Plane Frame Report

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ME20023: Solid Mechanics 4

Plane Frame Lab Report

UniversityofBath,DepartmentofMechanicalEngineeringBath BA2 7AY e:[email protected]:22/02/2013Dr.R.Butler

Mustapha Bello (1102748670)
mailto:[email protected]:[email protected]


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Table of Contents

Summary 2

Introduction. 2

Theory 3

FEM And CAE 3

Stiffness Matrix Method 4

Method 6

Member g Test 6

Frame Test 6

Results

6

The Stiffness Matrix Method 6

FrameTest 8

FEA Ansys 9

Member g 9Frame 10

Discussion 10

Conclusion 10

References 10

Mustapha Bello

Solid Mechanics 4 (ME20023) Plane Frame Lab Report


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Summary

This report describes an investigation of the behavior of a plane frame under loading. In almost all engineering scenarios, the

understanding of structural rigidity and the behavior of a framework or form under the influence of external loads is

imperative in order to ensure safe, efficient and adequate performance of a product.

In this Lab, a pin-jointed plane frame is analysed experimentally by means of a tensile test rig to find the buckling loads

associated with the structure, as well as the effect of buckling itself and structural redundancy. The model was also analysed

with the aid of a computer software simulation employing the method of Finite Element Analysis (FEA).

Each Method was carried out individually, and the findings of the experimental analysis, FEA software and of course the

theoretical predictions were collated. It was found that all three methods differed from each other, the reasons of which shall

be discussed further in this report.

Introduction.

In this class of engineering, where structural integrity is of paramount significance, stresses and deflection of structures are

kept to a minimum. This is not only to ensure safe and adequate functioning of any design, but also to ensure reliability and

perhaps improve the lifetime by minimising the effects of fatigue. Hence, the understanding of the structural relationships and

the stresses involved are just as important.

Simple Analytical Methods become extremely laborious and sometimes, heavily inaccurate as a result of oversimplification

when it comes to analysis of complex mechanics. This is largely due to an incredible number of assumptions that have to be

made, which may include over simplification of a shape or mould due to complicated contours or networks of struts. As in

most, if not all complex engineering problems, the equations governing the behavior of a complex structure or shape under

loading can be so complicated that approximations have to made, and those approximations, yet still elaborate, have to be

solved numerically in order to obtain a solution close enough to the actual.

There are a variety of techniques used in the analyses of such structures. For the purpose of this lab, the finite element

method shall be discussed.

The finite element method (FEM), is an analytical technique which employs the principle of simplifying a structure into simpler

idealised shapes which are all connected at particular points known as nodes to make up the required structure, this is

known as Meshing. The individual constituent meshes together form a collection of numerical equations that make up an

approximation for an otherwise incomprehensible structure. Hence FEA can he described as a group of numerical methods

for approximating the solution of governing equations of any continuum.

Problems that can be treated with FEA include Structural Analysis, Heat transfer, Fluid Flow, Mass Transport,Electromagnetic potential, Acoustics, Bioengineering etc.

An analytical Method that employs FEM in solving this is the Stiffness Matrix Method which the material stiffness properties

of these elements are compiled into a group of matrices which constitute a single matrix equation which governs the

behavior of the entire structure.

The computer software used in this Lab; Ansys, is a common CAE software that also employs FEM. Ansys, like all other

commercial and freeware FEA softwares operate in the following Manner:

1. Computer Modelling: Mesh Generation

2. Definition Of Material Properties

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3. Assembly of elements

4. Definition OF Boundary Conditions and Loads

5. Solution using the required solver

Figure 1a [1] Below shows the topology of the plane frame, indicating the position of strain gauges which are used in

measuring the strains in the respective members of the frame. In the Stiffness Matrix Analysis, the double symmetry of theframe is exploited and the a quarter frame model is used instead of the entire frame itself (Figure 1b [1] ).

Theory

FEMAndCAE

The finite element method is a numerical method for solving problems of engineering and mathematical physics. In FEA, the

continuum is divided into finite number of elements and the governing equations are represented in matrix form [2]. Method

for solutions developed to solve complex mathematical problems: Runge-Kutta,Gauss-Seidel,Galerkin,Rayleigh,Ritz,Forward

Difference, etc. [3]

In obtaining the approximate solution, the continuum is discretized into finite elements. This is useful for problems with

complicated geometries, loadings, and material properties where analytical solutions can not be obtained. The

computerisation of the method provides a lot of ease in which solutions may be obtained, however it is still also limited by

processing speeds, hence more often than not, models may also be simplified, which decreases accuracy, but allows for

faster speeds, which is of significant economical importance commercially. Figure 2 below Shows a sample problem where

simple analytical calculations are not sufficient. Fig 3. is an example of a CAE FAE software model.

