1. 1
Evaluating DOE and Optimization of Structural Performance in a
Nonlinear Regime: Energy-Absorbing Masonry Wall Retrofits Under
Blast Loading
Adam Cone, Adam Hapij,
Ka-Kin Chan, Raymond Daddazio
Applied Science Division
Weidlinger Associates Inc, New York
Supported by UCLA NSF VIGRE Grant
Abstract
We assess the utility of standard optimization and DOE algorithms for a highly nonlinear
problem. Namely, we simulate the behavior of a steel plate-stiffener masonry wall retrofit
during blast loading, and attempt to determine the retrofit design that absorbs the most
energy. Design variables under investigation for this study are plate thickness and stiffener
dimensions. We discretize the design space and evaluate every design to obtain a complete
response surface. We present DOE study results of absorbed energy sensitivity to changes
in general retrofit properties. Using the response surface data and original criteria for rating
optimization performance, we assess the effectiveness of a standard optimizer when
initialized at various designs. Finally, we describe our overall results, and discuss the
feasibility of comparable DOE and optimization schemes to other nonlinear problems.
Introduction
Two fundamental questions in a structural design problem are:
• What are the most influential design variables?
• What is the optimum design for a given purpose?
Design of experiments (DOE) and optimization tools are designed to answer these two
questions, respectively, when obtaining a complete response surface is impractical. DOE
and optimization algorithms are based on certain assumptions about the quality of local
linear approximations to the response surface, and can perform well when such assumptions
are met. Similarly, on a discontinuous or sufficiently oscillatory surface, the algorithms will
almost certainly fail, since the continuity assumptions are not met. Response surfaces in
applications, however, often lie between these two extremes, and it is in this middle-ground
where DOE and optimization tools are most useful—if they function on such a surface. We
consider a specific test problem for which we can and do obtain a complete, nonlinear
response surface, thereby obtaining the actual design variable sensitivities and absolute
optimal design. We then implement standard DOE and optimization tools and observe how
well their results match the results obtained from the complete response surface.
Our test problem involves a time-consuming finite-element solver, a nonlinear
response surface, and involved design feasibility constraints. We choose a complicated
problem to gauge how the tools respond to actual engineering problems where such
complications are present. For the infeasible designs, the objective function, strain energy, is
far higher than it is for feasible designs. This effectively gives us two response surfaces to
work on:
• The entire response surface, with “acute” nonlinearities at the infeasible designs
• The subset of the response surface corresponding to feasible designs

2. 2
The latter surface is less problematic because it lacks the “acute” nonlinearities of the former.
We use the DOE and optimization tools on each surface, and evaluate the results.
Physical Problem: Depending on the threat parameters: charge weight, and standoff
distance, even a heavily reinforced and grouted wall will have limited resistance to blast
pressures. Therefore, load-bearing masonry walls require structural redundancy to prevent
the initiation of a catastrophic progression of collapse. The design of retrofit systems (Figure
1) to withstand the effects of explosive loading is one way of achieving such redundancy.
“A study of conventional threats and their associated peak pressures and impulses
has indicated that the stiffened plate response is driven only by the impulse of the blast
loading on the masonry wall. The blast loading is assumed to obliterate the wall and
effectively sends a layer of mass projectiles onto the stiffened plate with an initial momentum,
in the worst case, equal to the blast impulse.” 1
Figure 1: Schematic of retrofitted masonry wall and charge threat.
Model Description
The objective of this study is to determine the specific design of a plate-stiffener retrofit that
maximizes energy absorption during blast loading, while maintaining stability and vertical
load carrying capacity. The design variables are (Figure 2):
• plate thickness (Tplate)
• stiffener depth (Dstiffener)
• stiffener wall thickness (Tstiffener)
• stiffener width (Wstiffener)
1
Development of Protective Design Analysis Tools for Stiffened Steel Plate Wall Retrofit, Smilowitz et
al, Weidlinger Associates, Inc. 2004

