Measurement Methods
and Calculations
to Determine
Internal Deposit Stress
Frank H. Leaman
Specialty Testing & Development, Inc.
York, PA
Methods for Deposit Stress
Determination

Bent Strip (simple beam theory)
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Spiral Contractometer
Simple Beam Theory
Where:
Then:
N = Thickness of the plated coating (inches)
T = Thickness of the test strip (inches)
D = Deflection of the strip due to bending (inches)
L = Length of the test section (inches)
E = Modulus of elasticity of the test strip (lb/in2)
I = Moment of inertia of the cross section of test strip about its neutral axis
S = Stress in plated layer (lb/in2)
S = 4E (N+T) D
3NTL
Bent Strip Method
(Initial Approach)
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During the application of a coating, one end
of the test piece is held in a fixed position
and the other end is free to move.
It is difficult to measure the value for D.
Bent Strip Method
(Different Approach)
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A test piece split into two legs spreads outward due
to the deposit stress
The deflection is easily read by placing the test piece
over a scale
Calculate the deposit stress value by using a simple
formula
Simple Beam Tensile and
Compressive Stress
Tensile
Compressive
Compressive and Tensile
Stress
Compressive
Tensile
Stress Evaluation Using the Bent
Strip Method
Test Strip in a
Plating Cell
In-Site 1 Plating Cell
Ideal for small
solution volumes and
lab studies,
particularly when
working with precious
metals
Bent Strip Test Piece Measuring
Stand
Stress Evaluation Using the Bent
Strip Method
Bent Strip
Plating Test Cell
Test Strip Plating Cell with
Accessories
Typical Deposit Stress Evaluation Plating
Set-Up
Deposit Stress Calculations for Test
Strips
The Stoney Formula:
 = E T² M δ  3 L² t
E = Modulus of elasticity of the substrate = 120,655 kg/cm².
T = Thickness of the substrate in millimeters = 0.05077 mm.
δ = 1/2 the distance between the test strip leg tips in mm.
Example: 0.540 inch spread ÷ 2 x 25.385 mm/inch = 6.85 mm.
 = Stress in megapascals, MPa. Note: MPa x 145 = PSI.
L = Length of substrate on which the deposit is applied in mm.
For Deposit Stress Analyzer test strips, this value is 76.2 mm.
t = Deposit average thickness in millimeters.
M = Correction for modulus of elasticity difference between
the deposit and substrate:
M = EDeposit ÷ ESubstrate = 206,900 ÷ 120,690 = 1.714
= E (.05077 mm) ² M ( δ mm) =
mm³=
MPa
3(76.2 mm)²(.002538 mm)
44.21 mm³
Deposit Stress in PSI = MPa x 145 = PSI
Note: MPa is Megapascals, kg/cm.²
Spiral Contractometer Existing
Design
•

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The test piece is a spiral.
One end of the spiral is held,
other end is free to move.
As the free end moves, a dial
registers the movement in
degrees.
The stress of the coating can
be calculated.
Spiral on an
Existing
Contractometer
Spiral Plated on Existing Type Contractometer for
Target Nickel Deposit Thickness of 500µ” in a
Semi-bright Bath after 20 Seconds Wood’s Nickel
Strike
Deposit Location
Outside Surface
Inside Surface
Thickness, µ”
410
85
Deposit stress over a 2 minute strike =
26.4% less than the New Design
Contractometer result
New Spiral Properties

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New design spirals are constructed from 0.010 inch
thick stainless steel and have a precise surface area
of 13.57 in2.
Spirals mount on the contractometer in a way that
the entire spiral plates from end to end and
deposition of metal on the inside of spirals is minimal
even if they are void of a masking material.
