A Finite Element Study of 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 ......................................................................................................... ix
KEYWORDS .................................................................................................................... xi
ABSTRACT ................................................................................................................... xiii
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 and Quasi-Static Inertia Analysis ... 6
2.3
FEA Model Optimization ................................................................................. 10
3. RESULTS AND DISCUSSION ................................................................................ 11
3.1
Analytical Results ............................................................................................. 11
3.2
Finite Element Analysis .................................................................................... 12
3.2.1
Finite Elements .................................................................................... 12
3.2.2
FEA Model Optimization..................................................................... 14
3.2.3
Material Properties ............................................................................... 15
3.2.4
Finite Element Model – Modal, Random, Inertial ............................... 16
3.2.5
Natural Frequencies ............................................................................. 22
3.2.6
Random Analysis ................................................................................. 24
3.2.7
Inertial Analysis ................................................................................... 28
3.2.8
BGA Stress Results .............................................................................. 31
3.3
Miner's Damage Index Calculations ................................................................. 40
3.4
Fatigue Life ....................................................................................................... 45
iii
4. CONCLUSIONS ....................................................................................................... 53
4.1
Future Work and Model Improvement ............................................................. 53
Appendix A...................................................................................................................... 57
Appendix B ...................................................................................................................... 59
Appendix C ...................................................................................................................... 69
iv
LIST OF TABLES
Table 1: Constant values for Steinberg calculations........................................................ 11
Table 2: Calculated first mode frequencies and PWB center displacements .................. 12
Table 3: BGA Mesh Size and Results ............................................................................. 15
Table 4: Material Properties Used ................................................................................... 16
Table 5: Random Analysis Acceleration and Displacement Results............................... 27
Table 6: Peak transmissibility calculations location 1 ..................................................... 28
Table 7: Peak transmissibility calculations location 2 and 3 ........................................... 28
Table 8: Inertial Analysis Loads and displacements ....................................................... 30
Table 9: Maximum Inertial Analysis Stress, RMS and Sinusoidal 60Sn-40Pb solder .... 36
Table 10: Maximum Inertial Analysis Stress, RMS and Sinusoidal 95.5Sn-4.0Ag-0.5Cu
solder........................................................................................................................ 37
Table 11: Average solder-joint stresses to cause high cycle fatigue - 60Sn-40Pb solder 37
Table 12: Average solder-joint stresses to cause high cycle fatigue – 95.5Sn-4.0Ag0.5Cu solder ............................................................................................................. 37
Table 13: PWB displacement per location and load ........................................................ 38
Table 14: Miner’s Damage Index calculation constants.................................................. 41
Table 15: Miners Damage Index calculation constants continued .................................. 42
Table 16: Miner’s Damage Index .................................................................................... 43
Table 17: High cycle fatigue values based on FEA mean solder join stress ................... 46
Table 18: Steinberg component constant C, MDI and S-N fatigue life interpolation ..... 49
Table 19: Optimized Steinberg component constant C, MDI and S-N fatigue life
calculation for a 6x9 inch PWB ............................................................................... 49
Table C-1: PWB mechanical properties .......................................................................... 69
Table C-2: BGA Die properties ....................................................................................... 70
v
LIST OF FIGURES
Figure 1: PWB BGA Placement ........................................................................................ 3
Figure 2: PWB and Interconnect Diagram ........................................................................ 4
Figure 3: Gaussian Distribution for Steinberg 3 Band Method [3] ................................... 5
Figure 4: RTCA DO-160 Vibration Levels for Fixed Wing Aircraft [1] .......................... 8
Figure 5: Wedge Lock Card Guides [3] ............................................................................ 9
Figure 6: BGA ball and solder joint figure [2] ................................................................ 13
Figure 7: BGA Ball Mesh Optimization .......................................................................... 14
Figure 8: Displacement of BGA ball in static analysis.................................................... 15
Figure 9: Finite element model, Modal and Random Analysis mesh.............................. 17
Figure 10: Finite element model, Inertial Analysis with BGA mesh .............................. 18
Figure 11: Finite element model, Inertial Analysis BGA mesh pad area ........................ 18
Figure 12: Finite element model, Inertial Analysis BGA ball mesh ............................... 19
Figure 13: Finite element model, Inertial Analysis BGA pad mesh................................ 20
Figure 14: Finite element model, Inertial Analysis BGA pad mesh close up ................. 21
Figure 15: First Mode 266.59 Hz .................................................................................... 22
Figure 16: Second Mode 341.07 Hz ................................................................................ 23
Figure 17: Third Mode 511.18 Hz ................................................................................... 23
Figure 18: Base Node Acceleration Curve C .................................................................. 25
Figure 19: Central Node RMS Displacement .................................................................. 25
Figure 20: Central Node RMS Acceleration (GRMS) ....................................................... 26
Figure 21: Maximum RMS Displacement ....................................................................... 26
Figure 22: Maximum RMS Acceleration ........................................................................ 27
Figure 23: Curve C Inertial Analysis Maximum Displacement ...................................... 30
Figure 24: Curve B Inertial Displacement ....................................................................... 32
Figure 25: Curve B inertial BGA ball stress field location 1 .......................................... 32
Figure 26: Curve D Inertial BGA ball stress field location 2 .......................................... 33
Figure 27: Curve B2 Inertial BGA ball stress field location 3 ........................................ 33
Figure 28: Curve B Inertial Analysis Maximum Displacement ...................................... 34
Figure 29: Solder ball stress distribution and crack initiation point ................................ 35
vi
Figure 30: Maximum Corrected Sine Stress vs. PWB displacement, lead and lead free
solder........................................................................................................................ 39
Figure 31: Mean Corrected Sine Stress vs. PWB displacement, lead and lead free solder
................................................................................................................................. 39
Figure 32: Allowable board displacement, Z [3] ............................................................. 41
Figure 33: S-N curve for 60Sn-40Pb solder .................................................................... 46
Figure 34: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 1 ................ 47
Figure 35: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 2 ................ 48
Figure 36: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 3 ................ 48
Figure 37: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location 1 50
Figure 38: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location 2 51
Figure 39: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location 3 51
Figure B-1: Maximum RMS Acceleration ...................................................................... 59
Figure B-2: Maximum RMS Displacement ..................................................................... 59
Figure B-3: Base Node Acceleration Curve B ................................................................ 60
Figure B-4: Central Node RMS Displacement ................................................................ 60
Figure B-5: Central Node RMS Acceleration (GRMS) .................................................. 60
Figure B-6: Maximum RMS Displacement ..................................................................... 61
Figure B-7: Maximum RMS Acceleration ...................................................................... 61
Figure B-8: Base Node Acceleration Curve B2 .............................................................. 62
Figure B-9: Central Node RMS Displacement ................................................................ 62
Figure B-10: Central Node RMS Acceleration (GRMS) ................................................ 62
Figure B-11: Maximum RMS Displacement ................................................................... 63
Figure B-12: Maximum RMS Acceleration .................................................................... 63
Figure B-13: Base Node Acceleration Curve B3 ............................................................ 64
Figure B-14: Central Node RMS Displacement .............................................................. 64
Figure B-15: Central Node RMS Acceleration (GRMS) ................................................ 64
Figure B-16: Maximum RMS Displacement ................................................................... 65
Figure B-17: Maximum RMS Acceleration .................................................................... 65
Figure B-17: Base Node Acceleration Curve D .............................................................. 66
Figure B-18: Central Node RMS Displacement .............................................................. 66
vii
Figure B-19: Central Node RMS Acceleration (GRMS) ................................................ 66
Figure B-20: Maximum RMS Displacement ................................................................... 67
Figure B-21: Maximum RMS Acceleration .................................................................... 67
Figure B-22: Base Node Acceleration Curve E ............................................................... 68
Figure B-23: Central Node RMS Displacement .............................................................. 68
Figure B-24: Central Node RMS Acceleration (GRMS) ................................................ 68
Figure C-1: Mechanical packaging 484 BGA [11].......................................................... 71
Figure C-2: Mechanical packaging dimensions 484 BGA [11] ...................................... 72
viii
NOMENCLATURE
BGA – Ball Grid Array
DOF – Degrees of Freedom
FEA – Finite Element Analysis
FEM – Finite Element Mesh
MDI – Miner’s Damage Index
MPC – Multi Point constraint
PSD – Power Spectral Density
PWB – Printed Wiring Board
RBE – Rigid Body Element
RMS – Root mean square
ROHS – Restriction of Hazardous Substances
fn - Natural Frequency
D – Plate Stiffness
p – Density
a – PWB length
b – PWB width
h – PWB thickness
E – Young’s Modulus
v – Poisson’s Ratio
G – Acceleration
P – Power Spectral Density
Q – Transmissibility
ZRMS – Dynamic single amplitude displacement, PWB center
GRMS – Root Mean Square Acceleration
Z – Steinberg allowable component deflection
 - Solder fatigue exponent
 - Gaussian probability distribution
 - Random
 - Gaussian probability
ix
 - Standard deviation
r – Steinberg position factor
X – BGA position along the length of the PWB
Y – BGA position along the width of the PWB
C – Steinberg component position factor
N - Number of cycles to failure
n – Number of cycles applied
L – Component length
Y - Actual PWB displacement based on random analysis
x
KEYWORDS
Ball Grid Array
Finite Element Analysis
Aerospace
Random Vibration
xi
ACKNOWLEDGMENT
Type the text of your acknowledgment here.
xii
ABSTRACT
This report summarizes a finite element modeling study to investigate common
aerospace random vibration environments for Ball Grid Array (BGA) components,
specifically the 484-ball package from ACTEL. The analysis looks at both 60Sn-40Pb
and 95.5Sn-4.0Ag-0.5Cu solder packages, different vibration environments from RTCA
DO-160, and printed wiring board component placement. The analysis results will show
how the random vibration level affects the solder ball joint stresses at various locations
for lead and lead free solder. High cycle fatigue is calculated using methods from
Steinberg and the S-N curve to determine a new Steinberg BGA component factor for a
6x9 inch PWB based off of the finite element analysis. The outcome is to be able to
optimize BGA placement earlier in the design process to eliminate expensive
development testing to prove out BGA placement.
xiii
1. INTRODUCTION
1.1 Background Information
The purpose of this project is to investigate some of the major issues with electronics
packaging in the aerospace industry today, specifically the application of Ball Grid
Arrays (BGAs) on rack style printed wiring boards (PWBs). There are many factors that
affect the performance of BGA components on the circuit board: placement on the PWB,
vibration level, temperature levels, and solder type. In addition to 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. BGAs 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.
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 fine pitch 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 1-2 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 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
1
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 [1]. Obviously one cannot test the electronics under the actual loads of the aircraft
because it may 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 vibration 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. A 1G input over the entire frequency
band is used, and is not meant to induce fatigue damage or failure. This data is 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 Wiring Board (PWB). 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. BGA
placements will include the center of the board, lower left corner and left center. This is
shown in Figure 1:
Location 2
Location 1
Location 3
Figure 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.
When developing a Finite Element Model, simplified analytical calculations are
invaluable to determine if the solution is reasonable. Equation (1) gives the first mode
natural frequency of a PWB that is fixed on 3 edges and has 1 free edge [3]. A fixed
boundary condition is one that constrains the three-translation axis and the three-rotation
axis, or 6 degrees of freedom. The FEM boundary conditions are modeled to be exactly
like a circuit card in a chassis that is shown in Figure 2. The fixed positions are from the
connectors at one end of the PWB that attaches to an interconnect, as well as the two
card guides on the long edges 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: PWB and Interconnect Diagram
fn 
where

