A Finite Element Approach to Ball Grid Array Components in
Common Aerospace Random Vibration Environments
by
Milan J Lucic
An Engineering Project Submitted to the Graduate
Faculty of Rensselaer Polytechnic Institute
in Partial Fulfillment of the
Requirements for the degree of
MASTER OF ENGINEERING IN MECHANICAL ENGINEERING
Approved:
_________________________________________
Ernesto Guiterrez-Miravete, Project Adviser
Rensselaer Polytechnic Institute
Hartford, Connecticut
December 2010
(For Graduation August 2011)
i
© Copyright 2010
by
Milan J Lucic
All Rights Reserved
ii
CONTENTS
LIST OF TABLES ............................................................................................................. v
LIST OF FIGURES .......................................................................................................... vi
NOMENCLATURE ........................................................................................................ vii
ACKNOWLEDGMENT .................................................................................................. ix
ABSTRACT ...................................................................................................................... x
1. INTRODUCTION ....................................................................................................... 1
1.1
Background Information ..................................................................................... 1
1.2
Lead vs. Lead Free Solder .................................................................................. 1
1.3
Random Vibration for Fixed Wing Aircraft ....................................................... 2
2. THEORY AND METHODOLOGY ........................................................................... 3
2.1
Analytical Model: PCB Normal Modes and Displacement ................................ 3
2.2
Finite Element Model: Random Vibration ......................................................... 6
2.3
FEA Model Optimization ................................................................................... 9
3. RESULTS AND DISCUSSION ................................................................................ 10
3.1
Analytical Results ............................................................................................. 10
3.2
FEA Model Optimization ................................................................................. 12
3.3
Finite Element Model ....................................................................................... 13
3.3.1
Finite Elements .................................................................................... 13
3.3.2
Material Properties ............................................................................... 14
3.3.3
Finite Element Model ........................................................................... 15
3.3.4
Natural Frequencies ............................................................................. 19
3.3.5
Random Analysis ................................................................................. 21
3.3.6
Inertial Analysis ................................................................................... 25
3.3.7
BGA Stress Results .............................................................................. 29
3.4
Miner's Damage Index Calculations ................................................................. 33
3.5
Fatigue Life ....................................................................................................... 34
iii
4. CONCLUSIONS ....................................................................................................... 36
4.1
Future Work and Model Improvement ............................................................. 36
Literature Cited ................................................................................................................ 37
Appendix A...................................................................................................................... 38
Appendix B ...................................................................................................................... 39
Appendix C ...................................................................................................................... 49
Appendix D...................................................................................................................... 50
iv
LIST OF TABLES
No table of figures entries found.
v
LIST OF FIGURES
No table of figures entries found.
vi
NOMENCLATURE
BGA- Ball Grid Array
PBGA – Plastic Ball Grid Array
PWB – Printed Wiring Board
MPC – Multi Point Constraint
FEM – Finite Element Mesh
FEA- Finite Element Analysis
RMS – Root Mean Square
**Add equations
vii
viii
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
ix
ABSTRACT
There are many analyses about electrical components concerning mechanical stresses in
electronics packaging for the aerospace industry. This approach is to investigate
common aerospace random vibration environments for Ball Grid Array (BGA)
components, specifically the 484-ball package. The Analysis includes looking at lead
vs. lead free solder packages, different vibration environments, and circuit card assembly
(CCA) placement. The result is not to find whether lead is better than lead free solder or
what placement is best for BGA components, but what combination of the two works
best in common vibration environments. The results will only take into account
mechanical stresses of each ball and the entire package to determine the optimized
placement and solder type.
x
1. INTRODUCTION
1.1 Background Information
The purpose of this project is to take a look at some of the major issues with electronics
packaging in the aerospace industry today. Namely, the application of Ball Grid Arrays
(BGAs) on rack style printed wiring boards (PWBs). There are many factors that affect
the BGA components on the circuit board: placement on the PWB, vibration level,
temperature levels, and solder type. In addition to all of these variables, every design is
slightly different with different resonances, and requirements. Rack card assemblies do
not have the ability to control the PWB resonances around the BGAs like other PWB
designs without significant design improvements; the goal is to keep the design simple.
The BGA that will be analyzed is the ACTEL 484 BGA. BGAs are used for
programmable logic devices as well as processors for high amounts of inputs and
outputs. These are essentially the brains of the electronics, and it is critical to make sure
they operate under all conditions. Lastly, BGA’s are unreliable in higher vibration
environments so proper understanding of how each of the above variables affects the
component is critical to having a good design the first time, and to not run development
testing under common environments that are shown in this paper.
Add information about steinberg and his book…
1.2 Lead vs. Lead Free Solder
The main reason for looking at lead free solder is to transform the aerospace electronics
to green lead free solder. Each of the packages in the ACTEL 484 BGA are built as lead
free. This is because these components are not only for aerospace applications, but also
for cell phones or computers too. The aerospace field is looking to go to lead free
components for many reasons including green compliance, ROHS compliance, and to
avoid obsolescence of key parts, mostly due to the fact that aerospace products life is up
to 20 years of service. The difficulty is that high volume products whose life is around 12 years, which means high risk or a part obsolescence, drive the electronics industry. In
addition to different mechanical and thermal properties, lead free solder has concerns
1
with tin whisker growth, which will not be explained in this paper. The last reason that
aerospace is behind in making the change to lead free solder is due to the different
techniques and procedures to assemble these parts to the PWBs. Lead based solder will
not be around forever, hence the importance of studying the differences of lead vs. lead
free solders and their mechanical responses in BGAs.
1.3 Random Vibration for Fixed Wing Aircraft
Electronic controllers are not structural members of the aircraft and thus cannot be
analyzed by simple static analysis. The mechanical stress analysis is governed by the
requirements from RTCA-DO-160, which is an aerospace standard for environmental
tests. Obviously one cannot test the electronics under the actual loads of the aircraft
because it would take years to find out a result. RTCA DO-160 accelerates the vibration
environments to test each axis in 1-5 hours on average. The random FEM analysis is
used primarily to solve frequency responses and find resonances/mode shapes of the
PWBs. Typically this analysis is matched with test data to calculate the number of cycles
to failure.
