Geometric Linear And Nonlinear Analysis Of Beam

Geometric Linear And Nonlinear Analysis Of Beam

ABSTRACT:-

Mr. Kashinath N. Borse1, ShailendrakumarDubey 2

1 M.E. Student, Civil Engg.Dept. S.S.V.P.S BSD College of Engg, Dhule, India

2 Associate Professor, Civil Engg.Dept. S.S.V.P.S BSD College of Engg, Dhule, India

The beams are structural elements with thickness smaller than other plan dimensions. These structural elements are used in vast varieties of structures. Hence analysis of beams becomes the topic of interest for civil, mechanical, aeronautical and marine engineer. Now a days steel is an economic and useful material, almost all the structural members are constructed by steel compare to timber and concrete. With the development of construction and manufacturing technology, beams of different shapes and varied sizes are demanded by designers. Analysis of these beams and thin plate attracted attention of many researchers. This paper is addressed to the review of advances, techniques and theoretical background of the non-linear analysis of steel beam. The formulation of beam element in bending has constituted the most exiting area in the development of the solution techniques. If the structure (beams) is made slender along, with bending, membrane action starts coming in picture .The aim of non linear analysis is to predict deflection of beam at various load stages. For present paper two nodes beam element is used for formulation of linear and geometric nonlinear analysis. In the present paper deflection of thin beam is obtained by finite element method in SAP software. The behavior of these flexure members in linear analysis and nonlinear analysis are compared. Some numeric examples are solved.

Keywords: – Finite element method, Steel beam, SAP2000.

INTRODUCTION

Structure is a free-standing, immobile outdoor construction. Typical examples include buildings and non-building structures such as bridges, dams, missile launching tower, transmission line towers. Most of structures are permanent though some structures are temporary, built for some events such as launching pads for spacecrafts, trade shows, conferences or theatre, and often dismantled after use. Temporary structures have fewer constraints relating to future use and durability thus these structures may be made slender and thinner. The flexure members of a structure, namely, beams and plates exhibit linear behavior till deflections are small compared to their thicknesses. As deflections increase, membrane forces are introduced and the external transverse load is supported by membrane- bending action. From this paper one can learn about the differences between linear and non-linear analysis and realize that there are optimum times to use one type of analysis versus the other.

Linear Analysis

Linear analysis (first order analysis) is also known as linear elastic analysis. The term of Elastic means that when the structure is unloaded it follows the same deformation path as when loaded. A linear FEA analysis is undertaken when a structure is expected to behave linearly, i.e. obeys Hooks Law. The stress is proportional to the strain, and the structure will return to its original configuration once the load has been removed. A structure is a load bearing member and can normally classified as a bar, beam, column or shaft. In linear elastic analysis, the material is assumed to be unyielding and its properties invariable and the equations of equilibrium are formulated on the geometry of the unloaded structure. It is assumed that the subsequent deflections will be small and will have insignificant effect on the stability and mode of response of the structure. The linear analysis of the beam and thin plate is done using stiffness method. In this approach the primary unknowns are the joint displacements, which are determined first by solving the structure equation of equilibrium. Then the unknown forces can be obtained through compatibility consideration.

Formulation Linear analysis of beam

A beam is a member predominantly supporting applied load by flexural strength of it. Fig.1 (a) shows a typical beam with its discretisation. Here beam is discretised in elements. The beam is discretised in four elements and having five nodes. Take a typical beam element shown in Fig. (b). It has two nodes, for generating formulation slope and deflection at each node is required.

Fig.1:Typical beam and element

In case of two dimensional structures, the displacement at any point can be expressed by its components w,

dw which are continues function of x.

dx

Therefore, degree of freedom per joint = 2 (i.e. w, )

The displacement within element at any point p(x) can be as follows

The variation of in plane displacement w, dw are prescribe using shape functions

dx

w N w

N 1 w

• N w

  N w

  p 1 1

  x

  2 2 2 x

  1 2

  N and N are Hermition shape function

  wp N1 N1 N2 N2

  w N e

  The linear strains resultant within element can be written as

  2 w

  X x2

  2 N

  1

  1

  2 N w

  1

  1

  2 N

  2

  2

  2 N w

  2

  2

  x2 w1

  1

  1

  x2

  x

  x2 w2

  2

  2

  x 2

  x

  2 N

  2 N

  2 N

  1

  2 N

  2

  1 1 2 2 e

  x2 x 2 x 2 x2

  X B e

  The linear stress resultant i.e. Moment within element can be express as

  M EI X

  M D X

  Strain energy within element is calculated using strain energy over tiny length within element.

  dU 1

  2

  X T

  M

  Then it is integrated over entire element. Thus the strain energy over entire length is

  2

  2

  L

  L

  U 1 X T

  0

  M dx

  L 1 T

  T

  2

  2

  U e

  0

  B D B

  e dx

  1 T

  L T

  U 2 e

  B

  0

  D B e dx

  From the above the element stiffness matrix can be extract as follows

  L

  Non-Linear Analysis

  Se

  0

  BT

  D B dx

  Typical geometric nonlinearity arises from mid plane stretching of a thin structure coupled with transverse vibrations or loading. This stretching leads to a nonlinear relationship between the strain and the displacement. In mathematics, non-linear systems represent systems whose behavior is not expressible as a sum of the behaviors of its descriptors. In particular, the behavior of non-linear systems is not subjected to the principle of superposition, as linear systems. Crudely, a non-linear system is one whose behavior is not simply the sum of its parts or their multiples.

