35/(2*F+1))+(-.3/(w+1))nsell,allsavesolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/soluantype,buckle !Perform eigenvalue buckling analysisbucopt,subsp,3 !Use subspace method, extract 3 modessavesolvefinish/soluexpass,onmxpand,3 !Expand 3 modoutres,all,allsolvefinish/post1set,listset,1,1 !Select 1 or 2 mode

69


Modal Analysis

BEAM 4

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!!Modal Analysisof a an I-shaped beam. Using Reduced Method!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Modal analysis of solid cantilever beam- BEAM4/prep7 /plopts,logo,0/plopts,date,0exx=30000 nu=0.3 A=7.61Gfactor=1Gmod=Gfactor*exx/(2*(1+nu))g=386sg=7.8wtden=sg*62.4/1728rho=wtden/g wtbeam=rho*A*g

70

L=120!@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Mass and moment of inertiam1=.25/gm2=.25/ghblock1=2wblock1=1hblock2=2wblock2=1Ic_block1=m1*(hblock1**2+wblock1**2)/12Ic_block2=m2*(hblock2**2+wblock2**2)/12I_block1=Ic_block1+m1*(hblock1/2)**2I_block2=Ic_block2+m1*(hblock2/2)**2et,1,beam4et,2,mass21,,,3!keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3mp,gxy,Gmodmp,dens,1,0! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyy,tf,twr,2,m1,I_block1r,3,m2,I_block2k,1,k,2,120l,1,2LESIZE,1,,,4lmesh,1midnode=node(L/2,0,0)type,2real,2

71

e,midnodetype,2real,3tipnode=node(L,0,0)e,tipnodeoutpr,nsol,alld,1,allsavefinish! Subspace Method/soluantype,modalmodopt,reduc,3mxpand,3m,4,uym,2,uytotal,2solve*get,ff1,mode,1,freq*get,ff2,mode,2,freq*get,ff3,mode,3,freqfinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape!!Modal Analysisof a an I-shaped beam. Using Subspace method!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam/title,Modal analysis of solid cantilever beam- BEAM4/prep7 /plopts,logo,0/plopts,date,0exx=30000 nu=0.3 A=7.61Gfactor=1Gmod=Gfactor*exx/(2*(1+nu))

72

g=386sg=7.8wtden=sg*62.4/1728rho=wtden/g wtbeam=rho*A*gL=120!@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Mass and moment of inertiam1=.25/gm2=.25/ghblock1=2wblock1=1hblock2=2wblock2=1Ic_block1=m1*(hblock1**2+wblock1**2)/12Ic_block2=m2*(hblock2**2+wblock2**2)/12I_block1=Ic_block1+m1*(hblock1/2)**2I_block2=Ic_block2+m1*(hblock2/2)**2et,1,beam4et,2,mass21,,,3!keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3mp,gxy,Gmodmp,dens,1,0! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyy,tf,twr,2,m1,I_block1r,3,m2,I_block2k,1,k,2,120l,1,2

73

LESIZE,1,,,4lmesh,1midnode=node(L/2,0,0)type,2real,2e,midnodetype,2real,3tipnode=node(L,0,0)e,tipnodeoutpr,nsol,alld,1,allsavefinish! Subspace Method/soluantype,modalmodopt,subsp,3mxpand,3solve*get,ff1,mode,1,freq*get,ff2,mode,2,freqfinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape

Transient Dynamic Analysis

BEAM 4

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@!!Transient dynamic analysis!!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/filnam,Solid Ibeam

74

/title,Transient dynamic analysis solid cantilever beam- BEAM4/prep7 /plopts,logo,0/plopts,date,0exx=30000 nu=0.3 A=7.61Gfactor=1Gmod=Gfactor*exx/(2*(1+nu))g=386sg=7.8wtden=sg*62.4/1728rho=wtden/g wtbeam=rho*A*gL=120!@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Mass and moment of inertiam1=.25/gm2=.25/ghblock1=2wblock1=1hblock2=2wblock2=1Ic_block1=m1*(hblock1**2+wblock1**2)/12Ic_block2=m2*(hblock2**2+wblock2**2)/12I_block1=Ic_block1+m1*(hblock1/2)**2I_block2=Ic_block2+m1*(hblock2/2)**2et,1,beam4et,2,mass21,,,3!keyopt,1,3,2mp,EX,1,exxmp,nuxy,1,0.3mp,gxy,Gmodmp,dens,1,0! GEOMETRY FORMATION! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120

75

a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyy,tf,twr,2,m1,I_block1r,3,m2,I_block2k,1,k,2,120l,1,2LESIZE,1,,,4lmesh,1midnode=node(L/2,0,0)type,2real,2e,midnodetype,2real,3tipnode=node(L,0,0)e,tipnodeoutpr,nsol,alld,1,allsavefinish! Subspace Method/soluantype,modalmodopt,subsp,3mxpand,3solve*get,ff1,mode,1,freq*get,ff2,mode,2,freqfinish/post1set,listset,1,1 !Select 1 or 2 mode/eshape,1pldisp,1 !Plot mode shape!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Two damping methods are provided below!@@@@@@@@@@@@@@@@@@@@@@/solu

