6 chapter 3

Chapter 3: Methodology for Finite Element Analysis 23

Chapter 3

METHODOLOGY FOR FINITE ELEMENT ANALYSIS

3.1 INTRODUCTION

Finite Element calculations more and more replace analytical methods

especially if problems have to be solved which are adjusted to specific tasks. In

many countries, a lot of efforts are carried out to get new code standards for the

calculation of ultimate load capacity of single steel angles under eccentric loadings.

All these calculation methods are based on linear descriptions of the material

behavior. Concerning the non-linear and time dependent characteristics of materials,

standard linear elastic finite element calculations in addition to code methods are

often not suitable.

Therefore, a new finite element model was developed to describe the real

(elastic-plastic) behavior of the single steel angle under eccentric edge loads.

Besides, an exact geometric modeling the description of the material behavior of all

components is very important for the quality of performed analysis. This applies to

analytical as well as to numerical methods. For components made of steel elastic or

elastic-plastic material laws are able to simulate the real behavior of those parts in

sufficient accuracy.

The actual work regarding the finite element modeling of a single steel angle

connected to end plates has been described in detail in this chapter. The

representation of various physical elements with the FEM (Finite Element

Modeling) elements, properties assigned to them, boundary conditions, material

behavior and analysis types have also been discussed. The various obstacles faced

during modeling, material behavior used and details of finite element meshing were

also discussed in detail.

3.2 THE FINITE ELEMENT PACKAGES

A large number of finite element analysis computer packages are available

now. They vary in degree of complexity and versatility. The names of few such


packages are:

- ANSYS (General purpose, PC and work stations)

- DYNA-3D (Crash / impact analysis)

- SDRC/I-DEAS (Complete CAD / CAM / CAE packages)

- NASTRAN (General purpose FEA on main frames)

- ABAQUS (Non-linear and dynamic analyses)

- COSMOS (General purpose FEA)

- ALGOR (PC and work stations)

- PATRAN (Pre / post processor)

- Hyper Mesh (Pre / post processor)

Of these packages ANSYS10.0 has been chosen for its versatility and

relative ease of use. ANSYS is capable of modeling and analyzing a vast range of

two- dimensional and three-dimensional practical problems. Buckling analysis of a

real structure (calculation of buckling loads and determination of the buckling mode

shape) can be performed quite satisfactorily by means of this software. . Both linear

(eigenvalue) buckling and nonlinear buckling analyses are possible

3.3 FINITE ELEMENT MODELING OF THE STRUCTURE

Figure 3.1: General sketch of a single steel angle with end plates at its both ends subjected to eccentric load.

End plate

Steel angle

Applied force


3.3.1 Modeling of Steel Angle and End Plates To facilitate the non-linear buckling analysis of the whole system, modeling

procedure has been simplified by eliminating bolts and considering end plates at the

two ends of the steel angle.

Since the whole modeling was performed in 3-dimension, the element used

here is 3-D in nature. For representing both the steel angle and the end plates,

SHELL-181(a 4 node structural shell element) has been used. Discussion about the

element is shown below in details:

SHELL181 Element Description

SHELL181 is suitable for analyzing thin to moderately-thick shell structures.

It is a 4-node element with six degrees of freedom at each node: translations in the x,

y, and z directions, and rotations about the x, y, and z-axes. (If the membrane option

is used, the element has translational degrees of freedom only). The degenerate

triangular option should only be used as filler elements in mesh generation.

SHELL181 is well-suited for linear, large rotation, and/or large strain

nonlinear applications. Change in shell thickness is accounted for in nonlinear

analyses. In the element domain, both full and reduced integration schemes are

supported. SHELL181 accounts for follower (load stiffness) effects of distributed

pressures.

Figure 3.2: SHELL181 Geometry


xo = Element x-axis if ESYS is not provided.

x = Element x-axis if ESYS is provided.

SHELL181 Input Data The geometry, node locations, and the coordinate system for this element are

shown in "SHELL181 ". The element is defined by four nodes: I, J, K, and L. The

element formulation is based on logarithmic strain and true stress measures. The

element kinematics allows for finite membrane strains (stretching).The thickness of

the shell may be defined at each of its nodes. The thickness is assumed to vary

smoothly over the area of the element. If the element has a constant thickness, only

TK(I) needs to be input. If the thickness is not constant, all four thicknesses must be

input.