The difference between sine vibration and random vibration is that for random vibration,
At any point in time, any frequency can be happening that will excite, or resonate,
multiple bodies at once. What this means is although one member of the chassis design
may have a low frequency, and another be high, they can both be resonating at the same
instant in a random vibration environment. The sine environment will run the input
curve over the frequency band from 20 to 2000 hertz that can be experienced in the
aircraft and only one member will resonate at a time, assuming of course two members
do not share a natural frequency. Most of the time a sine scan is run to determine what
the resonances of the chassis and circuit cards are. The input is a simple 1G input over
the entire frequency band, and is not meant to induce fatigue damage or failure. This
data is usually used to validate the FEA before certification or qualification testing
occurs.
2
2. THEORY AND METHODOLOGY
2.1 Analytical Model: PCB Normal Modes and Displacement
The analysis will be done based on a 6”x9” Printed Circuit Board (PCB). Of course there
are many sizes and configurations that can be used to optimize the board design, but
boards of this size are common in the lower fuselage electronics bays of aircraft.
Placements will include the center of the board, upper right corner, lower left corner and
left center. This is shown by the figure below:
Location 2
Location 1
Location 3
Figure 2.1-1: PWB BGA Placement
Each placement on the board will have different results based on the curvature of the
board and the board displacement at that point. This means that every single solder ball
will have a unique max stress based on the curvature and placement. This is one key
item to find in the analysis.
The first thing about doing a Finite Element Model, analytical calculations are
invaluable to determining if the solution is correct. There are some simple equations
from Steinberg’s Vibration Analysis for Electronic Equipment. The first equation is used
to solve for the first mode natural frequency of a PWB that is fixed on 3 edges and has 1
free edge (fixed means that the edge is controlled in all 6 degrees of freedom). The FEM
boundary conditions are modeled to be exactly like a circuit card in a chassis. The fixed
positions are from the connectors at one end of the PWB that connect to the
interconnect, and also the two card guides on the long edge of the PWB.
3
Interconnect
Fixed Edge (6DOF) (Card Guide)
b
Connectors
Fixed Edge (6DOF)
Free Edge
a
a
Fixed Edge (6DOF) (Card Guide)
Figure 2.1-2: PWB and Inteconnect Diagram
D
fn 

E * h3
[3]
12 * (1   2 )
 D .75
2
12 
   4  2 2  4  [3]
a
3 
a b b 
The first mode natural frequency is based upon the Young’s Modulous, thickness,

poisson’s ration, which is the plate stiffness D, the density and the length and width; a
and b respectively. The second equation is to solve for the displacement of the PWB
based off the first mode natural frequency at the center of the board. :
G

2
* P * Q* f n [3]
Z RMS 

9.81* G
[3]
fn2
The displacement is really a dynamic single amplitude response based on the first mode

natural frequency and G, or GRMS. GRMS is the response of the PWB based off the
4
transmissibility, Q, first mode natural frequency, and the power spectral density value
where the first mode is. The transmissibility is then calculated through the random
analysis and post processing the data. These results are then compared for accuracy
based on the FEM. Although the displacement calculated is accurate, it does not give
you the worst case. Steinberg uses a 3 band method approach to determine max
displacements of the PWB which is used in the equations to determine BGA high cycle
fatigue life. This is the point where maximum damage will occur in the electrical
components. Although based on this statistical approach will only occur 4.33% of the
time, it must be considered in the overall damage calculations.
Figure 2.1-3: Gaussian Distribution for Stienberg 3 Band Method [3]
The basic idea is that when the number of cycles reaches the number of allowed or
calculated cycles, the part will ideally fail. Traditionally, this calculation is used to
ensure that this never happens for any electrical component; BGA’s do not always
follow the same “simple calculations” and have relatively short cycle lives. There is a lot
that can go wrong with a single BGA: improper installation, solder joint failures due to
thermal or structural inputs, and for each of these failures, every solder ball can be
slightly different. Controlling this environment is crucial as well as the processes
involved in assembling the BGA’s to PWB’s. The assumption for this analysis is that all
the balls are the same and resemble an acceptable solder joint. The model of this is based
upon a report by David M. Pierce, Fatigue Life Prediction Methodology for Lead-Free
Solder Alloy Interconnects: Development and Validation described in chapter 3.3.3.
5
2.2 Finite Element Model: Random Vibration
The FEM will be created in PATRAN with MD enabled version 2010 and solved with
MD NASTRAN 2008. The FEM is composed of a few different types of elements to try
and decrease the model size. The PWB and BGA body consists of 2-D Quad-8 elements,
and 2-D Quad 4 elements. The Solder balls in the BGA will be modeled with 3-d Hex-20
elements for detail and accuracy. All of the elements are tied together using a glue
constraint for deformable bodies through PATRAN. The Glue constraint makes it much
easier to connect multiple bodies and greatly reduce the number of user input REB2 or
REB3 Multi point constraints.
The method that is used to solve the problem of ball grid arrays and common aerospace
random vibration environments is composed of multiple independent steps. The first is
to simply run a modal analysis of the PWB with no BGA attached. Having the BGA
attached to the PWB will not affect the stiffness to a degree of concern and the model
can now be much smaller. The modal analysis should line up with the calculated results;
this is the success criterion for this step. The next step involves developing a model for
random vibration cases. Again the simple PWB model used for the modal analysis is
used to keep the model size down during dynamic analysis. The result that is desired
here is to find the max displacement of the PWB and then compare this to the hand
calculations, again which should line up for each vibration case. The difference here is
that for each random vibration case, although the natural frequencies are going to be the
same as long as the boundary conditions are the same, the transmissibility will not be.
This is the term that is calculated from the room mean square (RMS) of the vibration
response of the board. From the modal analysis, it is easy to narrow down the area where
the Q will be the highest and thus the highest displacement. The hand calculations are
modified and again compared with the FEM results to validate this step. In the idea of
keeping the model as light as possible and the iterations as few as possible, it was not
possible to run the full model with the BGA on the PWB. Instead, taking the results from
the random acceleration analysis, determining the max displacement, it is just as easy to
run a static inertial analysis on the full FEM. This dramatically reduces the computation
time for the study at hand. Although this deviates slightly from the original plan of
6
running dynamic simulation for each case, the results are very close to one another. This
is a standard industry practice that is creative and very time forgiving.