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Px2

l x2

x1 Fig.4

1 2

Px1

Fig.2Example

Fig.3FEMMeshmodel[4]

StiffnessMatrixMethod

Many engineering problems in FEM will result in a set of simultaneous equations represented by [K] {x} = {P} where [K] is the

stiffness matrix of the entire structure and {x} is the vector of nodal displacements due to the applied loads {P}.

Consider the Below diagram in Figure 4.

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....................... Eq. 4

........................ Eq. 5

Where the bar is of length l, Cross sectional area A, and material of Elastic Modulus E, and 1 & 2 are nodes with thier

respective forces and deflections. From Hookes Law, we can say,

Eq. 1 & 2 can be represented in matrix form as:

i.e:

[P]={K}{x}

Where [P] is the load vector, {K} is the stiffness vector, and {x} is the displacement vector. Solving Equation 3 for {x} gives

the solution for the simultaneous Eq.1 and 2 In terms of other known values. In a more complicated system, comprising of a

large numb of elements, connected at numerous points, there will evidently be a large number of equations that are to be

solved. Exploiting Symmetry eases this in structures with some symmetry (as shown in Figure 1).

For the quarter frame shown in Fig. 1(b), Stiffness matrix analysis gives the below expressions for the nodal displacements

Where

For any pin jointed plane frame member of length l, connecting nodes i and j, with i


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And the Strain in the member as:

As regards to structural redundancy, for m members, r reactions and j joints, the frame is said to be:

Statically determinate if m+r=2j

A mechanism if m+r2j

Method

MembergTest

The member g shown in fig. 1 (a) was tested both in tension and compression. It was hooked up on a tensile test rig, with 2

strain gauges on either surface of the member. For The tension test, Loads were applied in steps of 1.0 kN up to 5.0kN and

each load step was recorded against their respective strains. This was recorded by a computer program attached to the

strain gauges. The results of the tension test of the member were used to obtain the Elastic modulus of the material of

member g (Aluminium).

The member was also tested using the same apparatus in compression at a maximum load of 600N in steps of 50N to

obtain sufficient data. The strains were also recorded, and the data was collected for analysis of the buckling loads of the

member.

FrameTest

In this test, All members of the plane frame where assembled and joined, then secured onto a tensile test rig. Members b, cand a all had strain gauges attached on either side. A dial gauge was also attached to member c, to record the out of plane

deflection of the member.

The frame was then loaded axially in steps of 0.25kN up to a maximum of 3.0kN, with all respective strains measured and

recorded and the dial gauge reading was also noted down.

Specific steps and instructions were given and followed in using the Ansys software to obtain the respective buckling modes

and loads for both the member g, and the frame as a whole.

Results

TheStiffnessMatrixMethod

From the tension test of the member g, the average strains were calculated. Having recorded the dimensions of the cross

section of the member, the cross sectional area was calculated and the stresses were calculated. A stress strain plot was

made to enable evaluation of the Elastic modulus of the material (Aluminium).

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................. Eq. 8


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As seen from the above graph, the Gradient of the Best fit line is 0.0688, implying the elastic modulus of the material is

68.8GPa.

Hence, the stiffness characteristics of the members in the frame are now all known, as shown in the table below.

Member Material ElasticModulus

E(GPa)

Widthw(mm) Thicknesst(mm) LengthL(mm)

a,d,e,f

b

c

g

Steel 200 20 4.9 2502

Aluminium 68.8 10 3.2 250

Aluminium 68.8 20 3.2 250

Aluminium 68.8 20 3.2 500

For a force P of 2.0 kN, inputting the figures from the table above, into Equations 4 and 5, x2 and y1 where calculated to

be 0.0685mm and 0.090mm respectively. And by using directional cosines (Equations 6, 7 & 8), The Strains in the members

Member cos sin (strain)

a

b

c

1/2 1/2 155

0 -1 360

-1 0 -274

-20

0

20

40

60

80

0 375 750 1125 1500

y = 0.0688x - 0.8302

Stress-StrainGraphofTensionTest(MemberG)

Stress(kPa)

Strain (strain)

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The results of the compression test of the Member g were used to obtain the buckling loads. A plot of the Axial load against

the axial Strain was made, and the asymptote of the curve was evaluated. This represents the buckling load, and was found

to be 651.875N,however the calculated theoretical Euler buckling load was 593.3N. Below is the graph of the axial load

against strain for the compression test of member g.