The average test deposit thickness is 500
Properties and Plating Conditions for Spiral
Contractometer Tests
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Spiral Material
Spiral Surface Area, in2
Square Feet
Amps per square foot
Amps
Stock Thickness, inches
Avg. Deposit Thickness, µ”
Plating Time, Minutes
Solution Temperature
Stainless Steel
13.57
0.0942
30
2.90
0.010
500
21
140° ± 1° F
A new geometry solves problems related to
an exposed interior that allows deposition of the
applied deposit to occur on the inside surface.
Interior deposits reverse the type of stress
and reduce calculated results as much as 30%.
Interior masking is critical.
The new design provides masking of the
interior surface by geometry and enables
spirals to be plated tip to tip so the plated
surface area is a constant value.
Other advantages:



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Stainless steel inserts
30% glass filled nylon
construction which prevents
thread damage and spiral
slipping
More accurate results
Saves time
Spiral Contractometer Equipment to
Determine Internal Nickel Deposit Stress
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Spiral Contractometer with calibration weights, support stand and spiral test
pieces. Container 4” diameter and 10” height for nickel strike anode basket and
bath (1750 ml)
Titanium Mesh Anode Basket 3.5” outside and 2.25” inside diameter, 8” high
with support contact tabs and cover for Wood’s nickel strike
Titanium Mesh Anode Basket 5” outside and 4” inside diameter with support
contact tabs and cover for the plating bath
Nickel anode buttons to fill the anode baskets
Pyrex beaker 4000 ml for a nickel plating bath
Support stand – designed to perfectly center over beaker
Magnetic stirrer hot plate, 115 volt
Digital temperature Controller pre-wired with probe to control ± 10 F
Power Supply constant current, constant voltage, 0-5 amps, 0-30 volts
Magnetic stirrer hot plate, 115 volt
Contractometer Stand, Anode Basket &
Beaker
Contractometer Plating Set-Up
Data Recording for Spiral Contractometer
Tests
Deposit weight in grams:
________
________
Kc degrees:
________
________
________
________
Kt degrees:
Degrees deflection caused by the deposit:
________
________
Spiral weight in grams:
________
________
Deposit weight in grams by subraction:
________
________
Average Deposit Thickness Calculation in
Inches
T=
_________W____________ = Inches
D (87.55 cm2) (2.54 cm/inch)
W = Grams of nickel
D = Density of nickel = 8.90 g/cm3, and
T = Deposit thickness in inches
For the new spirals plated on the new design contractometers,
the constant spiral plated surface area is 13.57 in2 and the
following shortened formula applies:
T=
W
= Inches
1979.2
Calculating Deposit Stress
Stress = 13.02 (D) (M) ÷ w x d = PSI
D = Degrees caused by the deposit,
M = Modulus of Elasticity of the deposit ÷ that of the substrate
= 206,897 ÷ 198,186 = 1.044 for nickel deposits over new
spirals that are 0.010 inch thick,
w = degrees Kt from spiral calibration if the stress is tensile or
degrees Kc if the stress is compressive, and
d = Deposit thickness in inches.
Calculation Example:
S = 13.02 (26) (1.04897) ÷ 33 (0.000536) = 20,073 PSI
Modulus of Elasticity Values
Stock Material
ES*
Stock Thickness, in
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Metal
ED**
Cadmium
31,720
Chromium 248,280
Cobalt
206,897 1.72
Copper
117,240 0.971
Gold
74,480
Nickel
206,900 1.71
Platinum
146,900
Rhodium 289,650
Silver
75,860
Zinc
82,760
Cu-Fe Alloy
Ni –Fe Alloy
120,690
0.0020
144,830
0.0015
179,310
0.0010
Values for M***
0.219
2.06
1.71
0.263
1.43
0.810
0.617
1.15
0.654
0.514
1.42
1.14
1.22
1.02
2.40
0.629
0.686
Ni-Fe Alloy
0.177
1.00
0.567
0.415
1.00
0.819
2.00
0.524
0.571
0.423
0.462
Pure Ni
206,900
0.0010
0.153
1.39
0.360
0.710
1.62
0.367
0.400
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1.20
ES*, modulus of elasticity of substrate material in the Stoney Formula.