A summary of the element input is given in below (Table 3.1).

Table 3.1: SHELL181 Input Summary

Element name SHELL181 Nodes

I, J, K, L

Degrees of Freedom

UX, UY, UZ, ROTX, ROTY, ROTZ if KEYOPT (1) = 0

UX, UY, UZ if KEYOPT (1) = 1

Real Constants TK(I), TK(J), TK(K), TK(L), THETA,

ADMSUA

E11, E22, E12, DRILL, MEMBRANE, BENDING

Material Properties

EX, EY, EZ, (PRXY, PRYZ, PRXZ, or NUXY, NUYZ, NUXZ),

ALPX, ALPY, ALPZ (or CTEX, CTEY, CTEZ or THSX, THSY, THSZ),

DENS, GXY, GYZ, GXZ


SHELL181 Assumptions and Restrictions:

Zero area elements are not allowed (this occurs most often whenever the

elements are not numbered properly).

Zero thickness elements or elements tapering down to a zero thickness at any

corner are not allowed (but zero thickness layers are allowed).

In a nonlinear analysis, the solution is terminated if the thickness at any

integration point that was defined with a nonzero thickness vanishes (within

a small numerical tolerance).

This element works best with full Newton-Raphson solution scheme.

The through-thickness stress, SZ, is always zero.

3.3.2 Material properties

Bilinear Kinematic Hardening

The Bilinear Kinematic Hardening (BKIN) option assumes the total stress

range is equal to twice the yield stress, This option is recommended for general

small-strain use for materials that obey von Mises yield criteria (which includes

most metals). It is not recommended for large-strain applications.

Figure 3.3: Bilenear kinematic hardening

T1

Strain,

y

E1

1

E2 1 y

Stre

ss,


Where,

y = Yield stress

y = strain corresponding to yield stress

E1 = Modulus of elasticity upto yield point

E2 = Modulus of elasticity after exceeding yield point

T1 = Temperature for material 1.

3.4 TYPES OF BUCKLING ANALYSES Two techniques are available for performing buckling analyses - nonlinear

buckling analysis and eigenvalue (or linear) buckling analysis. These two methods

frequently yield quite different results.

3.4.1 Nonlinear Buckling Analysis

Nonlinear buckling analysis is usually the more accurate approach. It is

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

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

which a structure becomes unstable. Using the nonlinear technique, models can

include features such as initial imperfections, plastic behavior, gaps, and large-

deflection response. In addition, using deflection-controlled loading, the post-

buckled performance of structures (which can be useful in cases where the structure

buckles into a stable configuration, such as "snap-through" buckling of a shallow

dome)can be obtained (Figure 3.4: “Buckling Curves” (a)).

3.4.2 Eigenvalue Buckling Analysis

The second method, eigenvalue buckling analysis, predicts the theoretical

buckling strength (the bifurcation point) of an ideal linear elastic structure. This

method corresponds to the textbook approach to elastic buckling analysis: for

instance, an eigenvalue buckling analysis of a column will match the classical Euler

solution. However, imperfections and nonlinearities prevent most real-world

structures from achieving their theoretical elastic buckling strength. Thus,

eigenvalue buckling analysis often yields unconservative results, and is not generally

used in actual day-to-day engineering analyses (Figure 3.4: "Buckling Curves” (b)).


(a) Nonlinear load-deflection curve (b) Linear (Eigenvalue) buckling curve

Figure 3.4: Buckling Curves

The Arc-Length Method:

One major characteristic of nonlinear buckling, as opposed to eigenvalue

buckling, is that nonlinear buckling phenomenon includes a region of instability in

the post-buckling region whereas eigenvalue buckling only involves linear, pre-

buckling behavior up to the bifurcation (critical loading) point (Figure 3.5).

Figure 3.5: Nonlinear vs. Eigenvalue Buckling Behavior

The unstable region above is also known as the “snap through” region, where

the structure “snaps through” from one stable region to another. To illustrate,

consider the shallow arch loaded (Figure 3.6) may be considered.