FrameTest

The average strains for each member was evaluated, and the plots of the out of plane deflection of member c against load

was used to evaluate the initial buckling loads of the member for each buckling mode. Below s the aforementioned graph.

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0

0.18

0.35

0.53

0.70

-150.00 -112.50 -75.00 -37.50 0

GraphOfLoad(kN)AgainstStrain(strain)forCompressiontestofmemberg

Load

(kN)

Strain (strain)

0.651875 kN

-3.00

-2.25

-1.50

-0.75

0

0 0.75 1.50 2.25 3.00

OutofPlaneDeflection(mm)ofMembercAgainstAxialLoad(kN)

Deflection(mm)

Load(kN)

Buckling load Mode 1

Bucklin load Mode 2


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From the above curve, it can be seen that there are 2 buckling modes observed. With mode 1 occurring at about 375N and

mode 2 at 2025N.

The Axial Strain against load plot for all the members a-c was also done, and was used to analyse the load carrying capacity

of the members and the effect of structural redundancy. The 2kN mark was highlighted to enable a comparison between the

strains obtained using the stiffness matrix method and those measured experimentally.

Member c is seen from the graph to have the lowest load carrying capacity of the three members, it experiences a very high

relative strain to the other members under the same load, member a has the highest. Structural redundancy is a factor that

affects the behaviour of the frame as seen from the graph. This though may be initially seen as a needless addition to the

structure, and economically, discarding the member will be advantageous, it can still be an advantage, as failure of any other

member during the duty of the structure will be compensated by the redundant member. hence it may act as a safety feature

in the structure.

FEAAnsys

The buckling loads obtained from the software Ansys and the nodal displacement was also recorded. They are all as follows.

Memberg

BucklingMode BucklingLoad(N)

1 603.45

2 1233.90

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Member a Member b Member c

-500

-250

0

250

500

750

1000

0 0.75 1.50 2.25 3.00

AxialStrain(strain)AgainstAxialLoad(kN)ForMembersa-cinPlaneFrame

Strain(stra

in)

Load(kN)


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Frame

BucklingMode BucklingLoad(N)

1 734.4

2 1458.8

The nodal displacements x2 and y1 were also recorded to be 0.060911mm and 0.19528mm respectively

Discussion

The obtained results from FEA with the software Ansys were markedly different from the Stiffness matrix results, albeit

maintained the same trend and relative magnitudes.

Under a 2kN load, The stiffness matrix values of the Strains in members a-c and the measured values were similar in

direction, but there was a significant difference in magnitudes. This may be as a result of imperfections/ slack in the pin joints

and as a result of the loading on the test rig not being on the neutral axes. Other problems may also include machine faults,

and the fact that the metals are not perfectly homogeneous and imperfections in grain structure.

A Transition from the two bucking modes was observed experimentally. The observed buckling mode under a particular load

range are preferred to others because they require less energy to be assumed.

The FEA software can be understood best as a fairly accurate representation of the behavior of structures under loading, but

it only calculates what it is told to, rather than what is required. In other words, the accuracy of FAE softwares is down to a

compromise between computational time and mesh/model complexity/simplicity. More accurate or even exact models can

be made and analysed, but the economical implications may be too great. Hence one tries to obtain a model that is as close

as possible to the required, but also notes the computational time and processor speed and a compromise usually gives an

accurate enough solution that can guide design decisions.

Conclusion

The FEA softwares are accurate means of analysis, however they are limited by computational speeds as a result of

increasing model complexity. They are efficient though in guiding us on the right direction as to the behavior of a system and

can serve as reasonable approximations for the otherwise laborious numerical methods.

Structural redundancy can be employed in design of structures under loading as a safety feature in times of failure of a

member comprising a framework, redundant structures maintain static determinacy.

References

[1] University Bath Solid Mechanics 4 ME20023 Notes

[2] G. Lakshmi Narasaiah, Finite Element Analysis, 2008

[3] Felippa, Carlos A. Introduction to Finite Element Method. Fall 2001. University of Colorado. 18 Sept. 2005

[4] University of Bath ME 20023: Plane Frame Lab. 2013

Mustapha Bello

http://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.phphttp://www.devdept.com/fem/books.php

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