 D .75
2
12 
   4  2 2  4 
a
3 
a b b 
(1)
Eh 3
D
12(1   2 )
(2)
The first mode natural frequency depends upon the Young’s Modulus E, plate thickness

h, Poisson’s ratio , the plate stiffness D, the density , and the length and width; a and b
respectively. Equation (3) gives the displacement ZRMS of the PWB based off the first
mode natural frequency at the center of the board.
9.81G
fn2
Z RMS 
G
where


2
PQf n

4
(3)
(4)
The displacement ZRMS is a dynamic single amplitude response based on the first mode
natural frequency and G, or GRMS. GRMS is the response of the PWB based on the
transmissibility, Q, the first mode natural frequency, fn, and the power spectral density
input, P, value where the first mode is. Transmissibility is a measure of the response of a
system; more specifically the ratio of the square root of the power spectral density, PSD,
output of the system over the PSD input from the random vibration curve.
The
transmissibility is calculated through dynamic analysis and post processing the data. The
calculated ZRMS results are compared to the FEA computed displacements. Although the
displacement calculated in equation (3) is accurate, it does not give the worst-case
displacement of the PWB. Steinberg uses a 3-band method approach to determine max
displacements of the PWB are used in the equations to determine BGA high cycle
fatigue life. This approach uses the statistical curve shown in Figure 3 to predict the
likelihood of a displacement to happen. The ZRMS value calculated only correlates to the
first Gaussian distribution, σ1, displacement that will only happen 68.2% of the time.
The second Gaussian distribution, σ2, displacement is 2 times the ZRMS and will only
happen 27.2% of the time, and the third Gaussian distribution, σ3, displacement is 3
times the ZRMS value that will happen only 4.33% of the time. The σ3 displacement is
where the maximum damage occurs.
Figure 3: Gaussian distribution for Steinberg 3 Band Method [3]
For the high cycle fatigue analysis, it is assumed that all of these conditions can exist and
must be taken into account for the calculations. This is the first step in determining the
stresses of the BGA solder balls and is necessary to be validated with the calculations
5
from equation (1) and (3). This leads directly into the random analysis where the
accelerations and displacements are determined from a dynamic approach.
2.2 Finite Element Model: Random Vibration and Quasi-Static Inertia
Analysis
The FEM model was 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 are modeled with 3-D
quad-20 elements for detail and accuracy. The elements are all tied together using mesh
matching that connects all of the surfaces together. This is not the most efficient method,
but it is the easiest and best method to get accurate results.
The method that is used to solve the problem of ball grid arrays in common aerospace
random vibration environments is composed of multiple independent steps. The first is
to run a modal analysis of the PWB with no BGA attached. Having the BGA attached to
the PWB will not significantly affect the stiffness and the model can be much smaller for
the initial analysis. The modal analysis results should line up with the results of the
simplified calculations; 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
desired result is to find the max displacement of the PWB and then compare it to the
hand calculations; this should also 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
6
the random acceleration analysis, determining the max displacement, it is just as easy to
run a quasi-static inertial analysis on the full FEM as opposed to a dynamic analysis on
the full FEM. The stresses that are of interest are the solder ball stresses because the
yielding properties of the PWB 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 quasi-static approach dramatically reduces the computation time for
the Finite element analysis. Since the stress results are equivalent RMS stresses that
would come out of a dynamic analysis of the BGA, there are correlation factors based on
the fatigue exponent of the solder type to relate the RMS stresses to sinusoidal stresses.
This factor allows the formulation of high cycle fatigue relationships.
Now that the thought process has been given for how the FEM model was be developed,
the boundary conditions must be analyzed. The most important boundary condition or
load case is the random vibration environments that are analyzed. Figure 4 from RTCA
DO-160 shows the vibration curves for fixed wing aircraft. The figure shows all 6
different dynamic inputs that an electronic controller environment may have for fixed
wing aircraft. Each curve represents a power spectral density acceleration input that
drives responses over a frequency band from 10-2000 Hz.
7
Figure 4: RTCA DO-160 Vibration Levels for Fixed Wing Aircraft [1]
The PWB boundary conditions are selected as described in [3] 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. Figure 5 is a pictorial
representation of what they are and how they work:
8
Figure 5: 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 much more
stiffness that will result in less displacements 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. 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 here that the mating conditions are fixed.
The last boundary condition for the model in this study 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, RBE2 or RBE3,
Mesh matching, or lastly the Glue constraint. The RBE2 MPC is used as a rigid or bolted
connection, which would be far too stiff for this application. The RBE3 element would
be ideal but very cumbersome, since about 968 connections to match the solder balls to
the PWB and the BGA body are required. Mesh matching is typically a very good
option, although for this case, the mesh is very fine (element size equal to .008 inches)
that results in a model, which is too large to solve for either dynamic or static modeling
with the computer hardware available. Lastly the Glue boundary condition can be
applied to the solid geometry to which the mesh is associated. This boundary sets up the
model with RBE3 connections that are ideal for this analysis case. These RBE3
connections are created when the solution is solved in NASTRAN rather 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.
9
Lastly the elements have all been chosen for a reason. The quad-8 shell elements are
used for all shell elements. The quad-8 elements allow the mesh on the PWB and the
BGA body to be coarser 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. Quad-8 elements were used for the
2-D mesh seed so a coarser mesh could be used and so that the element edges along the
spherical edge actually resemble a sphere and map to the surface much better. A very
fine quad-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 typical model size consists of
a few hundred thousand elements. 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. First, the solder balls must be detailed and accurate, but should not
slow down the model by having such a fine mesh that the model will not converge in a
reasonable time. The solder ball optimization will be done by looking at static cases of
single solder balls under static loading conditions. Computed displacements determine
the coarsest mesh that still yields highly accurate results.
10
3. RESULTS AND DISCUSSION
3.1
Analytical Results
The following analytical section is intended to validate the FEA model. The equations
given in section 2.1 will be used here. The first step is to calculate the first mode natural
frequency of the PWB. This is done using Steinberg’s equation for a thin plate with three
fixed edges. This solution is derived using the Rayleigh method and shown as equation
(1). Note that the length and width are not exactly 9x6 in 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 model exactly. Table 1 shows the values of all the constraints
that are used in equation (1).
Table 1: Constant values for Steinberg calculations
Board Properties
PWB Length, a (in)
PWB width, b (in)
Thickness, h (in)
Adjusted weight (lbs)
Gravity, g (in/sec^2)
Poisson's Ratio, v
Adjusted density, p (lbs s^2/in^3)
8.85
5.7
0.1
0.9819
386.4
0.3
5.04E-05
The next calculation consists of determining the maximum board displacement at the
center of the PWB as seen in Equation (3). 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 is expected 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 usually done with the random analysis
tool, so it may have to be subsequently modified 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 was be used. For each random category, the values for GRMS and
displacement, Z is different as seen in Table 2.
11
Table 2: Calculated first mode frequencies and PWB center displacements
PWB stiffness, D
Fn (Hz)
PSDIN, P (G^2/Hz)
Transmissibility, Q
G
ZRMS (in)
Curve B
265.570
266.400
0.002
20.00
4.09
0.00057
Curve B2
265.570
266.400
0.001
20.00
2.05
0.00028
Curve B3
265.570
266.400
0.002
20.00
4.09
0.00057
Curve C
265.570
266.400
0.020
20.00
12.93
0.00179
Curve D
265.570
266.400
0.040
20.00
18.29
0.00253
Curve E
265.570
266.400
0.080
20.00
25.87
0.00358
Based on a random analysis, PSD vs. frequency plots are generated to determine a value
for PSDOUT, which allows for a new accurate transmissibility to be calculated from
equation (5).
Q
PSDout
PSDin
(5)
This is the most reliable way of calculating the new transmissibility values for the PWB.
The PSD input from DO-160 at the first resonance is taken as the PSDin, and the PSDout