3. 3
In the stiffened plate system, the plate is restrained only along the stiffener length; the plate
is not restrained along the top or bottom edge, and hence does not contribute to the vertical
load carrying capacity of the system. This vertical load carried by the 10inch masonry wall is
180kips over a tributary wall width of 36in. We neglect any dynamic effects of the load
transfer from the masonry wall to the stiffeners and any energy dissipation associated with
the failure of the masonry system.
Figure 2: Bird’s-eye view schematic of reinforced masonry wall.
We did not represent the failure of the masonry system, so our simulated physical scenario is
reduced to just the retrofit and the explosive threat. All simulations are performed with plate
and stiffener heights of 11ft and 36in stiffener spacing (Figure 1). The structure supports a
500psi vertical load from above. The stiffeners are pinned at the bottom and restrained at
the top from moving in the horizontal plane.
Because the stiffened plate is designed to be
regular, we take advantage of symmetry and
only simulate a single stiffener with the 36in
tributary plate length.
The stiffened plate is loaded with a
100psi and 300psi-msec blast threat. Although
both charge weight and standoff distance of the
explosive can be varied, the set of charge
threats can be divided into equivalence classes
based on the pressure and impulse they deliver
to the structure. The pressure-impulse signal
we use for simulation is a linear decay from
100psi to 0psi over approximately 6ms seconds
(Figure 3). This does not correspond to a
Figure 3: Pressure-time plot of shock
wave from charge threat.

4. 4
unique charge weight / standoff. We assume an on face blast pressure loading, where the
extent of the stiffened plate is loaded with the same pressure; i.e., given realistic standoffs
associated with the pressure impulse combination, the shock wave is assumed to planar.
Finite Element Model: The plate is computationally represented with nonlinear plate
elements and the stiffeners with nonlinear beam elements. The finite element mesh
resolution was nominally 2-inches, square.
Our plate is 11ft in the vertical and 36inches in the lateral dimension, resulting in 1188
elements. There are 66 beam elements for each simulated stiffener, and the beam and plate
meshes are defined using general connectivity, where
the beam elements are defined with a fixed offset. The
plate and beam elements are simulated in EPSA using
elastic perfectly plastic constitutive models. The steel
material properties are listed in Table 1. Our integration
time step is 4.7x10-7
s, and the simulation is conducted
for a total of 42.5msec. The units for modeling are
pounds (lb) for mass, pounds per square inch (psi) for pressure, inches (in) for length, and
seconds (s) for time.
Design Space: Our design space for this project is 4-dimensional, and each design variable
has units of length, which is a continuous quantity. However, this does not imply that we
have infinitely many designs to choose from.
Only a fixed number of stiffener and plate models are manufactured; not any combination of
the design variable values corresponds to a design that can actually be obtained. It is from
these manufactured models (Table 2) that we must select a design. We could have assumed
the availability of custom-made components, but these are expensive and therefore not
considered.
Design Variable Manufacturing Lengths (in) Tolerance (in)
Tplate _ , 3/8, _ 1/32in
Dstiffener 6, 8, 10, 12 1/32in
Wstiffener 3/16, _, _ 1/32in
Tstiffener 4, 6, 8 1/32in
Table 2: Manufacturing lengths and tolerances. These tolerances are from AISC Manual of
Steel Construction: Allowable Stress Design.
Since each design variable is discretized, the design space is discretized (Figure 4), and
there are 108 designs. Since there are a discrete, finite number of designs, we generate a
complete response surface on a discrete design space.
There is an issue of manufacturing tolerances: a plate reported as _-inch thick is not
mathematically _ inches thick. Therefore, each aforementioned manufacturing length actually
implies a range of lengths centered at that manufacturing length.
Using these manufacturing tolerances for each design, we discretize the design
space into cells, each of which contains many possible designs (Figure 4). Now, we make
the assumption that all responses (e.g. absorbed energy) change insignificantly between any
two designs inside the same cell.2
Therefore, when evaluating points in our design space, we
need only choose one representative point from each cell. In this study, we choose the
2
Without this assumption, we would need some indication that intra-cell energy absorption variation is
insignificant. We could, for instance, evaluate multiple designs in the same cell and compute the
energy absorption variation for those designs.
Property Steel
Density (lb s
2
in
-4
) 7.339x10
-4
Elastic Modulus (psi) 29*10
6
Yield Strength (psi) 5*10
4
Poisson’s Ratio 0.3
Table 1: Material properties

