Dynamic Analysis of a structure by the finite element method

207

International Journal of Innovative and Emerging

Research in Engineering

e-ISSN: 2394 - 3343 p-ISSN: 2394 - 5494

Dynamic Analysis of a structure by the finite element

method

Shivanshi

_{, Ankush Kumar Jain}

_{, Pinaki}

a_{Assistant Professor, Department of Structural engineering, Poornima college of Engineering, Jaipur, India} b_{Assistant Professor, Department of Structural engineering, Poornima University, Jaipur, India}

c _{B.Tech.student, , Gurukul Institute of Engineering and Technology, Kota, India}

ABSTRACT:

The finite element method is used to solve problems by dividing the whole system in to coordinates. In the structural dynamics we study about the frequency and modes of the structure subjected to static and dynamic loading. It mainly concerned with finding out the behavior of a physical structure when subjected to force. So we study a structure for the static and dynamic loading. Dynamic analysis for simple structures can be carried out manually, but for complex structures finite element analysis can be used to calculate the mode shapes and frequencies. In this work, we would limit our scope to the analytical study of the structure considering beams and rigid frames by structural dynamics using finite element method.

Keywords: structural dynamics, finite element method, dynamic analysis, mode ,frequency

I. INTRODUCTION

A dynamic analysis is also related to the inertia forces developed by a structure when it is excited by means of dynamic loads applied suddenly (e.g., wind blasts, explosion, earthquake).A static load is one which varies very slowly. A dynamic load is one which changes with time fairly quickly in comparison to the structure's natural frequency. If it changes slowly, the structure's response may be determined with static analysis, but if it varies quickly (relative to the structure's ability to respond), the response must be determined with a dynamic analysis. A dynamic load can have a significantly larger effect than a static load of the same magnitude due to the structure's inability to respond quickly to the loading (by deflecting). The increase in the effect of a dynamic load is given by the dynamic amplification factor (DAF) or dynamic load factor (DLF) where u is the deflection of the structure due to the applied load.

DAF=DLF = umax / ustatic where u is the deflection of the structure due to the applied load

Graphs of dynamic amplification factors verses non-dimensional rise time (tr/T) exist for standard loading functions

(for an explanation of rise time, see time history analysis below). Hence the DAF for a given loading can be read from the graph, the static deflection can be easily calculated for simple structures and the dynamic deflection found.In FEM to solve the problem, it subdivides a large problem into smaller, simpler parts that are called finite elements. The simple equations that model these finite elements are then assembled into a larger system of equations that models the entire problem. FEM then uses variational methods from the calculus of variations to approximate a solution by minimizing an associated error function.

II. LITERATURE REVIEW

208 III. ANALYSIS

(A) OBJECTIVE

In this, the beams and rigid frames as the components of structure are being analytical studied for the structural dynamics behaviour by the Finite Element Method .The structural dynamics is the study of structure for the dynamic loading as well as static load applying on the structure. So in this we will focus on the relationship among the modes, frequencies of the structure for the dynamic loading and fragmenting requirements in Finite Element Method.

(B) DESCRIPTION

The typical beam element is shown in Fig. Note the orientation of axes is as per the right hand thumb rule.[1]

“Figure 1.Beam element “[2] The nodal variable vector is

{δ}T _{= [ δ}

1 δ2 δ3 δ4 ] = [w1 ϴ1 w2 ϴ2]

where wi–lateral displacement at node i.

ϴ

N1 = 1 – 3s2 + 2s3 N2 = le s(s – 1)2 N3 = s(3 – 2s) N4 = le s2(s – 1)

209 “Figure 2.Equivalent nodal loads”[3]

The reactions at supports are nothing but end equilibrium forces.

In this we will see the dynamic behaviour of structures here as beam i.e. that carry loads that are transverse to the longitudinal direction thus producing flexural stresses and lateral displacement Two approximate methods are presented to take in to account the inertial forces in structure by dynamic effects: One is Lumper mass method in which the assignment to the point masses includes rotational effects. The latter method is consistent with the static deflection of the beam.

“Figure 3.Beam segment showing forces and displacement at the nodal coordinates”[1]

A uniform beam element of cross sectional moment of inertia I, length Land material modulus of elasticity E is considered. We will establish the relationship between static forces and moments designated as P1 P2 P3 and P4 and

210

“Figure 4.Beam element showing static deflection curves due to a unit displacement at one of the nodal coordinates”[1]

Since N1(x) is the deflection corresponding to a unit displacement δ1

=1,the displacement resulting from an

arbitrary displacement δ

is

and so on, then the total deflection is u(x).For a uniform beam

element the evaluation of coefficients of stiffness matrix as[1]

The natural frequencies and mode shapes for uniform beams are there for the following cases[1]:

(a) Both ends simply supported (b) Both ends free(Free Beam) (c) Both ends fixed

(d) One end fixed and the other end free(Cantilever beam) (e) One end fixed and the other simply supported

211 IV. RESULTAND CONCLUSION

In this we have formulated the dynamic equations for beams in reference to a discrete number of nodal coordinates. These coordinates are translational and rotational displacements defined at joint between structural elements of the beam. The dynamic equations for a linear system can be written in matrix notations as:

|M|{ű}+|C|{ů}+|K|{u} = {F(t)}

Where F(t )is the force vector,[M],[K],[C] are mass, stiffness matrices and damping of the structure.[3]

The dynamic analysis of single span beams with distributed properties (mass and elasticity) and subjected to flexural loading is done. The results are particularly important in evaluating approximate methods. It has been found that stiffness method of dynamic analysis in conjugation with a rather discretization of structure.[5]

ACKNOWLEDGMENT

First of all I feel great pleasure in acknowledging my deepest gratitude to all the researchers who carried their work in this field.

I would also like to thank to entire members of IJIERE for providing me the best platform for successful completion of work and presenting the research work.

REFERENCES

[1] Friswell and mottershed,Swansea,Finite element model updating in structural dynamics 1994 [2] Mario paz and William leigh ,Theory and computation, fifth edition

[3] Manish shrikhande Finite element method and computational structural dynamics, [4] Paulo B.Lourenco ,Types of analysis:Linear dynamic and non linear static

[5] A.thorin,P.Deleziode and M legrand, non smooth modal analysis of piecewise linear impact occilators,preprint,2016

[6] T Belystscko,DF S cheoeberle,On the unconditional stability of an implicit algorithm for non linear structural dynamics , 1975