is the value measured from the response at the center of the board. The new
transmissibility value is then used in equations (3) and (4) for more accurate analytical
results. This calculation is done in section 3.2.6.
3.2
Finite Element Analysis
3.2.1
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
12
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 [2]. Figure 6 shows the
dimensions of the solder ball.
Figure 6: 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. While
Pierce models the complete BGA package and PWB, this has not been done in this
study. 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 fine enough mesh to achieve an
accurate 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-8 elements to help reduce the size of the model.
13
3.2.2
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 intent
of this analysis is to see how the mesh size affects the stiffness of the BGA ball. The
mesh size is optimized to have a balance between the number of elements and the
stiffness of the ball. The analysis results, mesh sizes and change in displacement are
reflected in Figure 7 and Table 3.
Figure 7: BGA Ball Mesh Optimization
14
Table 3: BGA Mesh Size and Results
Case
Elements
270
390
570
750
1
2
3
4
Nodes
1053
1569
2343
3117
Elements in Entire
Model (For BGA
Balls)
130680
188760
275880
363000
Displacement
4.99E-07
4.71E-07
4.52E-07
4.43E-07
Displacement vs. Optimization case
5.10E-07
Displacement, in
5.00E-07
4.90E-07
4.80E-07
4.70E-07
4.60E-07
4.50E-07
4.40E-07
1
2
3
4
Optimization Case
Figure 8: Displacement of BGA ball in static analysis
The displacement of the BGA ball in this analysis shows that results of case 3 and 4 are
within 2% of one another, which means that both meshes are accurate. Going from a
BGA ball with 750 elements to 570 elements, there is a reduction of 87120 quad-20
elements, so the mesh in case 3 is chosen for the BGA analysis. A finer mesh would
have been used, but the wedge elements at the center of the ball failed and therefore 750
elements is the finest mesh size attainable while keeping the same surface mesh.
3.2.3
Material Properties
The material properties used are listed below in Table 4. The properties for 60Sn-40Pb
[9] and 95.5Sn-4.0Ag-0.5Cu [8] solder come from Matweb. The type of lead free solder
used is based on the work of Erinc [10], as it is one of the most common types of lead
15
free solder for electronic assemblies. The properties for the PWB and BGA die are
derived based on calculations in Appendix C.
Table 4: Material Properties Used
Lead based solder
(60Sn-40Pb)
Youngs
Modulous, E (Psi)
Poisson's Ratio
Density (lbs/in )
3
Lead free solder
4.35E+06
2.60E+06
PWB
2.90E
+06
0.4
0.4
0.3
0.28
0.311
0.267
0.157
0.071
(95.5Sn-4.0Ag-0.5Cu)
BGA Die
4.53E+06
The PWB properties are based upon a .1 inch thick PWB with 16 layers. The stack-up of
the PWB as well as the thicknesses of the layers can be found in Appendix C. The PWB
also has .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 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.
3.2.4
Finite Element Model – Modal, Random, Inertial
There are various different 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 case would consist of a very large number of elements. 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 proved to be the best for what was being investigated, namely 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. Figure 9 shows the FEM model used for the modal analysis and random
analysis. It is a very simple model that consists of a single plate with a paver quad-8
mesh. An RBE2 MPC connected to an independent node fixes the edges to satisfy the
boundary conditions. The RBE2 element is chosen since this is a completely rigid body
element. This means that whatever boundary condition that is applied to the independent
16
node, the same condition is applied to all other nodes connected to the RBE2 MPC. The
random model has a different constraint on the independent node, but the same model is
used. The total number of elements used for this model was 21600.
Figure 9: Finite element model, Modal and Random Analysis mesh
The quasi-static inertial analysis has a much more complicated mesh because it
incorporates the BGA into the model. Figure 10 shows the FEM used for the inertial
analysis. The board mesh uses a paver quad-8 surface mesh that is .008 inches in size.
With a larger mesh size, the elements of the PWB around the BGA part fail the
PATRAN element test; this was the largest mesh size that the model passed with.
17
Figure 10: Finite element model, Inertial Analysis with BGA mesh
Figure 11 shows a close up of the BGA part and the area of the PWB around the BGA.
All 484 Balls are visible in this model.
Figure 11: Finite element model, Inertial Analysis BGA mesh pad area
18
Each ball is modeled in 3-D quad-20 elements and each plate, PWB and Die, is modeled
with 2-D quad-8 elements. This is seen in Figure 12 and how each plate is oriented with
respect to the BGA. Each plate is oriented directly on top and bottom of the BGA, which
allows the meshes to match. In order to do this correctly the plates must also have a user
defined offset applied in the element properties to accurately model the plate. The offset
is half of the plate thickness, because the plate is modeled after the mid-plane of the 3-D
solid.
Figure 12: Finite element model, Inertial Analysis BGA ball mesh
The next important mesh is the surface of the PWB where the solder balls connect the
BGA to the PWB. This is shown in Figure 13.
19
Figure 13: Finite element model, Inertial Analysis BGA pad mesh
Figure 14 is a zoomed in view of Figure 13 to show the points where the solder balls
connect the PWB in more detail; this is called the solder joint. Each solder balls lines up
exactly with the mesh on this surface in order to have 100% nodal connection. The mesh
for the BGA die is the same mesh as the PWB pad, and therefore is not shown here.
20
Figure 14: Finite element model, Inertial Analysis BGA pad mesh close up
In section 2.2 the different types of boundary conditions were described and it was
pointed out that 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. MD NASTRAN writes the RBE3 MPCs for all of the deformable
body connections based on the user inputs. Unfortunately, this method did not 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 examined in the future for more detailed analysis. Instead of using the Glue
boundary condition, 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 large numbers of MPCs. Although the pre processing of the
21
model takes much longer, the end result is a more reliable solution with much less
debugging.
3.2.5
Natural Frequencies
The first analysis consisted of calculating the first mode natural frequency. This analysis
was done without the BGA for the reason that the BGA does not have enough mass to
affect the result. The FEA was meshed with a quad-8 shell element. The PCB was
modeled as 2-D thin plate. The small displacements and .1 inch plate thickness makes
this an appropriate modeling decision. Figures 15 through 17 show the first 3 modes for
the PCB. The PWB was constrained in 6 degrees of freedom as shown in Figure 2.
Figure 15: First Mode 266.59 Hz
The computed first mode natural frequency nearly exactly matches the hand calculations
from chapter 3.1. Although the first mode natural frequency is the only mode of concern
for this report, Figures 3.2.5-2 and -3 show the second and third modes respectively.
22
Figure 16: Second Mode 341.07 Hz
Figure 17: Third Mode 511.18 Hz
The second and third modes are not used in any calculations in this report. The intent is
to look at the maximum displacements and the maximum PWB curvatures, which can all
be determined by the first mode shape. When doing RMS, or room mean square,
calculations, the entire spectrum of results from 10 to 2000 Hz must be taken into
account. This is talked about in the next chapter about the Random Analysis.
23
3.2.6
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 reduce calculation time. The random analysis, or frequency response, is done
again for each case. The results show peak responses, which are the modes of natural
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 a node of interest to calculate the relative
displacement of the point of interest. For this the nodes of interest are of course the node
at the center of the board and the node with the maximum displacement. The
displacements were first used to validate the hand calculations in section 3.1, and are
later used for Miner’s Damage Index calculations. Following the routine used earlier
with the hand calculations, the random analysis for curve C of RTCA DO-160 will be
shown below.
The modal analysis model was used for the random analysis with some minor boundary
condition changes. The RBE2 MPC that was used to constrain the model is slightly
different in that all six degrees of freedom are not fixed. For this model, the axis of
interest is the axis perpendicular to the board area, the z-axis. This will give the
maximum response, and most displacement so it is the limiting case. The input node is
fixed in five degrees of freedom where the z-axis is left free since it is the direction the
PSD input from RTCA DO-160 is being applied. Other model constraints are a 2%
material dampening factor for the entire model that Steinberg suggests to use as a
conservative value [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. The
24
acceleration frequency response in Figure 18 should be the same as the input curve C,
and this is indeed the case.
Figure 18: Base Node Acceleration Curve C
To next validation is the displacement of the center of the board. This is done by the
relative displacement of the centermost node and the fixed PWB edge; this is a special
PATRAN post-processing function. Taking the centermost node, the RMS displacement
is shown in Figure 19 and the acceleration at this point is shown in Figure 20.
Figure 19: Central Node RMS Displacement
25
Figure 20: Central Node RMS Acceleration (GRMS)
The RMS relative displacement is .00209 inches. This compares well to the calculated
displacement of .0018 inches using Steinberg’s method in section 3.1. This is used to
validate the random analysis model, but there is still one value that we do not know
precisely, the transmissibility at the center node. With Figure 20 and equation (5), the
new values for transmissibility can be calculated. This calculation is summarized in
Table 6 for location 1 and Table 7 for locations 2 and 3. 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 in Figure 21 for Curve C:
Figure 21: Maximum RMS Displacement
26
Figure 22: 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 those in the BGA balls are, the