Now that the thought process is given for how the FEM will be solved, the boundary
conditions must be analyzed. The most important boundary condition or load case is of
course the random vibration environments that are analyzed. The figure below from
RTCA DO-160F shows the vibration curves for fixed wing aircraft:
Figure 2.2-1: RTCA DO-160 Vibration Levels for Fixed Wing Aircraft [1]
7
The PWB boundary conditions are simple as described in Steinberg for the 3 fixed edge
case. This approach is chosen after a popular robust design in aerospace rack card
assemblies. This design includes the use of wedge lock card guides. Below is a picture
representation of what they are and how they work:
Figure 2.2-2: Wedge Lock Card Guides [3]
The wedge lock card guide essential keeps the board edge fixed (all 6 degrees of
freedom) for input loads of 8G’s or less [3]. In turn this gives the designer a much more
stiffness that will have less displacement at the board center due to higher first modes.
The last edge that is fixed comes from connectors on the daughterboard connecting to
the motherboard as seen in Figure 2.1-2. Although this connection may not always be
completely fixed due to tolerances in the rack assembly and improper mating with the
interconnect board, it is assumed that the mating conditions are fixed.
The last boundary condition for this model is the mating condition of the solder balls to
the PWB and the solder balls to the BGA package. There are several ways to do this
which include user defined Multi Point Constraints (MPCs), REB2 or REB3, Mesh
matching, or lastly the Glue constraint. The REB2 MPC is used as a rigid or bolted
connection which would be far to stiff for this application. The REB3 element would be
idea but very cumbersome, about 968 connections to match the solder balls to the PWB
and the BGA body. Mesh matching is typically a very good option, although for this
case, the mesh is very fine, .008 inches, which results in a model too large to solve for
either dynamic or static modeling with the computer hardware available. Lastly the Glue
boundary can be applied to the solid geometry to which the mesh is associated. This
boundary sets up the model with REB3 connections that are ideal for this analysis case.
These REB3 connections are created when the solution is solved in NASTRAN rather
8
than PATRAN. The tolerances of these connections are set within PATRAN and are
usually a percentage of the smallest shell or solid element. More detail will be discussed
in the results chapter about these boundary conditions.
Lastly the elements have all been chosen for a reason. The hex-8 shell elements are used
for all shell elements. The hex-8 elements allow the mesh on the PWB and the BGA
body to be much more course due to the larger number of nodes. This is necessary to
keep the model size down. For the 3-d solid solder balls, the mesh was done by taking a
section view of the solder ball and doing a surface mesh seed. This 2-d mesh seed was
then revolved to get the spherical shape of the solder ball. Hex-8 elements were used for
the 2-d mesh seed so a more course mesh could be used and the element edges that are
along the spherical edge actually resemble a sphere and map to the surface much better.
A very fine hex-4 can be used to get the same result, but is much heavier in the model. A
detailed description of the FEM will be described in the results section.
2.3 FEA Model Optimization
FEA model optimization is crucial for this problem as the model size is a few hundred
thousand elements and many more nodes. Since there are only three different bodies in
this analysis, the PWB, BGA solder balls, and the BGA body, there are a few different
mesh optimization ideas. Firstly the solder balls must be detailed and accurate, but
cannot slow down the model and have such a fine mesh that the model will not even
converge in a reasonable time. This will be done by looking at static cases of single
solder balls under static loading conditions. Displacement and stress gradients are
observed to get the most course mesh yet highly accurate results. When using such a fine
mesh on the balls, the model must also have a very fine mesh. Since this is the case, the
PWB need not be optimized for mesh size, but for amount of total elements. This will be
done as the analysis takes place and if the model has trouble running with so many
elements.
9
3. RESULTS AND DISCUSSION
3.1
Analytical Results
The analytical section for the results is not intended to exactly calculate results but to
validate the FEA model. This is talked about in section chapter 2.1 under the analytical
model methodology. The first step is to calculate the first mode natural frequency of the
PWB. This is calculated using Steinberg’s equation for a thin plate with three fixed
edges. This solution is derived using the Raleigh method and shown as equation 1. Note
that the length and width are not the 9in by 6in as stated above. This is due to the wedge
lock card guides and the constraints that are made on the FEM. The length and width
match the FEM exactly.
PWB length, a = 8.85 in
PWB width, b = 5.7 in
Thickness, h = .1 in
Adjusted Weight, W = .9819 lbs
Gravity, g = 386.4 in/sec2
Poisson’s Ratio, v = .3
PWB Stiffness, D = 265.57 Lbs in
Adjusted Density, p (W/gab) = 5.0375e-5 lbs s2/in3
fn = 266.4 Hz
The next hand calculation is to calculate the maximum board displacement at the center
of the PWB. For this model, this is not the maximum displacement due to the fact that
only 3 edges are fixed. If 4 edges were fixed or simply supported, the maximum
displacement would be directly at the center. The maximum displacement will really
reside at the center of the unsupported edge. This value will be found through the
random analysis of the PWB. To calculate the displacement at the center of the board,
the GRMS must be calculated. This is something that is usually calculated with the
random analysis tool, so it may have to be modified after to validate the results. Also the
transmissibility, Q, will need to be modified after the random analysis is run, so an initial
value of 20 will be used. For each random category, there is a different value for GRMS
and Z, displacement, will be different. The calculations for each of the other cases will
10
be shown in Appendix 1. The calculation shows below is based on Curve C in RTCA
DO-160.
PSD, Power Spectral Density, P = .2 (at first resonance)
Transmissibility, Q = 20 (initial assumption)
fn= 266.4
Equation 3, GRMS= 12.94
Equation 4, ZRMS= .0018 in
The first mode natural frequency is found from doing a modal analysis on the PWB and
compared to the calculated value above. The transmissibility is recalculated using the
FEM to adjust this calculation and recalculate the single amplitude displacement
response from the random analysis tool and finally used to validate the accuracy of the
finite element model. The results from the other 5 vibration curves are in Table 3.1-1.
