The Lambda Factor

The Lambda Factor
By
Paul Geck, Richard Patton and Glen
Prater
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• The lambda factor methodology was
developed as a unified theory for
calculating the weight benefits of
converting from one material to another
(e.g. converting from steel to aluminum).
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Overview

• Concept
• Geometric Effects
• Curved Beam Effects
• Joints
– Analysis
–Results
– Improvement
• Generalization of the Theory
• Next Steps
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Outline

Theory – The Lambda Concept and Material
Selection
• In general, a mechanical response function (e.g.,
stiffness) can be approximated (or for simple
shapes – exactly) by a coefficient times thickness
to some power (C*t λ )
• The magnitude of λ is a function of the
architecture being studied and the specific
response being solved for.
• Generally speaking, the response to an increase in
metal thickness is the most important criterion for
weight savings via material substitution
– A linear response favors steel
– A non-linear response favors aluminum
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• Flat plate has bending
stiffness dependent on
inertia
• I=1/12*b*t 3
• λ = 3
• Thin-walled tube has
torsional stiffness
dependent on polar
moment of inertia
• J~2*π*r 3 *t 1
• λ = 1
Stiffness of a Flat Plate vs. a Tube
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• A flat plate can be
considered as the
extreme of a
continuum of forms
• A tall beam in bending
is more efficient than a
flattened out beam.
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Beams in bending

Beam Geometric Non-linearities
• Beam stiffness does not necessarily vary linearly
with metal thickness
• These non-linearities favor aluminum over steel
• In the following plots, 2*t/h is the abscissa, where
– t is the material thickness
– h is the height between the top and bottom flanges
• h = 0 for a flat plate
• h = overall height – 2*t for all other forms
• The ordinate is the beam thickness exponent - λ
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Beam thickness exponent, λ
3
2.8
2.6
2.4
2.2
2
1.8
1.6
1.4
1.2
1
10 -2
Hollow Beam
Mid-rails
10 -1
Transition
Roughened
sheet
Corrugations
10 0
2*t/h (steel)
Flat Plate
10 1
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Beam thickness exponent, λ
10 2

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Beam thickness exponent, λ
• Determines weight saved by going to
aluminum from steel
•Formula is:
–m ratio = (ρ al /ρ steel )*(E steel /E al ) 1/λ
– % Weight saved = (1 – m ratio )*100

% Weight Saved
60
50
40
30
20
10
0
-10
10 -2
Aluminum vs. Steel Weight Savings: Geometric
Beam Effects
Hollow Beam Transition Flat Plate
< 8% saved 8% < saved < 46% 46% < saved < 51%
10 -1
10 0
2*t/h (steel)
10 1
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10 2

Summary of λ and Beam Geometric Effects
• λ can be used to categorize different structures
• Different beam geometries result in different λs
• Generally speaking, the more efficient the
structure, the lower λ will be
• λ can be computed using finite elements, by
making 2 runs, identical except for changing the
thickness of the elements, from the formula:
– Response = C*t λ
– In these runs, the material should be kept the same
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• Thin walled curved beams can also be
treated using λ
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Curved Beam Effects
• This includes all joints in the vehicle, as
well as curved beams
• A T-joint can be considered a double curved
beam with two inside flanges

1mm
thickness
λ = 1.3
6.25mm
thickness
λ = 1.1
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Curved Beam Bending-Results
From these results, we
know that the stresses
along the inside flange
layer distribute more
evenly when the
thickness becomes
larger.

• Example: A-pillar to roof
• Two loads are applied
– up/down
– In/out
• Rest of joint is grounded
as shown
• Displacements are found
and stiffness calculated
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Joint Modeling

• Equations:
• θ = (δ 1 - δ 2 )/D
• M = δ 1 /θ
• K = F*M/θ
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Stiffness Calculation

Joint
B/Roof
B/Rocker
Hinge/
Rocker
Load
F/A
I/O
F/A
I/O
F/A
I/O
1.39
1.54
1.33
1.46
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λ
1.58
2.00
Joint Analysis Results
% Weight
Saved
32%
41%
25%
30%
22%
28%

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Changing Lambda - Joints
• Lambda can be changed through joint
reinforcement
• The reinforcement reduces the stress-relieving
displacement which occurs in curved beams
• As a result, the metal carries the stresses properly,
which increases the stiffness and reduces the stress
• Because the flange metal is carrying stress, λ is
reduced, which in turn reduces the weight-saving
potential value of going to alternate materials

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B-Pillar to Rocker –
Original and Reinforced

Inboard-outboard
loading
Original Model
Model with
Reinforcement
Fore-aft loading
Original Model
Model with
Reinforcement
Reinforcement Results
1.5474
1.1517
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λ
λ
1.3893
1.144

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Joint Analysis - Conclusions
• The response of a structure to changes in wall
thickness can be used to predict the weight savings
from material substitution
• λ is a very good predictor of weight savings for
equivalent stiffness
• Average λ = 1.51
– Average weight savings of 30% in joints for steel to
aluminum material substitution
• Variations in geometry between joints cause the
changes in λ for the different joints
• λ for a joint can be changed through redesign and
reinforcement

• Gauge sensitivity can be assessed for individual components, cross sections, or full vehicle
body structures.
• The parameter is very well suited for evaluating the weight changes associated with
material, architecture, and/or gauge changes.
• For the negligible-web beam, l ranges between 1.0 (uniform internal load distributions,
very efficient structure) to 3.0. (non-uniform internal load distribution, less efficient
structure).
• Other effects that have effect of increasing λ include torsional loads, shear lag, elastic
instability associated with compressive buckling.
• For structures with low λ values, the mass savings generated by a constant-stiffness
material substitution is proportional to the ratio of specific elastic moduli:
Δ
m
⎡ E1
/ ρ
= 100⎢1−
⎣ E2
/ ρ
1
2
⎤
⎥
⎦
Generalization of the Theory
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Future Work - Global Gauge
Sensitivity Indices
• Investigate the possibility of defining a global λ based upon forced dynamic response,
including compliance frequency response, and crash.
• Develop the analytical background (calculation methodology, weight savings
predictions) for a global λ based upon a variety of body stiffness metrics (e.g., closure
distortions).
• Use CARS/GAS and other tools to continue to investigate the influence of thin-wall
effects on λ behavior and interpretation.
• Conduct comparative global gauge sensitivity studies for a variety of passenger car and
truck bodies.
• Determine the nature and magnitude of the errors induced by the assumption that
responses are exactly proportional to a constant times “t” to the “λ”(C*t λ ).
• Continue to investigate the use of global λ curves to assess the effectiveness of local
architecture changes.
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