stress calculations are performed with the inertial analysis instead to save on calculation
time. The random analysis results are summarized in Table 5.
Table 5: Random Analysis Acceleration and Displacement Results
Curve
Acceleration
at Max
Displacement
(Gs)
Maximum
Displacement
(in)
Acceleration
at PWB
Center (Gs)
Displacement
at PWB
Center (in)
B
11.97
1.37E-03
6.14
6.75E-05
B2
5.986
6.86E-04
3.07
3.37E-04
B3
11.98
1.38E-03
6.16
6.82E-04
C
37.77
4.31E-03
19.26
2.09E-03
D
69.28
6.43E-03
37.1
3.17E-03
E
75.47
8.54E-03
38.41
4.16E-03
At this point, new values for Q can be calculated. By referring to Figure 20, the first
large peak resembles the first mode natural frequency. Since the first mode was used as
the shape for the calculations in section 3.1, the PSDOUT from this peak is used. With
equation (5), it is easy to calculate the new value for Q. The value of the peak at 266.6
Hz is the PSDOUT term in the equation and the PSDIN, P, from the DO-160 Curve in
Figure 4 at the same frequency; also shown Table 1. Table 6 shows all of these values
27
for location 1 and the new Q value, new GRMS value and ZRMS for the center of the PWB
using equations (5), (4) and (3) respectively.
Table 6: Peak transmissibility calculations location 1
Location 1
PSDOUT
Q
GRMS
ZRMS
Curve B
1.06
23.08
4.39
0.0006
Curve B2
0.27
23.06
2.20
0.0003
Curve B3
1.06
23.08
4.39
0.0006
Curve C
10.65
23.08
13.89
0.0019
Curve D
22.77
23.86
19.98
0.0028
Curve E
42.58
23.07
27.78
0.0038
The ZRMS values are compared to the values in Table 5 to validate the accuracy of the
random analysis. The new transmissibility is now used in section 3.4 for the Miners
Damage Index fatigue life calculations. The peak transmissibility values for location 2
and location 3 of the ball grid array are not the same as location 1 at the center of the
board. This process is done again for locations 2 and 3 to find the peak Q value at 266.6
Hz and recorded in Table 7. Note that this calculation is only done for one curve since
the Q does not change with different inputs as shown for the location 1 calculations.
ZRMS is not calculated at these points because they are not at the center of the PWB.
Table 7: Peak transmissibility calculations location 2 and 3
Curve B
Location 2
PSDOUT
Q
Location 3
3.21
1.41E-12
40.00
2.66E-05
Notice that the Q for location 2 is about twice that of location 1 and Q for location 3 is
almost non-existent. These values make sense due to the fact that location 2 has a higher
displacement than location 1 and location 3 is near the card guide edge where the
displacement is the smallest. The next section shows the actual acceleration loads
required to achieve said displacements and will flow into the damage caused to the
BGA’s.
3.2.7
Inertial Analysis
Inertial analysis is a static analysis that enables the user to solve dynamic problems for
stresses with much less calculation time and power. This scheme only works if the first
mode is the mode of interest, which is the case in the present study. This is due to the
shape of the board and the applied constraints having the edges fixed with six degrees of
28
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 one is interested in the maximum displacement and the 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 without compromising the
results. The PWB is broken up in to 4 pieces, the main PWB and the three different
mounting positions for the BGA. The analysis is first done is with just the PWB and no
BGA. An initial value of 1G is applied to the model, or 1 times the acceleration of
gravity, to calculate the initial displacement, which turns out to be 2.44e-4 inches. This
initial 1G displacement is used as a baseline or starting point for the iterations. The ratio
of the displacement to the applied inertial acceleration is linear thus the maximum
displacement can be interpolated from the initial analysis and only two iterations are
needed. 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 in Figure 23 for the maximum displacement case. Note that the curvature of the
PWB looks exactly like the first mode natural frequency. The summary of all results for
the iterative calculations is in Table 8.
29
Figure 23: Curve C Inertial Analysis Maximum Displacement
Table 8: Inertial Analysis Loads and displacements
Random
Curve
Wanted
Displacement
(in)
Displacement
iteration 1
(in)
Iteration 1
(G's)
Iteration 2
(G's)
Displacement
iteration 2 (in)
B
1.37E-03
5.62E+00
1.18E-03
6.54E+00
1.38E-03
B2
6.86E-04
2.81E+00
5.92E-04
3.26E+00
6.87E-04
B3
1.38E-03
5.64E+00
1.19E-03
6.52E+00
1.37E-03
C
4.31E-03
1.76E+01
3.71E-03
2.05E+01
4.32E-03
D
6.43E-03
2.63E+01
5.54E-03
3.05E+01
6.43E-03
E
8.54E-03
3.50E+01
7.38E-03
4.05E+01
8.54E-03
Column 1 represents the random curve from RTCA DO-160 inputted for the random
analysis. The second column represents the maximum displacements from the random
analysis taken from the maximum displacement column in Table 5. This is where the
analysis becomes quasi static due to running a static analysis to determine displacements
from a random input analysis. The 1G-acceleration displacement is used in the ratio of
the displacement over the inertial load 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.
The inertial model is taken one step further, and the BGA is finally added to the FEM
shown in Figure 10. To run a dynamic model, or random analysis, the calculation time is
30
very extensive and sometimes will not converge due to the number of nodes and
elements; therefore a quasi-static inertial analysis was run. The PWB already has a mass
of electrical components spread over the surface to add some stiffness so the analysis is
more realistic. The addition of the BGA, since the mass is so small, does not
significantly 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
acceptable due to the displacements being the same as well as the PSD level at which the
first mode natural frequency occurs. As a result, 5 curves are run to examine the stress in
the solder balls of the BGA. The details and results of the BGA stresses will be
described in section 3.2.8.
3.2.7.1 Hardware Characteristics
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 was used. For this
analysis, a super computer is used to speed up calculations since many analysis cases
were 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. This analysis was done with 4 Processor Type
processors and 4 gigabytes of memory. The total analysis time was around 3 hours per
case, and used a total of SWAP MEMORY.
3.2.8
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 mean stress areas on
the solder joints. When examining the results, the max stress was found at the edge the
solder joint modeled in the FEM. Before the stress is analyzed, the FEM must have the
correct displacement from the inertial analysis shown in Table 8. For each case, this is
true at each of the different BGA locations. The displacement corresponding to Curve B
from the inertial analysis with the BGA at the board center, location 1 is shown in Figure
24.
31
Figure 24: Curve B Inertial Displacement
The stresses in the BGA balls resemble the stress from a random vibration RMS
displacement. Figures 25 through 27 show the BGA ball stress fields for locations 1, 2
and 3.
Figure 25: Curve B inertial BGA ball stress field location 1
32
Figure 26: Curve D Inertial BGA ball stress field location 2
Figure 27: Curve B2 Inertial BGA ball stress field location 3
33
The results show that the stress on the balls follows the curvature of the board. This is a
basic check that the stress distribution is qualitatively correct. The next thing to realize is
that since there is little to no displacement at the center of the BGA part relative to the
PWB, the balls closest to the center of the BGA die will have the lowest stress values.
As the balls get further away from the center of the BGA part, the stress increases and
the stress is 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 for locations 1 and 2.
The stress field in location 3 is a little different than the other two as shown in Figure 27
due the different curvature in the board. The center of the board has the most curvature
and thus the highest stress fields in location 1 and 2. Figure 28 shows where the actual
failure points are and the stress distribution of the BGA solder ball.
Figure 28: 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 may be and is consistent with the original assumption. Corresponding
figures are not shown for all the cases as the stress induced from the inertial analysis
increases linearly with the increased board displacement. A very important thing to note
is that the high stress area is at a single point where the rest of the solder joint is at a
much lower stress. Figure 29 shows this stress distribution.
34
Solder Joint
region
Mean stress
region
`
High stress area,
crack formation
Figure 29: Solder ball stress distribution and possible crack initiation point
The high stresses will likely result in a crack in the solder joint rather than a failure due
to solder’s non-linear characteristics at high stresses. This high stress region will likely
cause a crack in the solder joint and grow as the number of cycles increases until failure.
This crack initiation would greatly reduce the high cycle fatigue life. This is very
difficult to test because of the microscopic nature of the solder balls being studied and
the various mixtures of solder that are available. Actual testing rather than analysis must
be done in order to determine such a factor. A lot of testing has been done for thermal
crack growth, but not in vibration high cycle fatigue crack growth. To properly match
the analysis stress data, the average stress of the solder joint is used to determine the
fatigue life for the solder ball. The solder ball with the largest mean stress in the solder
joint was used, as this will again be the limiting stress value for high cycle fatigue. The
35
solder ball with this highest mean stress turned out to be the same solder ball with the
highest single point stress.
The quasi-static inertial analysis yields an equivalent RMS stress that would be induced
from the random vibration input. In order to study the high cycle fatigue damage of the
random vibration environment, the RMS stress needs to be corrected and modeled as a
sinusoidal stress. The sinusoidal correction factor comes from Steinberg and
incorporates the 3-band method described in chapter 2.1 [3]. This method uses equations
(6) and (7) to define the correction factor  that incorporates the percentage of
likelihood, , the sigma factor, , and the fatigue exponent for the lead based solder
under study, :
  11  2 2  3 3
(6)
  