Table 3.1-1: Calculated first mode frequencies and PWB center displacements
D
Fn (Hz)
P (G^2/Hz)
Q
G
Z_RMS
(in)
Curve B
265.5
266.4
0.002
20.00
4.09
Curve B2
265.5
266.4
0.001
20.00
2.05
Curve B3
265.5
266.4
0.002
20.00
4.09
Curve C
265.5
266.4
0.020
20.00
12.93
Curve D
265.5
266.4
0.040
20.00
18.29
Curve E
265.5
266.4
0.080
20.00
25.87
0.00057
0.00028
0.00057
0.00179
0.00253
0.00358
Once the random analysis is done, a new value for Q can be calculated through the
graphs. The equation is as follows:
Q
PSDout
PSDin
This is the most reliable way of calculating the new transmissibility values for the PWB.
Basically the PSD input from DO-160 at the first resonance is taken as the PSDin. The

PSDout is the value measured from the response at the center of the board. This is not the
GRMS value that is calculated; that value is calculated from the area under the curve. The
Y-axis of the frequency response graph will give this value. The Q is then input back
into the hand calculations to estimate the board deflection once again. These values
match and can be seen in Table 3.3.5-1.
11
3.2
FEA Model Optimization
Optimization of the FEM is important to ensure accuracy yet reduce calculation time and
model size. This is especially important in this analysis, as there are 484 balls with a
very fine mesh. In order to help reduce the size of this model, a single BGA ball is
modeled with several different mesh sizes. The intent is to see how well the BGA ball
responds to tension and compression loads. This is important because due to small
displacements in the analysis, there is little to no shearing effects and just compression
and tension as the board flexes. The model is setup very simple with a fixed boundary
condition on the base of the ball and a distributed load on the top of the ball. The values
of this load do not matter but the differences in results do. The analysis results, mesh
sizes and change in displacement are reflecting in Figure 3.2-1 and Table 3.2-1.
Figure 3.2-1: BGA Ball Mesh Optimization
Table 3.2-1: BGA Mesh Size and Results
Case
1
2
3
4
Elements
750
570
390
270
Nodes
3117
2343
1569
1053
12
Elements in Entire
Model (For BGA
Balls)
363000
275880
188760
130680
Displacement
4.43E-07
4.52E-07
4.71E-07
4.99E-07
Figure 3.2-2: Displacement of BGA ball in static analysis
The displacement of the BGA ball in this analysis shows that case 1 and 2 have good
meshes and have little change in the overall result of the ball deformation. By going
from a BGA ball with 750 elements to 570 elements, 87120 Quad-20 elements are
reduced in the mode size. An even smaller mesh would have been used, but the wedge
elements at the center of the ball failed and therefore 750 elements is the smallest mesh
size attainable by keeping the same surface mesh.
3.3
Finite Element Model
3.3.1
Finite Elements
The finite element model is intended to be very detailed. The solder balls on the BGA
are the focus of the analysis. There are many different types of elements that can be used
like TRI elements, Quad elements or even BAR elements. Since stress distribution is the
point of interest, Quad elements are the best option. TRI elements, although mesh very
nicely around complicated geometry, do not have a central node and thus can have an
uneven stress distribution or no stress at the element center. For the BGA balls, a Quad-8
element was used as a surface mesh seed on a cross section of the ball. The Quad-8
element was used because the elements have a node not only at the ends of the square
element but also at the midpoints of the four sides. This gives a much more true shape to
the curved surface and allows the use of half as many elements as the Quad-4 to get the
13
same curved shape. When revolved around the center of the BGA ball, this mesh creates
a spherical shape that makes up the BGA ball and the solder joints. The BGA ball shape
is the most important part of this analysis and must be modeled correctly, both the ball
and the solder joint. To model this, a design taken from David M. Pierce’s paper is used
since his model has been validated with actual testing. The figure of this is shown below:
Figure 3.3.3-1: BGA ball and solder joint figure [2]
The dimensions are used right from the paper and modeled within the FEM used for this
analysis. Note that the ball is not completely spherical. This is because when the solder
bonds to the respective surfaces, a solder joint forms that is shown by the y1 and y2
dimensions. Although there can be a lot of variation in these values due to the
production of these parts and also assembly methods, this is assumed to be the average
case. Variations in the solder balls will not be investigated within this analysis. Although
Pierce models the complete BGA package and PWB, for this analysis it is not necessary
for the accuracy of the results. The Quad-8 2-d surface element is now transformed into
a 3-D Quad-20 element that makes up the sphere shape. The intent is to have a very fine
mesh to achieve an even stress distribution, which makes this element the prime choice.
For all other surfaces of the model, the PWB, and the BGA body composed of an inner
and outer die that is modeled with Quad-4 elements so help reduce the size of the model.
3.3.2
Material Properties
The material Properties are listed below in Table 3.3.2-1. The material properties come
as well as calculated values for the PWB and BGA die as shown in Appendix D. The
14
properties for solder are defined within IPC-2221 as a standard solder for electronic
components. The lead free solder that is used for this analysis comes from the source
Lead Free Solder. The material properties for the two types of solder are found from the
website Matweb.
Table 3.3.2-1: Material Properties
Youngs Modulous, E
Poisson's Ratio
Density
Lead based
solder
(60Sn-40Pb)
4.35e6
.4
8.05e-4
Lead free
solder
TBD
TBD
TBD
PWB
2.9e6
.3
4.07e-4
BGA Die
4.53e6
.28
1.857e-4
The PWB properties are based upon a .1 inch thick PWB with 16 layers. The stackup of
the PWB as well as the thicknesses of the layers can be found in appendix D. The PWB
also has roughly .3 lbs of electrical components to better simulate real life conditions. A
heavier board will result in lower frequencies and thus more displacement. The BGA die
is made up of two components; the inner silicon die and an outer epoxy resin die similar
to that of the PWB material. These two materials are combined into one just the like
PWB layers are combined into a single shell to greatly simplify the model.