5. 5
nominal, central values. Designs are referenced with the syntax (Tplate,Dstiffener,Tstiffener,Wstiffener).
For instance, the representative point of the cell containing (1/2, 8, 4, 1/4) is (1/2, 8, 4, 1/4).
Figure 4: Schematic of 2-D design space cell discretization. A, B, X, and Y are manufacturing
values of their respective design variables, and TA is the manufacturing tolerance of A.
Response: The responses of the stiffened plate systems vary, given the selected
configuration of stiffener size and plate thickness. For the selected range of potential
designs, the specified threat level offers a good distribution of systems that are destabilized
and systems that remain stable. Figure 5 illustrates the evolution of a feasible design.
Figure 5: Snapshot of unscaled displacements of finite element mesh. The highlighted green
represents the stiffener.

6. 6
We are interested in the total energy absorbed by the stiffened plate system over the course
of the blast event. In this effort total energy is quantified as the total internal energy absorbed
by the plate and beam finite elements.
Figure 6: (L) attenuating velocity of a stable retrofit; (R) velocity of an unstable retrofit
If a retrofit design buckles or becomes unstable under the combined vertical-blast load, it is
considered infeasible. To determine whether an instability occurred during a simulation, we
evaluate vertical velocity of the center of gravity. If this signal attenuates, we assume that the
Figure 7: Illustration of stiffener (_s) and plate (_p) rotation angles, taken as the maximum
values over time. The two illustrated plate rotation angles are equal by symmetry of both the
charge and the plate-stiffener structure. In general, _s ≠ _p. The plate is not restrained at the top
or pinned at the bottom.

7. 7
retrofit retained structural integrity. If the signal does not attenuate, we assume that the
retrofit failed (Figure 6). If this attenuation constraint is not met, the stiffened plate responses
are not considered. We can only meaningfully analyze the energy absorption of a feasible
design, but we do not know which designs are feasible until after we evaluate them.
However, after we obtained the response surface, we identified simpler design
feasibility criteria. We programmed HyperStudy to distinguish between feasible and
infeasible designs using two additional responses: plate rotation and stiffener rotation
(Figure 7). Based on the complete response data set a design is feasible if and only if its
stiffener and plate rotations are less than 10_
. Functionally, HyperStudy can now distinguish,
without human intervention, between feasible and infeasible designs.
Design of Experiments
Analysis of Variance (ANOVA) is a statistical significance test for estimating response
sensitivity to changes in design variables, or functions of the design variables. To test the
utility of the DOE ANOVA, there were two separate issues to address:
1. When all the response data is available, do the computed sensitivities make sense?
2. When only incomplete response data is available, do we obtain good approximations
of the actual sensitivities?
Because of the stiffener geometry, it is
difficult to intuit sensitivities for design
variables Tplate, D stiffener, T stiffener, and
Wstiffener. Therefore, we cannot assess the
ANOVA results for strain energy on these
design variables. However, if we phrase
the problem in terms of stiffness
properties, instead of lengths, we can
make sensitivity predictions based on the
following reasoning. The stiffened steel
plate system design has a dual purpose:
1) to pose as a redundant load bearing
system and 2) to limit the blast energy
imparted to the supported structure. From
the blast resistance perspective, effective
designs are those that absorb a significant
amount of energy and remain stable. The steel stiffener is the sole means by which the
vertical load bearing redundancy can be effectively achieved. The steel plate, spanning
between axially loaded stiffeners, is attached along only two out of four edges. This
attachment scheme enables it to respond as a catch system, directly transmitting the loads to
the stiffeners. Because the stiffeners are axially loaded, any plasticity in the cross section
results in an erosion of axial bearing capacity, often leading to an instability. Therefore, it is
through the inelastic deformations of the plate spanning between two adjacent stiffeners that
much of the blast energy is absorbed, with the system maintaining stability.
We obtain the ANOVA results of absorbed energy to Tplate, and the stiffener properties
moment of inertia I, cross-sectional area A, radius of gyration r, and plastic strain modulus Z.
The DOE was restricted to the feasible subset of the design space, so the response surface
had less variation. These results confirm that absorbed energy is most sensitive to Tplate, and
that the stiffener properties are relatively unimportant (Figure 8). This indicates that the
ANOVA computed sensitivities do indeed make sense, at least in the best case scenario of
total response data availability on a smooth response surface.
Figure 8: ANOVA for I, A, r, Z, and Tplate