(7)
 of 1.95 as shown in Appendix A. The value of  is simply
Solving for  yields a value

the Gaussian probability distribution, which incorporates the three σ’s and their
respective percentage of occurrence [3]. Tables 9 and 10 show the maximum RMS stress
values and the equivalent sinusoidal stress values for all vibration curves and BGA
locations for both lead and lead free solder.
Table 9: Maximum Inertial Analysis Stress, RMS and Sinusoidal 60Sn-40Pb solder
Curve:
B2
B
C
D
E
Max Stress Location 1 (PSI)
844
1700
5310
7910
10500
Corrected Sine Stress (PSI)
1645.8
3315
10354.5
15424.5
20475
Max Stress Location 2 (PSI)
900
1810
5650
8430
11200
Corrected Sine Stress (PSI)
1755
3529.5
11017.5
16438.5
21840
Max Stress Location 3 (PSI)
550
1260
3950
5900
7830
Corrected Sine Stress (PSI)
1072.5
2457
7702.5
11505
15268.5
36
Table 10: Maximum Inertial Analysis Stress, RMS and Sinusoidal 95.5Sn-4.0Ag0.5Cu solder
Curve:
B2
B
C
D
E
Max Stress Location 1 (PSI)
614
1230
3870
5690
7630
Corrected Sine Stress (PSI)
1197.3
2398.5
7546.5
11095.5
14878.5
Max Stress Location 2 (PSI)
661
1330
4210
6190
8210
Corrected Sine Stress (PSI)
1288.95
2593.5
8209.5
12070.5
16009.5
Max Stress Location 3 (PSI)
451
905
2890
4230
5610
Corrected Sine Stress (PSI)
879.45
1764.75
5635.5
8248.5
10939.5
The maximum stresses in the two tables resemble the stresses that will case the crack in
low cycle fatigue and are not used in the high cycle fatigue life calculations. To
accurately depict the stresses for high cycle fatigue, the average stresses of the BGA
solder ball at the solder joint were recorded. These values are in Tables 11 and 12.
Notice that most of the stresses in the solder balls differ from the max stress area by an
order of magnitude.
Table 11: Average solder-joint stresses to cause high cycle fatigue - 60Sn-40Pb
solder
Curve:
Average Stress Location 1 (PSI)
Corrected Sine Stress (PSI)
Average Stress Location 2 (PSI)
Corrected Sine Stress (PSI)
B2
C
D
E
123
248
776
1160
1530
239.85
483.6
1513.2
2262
2983.5
129
258
806
1200
1600
251.55
503.1
1571.7
2340
3120
78
179
561
838
1110
152.1
349.05
1093.95
1634.1
2164.5
Average Stress Location 3 (PSI)
Corrected Sine Stress (PSI)
B
Table 12: Average solder-joint stresses to cause high cycle fatigue – 95.5Sn-4.0Ag0.5Cu solder
Curve:
Average Stress Location 1 (PSI)
Corrected Sine Stress (PSI)
Average Stress Location 2 (PSI)
Corrected Sine Stress (PSI)
Average Stress Location 3 (PSI)
Corrected Sine Stress (PSI)
B2
B
C
D
E
91
183
574
855.22
1130
177.45
356.85
1119.3
1667.68
2203.5
96.4
193
614.05
902
1200
187.98
376.35
1197.4
1758.9
2340
65.4
131
411.8
613
813
127.53
255.45
803.02
1195.35
1585.35
37
The mean solder joint stresses are compared to the test results and FEA done by David
M. Pierce and are of the same order of magnitude for similar PWB displacements [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 section 3.3, the MDI will be calculated using Steinberg’s
method for the BGA analyzed and will be compared to the S-N curve for 60Sn-40Pb
solder.
The lead free solder was not analyzed for high cycle fatigue because a vibration fatigue
exponent was not available for 95.5Sn-4.0Ag-0.5Cu solder. Further testing and
validation of this type of lead free solder must be done in order to properly model the
FEM to properly calculate the high cycle fatigue. Even though the fatigue exponent is
not readily available, this version of lead free solder can be looked at and some
conclusions can be made from a simple correlation of the two solders and their stresses.
Keeping the fatigue exponent the same for both solders, the correction factor, , gives
us corrected stress values as shown in Tables 10 and 12. These values are then plotted
against the locations displacement for each input vibration load as shown in Table 13.
Figure 30 shows the relationship between the max stresses on the solder joint of the
BGA solder ball vs. the PWB displacement per location. Figure 31 shows this same
relationship, but for the mean stress region of the solder joint.
Table 13: PWB displacement per location and load
Curve
Displacement
Location 1
Displacement
Location 2
Displacement
Location 3
B2
3.36E-04
5.24E-04
1.45E-04
B
6.79E-04
1.04E-03
2.89E-04
C
2.10E-03
3.27E-03
9.44E-04
D
3.16E-03
4.82E-03
1.39E-03
E
4.16E-03
6.51E-03
1.80E-03
38
BGA Max Stress vs PWB Displacement
25000
Location
1
Location
2
Location
3
Location
1 LF
Location
2 LF
Location
3 LF
Stress, PSI
20000
15000
10000
5000
0
0
0.001
0.002
0.003
0.004
0.005
0.006
0.007
Displacement, in
Figure 30: Maximum Corrected Sine Stress vs. PWB displacement, lead and lead
free solder
BGA Mean Stress vs PWB Displacement
3500.00
3000.00
Location
1
Location
2
Location
3
Location
1 LF
Location
2 LF
Location
3 LF
Stress, PSI
2500.00
2000.00
1500.00
1000.00
500.00
0.00
0
0.002
0.004
0.006
0.008
Displacement, in
Figure 31: Mean Corrected Sine Stress vs. PWB displacement, lead and lead free
solder
39
This stress relationship shows that when the stiffness of the material is less, the resultant
stress is less, but not proportionally less. The results of the lead free solder show that it
merits a further investigation as to what the fatigue life of the solder is to determine
whether lead free solder is better at low or high cycle fatigue and at which points the are
comparable. The mean stress plot has the same conclusions, but the magnitudes of the
stresses are proportionally less. The average stress will be used in chapter 3.4 to
determine the high cycle fatigue based on the finite element analysis.
3.3
Miner's Damage Index Calculations
The Miner's Damage Index is a cumulative fatigue damage ratio based upon the number
of actual cycles done, n, over the number of cycles to failure, N, equation (12). This
calculation is represented below:
Rn 
n1 n 2 n 3


