Reinforced Concrete Analysis and Design with
Truss Topology Optimization
Cris Moen
1
and Jamie Guest
2
1
Dept. of Civil and Environmental Engineering, Virginia Tech
www.moen.cee.vt.edu
2
Dept. of Civil Engineering, Johns Hopkins University
www.ce.jhu.edu/jguest
EAD16

Engineering Analysis: Part II
The 3
rd
International
fib
Congress and Exhibition
Washington, D.C.
Monday, May 31, 2010
Historical development
Mörsch
and Ritter
•
Cracked concrete member behaves as a truss
•
Facilitates visualization of the flow of forces and aids design
•
Implemented truss model design as an analogy to elastic stress trajectories
Mörsch
, E. (1909).
Concrete

steel construction
, E. P. Goodrich, translator, McGraw

Hill, New York.
Ritter, W. (1899). “Die
bauweise
hennebique
.”
Schweizerische
Bauzeitung,33(7), 59
–
61
.
Supporting theory
Schlaich
, Marti, Collins & Mitchell, Breen,
Kuchma
, and
others….
Marti, P. (1980). "On Plastic Analysis of Reinforced Concrete, Report No. 104."
Institute of Structural Engineers, ETH, Zurich.
Schlaich
, J., Schaefer, K., and
Jennewein
, M. (1987). "Toward a consistent design of
structural concrete."
PCI Journal, 32(3), 74

150.
Marti relates truss model concept to a lower bound plasticity theory
•
Optimum truss has compressive struts and tension ties coincident with the
elastic stress trajectories
•
Stiffer trusses offer improved ductility and performance at ultimate limit
state
Schlaich
offers guidelines for proportioning and developing truss model to
achieve such performance:
•
Stiffest trusses produce the safest load
‐
deformation response
•
Minimize the reinforcing steel’s total strain energy (stretching) to prevent
large plastic deformations
Research Motivation
Modern Implementation
Research on truss models has confirmed their viability and led to code
provisions and design guidelines
Sometimes the flow of forces is difficult to identify…
Project Goals
Develop topology optimization as a visualization tool and design aid in
reinforced concrete design
Develop open

source software program to promote this tool
Automated Truss Models
Several papers on automating truss models via generating principal
stress trajectories or continuum topology optimization.
We propose a rigorous topology optimization approach using state

of

the

art algorithms with constructability considerations…
Anderheggen
, E., and
Schlaich
, M. "Computer

aided design of reinforced concrete structures
using the truss model approach." Swansea, UK, 1295

306.
Biondini
, F.,
Bontempi
, F., and
Malerba
, P. G. (1999). "Optimal strut

and

tie models in reinforced
concrete structures."
Computer Assisted Mechanics and Engineering Sciences, 6(3), 279

293.
Ali, M. A., and White, R. N. (2001). "Automatic generation of truss model for optimal design of
reinforced concrete structures."
ACI Structural Journal, 98(4), 431

442.
Liang, Q. Q.,
Xie
, Y. M., and Prentice Steven, G. (2000). "Topology optimization of strut

and

tie
models in reinforced concrete structures using an evolutionary procedure."
ACI Structural
Journal, 97(2), 322

330.
Tjhin
, T. N., and
Kuchma
, D. A. (2002). "Computer

based tools for design by strut

and

tie
method: Advances and challenges."
ACI Structural Journal, 99(5), 586

594.
Bruggi
, M. (2009). "Generating strut

and

tie patterns for reinforced concrete structures using
topology optimization."
Computers and Structures, (in press).
5
Topology Optimization
Topology optimization is a free

form structural design tool capable of
introducing holes and changing connectivity.
Very useful for discovering new design ideas…
Design problem: reduce the beam weight
Conventional low