8. 8
To address the second issue, we must allow the DOE algorithm to select designs
from the design space. HyperStudy’s algorithm can do this only if the design variables can be
varied independently. In the stiffness design variables I, A, r, Z, and Tplate, this is impossible,
so we revert back to the original design variables, which can be independently varied.
To assess DOE ANOVA performance on a subset of the design space, we ran the
same analysis with the standard design variables for 10% (11 runs), 20% (22 runs), and
100% (108 runs) of the design space. We used Latin Hypercube design sampling.
HyperStudy’s DOE algorithm does not distinguish between feasible and infeasible designs,
and we ran the analyses both for the feasible subset of the design space and the entire
design space. If the infeasible designs are not excluded from the design space, the strain
energy response surface is more difficult to approximate with a small number of runs, and
local linear approximations are less valid. This may explain why neither ANOVA run on a
subset of the entire design space resembled the analysis for the whole design space.
However, in HyperStudy, the user can, manually, identify the infeasible designs after
the sampling and evaluation is complete, and exclude these infeasible runs from the ANOVA.
After removing the infeasible designs, we observed that the analyses for the design space
subsets closely resemble the analysis for the whole design space. The response surface,
when restricted to the subset of feasible designs, is smoother, and thus is better
approximated both by linear fits and by fewer solver runs.
The answer to the second question is “yes” if the response surface is smooth, and
“no” if the design space is more nonlinear.
Optimization
We used the two optimization methods available in HyperStudy: Sequential Response
Surface (SRS) and Method of Feasible Directions (MFD). For each method, HyperStudy
works on either a continuous or a discrete design space.
We introduce an optimization performance ratio R to quantify the effectiveness of an
optimization on a discrete design space, given some objective function. For the purposes of
definition, we will take the convention that our objective function is to be maximized. Let us
make the following definitions:
• L(A): number of feasible designs with objective function values not greater than A.
• I: number of infeasible designs
• n: number of designs that the optimizer evaluated
• N: total number of designs in the design space
Then if an optimization yields a maximum objective function value A,
n
N
IN
AL
nAR
−
=
)(
),( .
The significance R is that it gauges how close the optimizer got to the absolute optimum
(L/N-I) and normalizes by the proportion of the design space the optimizer explored (N/n). In
general, when running a discrete optimization, none of these values are known.3
In our case,
however, we have a complete response surface, and thus can evaluate R for each
optimization.
However, such a number is not sufficient to determine the optimizer’s performance.
Consider an optimizer that simply evaluates n arbitrarily chosen designs from the design
space, and reports the best design it found. The average performance ratio Rarb of such an
optimizer that samples n designs is given by
3
If they were, the absolute optimum would be known, and there would be no need to use an optimizer.