76

antype,transtrnopt,msup,3,,1,yes ! Output modal coordinates!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Modal coordinates are printed in file .mcf ! Import this data to excel and plot contributions! associated with each mode. Observe different! frequencies and magnitudes of each mode.!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@

! Damping method #2! Calculation for proportional dampingf1=ff1f2=ff2zeta1=.1zeta2=.2PI=ACOS(-1)*dim,coeff,,2,2*dim,rhs,,2,1*dim,alfbeta,,2,1coeff(1,1)=1/(4*PI*f1),1/(4*PI*f2)coeff(1,2)=PI*f1,PI*f2rhs(1,1)=zeta1,zeta2*moper,alfbeta(1,1),coeff(1,1),solve,rhs(1,1)alpha=alfbeta(1)beta=alfbeta(2)alphad,alfbeta(1)betad,alfbeta(2)!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@f,3,fy,5kbc,1time,.01ITS=1/(20*f2)deltim,ITSoutpr,all,alloutres,all,all

77

lswrite,1time,.02f,3,fy,0lswrite,2time,0.6lswrite,3/output,file,datlssolve,1,3,1finish/output/post26file,,rdsptime,.005,.25nsol,2,4,u,y,middispnsol,3,2,u,y,tipdisp/axlab,y,Displacement, in/axlab,x,Time, secplvar,2,3

!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@finish/soluexpass,onnumexp,20,0.0,0.1outres,all,allsolvefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@/post1/eshape,1set,listset,1,1pldisp,2

78

Hammer Strike Case

!@@@@@@@@@@@@@@@@@@@@@@! A cantilever beam has attached masses. One mass is freee to "hop" or! leave the surface of the beam!! Two concentrated masses are affixed to a cantilever beam. A third! mass is resting on the top of the beam. A hammer strike is applied! at the beam free end. The load is of the form of a triangle. The! object is to investigate the possibility of a "hop" of the third! mass which is connected to the beam using contact12 elements. ! Spring elements, combin14, are used to simulate the attachment to ! beam of the "hopping" mass. This simulates something like an! accelerometer attached to the beam with magnets.!!@@@@@@@@@@@@@/title, Transient Dynamic Analysis of Cantilever Beam With Added Masses/filnam,hopc/prep7/plopts,logo/plopts,date! System parameter definitionsacy1=.5 !Radius of cylinder 1 acy2=.5 !Radius of cylinder 2acy3=.5 !Radius of cylinder 3Lcy1=1 !Length of cylinder 1Lcy2=1 !Length of cylinder 2Lcy3=1.5 !Length of cylinder 3

A=7.61! A W10x26bf=5.75dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1

79

Iyy=144r,1,a,Izz,Iyy,tf,twrho=0.000728 !Beam material densityrho1=sqrt((3*(acy1**2)+(Lcy1**2))/12) !Mass radius of gyration 1rho2=sqrt((3*(acy2**2)+(Lcy2**2))/12) !Mass radius of gyration 2rho3=sqrt((3*(acy3**2)+(Lcy3**2))/12) !Mass radius of gyration 3g=386w=rho*A*gL=120hcy1=4.5 !Height of cylinder1 cg above beamhcy2=4.5 !Height of cylinder2 cg above beamhcy3=6 !Height of cylinder3 cg above beamm1=.2/g !Mass of accelerometer#1m2=.2/g !Mass of accelerometer#2m3=.4/g !Mass of "free" objectIz1=m1*(rho1**2)Iz2=m2*(rho2**2)Iz3=m3*(rho3**2)c***Element Definitionset,1,beam4!keyopt,1,3,2et,2,mass21,,,3et,3,contac12et,4,combin14,,2 !UY dofet,5,combin14,,1 !UX dofkeyopt,3,7,1mp,ex,1,30e6mp,ex,2,100e6 !For very stiff mass structures mp,dens,1,rhomp,damp,1,0.005r,1,a,Izz,Iyy,tf,twr,2,m1,Iz1 !Accelerometer#1r,3,m2,Iz2 !Accelerometer#2r,4,m3,Iz3 !Free cylinderr,5,,1e4 !Contact elementr,6,1e5 !Spring to attach accelerometers to beamr,7,0.1 !Spring for fastening free cylinder to beamk,1k,2,120l,1,2c*** Beam Construction

80

!Set beam node locations at 1 inch intervals along the beamlesize,1,,,20type,1mat,1real,1lmesh,1c*** Construction of accelerometers and free cylinder/com Locations are at positions determined by inch locations/com following from the lmesh operation of the beam!Accelerometer #1type,1mat,2real,1k,3,60k,4,66k,5,63,-hcy1l,3,4l,4,5l,5,3lsel,s,,,2,4,1lesize,all,,,1lmesh,2,4,1!Accelerometer #2k,6,108k,7,114k,8,111,-hcy2l,6,7l,7,8l,8,6lsel,s,,,5,7,1lesize,all,,,1lmesh,5,7,1!Free cylinderk,9,102k,10,108k,11,105,hcy3l,9,10l,10,11l,11,9lsel,s,,,8,10,1lesize,all,,,1

