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
Chapter 3: Methodology for Finite Element Analysis 24
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
Chapter 3: Methodology for Finite Element Analysis 25
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
Chapter 3: Methodology for Finite Element Analysis 26
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
Chapter 3: Methodology for Finite Element Analysis 27
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,
Chapter 3: Methodology for Finite Element Analysis 28
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)).
Chapter 3: Methodology for Finite Element Analysis 29
(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.
Chapter 3: Methodology for Finite Element Analysis 30
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
Chapter 3: Methodology for Finite Element Analysis 31
+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
Chapter 3: Methodology for Finite Element Analysis 32
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.
Chapter 3: Methodology for Finite Element Analysis 33
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)
Chapter 3: Methodology for Finite Element Analysis 34
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
Chapter 3: Methodology for Finite Element Analysis 35
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.