3.3.3
Finite Element Model
There are a few approaches that can be used to create the finite element model. One
method would be to take the full 3-D model and mesh with 3-D elements, which for this
would be very large. The second would be to mesh the Balls as 3-D elements but mesh
the PWB and BGA body with 2-D shell elements and assume thin plate theory. The last
option would be to mesh the BGA balls with 1-D bar elements, and mesh the PWB and
BGA body with 2-D shell elements. The second option proves to be the best for what is
being looked for which is the detailed stress of each solder ball in the BGA package. The
third option is good if the stress results do not need to be completely accurate and the
general stress distribution is needed. Figures 3.3.3-2 through 3.3.3-8 show the different
FEA models used in this analysis.
15
Figure 3.3.3-2: Finite element model, Modal Analysis mesh
Figure 3.3.3-3: Finite element model, Random Analysis mesh
Figure 3.3.3-4: Finite element model, Inertial Analysis with BGA mesh
16
Figure 3.3.3-5: Finite element model, Inertial Analysis BGA mesh pad area
Figure 3.3.3-6: Finite element model, Inertial Analysis BGA ball mesh
17
Figure 3.3.3-7: Finite element model, Inertial Analysis BGA pad mesh
Figure 3.3.3-8: Finite element model, Inertial Analysis BGA pad mesh close up
Table 3.3.3-1: FEA model summary
Modal
Random
Inertial
Elements
21600
21600
402813
18
Nodes
65402
65402
1278711
In chapter 2.2 the different types of boundary conditions are talked about and the ideal
option for this large model would be to use the Glue boundary condition, which allows a
user to have a very fine mesh and a relatively coarse mesh next to each other. What
happens is MD NASTRAN writes the REB3 MPCs for all of the deformable body
connections based on the user inputs. Unfortunately, this method did not prove to work
well with the amount of MPCs being written in NASTRAN along with the model size.
This may have been a limitation to the code or the hardware being used for this analysis,
and may be looked at in the future for more detailed analysis’. Instead of using the Glue
boundary condition, the tried and true mesh matching was used to mesh the model as
shown in the above pictures. The two plates are connected to the BGAs by their
respective mesh. Even though a mid-plane is being used for this analysis, and one would
think that the plates should not be touching the balls, the plate offset option is used to
make the meshing possible without using loads of MPCs. Although the pre processing of
the model takes much longer to do, the end result is a more reliable solution with much
less Fortran debugging.
3.3.4
Natural Frequencies
The first analysis that was done is the to calculate the first mode natural frequency. This
analysis was done without the BGA for the reason that the BGA does not have enough
mass to impact the analysis. The FEA was meshed with a Quad 8 shell element. The
PCB was modeled as 2-D and considered a thin plate. The small displacements and .1
inch thickness makes this a smart modeling decision. This could have been done with
quad 4 elements, but would not have made any noticeable calculation time savings. The
following figures below show the first 3 modes for the PCB. The PCB was constrained
with a 6 degree of freedom MPC that is constrained by a single node in space. This node
has the 6 DOF constraint applied.
19
Figure 3.3.4-1: First Mode 266.59 Hz
It is obvious to see that this first mode natural frequency nearly exactly matches the hand
calculations from chapter 3.1. Although the following modes are not directly calculated,
the following figures show what the next two mode shapes look like. The remaining
natural frequencies are calculated using MSC PATRAN and located in Appendix 2.
Figure 3.3.4-2: Second Mode 341.07 Hz
20
Figure 3.3.4-3: Third Mode 511.18 Hz
The second mode and third mode are not used for the calculations in this paper. The
intent is to use the first mode as the basis of comparison for a few reasons; the first being
that the initial natural frequency will give the largest displacements and thus the most
curvature to the board. This more than anything is the primary reason for looking at the
first mode. Board curvatures and displacements are the name of the game for electronics
packaging engineers as we try to optimize the Ball Grid Array’s. The later modes, 2 and
on, are not of concern due to their displacements. As shown in the hand calculations, the
displacements expected are to be around .002 inches at the center of the PWB. The
second mode, at a much higher frequency, will yield much less displacement and less
curvature. Of course when doing RMS, or room mean square, calculations, it does take
into account the entire spectrum of results, which for this is 10-2000 Hz. This is talked
about in the next chapter about the Random Analysis.
3.3.5
Random Analysis
Initially, the random analysis was going to be used to not only calculate displacements,
and accelerations but stresses. After doing some optimization, it was obvious that the
calculations to do the dynamic analysis would take far too long based on some test cases.
For the stress values of the BGA balls, the inertial, static, analysis is being done to
drastically cut calculation time. The random analysis, or frequency response, is done
again for each case. The results show peak responses, which are your modes of natural
21
frequency shown as peaks on the graphs. The displacements are also calculated by using
a relative displacement technique in PATRAN. This is done by taking a node of known
displacement, which is the fixed edge, and then the node of interest. This will give the
relative displacement of the board at a particular point. For this the nodes of interest are
of course the node at the center of the board and the node with the maximum
displacement. This is first used to validate the hand calculations in chapter 3.1, and then
later used for miner’s damage index calculations. Following the routine used earlier with
the hand calculations for curve C, the random analysis for curve C will be shown below.
The Analysis is done by taking the same model used for the normal mode analysis and
creating a new node, independent of the PWB FEM. This node will be used as the input
node for the frequency response. For this model, the axis of interest is the axis
perpendicular to the board area. This will give the maximum response, and most
displacement so it is the limiting case. The input node is then fixed in five degrees of
freedom where the z-axis is left free for the frequency response. This node is then
connected to the PWB with an REB2 element, which connects the fixed edges to the
base node and implements all the necessary boundary conditions. This is shown in
chapter 3.3.3 Finite Element Model Figure 3.3.3-1. Other model constraints are a 2%
dampening factor for the entire model, which is rather conservative and a real value can
only be measured through testing [3]. The last condition is the actual PSD Curve, which
is input into a non-spatial frequency domain field. Each of the curves is input this way so
the model may be run once with different sub cases so one model may be used
throughout and one analysis may be performed. Once the analysis is complete, to
validate if the input is correct, the base node acceleration is plotted and compared to the
PSD curve. These curves should be identical as shown in the figure below.