  Formulation of non-linear analysis of beam

  The non-linear analysis of beam is due to the bending of beam, and due to thin thickness of beam the neutral axis of beam is stretched due to this additional axial force is induced in the beam Fig.2 Shows a typical element having length L. it has same thickness throughout its length. It has two nodes and also shows the displacement in x direction and y direction are u and w respectively.

  Fig. 2: Typical nonlinear beam element

  In nonlinear analysis each node of a beam element has three degree of freedom in x direction and z direction

  Degree of freedom per joint = u, w,

  w,

  displacement due to bending

  1

  2

  = 1 , 1 , , 2, 2 ,

  The displacement within element at any point p(x) can be as follows.

  The variations of in plane displacement u, w, are prescribing using shape functions. Fig. 3 shows u1 and u2 displacements in x direction

  Fig. 3: Typical Non-Linear beam element

  = 1 1 + 2 2

  w N w

  N 1 w

• N w

N w

p 1 1

x

2 2 2 x

Where,

1

2

l1and l2 are Lagrangian shape function in x direction

1 = 1 and2 =

The Non-Linear strains resultant within element can be written as

= 0 + 1 2

2

The strains resultant is taken from the theory of elasticity, due to stretching of neutral axis the point p is also displace in z direction. The displacement of a point at distance

z from the middle plane can be as fig.4 shows the displacement of point p in u and z direction.

Fig.4: Displacement of point p in x and z direction

=

=

Now substitute the nodal displacements in strain resultant

2 1 2

=

2 + 2

Where,

is constant across the thickness of beam, 2

2

2

is varies with Z distance

Strain resultant due to axial force within element

=

1 2

+ 2

Strain resultant due to bending within element

= 2

2

2

Therefore

= () 2

2

2

The linear stress resultant i.e. Moment within element can be express as Axial force within element

{N} = [EA] {p}

Moment within Element

{} =

Therefore stress resultant is written in the matrix form is as follows.

= 0

0

NUMERICAL EXAMPLES

To check the validity of the present formulation, some examples are solved by using computer program i.e. Simply- supported beam subjected to a center point load, the beam has a length (L) 700mm, width (b)50mm and depth (h) 10mm,05mm.

Behavior of geometric nonlinear Example-

W KN

b

L/ L/ h L

1. Simply supported beam (b) cross section of beam

Fig.5: Simply supported beam and its cross section

The simply supported beam over uniformed distributed load is increase gradually in this cases non-linearity so very large it shows following cases

The deflection of beam by theoretical calculated formula is given below

wL3

48EI

The above formula is most suitable for calculating deflections of linear analysis for thick beam. For thin beams this formula is not suitable. Hence we use SAP software for calculated nonlinear analysis.

Case 1- Thickness of beam = t = 10 mm

Table 1: Loading and deflection for thickness 10 mm

Case 1- Thickness of beam = t = 5 mm

Table 2: Loading and deflection for thickness 5 mm

Fig. 6 Load Vs Deflection curve theoretical, SAP and nonlinear (10mm)

1400

1200

1000

DEFLECTION IN MM

DEFLECTION IN MM

800

600

By Theoretical Linear By SAP NonLinear

By SAP Linear

400

200

0

0 2 4 6 8 10 12 14 16 18 20

LOAD IN KN

Fig.7: Load Vs Deflection curve theoretical, SAP linear and nonlinear (5mm)

CONCLUSIONS:-

The general purpose finite element software SAP2000 was used to conduct the linear and nonlinear analysis. Certain examples are analysed by SAP2000 and the results are compared with theoretical calculation.The studies on Software and theoretical results associated with them lead to the following conclusions:

1. When loads intensity is small there is a very small (Negligible) variation between theoretical and SAP 2000in linear deflection.

2. P-delta value shows the significant geometric nonlinear analysis as compared to the linear analysis in this study.

3. Geometric nonlinearity is not induced in the thick beam when the load intensity is small and it is induced in the thick beam when the big load intensity goes on increasing.

4. Geometric nonlinearity is induced in the beam because of its lesser thickness (thinner). it is produced more in the thinner beam, when load is increased.

5. As the stretching of middle plane (Neutral axis) starts, the stiffness of structure increases (axial stiffness is added with bending stiffness).Thus the beam becomes stiffer progressively. It is a positive aspect of geometric nonlinearity.

References

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   ,1963.

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