Figure 3.6: “Snap Through” Buckling

For most nonlinear analyses, the Newton-Raphson method is used to

converge the solution at each time step along the force deflection curve. The

Newton-Raphson method works by iterating the equation [KT]{u}={Fa}-{Fnr},

where {Fa} is the applied load vector and {Fnr} is the internal load vector, until the

residual, {Fa}-{Fnr}, falls within a certain convergence criterion. The Newton-

Raphson method increments the load a finite amount at each substep and keeps that

load fixed throughout the equilibrium iterations.

Because of this, it cannot converge if the tangent stiffness (the slope of the

force-deflection curve at any point) is zero (Figure 3.7).

Figure 3.7: Newton - Raphson Method

To avoid this problem, the arc-length method should be used for solving

nonlinear post-buckling. To handle zero and negative tangent stiffness, the arc-

length multiplies the incremental load by a load factor, λ, where λ is between -1 and


+1. This addition introduces an extra unknown, altering the equilibrium equation

slightly to [KT]{u} = λ{Fa}-{Fnr}. To deal with this, the arc-length method imposes

another constraint, stating that

7 (3.1)

throughout a given time step, where is ℓ the arc-length radius. Figure 3.8

illustrates this process.

Figure 3.8: Arc-Length Methodology The arc-length method therefore allows the load and displacement to vary

throughout the time step as shown (Figure 3.9).

Figure 3.9: Arc-Length Convergence Behavior


3.5 FINITE ELEMENT MODEL PARAMETERS In this analysis small deflection and plastic materials properties (material

nonlinearity) are considered. The following properties are used in the modeling.

Table 3.2: Various input parameters

Parameter Reference value

Angle dimension in X-direction 102 mm

Angle dimension in Y-direction 102 mm

Thickness of angle 6 mm

Thickness of end plate 25 mm

Corner dimension of end plate excluding the

angle (on each side)

25 mm

Center of gravity of bolt pattern 38 mm

Young’s modulus of elasticity 200 KN/mm2

Yields stress for the angle .3259 KN/mm2

Poison’s ratio .3

Applied load Slightly greater than critical

buckling load for the steel

angle

3.6 MESHING

3.6.1 Meshing of the End Plate The end plate is divided along both of its axes (X-axis and Y-axis in the

global co-ordinate system). Individual division is rectangular. Number of division is

chosen in such a way that the aspect ratio is of the element is reasonable.

3.6.2 Meshing of the Steel Angle

The length of the steel angle (in Z-direction) perpendicular to the axis of the

end plate is divided into sufficient number of divisions. Individual division is

rectangular. Number of division is chosen in such a way that the aspect ratio is of the

element is reasonable.


3.7 BOUNDARY CONDITIONS

3.7.1 Restraints In case of the bottom end plate, corresponding node at the location of center

of gravity of the bolt pattern, which simulate the contact, is kept restrained in all

direction(in X,Y and Z-directions). But, in case of the the top end plate, the node at

the same location, was restrained in X and Y direction only to allow the free

deflection in the Z direction (along the length of the steel angle).At the right most

corner of the top end plate, only the deflection in the X direction has been kept

restrained. In all cases, the whole model is kept unrestrained against rotation. These

options are allowed to facilitate the non-linear buckling analysis of the system.

3.7.2 Load

In this case, the steel angle is given an applied load slightly higher than its

critical buckling load at node at the location of center of gravity of the bolt pattern

on the top end plate along the length of the steel angle.

Figure 3.10: Preliminary model of a single steel angle connected to end plates at its both ends (prior to meshing)


Figure 3.11: Finite elements mesh of the steel angle with end plates at its both ends

Figure 3.12: Finite elements mesh with loads and boundary conditions


Figure 3.13: Typical deflected shape of the model

0

50

100

150

200

250

300

350

400

450

0.00 1.00 2.00 3.00 4.00 5.00 6.00

Displacement, mm

Load

, kN

10

20

40

90

110130

l/r

Figure 3.14: Typical load vs deflection curve for different slenderness ratio obtained from non-linear buckling analysis of L102x102x6 steel angle.

6 chapter 3

Documents

top end plate

general purpose

finite element

single steel

finite elements

eigenvalue

critical buckling

finite element