22
Figure 3.3.5-1: Base Node Acceleration Curve C
This figure should compare exactly to the RTCA DO-160 specification, which it does.
To next validation is the displacement of the center of the board. This is done by the
relative displacement scheme described earlier, which is built into PATRAN. Taking the
centermost node and calculating the RMS displacement, the result is shown below.
Figure 3.3.5-2: Central Node RMS Displacement
Insert Figure!!!
23
Figure 3.3.5-3: Central Node RMS Acceleration (GRMS)
This value is .00209 inches maximum. This compares to the calculated displacement of
.0018 inches using Steinberg’s method in chapter 3.2. This is used to validate the
random analysis model, but there is still one value that we do not know completely.
Since we can calculate the GRMS at the center of the board, the hand calculations can be
further refined with the correct value. For this calculation, Q is not needed but it will be
needed later for the miner’s damage index calculation. Using the new value for GRMS the
new ZRMS for single amplitude center of the board displacement is revealed along with
the new value for Q, which will be used in chapter 3.4.
The maximum displacement for the board is also calculated using the relative
displacement technique as well as the acceleration at this point. The results are shown
below for Curve C:
Figure 3.3.5-4: Maximum RMS Displacement
24
Figure 3.3.5-5: Maximum RMS Acceleration
The maximum values are determined from the maximum amplitude from the first mode
natural frequency. Typically one can solve for the RMS stresses with the same analysis,
but since the stresses in the PWB are not of concern, and the BGA balls are, the stress
calculations are solved with the inertial analysis instead to save on calculation time. The
rest of the curves results are summarized in the table below and the graphs are located in
Appendix B.
Table 3.3.5-1: Random Analysis Acceleration and Displacement Results
Curve
B
B2
B3
C
D
E
3.3.6
Acceleration at
Max Displacement
11.97
5.986
11.98
37.77
69.28
75.47
Maximum
Displacement
1.37E-03
6.86E-04
1.38E-03
4.31E-03
6.43E-03
8.54E-03
Acceleration
at PWB
Center
Displacement
at PWB Center
6.75E-05
3.37E-04
6.82E-04
2.09E-03
3.17E-03
4.16E-03
Inertial Analysis
Inertial analysis is a completely static analysis that enables the user to solve for stresses
with much less calculation time and power. This scheme only works easily if the first
mode is the mode of interest. This is due to the shape of the board when the inertia is
25
applied having the edges fixed with six degrees of freedom like in the modal analysis. It
is important to note that you will not get the same shape if the second or any higher
modes are of interest, and a dynamic analysis must be done. The first mode works for
this particular case and most other electronic component analysis since the biggest
interest is maximum displacement and shape of the PWB where the component resides.
Obtaining the maximum displacement using the inertial method is an iterative method.
The bare board again is taken for the first few iterations to obtain the maximum
displacement for each curve. This is done to save time and is fairly accurate since the
BGA part does not have a large mass or stiffness to influence the curvature of the board
much. The last iteration is done to obtain the max displacement of the PWB with the full
detailed BGA model.
The model is setup just like the modal analysis with all 6 degrees of freedom fixed, and
the inertial load is applied to the axis perpendicular to the PWB area. This is the basis for
all inertial analysis that is being done for this project. The PWB is setup slightly
different than the modal analysis and the random analysis, but will not compromise the
results. The PWB is broken up in to 4 pieces, the main PWB and the three different
mounting positions for the BGA. The way the analysis is first done is with the PWB
having the same mach across the entire surface. An initial value of 1G is applied to the
model, or 1 times the acceleration of gravity, to see the initial displacement. Luckily, the
displacement relation to the amount of inertia applied is linear thus the maximum
displacement can be interpolated easily from the initial analysis so only two runs can be
done. The last iteration will give the maximum board displacement, shape and stresses
for the BGA balls. The inertial analysis is done for each of the 6 PSD curves, each of the
three PWB positions, and also for lead and lead free solder. The results for curve C are
shown below for the maximum displacement case. Note that the curvature of the PWB
looks exactly like the first mode natural frequency. The results for all of the iterative
calculations are shown in Table 3.3.6-1.
26
Figure 3.3.6-1: Curve C Inertial Analysis Maximum Displacement
Table 3.3.6-1: Inertial Analysis Loads and displacements
Random Curve
B
B2
B3
C
D
E
Wanted
Displacement
(Inches)
1.37E-03
6.86E-04
1.38E-03
4.31E-03
6.43E-03
8.54E-03
Iteration 1 G's
5.62E+00
2.81E+00
5.64E+00
1.76E+01
2.63E+01
3.50E+01
Displacement
iteration 1
1.18E-03
5.92E-04
1.19E-03
3.71E-03
5.54E-03
7.38E-03
Iteration 2
G's
6.54E+00
3.26E+00
6.52E+00
2.05E+01
3.05E+01
4.05E+01
Displacement
iteration 2
1.38E-03
6.87E-04
1.37E-03
4.32E-03
6.43E-03
8.54E-03
Now that we have the results, what do they mean? Colunm 1 represents the random
curve from RTCA DO-160 being input from the random analysis. The second column
represents the maximum displacements from the random analysis shown in Table 3.3.51. This is where the static analysis becomes quasi static due to running a stating analysis
to determine displacements from a random input analysis. There is an initial iteration
done to baseline these calculations, which is applying a 1G inertial load to the model.
The displacement from this analysis comes to 2.44e-4 inches. From here the ratio of the
displacement over the inertial load is used to determine the wanted displacement.
Although this calculation is meant to be linear, is does not always work out on the first
iteration. For this model, each analysis took two iterations to get the wanted
displacement from the random analysis to be represented in an inertial model.