81

lmesh,8,10,1/pnum,node,1!Accelerometers!Accel Massestype,2real,2e,24real,3e,27!Accel springstype,4 !Uy dofreal,6e,12,22e,13,23e,20,25e,21,26type,5 !Ux dofreal,6e,12,22e,13,23e,20,25e,21,26!Free Cylinder!Masstype,2real,4e,30!Springstype,4 !Uy dofreal,7e,19,28e,20,29type,5 !Ux dofreal,7e,19,28e,20,29!Contact elementstype,3real,5e,19,28e,20,29

82

savefinish!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Solution Commands/soluoutres,all,alloutpr,nsol,1! Boundary Conditionsksel,s,,,1nslk,sd,all,allc*** Analysisallsel,all,all!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@antype,translumpm,on !Lumped mass optiontintp,.05 !Transient integration parametersnlgeom,onpred,on!autots,onkbc,1timint,offnsubst,50time,.001acel,,g !Apply accelerationlswrite,1timint,on !Beginning of loading commandskbc,0nsubst,40time,0.01f,2,fy,300 !Hammer load applied at beam tiplswrite,2time,0.02f,2,fy,0 !Hammer load removedlswrite,3time,0.5nsubst,100lswrite,4

83

lssolve,1,4,1finish/output!@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@@! Note: Beamleft and Cyleft represent the left! side of the hopping mass and beam.! BeamRite and Clyrite represent the right! hand side of the hopping mass./post26nsol,2,24,u,y,MidAccelnsol,3,27,u,y,TipAccelplvar,2nsol,4,19,u,y,BeamLeftnsol,5,28,u,y,Cyleftnsol,6,20,u,y,BeamRitensol,7,29,u,y,Clyriteplvar,2,3

!Block falls and strikes a beam. Based on WB11. Version 5.0.!/output,xbounce,txt!@@@@@@@@@@@@@@@@@/prep7/plopts,logo/plopts,dateet,1,beam4et,2,plane42,,,3et,3,conta171et,4,targe169!keyopt,1,3,2keyopt,3,2,1!@@@@@@@@@@!Beam characteristics! GEOMETRY FORMATION! A W10x26bf=5.75

84

dg=10.3tf=0.44tw=0.26lb=120a=7.61Izz=14.1Iyy=144r,1,a,Izz,Iyy,tf,twL=120!Block characteristicsblkthk=1 !Block thicknessblklen=2 !Block side lengthhblock=1 !Block heightadist=L/2 !Distance to block corner!Contact element propertiescstiff=.05 !Contact stiffness FKN Default 1, reduce for convergence!@@@@@@@@@@@@@@@@@@@!Realsr,1,a,Izz,Iyy,tf,twr,2,blkthkr,3,blklen,blklen,cstiff!@@@@@@@@@@@@@@@@@@@@@@!Material propertiesmp,ex,1,30000mp,nuxy,1,.3mp,dens,1,.001 !Mass density!@@@@@@@@@@@@@@@@@!Model buildingk,1k,2,Ll,1,2lesize,1,,,20mat,1lmesh,1/pnum,elem,10eplot/eshape,1eplotrectang,adist,adist+blklen,hblock,hblock+blklenesize,,1type,2

85

real,2mat,1amesh,alleplotlsel,s,,,2nsll,s,1type,3real,3esurflsel,s,,,1nsll,s,1type,4esurfnsel,allesel,allfinish!@@@@@@@@@@@@@@/soluantype,transtrnopt,fullnropt,full,,onlnsrch,onnlgeom,onlumpm,onfcr=0.1 !Force convergence valueldcr=0.01 !Large displacement criteriansbstp=10000 !Number of substepsneq=100 !Number of equilibrium iterations!@@@@@@@@@@@@@@@@ksel,s,kp,,1,2nslk,sd,all,all,0nsel,allesel,s,ename,,42nsle,sd,all,all,0nsel,allesel,allacel,,386time,.0002deltim,.0001

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kbc,1betad,.0003timint,offoutres,all,lastsolveesel,s,ename,,42nsle,sddele,all,allnsel,allesel,alltf=0.4time,tfcnvtol,f,,fcrcnvtol,u,,ldcrcnvtol,stat!deltim,.0002,.00002,.02 !Initial, minimum and maximum delta timesneqit=neqnsubst=nsbstpsolcontrol,onautots,ontimint,onpred,on !Used to predict the start of the next substepoutres,all,allsolvetime,2*tfsolvefinish/pnum,node/pnum,elem/output!@@@@@@@@@@@@@@@@/post26nsol,2,12,u,y,uy12nsol,3,14,u,y,uy14nsol,4,7,u,y,uy7nsol,5,8,u,y,uy8plvar,2,3,4,5

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Web viewλr = obtained from AISC manual depending on the beam is compact section or non compact section. λ = Width/ Thickness ratio