N1 N 2 N 3
(8)
where

ni  f n 3600i
(9)
The assumptions are that the damage is linear and is a simple function of the load as well
 to the loading sequence. The Damage index is designed
as the failures not being related
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 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 for a
particular displacement. This value of fatigue cycles corresponds with the single
40
amplitude value of displacement for each particular component that is limited to Z, to
meet the 10 million cycles. Figure 32 represents how Z relates to board placement.
L
Large
relative
motion
Lead
Wire
Component
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 32: Allowable board displacement, Z [3]
The important values that can change are C and r. C is based on the component, which is
derived from test data or analysis data and r is based on where the component is placed
with respect to the board center. For this analysis, the components are placed as shown
in Figure 2.1-1. Steinberg’s method is used for the initial calculation of the MDI. These
values are 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 mean stress FEA data. For
the analysis, there are three positions where r is equal to 1 for location 1, .5 for location
2, and .25 for location 3 [3]. Tables 14 and 15 are a list of constants that do not change
for any of these calculations.
Table 14: Miner’s Damage Index calculation constants.
Location
Curve
fn (Hz)
Natural
Frequency
GRMS
Q
Y1
Y2
Y3
PSDIN
B
Board
dim "X"
1
B
266
23
0.002
4.4
0.00068
0.0014
0.002
6
1
B2
266
23
0.001
3.1
0.00034
0.0007
0.001
6
1
C
266
23
0.02
13.9
0.0021
0.0042
0.0063
6
1
D
266
23
0.04
19.6
0.00316
0.0063
0.0095
6
41
1
E
266
23
0.08
27.8
0.00416
0.0083
0.0125
6
2
B
266
40
0.002
5.8
0.00104
0.0021
0.0031
6
2
B2
266
40
0.001
4.1
0.00052
0.001
0.0016
6
2
C
266
40
0.02
18.3
0.00327
0.0065
0.0098
6
2
D
266
40
0.04
25.9
0.00482
0.0096
0.0144
6
2
E
266
40
0.08
36.6
0.00651
0.013
0.0195
6
3
B
266
0
0.002
0
0.00029
0.0006
0.0009
6
3
B2
266
0
0.001
0
0.00015
0.0003
0.0004
6
3
C
266
0
0.02
0
0.00094
0.0019
0.0028
6
3
D
266
0
0.04
0
0.00139
0.0028
0.0042
6
3
E
266
0
0.08
0
0.0018
0.0036
0.0054
6
Table 15: Miners Damage Index calculation constants continued
Location
Curve
h
Board
Thick.
L
Comp.
Length
r
Location
Factor
Beta
Fatigue
Exp.
Test Time
(min)
n1
n2
n3
1
B
0.1
1.06
1
6.5
60
6.54E+05
2.60E+05
4.15E+04
1
B2
0.1
1.06
1
6.5
60
6.54E+05
2.60E+05
4.15E+04
1
C
0.1
1.06
1
6.5
60
6.54E+05
2.60E+05
4.15E+04
1
D
0.1
1.06
1
6.5
60
6.54E+05
2.60E+05
4.15E+04
1
E
0.1
1.06
1
6.5
60
6.54E+05
2.60E+05
4.15E+04
2
B
0.1
1.06
0.707
6.5
60
6.54E+05
2.60E+05
4.15E+04
2
B2
0.1
1.06
0.707
6.5
60
6.54E+05
2.60E+05
4.15E+04
2
C
0.1
1.06
0.707
6.5
60
6.54E+05
2.60E+05
4.15E+04
2
D
0.1
1.06
0.707
6.5
60
6.54E+05
2.60E+05
4.15E+04
2
E
0.1
1.06
0.707
6.5
60
6.54E+05
2.60E+05
4.15E+04
3
B
0.1
1.06
0.25
6.5
60
6.54E+05
2.60E+05
4.15E+04
3
B2
0.1
1.06
0.25
6.5
60
6.54E+05
2.60E+05
4.15E+04
3
C
0.1
1.06
0.25
6.5
60
6.54E+05
2.60E+05
4.15E+04
3
D
0.1
1.06
0.25
6.5
60
6.54E+05
2.60E+05
4.15E+04
3
E
0.1
1.06
0.25
6.5
60
6.54E+05
2.60E+05
4.15E+04
Many of the constants in the two tables are given values and constraints set on the
problem. The natural frequency is a calculated value from both equation (1) and analysis
results. Q is also a calculated value from the random RMS analysis. The PSDin comes
from the point on the RTCA DO-160 curve at which the first natural frequency occurs,
and from equation 3, the new GRMS value is calculated. Y1 comes from the displacement
at the location of the BGA from Table 13, and Y2 is the σ2 displacement, Y1 multiplied
by 2, and Y3 is the σ3 displacement, Y1 multiplied by 3. The board width b, thickness h,
component length L is constant values for each case. The location factor, r, resembles
42
the position on the PWB with respect to the center of the PWB. Notice this changes with
the location based on the following equation where X and Y resemble the location of the
component on the PWB, and the length and width of the PWB are represented as a and b
in equation (10) [3].
X  Y 
r  sin   sin   
 a  b 
(10)
Equation (10) gives a BGA placement factor for the allowable displacement Z, based on
the curvature of
the board. The fatigue factor beta is constant for the type of solder used,
and the test time is considered 60 minutes, as this is a standard requirement for these
types of vibration levels. All of this data enables us to calculate the σ1, σ2, and σ3 cycles
being done during a 60 minute test shown by n1, n2, and n3. This is a simple calculation
where the natural frequency is multiplied by the test time, in seconds, times the percent
occurrence of one σ, two σ and three σ occurrences, 68.26%, 27.18%, and 4.33%
respectively, equation (9). Table 16 shows the results of the MDI calculations.
Table 16: Miner’s Damage Index
Location
Curve
Steinberg
Factor
C
Comp.
Coeff.
Z
Allow
Deflect
N1
10Mil cycles
N2
10Mil cycles
N3
10Mil cycles
MDI
1
B
2.25
18
0.0043
1.56E+12
1.72E+10
1.24E+09
0
1
B2
2.25
18
0.0043
1.50E+14
1.66E+12
1.19E+11
0
1
C
2.25
18
0.0043
1.00E+09
1.11E+07
7.93E+05
0.076
1
D
2.25
18
0.0043
7.06E+07
7.80E+05
5.59E+04
1.085
1
E
2.25
18
0.0043
1.19E+07
1.31E+05
9.42E+03
6.436
2
B
2.25
18
0.0085
8.62E+12
9.52E+10
6.82E+09
0
2
B2
2.25
18
0.0085
7.62E+14
8.42E+12
6.03E+11
0
2
C
2.25
18
0.0085
5.20E+09
5.74E+07
4.12E+06
0.015
2
D
2.25
18
0.0085
4.17E+08
4.61E+06
3.30E+05
0.184
2
E
2.25
18
0.0085
5.87E+07
6.49E+05
4.65E+04
1.304
3
B
2.25
18
0.0034
9.43E+13
1.04E+12
7.47E+10
0
3
B2
2.25
18
0.0034
8.34E+15
9.22E+13
6.61E+12
0
3
C
2.25
18
0.0034
4.30E+10
4.75E+08
3.40E+07
0.002
3
D
2.25
18
0.0034
3.47E+09
3.84E+07
2.75E+06
0.022
3
E
2.25
18
0.0034
6.47E+08
7.15E+06
5.13E+05
0.118
The Steinberg component factor C equal to 2.25 is used based off of his test data. Note
that this value changes for all different part types, and is determined by extensive testing.
43
The component coefficient is based on Steinberg’s factor multiplied by 8, which is used
in a second method to calculate Z by Calvin [3]. Either factor can be used, based on the
equation to calculate the allowable displacement Z. The first equation is shown in Figure
32 derived by Steinberg. The second equation derived by Calvin uses the same idea but
the board width is squared. Equation (11) is Calvin’s method used in this report as it
gives a slightly more accurate result for this analysis. Calvin derived equation (11) with
a PWB that was 8x8 inches where Steinberg based his relationship for Z on a 1x1 inch
PWB. The calculation in Appendix A shows how the two component constants are
related.
Z
.00022B 2
8Chr L
(11)
Z is the allowable displacement for the BGA part at each location shown in Table 16.
Since the displacement, Z, is known for the part to reach 10E7 cycles, the high cycle