27
The inertial model is taken one step further, and the BGA is finally added to the FEM
shown in Figure 3.3.3-4 and 3.3.3-5. This is where the purpose of the quasi-static
analysis comes into play. To run a dynamic model, or random analysis, the calculation
time is very extensive and sometimes will not be able to converge due to the number of
nodes and elements. To run a static analysis and get the results of a random analysis
some assumptions have to be made. First and foremost is that the curvature of the board
is only going to represent that of the first mode of the PWB. This mode shape for
packaging engineers, tends to be the most important as it gives the maximum
displacement and most severe board curvature. This comes back to one of the reasons
this project is being done, which is to reduce errors in design during the early stages of
design where one may not have the time or be able to run a structural analysis. Knowing
which position to place the BGA or multiple BGAs to get the most reliability is crucial
for the design and life of the product. Although the center of the board is typically a very
bad place to put a BGA mechanically, it can be very beneficial to an electrical designer
as this part can act as a processor or a programmable logic device. This is exactly why
multiple positions are being looked at as well as different random vibration
environments.
The PWB already has a mass of electrical components spread over the surface to add
some stiffness so the analysis is more like real life, which is described back in chapter
3.3.2. The addition of the BGA, since the mass is so small, will not impact the curvature
of the PWB or the displacement with the already derived inertial load to simulate the 6
vibration input curves. Since curve B and B3 have the same maximum displacement,
only curve B will be run. This is assumed to be ok due to the displacements being the
same as well as the PSD level at which the first mode natural frequency occurs. As a
result, the 5 curves are run to namely examine the stress in the solder balls of the BGA.
The PWB stresses are not of concern as their yielding properties are much higher than
that of solder. Solder is thus the limiting case for the small deformations that are seen
and also the limiting case for high cycle fatigue. The figures below show the FEM with
the BGA, the Stresses of the BGA die, stresses of the BGA PWB pad, and of course the
28
stresses in the BGA ball itself. The details and results of the BGA stresses will be
described in the next chapter.
3.3.6.1 Computer Computation Characteristics
In order to solve this detailed model, a normal computer cannot be used. The modal
analysis and simplified random analysis can easily be solved because the size of the
model is small, but for the large model a super computer is sometimes needed. For this
analysis, a super computer is used to speed up calculations since many analysis cases are
needed. The super computer is setup on a special server where the analysis deck is
loaded. The parameters that can be set are the number of processors 1, 2, or 4 and the
amount of memory that is needed.
Get data on processor type, swap memory need to calculate results, time, etc.
3.3.7
BGA Stress Results
The whole purpose of the modal analysis, random analysis and finally the inertial
analysis is to determine the stress in each solder ball, namely the max stress areas. The
primary reason for BGA failure is within the solder joint for vibration environments.
This means when looking at the results, the max stress should reside within the solder
joint modeled in the FEM; the results depict this exactly. Before the stress is analyzed,
the FEM must have the correct displacement from the inertial analysis as shown in Table
3.3.6-1. For each case, this is true at each of the different BGA locations. The Curve B
displacement from the inertial analysis with the BGA at the board center, location 1, is
shown in Figure 3.3.7-1.
29
Figure 3.3.7-1: Curve B Inertial Displacement
Now that the displacement is verified, the stress will then resemble the stress from a
random vibration RMS displacement. Figure 3.3.7-2 resembles the BGA ball stresses.
Figure 3.3.7-2: Curve B inertial BGA ball stress
Notice the stress on the balls follows the curvature of the board. This is a basic check to
know that the stress distribution is correct. The next thing to realize is that since there is
30
little to no displacement at the center most balls of the grid relative to the edge of the
part the stress is minimal. As the balls get further away from the center of the BGA part,
the stress values increase where the stress is the largest at the edges of the BGA. The
max stresses are at the point of greatest deflection relative to the part center which is at
the four corners. Figure 3.3.7-3 shows the BGA ball with the maximum stress and at
which point of the ball this occurs.
Figure 3.3.6-2: Curve B Inertial Analysis Maximum Displacement
The high stress regions of the solder ball are clearly at the solder joint where the rest of
the ball has an order of magnitude less stress. This figure clearly depicts where the
failure point is and satisfies the original assumption. The whole point to find the
maximum stress is to determine the point at which the part will fail.
Although this analysis give the equivalent RMS stress from the random analysis via an
inertial analysis, an adjustment factor must be applied to get the sinusoidal stress which
is needed for the high cycle fatigue analysis. The sinusoidal correlation factor comes
from Steinberg and incorporates his 3 band method described in chapter 2.1. This
method uses the following equation to define the correction factor G:
D  N 1 S1  N 2 S 2  N 3 S 3
b
b
31
b
Gb  D
Solving for the correction factor G yields a correction factor of 1.95 as shown in
Appendix A. The value of D is simply the Gaussian probability distribution which
incorporates the three sigmas and their respective percentage of occurrence. The factor b
is defined as the fatigue exponent for solder from Steinberg [3]. Table 3.3.7-1 shows the
RMS stress values and the equivalent stress values for all vibration curves and BGA
locations.
Table 3.3.7-1: Inertial Analysis Stress, RMS and Sinusoidal 60Sn-40Pb solder
Curve:
Max Stress Location 1 (PSI)
Corrected Sine Stress (PSI)
Max Stress Location 2 (PSI)
Corrected Sine Stress (PSI)
Max Stress Location 3 (PSI)
Corrected Sine Stress (PSI)
B2
B
1700.00
3315.00
1670.00
3256.50
832.00
1622.40
C
D
7910.00
15424.50
7800.00
15210.00
E
10500.00
20475.00
10300.00
20085.00
Table 3.3.7-2: Inertial Analysis Stress, RMS and Sinusoidal Lead-Free solder
Curve:
Max Stress Location 1 (PSI)
Corrected Sine Stress (PSI)
Max Stress Location 2 (PSI)
Corrected Sine Stress (PSI)
Max Stress Location 3 (PSI)
Corrected Sine Stress (PSI)
B2
B
C
0
0
0
0
0
0
D
E
0
0
0
0
0
0
0
0
This data is compared to the test results and FEA done by David M. Pierce and is in the
same order of magnitude per displacement [2]. Comparing the data from his paper on
life estimations to the results from the BGA FEA within this paper, gives one level of
validation for the model and the results herein. The second level of validation is to
calculate the Miner’s Damage Index and see how the calculated life based on the BGA
corresponds to the fatigue life of the solder balls based upon their sinusoidal stress. In
chapter 3.4, the MDI will be calculated using Steinberg’s method for the BGA analyzed
and then compared to the S-N curve for solder.