ED**, Modulus of elasticity of deposit for use in modified Deposit Stress Analyzer and
Stoney formulas.
M***, modulus of elasticity of deposit ÷ modulus of elasticity of substrate for deposit
stress determinations using the modified Deposit Stress Analyzer and Stoney Formulas.
1.400
A Frequent Mistake in Test Procedure
Spiral
1
Deposit Thickness
To Stock Ratio
Stock Thickness, Inches
Deposit Thickness, µ Inches
Minutes Plated
Current Density, ASF
Deposit Stress, PSI
1:20
0.010
500
20
30
14,060
Test Strips
2
1:20
0.002
500
4
30
14,127
3
1:5
0.002
100
20
30
6,865
Note: Extra thick deposits of the harder metals increases the degree of stiffness which
results in lower proportional test strip spread.
Formulas for Bent Strip with One End Stationary*
Bent Strip Stress Curve
For the comparison of equations that follow that apply to
calculating the internal deposit stress of applied metallic coatings
over various substrate materials, the value of U = 8.5 units = 0.780
inch will consistently be used as a basis. It will be noted that the
calculated internal deposit stress values vary from equation to
equation, particularly where the equation fails to address Modulus
of Elasticity differences between the substrate and the deposit.
Relationship between δ and Z. Example: For a given test strip, U = 8.5
units = 0.780 inch, and δ = U in inches x 25.385 mm/inch ÷ 2, so in this case
δ = 9.90 mm.
δ = 4Z
Z=δ÷4
L = 76.155 mm
Using δ = 9.900 mm, Z = 9.90 mm ÷ 4 = 2.475 mm
R = L² + 4Z²
8Z
= 5824.1 = 303.34 mm
19.2
*Note: These formulas only work for bent strip applications
Stoney Formula Without and With Correction
for Modulus of Elasticity Differences Between
the Deposit and the Substrate
Example: For a Cu-Fe test strip, U = 8.5 units = 0.780 inch
δ = U in inches x 25.385 mm/ inch ÷ 2 = 9.900 mm
WITHOUT
σ = 4ET²Z = ET² δ = 91.137 MPa = 13,214.9 PSI
3L²t
3L²t
L = test strip plating length = 76.2mm,
T = Stock thickness = 0.05077mm and
t = Deposit thickness = 0.000075 inch = 0.001904mm
WITH
M = Edeposit ÷ Esubstrate = 206900 ÷ 120690 = 1.714
σ = ET² δM = 120690(0.05077)²(9.900mm)(1.715) = 156.30 MPa
3 L²
3(76.2mm)²(0.00194mm)
σ = MPa (145 PSI/MPa)
σ = 22,663.5 PSI
Other Bent Strip Formulas to Determine
Internal Deposit Stress in Applied Metallic
Coatings
Barklie and Davies Formula
σ=
ET²
6Rt (1 – t/T)
Heussner, Balden and Morse Formula
σ=
4ET²Z
3t (T + t) L
Brenner and Senderoff Formulas
σ = ET(T+ ᵦt)
ᵦ = Edeposit ÷ Esubstrate
6Rt
σ=
E (t + T)³
3Rt (2T + t)
Brenner and Senderoff Formula for Bent
Strip Applications
Brenner and Senderoff Formula
σ = ET(T+ ᵦt)
ᵦ=
Edeposit ÷ Esubstrate = 1.714
6Rt
σ = 120690 MPa (.05077mm)(.05077mm +1.714(.001904mm) = 95.538 MPa
6(303.34mm)(.001904mm)
95.538 MPa x 145PSI/MPa = 13,853 PSI
Note: This formula doesn’t correct for large differences in
Modulus of Elasticity values. The uncorrected Stoney
result was 13,215 PSI. To be correct, this Brenner and
Senderoff formula requires modification as follows:
σ=
ET²ᵦ = 22,310 PSI where
6Rt
ᵦ = 1.714