fatigue of the parts is calculated; N1, N2, and N3 for σ1, σ2 and σ3 displacements
respectively. N1, N2, and N3 are the values at which the part will fail at Y1, Y2 and Y3
respectively. Equation (12) below represents the calculation method in the MDI to find
the number of cycles to failure:
Yi  
N i  10E 7 
Z 
(12)
This equation has the ratio of actual PWB displacement over allowable component
displacement, raised to the fatigue exponent of solder, which for this analysis is 60Sn
40Pb lead based solder. This same approach can be used to the lead free solder but the
fatigue exponent is necessary for this relationship.
The last part of this calculation is the actual Miner’s Damage Index calculation, which is
expressed as Rn at the beginning of the section. This is calculated by taking the number
of cycles run, ni, over the number of cycles to failure, Ni. Since the Steinberg 3 band
method is being used, there are three ratios that are added together to achieve the real
Cumulative Damage Index represented in equation (8). The MDI values from Table 16
in green are considered to be acceptable, or lower than .5 where the MDI values in red
are above .5 and are considered unacceptable. Based off of the analysis, it shows that
44
location 1 is only be acceptable for curves B, B2, B3 and C. Curve B3 does not show up
in this calculation as it is the same as curve B for the natural frequency case being
studied. Location 2 can be used for all curves but curve E, and lastly location 3 is
acceptable for all curves. Location 3 is the optimal location for the ACTEL 484 BGA
with the random vibration inputs herein. This may not be true for all BGA’s, as a smaller
package may be acceptable in any of these locations. Additional analysis of different
package types must be done to validate this conclusion. The next section describes the
optimization of the MDI calculation through the finite element data in this report. The
fatigue life data from the analysis will be compared to the fatigue life data calculated
using Steinberg’s component factor C.
3.4
Fatigue Life
Steinberg’s component factor C is one of the key values used to calculate fatigue life.
Using the stress data from the analysis, a new value for C was calculated which gives
much more accurate results for the MDI. The fatigue life curve for the 60Sn-40Pb solder
is looked at and compared to the sinusoidal stresses in the BGA balls. This gives a
fatigue life based on the analysis data that can be compared to the high cycle fatigue life
in the MDI calculation. Ideally these values should be close to each other, but the
component C factors for the BGA components are based on a board that is 8x8 inches.
Since the board in this analysis is 6x9 inches, the value needs to be adjusted and
optimized. Figure 33 shows the fatigue S-N curve for solder with a fatigue exponent of
6.5.
45
S-N curve 60Sn-40Pb
Stress, PSI
10000
1000
S-N curve 60Sn40Pb
100
1.00E+03
1.00E+06
1.00E+09
1.00E+12
1.00E+15
Number of cycles, N
Figure 33: S-N curve for 60Sn-40Pb solder
By taking the stress values from the average solder joint stress on the solder ball in Table
11, a fatigue life can be easily calculated by using the stress to find the number of cycles
to failure by the figure above or equation (12):
ave  
N failure  N 0  
 0 
(13)
N0 and 0 are equal to 10e3 cycles and 6500 PSI respectively and are defined in [3]. The
derivation of this equation
 is shown in Appendix A, which comes from the slope of a
line on a log-log graph. The number of cycles to failure for each curve and location
based on the S-N curve for 60Sn-40Pb solder is in table 17.
Table 17: High cycle fatigue values based on FEA mean solder join stress
Curve
B2
B
C
D
E
Location 1 Cycles
2.06E+13
2.16E+11
1.30E+08
9.54E+06
1.58E+06
Location 2 Cycles
1.51E+13
1.67E+11
1.02E+08
7.66E+06
1.18E+06
Location 3 Cycles
3.98E+14
1.80E+12
1.07E+09
7.90E+07
1.27E+07
46
The number of cycles to failure was plotted against the mean solder ball stress based on
the two high cycle fatigue calculation methods. The high cycle fatigue calculation for the
MDI is the proper way to compare these values. The number of cycles to failure, N,
comes from two different methods, where the only thing in common is the fatigue
exponent. MDI uses the ratio of the actual displacement over allowable displacement to
compute the total number of cycles to failure where the FEA approach uses the fatigue
curve for 60Sn-40Pb solder and equation (13). These two different correlations are
comparable as shown in Figure 34 through 36 for locations 1 2 and 3 respectively.
Number of cycles vs. BGA Ball Stress
Location 1
1.00E+14
Number of cycles, N
1.00E+13
1.00E+12
Location 1 MDI
1.00E+11
1.00E+10
1.00E+09
Location 1 SN
curve
1.00E+08
1.00E+07
1.00E+06
1.00E+05
100
1000
10000
Stress, PSI
Figure 34: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 1
47
Number of cycles vs. BGA Ball Stress
Location 2
1.00E+14
Number of cycles, N
1.00E+13
1.00E+12
1.00E+11
Location 2 MDI
1.00E+10
1.00E+09
1.00E+08
Location 2 SN
curve
1.00E+07
1.00E+06
1.00E+05
100
1000
10000
Stress, PSI
Figure 35: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 2
Number of cycles, N
Number of cycles vs. BGA Ball Stress
Location 3
1.00E+16
1.00E+15
1.00E+14
1.00E+13
1.00E+12
1.00E+11
1.00E+10
1.00E+09
1.00E+08
1.00E+07
1.00E+06
Location 3 MDI
Location 3 SN
curve
100
1000
10000
Stress, PSI
Figure 36: Log-Log plot of MDI vs. S-N curve high cycle fatigue location 3
For each location, the two methods compare closely, but C needs to be optimized so the
MDI method matches the FEA data. C is modified in the MDI calculation until the
number of cycles to failure matches the number of cycles to failure based on the S-N
curve. All of these values are averaged and the best-fit value will be the new Steinberg
component constant. The interpolation of the data is shown in Table 18.
48
Table 18: Steinberg component constant C, MDI and S-N fatigue life interpolation
Location
Steinberg
Factor
Curve
C
Comp.
Coeff.
N1
10Mil
cycles
N2
10Mil
cycles
N3
10Mil
cycles
MDI
N
Total number of
cycles to failure
1
B
3.056
24.45
2.13E+11
2.35E+09
1.69E+08
1
B2
3.059
24.47
2.04E+13
2.25E+11
1
C
3.085
24.68
1.29E+08
1.42E+06
1
D
3.066
24.53
9.44E+06
1
E
3.074
24.59
2
B
4.138
2
B2
4.125
2
C
2
N_FEA
Cycles to
failure based
on FEA
0
2.16E+11
2.16E+11
1.62E+10
0
2.06E+13
2.06E+13
1.02E+05
0.595
1.30E+08
1.30E+08
1.04E+05
7.48E+03
8.112
9.55E+06
9.54E+06
1.57E+06
1.73E+04
1.24E+03
48.897
1.58E+06
1.58E+06
33.1
1.64E+11
1.82E+09
1.30E+08
0
1.66E+11
1.67E+11
33
1.48E+13
1.64E+11
1.17E+10
0
1.50E+13
1.51E+13
4.138
33.1
9.91E+07
1.10E+06
7.85E+04
0.772
1.00E+08
1.02E+08
D
4.169
33.35
7.57E+06
8.37E+04
6.00E+03
10.113
7.66E+06
7.66E+06
2
E
4.113
32.9
1.16E+06
1.29E+04
9.22E+02
65.754
1.18E+06
1.18E+06
3
B
4.144
33.15
1.78E+12
1.97E+10
1.41E+09
0
1.80E+12
1.80E+12
3
B2
3.6
28.8
3.93E+14
4.34E+12
3.11E+11
0
3.98E+14
3.98E+14
3
C
3.975
31.8
1.06E+09
1.17E+07
8.42E+05
0.072
1.08E+09
1.07E+09
3
D
4.035
32.28
7.80E+07
8.61E+05
6.17E+04
0.982
7.89E+07
7.90E+07
3
E
4.144
33.15
1.22E+07
1.35E+05
9.68E+03
6.265
1.24E+07
1.27E+07
After the number of cycles to failure for the two methods match, there are a lot more
MDI values that do not satisfy the condition of being less than .5. This means that the
original calculations did not accurately depict where the failures would occur and the
new C value adds more restriction for part placement. Table 19 shows the MDI
calculation for the new component factor C equal to 3.728.
Table 19: Optimized Steinberg component constant C, MDI and S-N fatigue life
calculation for a 6x9 inch PWB
Location
Curve
Steinberg
Factor
C
Comp.
Coeff.
N1
10Mil
cycles
N2
10Mil cycles
N3
10Mil
cycles
MDI
1
B
3.728
29.823
5.8573E+10
6.4715E+08
4.6389E+07
0.001
1
B2
3.728
29.823
5.6381E+12
6.2292E+10
4.4652E+09
0.000
1
C
3.728
29.823
3.7589E+07
4.1530E+05
2.9769E+04
2.037
1
D
3.728
29.823
2.6504E+06
2.9283E+04
2.0991E+03
28.888
1
E
3.728
29.823
4.4675E+05
4.9360E+03
3.5382E+02
171.383
2
B
3.728
29.823
3.4047E+10
3.7617E+08
2.6964E+07
0.002
2
B2
3.728
29.823
3.0096E+12
3.3252E+10
2.3836E+09
0.000
2
C
3.728
29.823
2.0536E+07
2.2690E+05
1.6264E+04
3.728
2
D
3.728
29.823
1.6472E+06
1.8199E+04
1.3046E+03
46.482
2
E
3.728
29.823
2.3193E+05
2.5625E+03
1.8368E+02
330.131
3
B
3.728
29.823
3.5408E+12
3.9121E+10
2.8042E+09
0.000
49
3
B2
3.728
29.823
3.1336E+14
3.4622E+12
2.4818E+11
0.000
3
C
3.728
29.823
1.6130E+09
1.7821E+07
1.2774E+06
0.047
3
D
3.728
29.823
1.3042E+08
1.4409E+06
1.0329E+05
0.587
3
E
3.728
29.823
2.4303E+07
2.6852E+05
1.9248E+04
3.150
This new value of C is the best-fit value for MDI calculations for BGA’s when using a
6x9 inch PWB with 3 fixed edges. Each variation of the boundary conditions results in a
slightly different factor for C, thus additional work should be done for all different edge
conditions; simply supported, all fixed, etc. Doing this analysis more accurately defines
this factor for all boards with a 6x9 inch PWB. The final results for this are shown in the
next three Figures, 37 through 39, how the new component C factor influences the total
high cycle fatigue life of the BGA.
Number of cycles, N
Number of cycles vs. BGA Ball Stress Location 1
1.00E+14
1.00E+13
1.00E+12
1.00E+11
1.00E+10
1.00E+09
1.00E+08
1.00E+07
1.00E+06
1.00E+05
Location 1 MDI
Location 1 SN curve
Location 1 MDI
Corrected
100
1000
10000
Stress, PSI
Figure 37: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location
1
50
Number of cycles vs. BGA Ball Stress Location 2
1.00E+14
Number of cycles, N
1.00E+13
1.00E+12
1.00E+11
1.00E+10
Location 2 MDI
1.00E+09
Location 2 SN curve
1.00E+08
Location 2 MDI
corrected
1.00E+07
1.00E+06
1.00E+05
1.00E+04
100
1000
10000
Stress, PSI
Figure 38: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location
2
Number of cycles vs. BGA Ball Stress Location 3
1.00E+16
1.00E+15
Number of cycles, N
1.00E+14
1.00E+13
1.00E+12
Location 3 MDI
1.00E+11
Location 3 SN curve
1.00E+10
Location 3 MDI
Corrected
1.00E+09
1.00E+08
1.00E+07
1.00E+06
100
1000
10000
Stress, PSI
Figure 39: Log-Log plot of corrected MDI vs. S-N curve high cycle fatigue location
3
51
The new component factor C makes the MDI fatigue life fit the FEA results for all three
locations. This method can be duplicated for the lead free solder as long as a fatigue
exponent is available. The next steps after the FEA is to do real testing to validate the
results of the FEA and Steinberg’s component factor C as well as determine a fatigue
exponent for the lead free solder.
52
4. CONCLUSIONS
The original scope of this project was to determine the effects of lead free solder, BGA
location, and the vibration input curve in terms of high cycle fatigue and solder ball
stress. Using Steinberg’s approach, after optimization of the component factor C, the two
methods for calculating high cycle fatigue match. In terms of lead, 60Sn-40Pb, vs. lead
free, 95.5Sn-4.0Ag-0.5Cu, solder, the analysis shows that there is a large difference in
the maximum and mean stresses in the solder ball and solder joint. Conclusions can be
made for the lead based solder but not the lead free solder due to the lack of available
data to support a valid fatigue exponent for lead free solder. The results in the report
validate the investigation of the lead free solder with testing to determine a vibration
fatigue exponent for solder. Once this is determined, a clear description of the
differences of lead vs. lead free solder can be made in terms of high cycle fatigue with
random vibration input. The BGA location plays a large role in the high cycle fatigue
life of the BGA as shown in the MDI calculations. For the central location, only curves
B, B2 and B3 have allowable stresses to warrant the use of the BGA at this position. The
second location shows the same results as location 1 but the high cycle fatigue life is
slightly less due to higher displacements and responses from the random vibration
environment. The last location shows to be ideal for curves B, B2, B3 and C from RTCA
DO-160. Curves D and E show that no matter the position, they cannot be used for any
kind of high cycle life due to the large stresses. Other means of stiffening the PWB, or
bonding the part down, are necessary to achieve any kind of high cycle life in the two
environments. The optimization of Steinberg’s component factor C, worked out very
well from the FEA data and the two high cycle fatigue calculation schemes. This new
factor better predicts fatigue damage of BGA components for 6x9 inch PWBs. With the
data in this report, the BGA placements can be optimized earlier in the design phase with
the corrected component factor C.
4.1 Future Work and Model Improvement
Although there was much success in being able to model the BGA in a very detailed
FEA to generate accurate and detailed stresses in the BGA solder ball, testing of real
parts is needed to validate any results in this report. Even though Steinberg’s component
53
factor, C, has been optimized to match the FEA results, test data is needed to validate the
fatigue life. The lead free solder has potential to perform better than the lead based
solder, but without a fatigue exponent, conclusions this large cannot be made from the
outcome of this report. Testing must be done to get the fatigue exponent of the lead free
solder to correctly compare the two solders.
54
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., Sheri D. Sheppard, Paul T. Vianco, Jerome A. Regent, and J.
Mark Grazier. "Validation of a General Fatigue Life Prediction Methodology
for Sn–Ag–Cu Lead-Free Solder Alloy Interconnects." Journal of Electronic
Packaging 130.1 (2008): 011003. 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.
[8] "95.5Sn-4.0Ag-0.5Cu Lead-Free Solder." Online Materials Information Resource MatWeb. Web. 05 Dec. 2010. <http://www.matweb.com/>.
[9] "60Sn-40Pb Tin-Lead Solder." Online Materials Information Resource - MatWeb.
Web. 05 Dec. 2010. <http://www.matweb.com/>.
[10] Erinc, M., P. Schreurs, and M. Geers. "Intergranular Thermal Fatigue Damage
Evolution in SnAgCu Lead-free Solder." Mechanics of Materials 40.10 (2008):
780-91. Print.
[11] Package Mechanical Drawings. Mountain View, CA: ACTEL Corporation, Aug.
2010. PDF.
55
56
Appendix A
Calculation of Sinusoidal stress correction factor using [3]
Equation (6)
  11  2 2  3 3
  (.683)(1)6.4  (.271)(2)6.4  (.043)(3)6.4  72.55