Add Discussion of Lead Free solder.
32
3.4
Miner's Damage Index Calculations
The Miner's Damage Index is a cumulative fatigue damage ratio based upon the number
of actual cycles done over the number of cycles to failure. This calculation is represented
below:
Rn  
n1 n 2 n 3


.....
N1 N 2 N 3
The assumptions are that the damage is linear and is a simple function of the load as well
as the failures not being related to the loading sequence. The Damage index is designed

to look at the wire leads of components, solder joints for non wire leaded parts, and of
course the vibration environment that the part is located [3]. In industry it is common to
do this calculation for every component in the worst-case condition, which is the largest
displacement and curvature area. The value for Rn should be less than .5, which is
representative of a factor of safety of 2.
Before the calculation for the number of cycles that are caused by the vibration
environment, it is important to understand some testing that has been done for electronic
components. Steinberg has concluded that through vibration testing and FEA studies of
electrical components can be related to the dynamic displacements developed by the
PWBs during vibration [3]. The data shows that the components can achieve a fatigue
life of approximately 10 million stress reversals in a sinusoidal environment. This value
of fatigue cycles comes directly with the single amplitude value of displacement for each
particular component is limited to Z, to meet the 10 million cycles. Figure 3.4-1
represents how Z relates to board placement.
33
Large
relative
motion
L
Component
Lead
Wire
Co
mp
h
Z
Small
relative
motion
on
en
t
B
Z  0.00022 B
Chr L
Z = allowable single-amplitude displacement for 1 x 107 cycles
B = length of PCB edge parallel to component
L = length of electronic component
h = thickness of PCB
C = constant related to type of component
r = scale factor related to the location of the component on the PCB
Figure 3.4-1: Allowable board displacement, Z [3]
The important values that can change are C and r. C is based on the component, which is
based on test data or analysis data and r is based on where the component is placed. For
this analysis, the component is placed as shown in Figure 2.1-1. The first way the MDI is
calculated is by Steinberg’s method. These values will be compared to the fatigue cycles
based on the stress in the analysis. The third comparison is with a new value for C that is
based on the test data. For the analysis, there are three positions where r is equal to 1,
.707, and .5.
Table 3.4-1: Miner’s Damage Index
Insert Table (Will be updated by Tuesday)
3.5
Fatigue Life
Steinberg’s C factor is one of the key factors used to calculate fatigue life. Using the
stress data from the analysis, a new value for C can be calculated which should give
much better results for the MDI. The first that that is looked at is the S-N curve for
solder and plot the sinusoidal stress values. This will give a fatigue life based on the test
data that can be compared to the fatigue life from the MDI. Ideally these values should
be close to each other, but the C factors for the BGA components are based on a board
34
that is 8x8 inch board. Since the board in this analysis is 6x9, the value needs to be
adjusted.
To be continued when all the data is gathered. Analysis still running
35
4. CONCLUSIONS
4.1
Future Work and Model Improvement
36
Literature Cited
[1] RTCA, Incorporated. “Environmental Conditions and Test Procedures for Airborne
Equipment.” RTCA, Incorporated., Washington, DC. SC-135, Dec. 2007.
[2] Pierce, David M., and Sheri D. Sheppard. Fatigue Life Prediction Methodology for
Lead-Free Solder Alloy Interconnects: Development and Validation. Tech.
Stanford, CA: Stanford University. Print.
[3] Steinberg, Dave S. Vibration Analysis for Electronic Equipment. New York: John
Wiley & Sons, 2000. Print.
[4] Chen, Y. S. "Combining Vibration Test with Finite Element Analysis for the
Fatigue Life Estimation of PBGA Components." (2007): 638-644. Science
Direct. Web. Aug.-Sept. 2010.
[5] Amy, Robin A. “Accuracy of Simplified Printed Circuit Board Finite Element
Models.” (2009): 1-12. Science Direct. Web. Aug-Sept. 2010.
[6] Bieler, T. R. “Lead Free Solder.” (2010): 1-12. Science Direct. Web. Aug-Sept
2010.
[7] Arulvanan, P., Zhong, Z. W. “Assembly and reliability of PBGA packages on FR-4
PCBs with SnAgCu solder.” (2006): 2462-2468. Science Direct. Web. AugSept. 2010.
37
Appendix A
Calculated values
38
Appendix B
Curve B Plots:
Figure B-1: Maximum RMS Acceleration
Figure B-2: Maximum RMS Displacement
39
Figure B-3: Base Node Acceleration Curve B
Figure B-4: Central Node RMS Displacement
Figure B-5: Central Node RMS Acceleration (GRMS)
40
Curve B2 plots:
Figure B-6: Maximum RMS Displacement
Figure B-7: Maximum RMS Acceleration
41
Figure B-8: Base Node Acceleration Curve B2
Figure B-9: Central Node RMS Displacement
Figure B-10: Central Node RMS Acceleration (GRMS)
42
Curve B3 plots:
Figure B-11: Maximum RMS Displacement
Figure B-12: Maximum RMS Acceleration
43
Figure B-13: Base Node Acceleration Curve B3
Figure B-14: Central Node RMS Displacement
Figure B-15: Central Node RMS Acceleration (GRMS)
44
Curve D plots:
Figure B-16: Maximum RMS Displacement
Figure B-17: Maximum RMS Acceleration
45
Figure B-17: Base Node Acceleration Curve D
Figure B-18: Central Node RMS Displacement
Figure B-19: Central Node RMS Acceleration (GRMS)
46
Curve E plots:
Figure B-20: Maximum RMS Displacement
Figure B-21: Maximum RMS Acceleration
47
Figure B-22: Base Node Acceleration Curve E
Figure B-23: Central Node RMS Displacement
Figure B-24: Central Node RMS Acceleration (GRMS)
48
Appendix C
Inertial Stress figures and calculations
49
Appendix D
Material Property Calculations
50