9. 9
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For an optimization to have been worthwhile, it must have a higher performance ratio than
what we could achieve by arbitrarily sampling the design space. Therefore, we must have
R>Rarb(n) to consider an optimization efficient. We now introduce a relative performance ratio
arb
rel
R
R
R = .
To test the SSR optimizer, we initialized the optimizer at 8 designs4
and evaluated each
optimization’s performance ratio. The optimizer rejects designs that do not meet the rotation
angle constraints mentioned above.
Initial Design n R Rarb Rrel
(1/4, 6, 3/16, 4) 5 0.00 16.59 0.00
(1/2, 12, 1/2, 8) 16 4.56 6.26 0.73
(1/4, 12, 3/16, 8) 6 15.22 14.46 1.05
(3/8, 10, 1/4 6) 12 8.39 8.10 1.04
(1/2, 6, 1/2, 4) 15 7.20 6.63 1.09
(3/8, 8, 1/4, 4) 25 4.20 4.13 1.02
(1/4, 8, 3/16, 6) 11 7.96 8.75 0.91
(1/2, 10, 1/2, 6) 15 5.55 6.63 0.84
Table 3: Results for SRS optimizations.
For the first optimization (Table 3), the optimizer evaluated 5 infeasible designs and
stopped. In the optimization initialized at (3/8, 8, 1/4, 4), the design (1/2, 8, 3/16, 6) was
evaluated 10 times, and (3/8, 10, 3/16, 6) was evaluated 9 times—repeated evaluation of a
design should never occur in optimization. Furthermore, no optimization found the absolute
optimal design: (1/4, 8, 1/4, 8).
To test whether the nonlinearities and infeasible designs were problematic (as we
conjecture they were for ANOVA), we restricted the optimizer to work in a subset consisting
of the 18 largest designs with fixed plate thickness 0.5in, all of which are feasible. This is the
largest rectangular subset of feasible designs in the design space. Here, the optimizer had
an uninterrupted, smoother surface to search. Initializing the optimizer away from the optimal
design (0.5, 10, 0.25, 4), we computed performance ratios of R = 2.57, Rarb = 2.38, and
Rrel=1.08.
MFD evaluated designs that were outside of the space, and was therefore ineffective
in optimizing. It is possible that this was due to our design space discretization, but we found
no indications in the help files that this discretization would be an issue. Therefore, it is
difficult to conjecture why MFD did not function as we expected.
Conclusion and Discussion
Of the 108 designs in our design space (1/4, 8, 1/4, 8) absorbs the most energy: 1,260,530 lb
in. However, there are other considerations in choosing an appropriate design. The optimal
design is the second lightest of the 74 feasible designs at 574.5 lb; the lightest feasible
design (1/4, 12, 3/16, 6) weighs 535.4 lb and absorbs the second most energy: 1,028,070 lb
4
The optimization results are sensitive to the initial design,

10. 10
in. Also, although buckling only occurred at plate and beam rotations >10_
, _p < 5_
and _s <
4_
should be enforced, in practice, to provide a margin of safety. The lightest design that
absorbs the most energy, and meets the rotation safety constraints is (1/4, 10, _, 8): 707,067
lb in, 608.8 lb, _p = 4.60_
and _s < 3.75_
. This last design is, practically speaking, the best
design, although it is only 63 of 74 in absorbed energy.
The strain energy response surface for the retrofit problem has relatively acute
nonlinearities at the interface between feasible and infeasible designs. For example, the
feasible design (3/8, 8, 1/2, 4) absorbs 476,361lb in, but the infeasible design (3/8, 8, 3/16, 4)
absorbs 18,239,800 lb in. In other words, two designs that differ in only one dimension can
differ by a factor of 38 in absorbed energy, and differ completely in functionality (the former
design withstands the blast; the latter buckles). These nonlinearities seemed problematic for
both DOE and optimization.
For both DOE and optimization, the tools were conclusively ineffective when working
on the unrestricted design space. DOE yielded inaccurate ANOVA results when only allowed
to sample the space. The average SRS relative performance ratio for the optimizations on
the whole design space is 0.84, indicating performance significantly worse than arbitrarily
choosing designs, while MFD could not stay within the design space.
The DOE was more effective on the feasible subspace, even using only %10 of the
design space for analysis. The SRS optimization was significantly more effective when
restricted to a smooth subset of the design space, but the relative performance ratio, 1.08, in
this case is not particularly impressive—it just indicates that the optimization was slightly
more effective than choosing designs arbitrarily.
Despite these limitations, HyperStudy was helpful in conceptualizing the design
process. HyperStudy treated our finite element solver like a black-box, which conceptually
partitioned the simulation and design processes. Another advantage is the ease with which
the user can automate solver runs and consolidate solver input and output data.
Overall, our experience using standard DOE and optimization tools on the EPSA-
generated, nonlinear response surface was mixed. Performing DOE studies, automating
solver runs, and conceptualizing the design process. However, the optimization tools should
only be used if the user is confident that the response surface is smooth and contains few, if
any, infeasible designs or acute nonlinearities.