Equation (7)

  
  (72.55)1/ 6.4 1.95


Steinberg component factor CS vs. Calvin component factor CC
Steinberg Equation [3]:
Z
.00022B
Cshr L
Calvin Equation [3]:

Z
.00022B 2
Cc hr L
Setting the two equations equal to each other:

B B2

Cs Cc

Where Cs is the Steinberg component factor and Cc is the component coefficient from
Calvin’s equation. Calvin did his testing with an 8x8 in PWB where Steinberg used a
1x1 in PWB. Using 8 for a board width value, B:
CC=8CS or the Calvin component coefficient is 8 times Steinberg’s component
coefficient. Either case will give the same answer; it just has a different value for C.
57
Calculation for the S-N curve based on a log-log plot
The slope of the curve is the fatigue exponent.
 
log(  i )  log(  0 )
log( N i )  log( N 0 )
solving for Ni yields equation (13)

AVE  
N i  N 0 

 0 

58
Appendix B
Curve B Plots:
Figure B-1: Maximum RMS Acceleration
Figure B-2: Maximum RMS Displacement
59
Figure B-3: Base Node Acceleration Curve B
Figure B-4: Central Node RMS Displacement
Figure B-5: Central Node RMS Acceleration (GRMS)
60
Curve B2 plots:
Figure B-6: Maximum RMS Displacement
Figure B-7: Maximum RMS Acceleration
61
Figure B-8: Base Node Acceleration Curve B2
Figure B-9: Central Node RMS Displacement
Figure B-10: Central Node RMS Acceleration (GRMS)
62
Curve B3 plots:
Figure B-11: Maximum RMS Displacement
Figure B-12: Maximum RMS Acceleration
63
Figure B-13: Base Node Acceleration Curve B3
Figure B-14: Central Node RMS Displacement
Figure B-15: Central Node RMS Acceleration (GRMS)
64
Curve D plots:
Figure B-16: Maximum RMS Displacement
Figure B-17: Maximum RMS Acceleration
65
Figure B-17: Base Node Acceleration Curve D
Figure B-18: Central Node RMS Displacement
Figure B-19: Central Node RMS Acceleration (GRMS)
66
Curve E plots:
Figure B-20: Maximum RMS Displacement
Figure B-21: Maximum RMS Acceleration
67
Figure B-22: Base Node Acceleration Curve E
Figure B-23: Central Node RMS Displacement
Figure B-24: Central Node RMS Acceleration (GRMS)
68
Appendix C
Table C-1: PWB mechanical properties
Layer
Material
1
Copper
2-Jan
GFG
2
Copper
3-Feb
GFG
3
Copper
4-Mar
GFG
4
Copper
5-Apr
GFG
5
Copper
6-May
GFG
6
Copper
7-Jun
GFG
7
Copper
8-Jul
GFG
8
Copper
9-Aug
GFG
9
Copper
10-Sep
GFG
10
Copper
11-Oct
GFG
11
Copper
12-Nov
GFG
12
Copper
13-Dec
GFG
13
Copper
13-14
GFG
14
Copper
14-15
GFG
15
Copper
15-16
GFG
16
Copper
Thickness
(Signal
Layers)
[in]
Cap
Thickness
(Plane
Layers)
[in]
Thickness
(Dielectric
Layers) [in]
0.0021
0.006
Plane
0.0014
0.006
Signal
0.0007
0.006
Signal
0.0007
0.0045
Plane
0.0007
0.006
Signal
0.0007
0.006
Plane
0.0014
0.004
Plane
0.0014
Plane
0.0014
0.005
0.004
Plane
0.0014
0.006
Signal
0.0007
0.006
Plane
0.0007
0.0045
Signal
0.0007
0.006
Signal
0.0007
0.006
Plane
0.0014
0.006
Cap
0.0021
0.0084
0.0098
0.082
Total Board Thickness [in] =
0.1002
Effective Area [in²] =
0.1
0.9
1
t = Effective Thickness [in]=
0.0008
0.0088
0.0795
69
Percentage
Copper
Copper Weight [lb]
GFG
Weight
[lb]
0.15
0.0065
0.0077
0
0
0.026
0.9
0.0261
0.0006
0
0
0.026
0.15
0.0022
0.0026
0
0
0.026
0.15
0.0022
0.0026
0
0
0.0195
0.9
0.013
0.0003
0
0
0.026
0.15
0.0022
0.0026
0
0
0.026
0.9
0.0261
0.0006
0
0
0.0173
0.9
0.0261
0.0006
0
0
0.0217
0.9
0.0261
0.0006
0
0
0.0173
0.9
0.0261
0.0006
0
0
0.026
0.15
0.0022
0.0026
0
0
0.026
0.9
0.013
0.0003
0
0
0.0195
0.15
0.0022
0.0026
0
0
0.026
0.15
0.0022
0.0026
0
0
0.026
0.9
0.0261
0.0006
0
0
0.026
0.15
0.0065
0.0077
0
0.3551
Soldermask Weight
[lb] =
0.0310224
PWB Weight [lb] =
0.7019
PWB Density
0.1084
[lb/in³] =
E = Young's Modulus [lb/in²]
=
E * t [lb/in] =
PWB
Area
(in2)=
Overall Young's Modulus
(E*t/total board thickness)
[lb/in²] =
1.50E+07
1.50E+07
2.00E+06
0
132300
159000
64.63
2.91E+06
Poisson's Ratio =
0.3
Printed Wiring Board Density [lb/in³]
Main Board =
0.1572
Table C-2: BGA Die properties
Density
Young’s
(lbs/in3)
Modulus (Psi)
Poisson’s Ratio
Inner Die
.069
6.18e6
.28
Outer Die
.073
2.26e6
.28
Equivalent Die
.071
4.53e6
.28
The Equivalent Die is based upon the mechanical dimensions in Figures D1 and D2
70
Figure C-1: Mechanical packaging 484 BGA [11]
71
Figure C-2: Mechanical packaging dimensions 484 BGA [11]
72