weight design
Topology optimized design:
~42% stiffer for same weight
Topology optimized design:
~48% lighter for same (elastic) stiffness
?
6
Forms of Topology Optimization
Continuum Topology Optimization
Discretize
the continuum design domain and define an element
volume fraction
e
such that:
e
(
x
)
1
if
x
solid
element
0
if
x
void
element
Discretize
the design domain
Determine whether each element
is a solid or void
Truss Topology Optimization
Mesh the design domain with a dense
ground structure and optimize
areas
e
.
Inefficient members (
e
<
threshold
) are
removed from the structure.
7
Problem Formulation
design variable bounds
min
e
F
T
d
d
T
K
(
e
)
d
such
that
K
(
e
)
d
F
e
v
e
e
V
0
e
max
e
equilibrium
volume (mass) constraint
External work (= Internal strain energy)
Minimum Compliance (maximum stiffness) formulation
Design variables
Continuum Solution Strategy
•
Optimizer
: Method of Moving Asymptotes (MMA) (Svanberg 1987)
•
Sensitivities
: computed using adjoint method
•
Interpolation model
: SIMP material model (Bendsøe 1989)
•
Stabilization
: length scale
control
via Heaviside Projection Method (HPM)
(Guest 2009)
Examples
–
truss topology
Optimized Model
Traditional Model
Topology optimized design
•
Steel ties
bridge the principal tension cracks
•
Stiffer truss structure
•
Less steel in this example
Reinforced concrete deep beam
Experimental results by
Nagarajan
and
Pillai
(2008)
Red is tension
Black is compression
Examples
–
truss topology
Deep beam with opening
Optimized Model
Traditional Model
Visualizing the flow of forces in beams with openings is often challenging
•
Vertical stirrups are eliminated
•
Stiffer truss structure
Examples
–
truss and continuum topologies
Shear wall with opening
Similar topologies but generally more freedom in continuum
Continuum
Topology
Truss
Topology
Examples
–
truss topology
Hammerhead Pier
Optimized Model
Traditional Model
Topology optimized design
•
Stiffer truss structure
•
Follows stress trajectories but a more complicated pattern
12
Truss Topology Optimization
•
Reduce ground structure complexity
•
Increase the smallest allowable area (threshold for element removal:
e
<
threshold
)
•
Include a constraint/penalty on number of connections (at the cost of convexity)
Improving constructability
Continuum Topology Optimization
Challenge:
•
Control the minimum length scale of features (Heaviside Projection Method)
•
Features (struts & ties) are created by the union of elements with different material
properties
Can we reign
in
design freedom in a
physically meaningful
way?
13
Simply Supported Beam Example
HPM:
Controlling length scale
d
min
=
4.0 ft.
d
min
=
2.0 ft.
d
min
=
1.5 ft.
d
min
=
1.0 ft.
Control minimum dimension of truss!
Examples
–
continuum topology
Hammerhead Pier
r
min
= 0.5 ft.
Design complexity decreases with increasing minimum allowable
length scale…
r
min
= 2 ft.
r
min
= 3 ft.
Implementation in research and practice
Dapped concrete beam for a parking garage
Does this reinforcing steel pattern follow the flow of forces?
Implementation in research and practice
Dapped concrete beam for a parking garage
What is the effect of boundary conditions on stress distribution?
Implementation in research and practice
Dapped concrete beam for a parking garage
A convenient tool for visualizing how the beam is working.
Implementation in research and practice
Dapped concrete beam for a parking garage
Reinforcing steel could be placed better to bridge cracks…
19
Topology Optimization Software
In development, but preliminary continuum design engine with GUI interface is
available…
Concluding Remarks
20
Topology optimization is a powerful tool for visualizing the flow of forces
through concrete structures and for generating new design ideas.
Truss Topologies
•
Convex formulation of minimum compliance problem allows for fast and
stable convergence to a global minimum
•
Members are straight and constructability may be enhanced by restricting
design freedom in ground structure and minimum member areas
Continuum Topologies
•
Full design freedom leads to high performance designs
•
Restricting minimum length scale reduces design complexity and such
control is implicit with HPM (
no constraints
)
Future Work
•
Maximum length scale for continuum
•
Hybrid truss

continuum models
•
Implementation into practice!
Concluding Remarks
21
The topology optimization program may be downloaded at
www.ce.jhu.edu/jguest/topoptprogram
This research was supported in part by the IABSE Foundation Talent
Support
Programme
Concluding Remarks
22
Problem Formulation
design variable bounds
min
e
F
T
d
d
T
K
(
e
)
d
such
that
K
(
e
)
d
F
e
v
e
e
V
0
e
max
e
equilibrium
volume (mass) constraint
External work (= Internal strain energy)
Minimum Compliance (maximum stiffness) formulation
Design variables
Continuum Solution Strategy
•
Optimizer
: Method of Moving Asymptotes (MMA) (Svanberg 1987)
•
Sensitivities
: computed using adjoint method
•
Interpolation model
: SIMP material model (Bendsøe 1989)
•
Stabilization
: length scale
control
via Heaviside Projection Method (HPM)
(Guest 2009)
Truss Solution Strategy
•
Convert to convex, displacements only formulation (e.g. Bendsøe et al. 1994)
Forms of Topology Optimization
23
Continuum Topology Optimization
Advantages
Design freedom
–
truly free form
Disadvantages
Discrete and
nonconvex
Numerical instabilities (e.g. checkerboards)
Too much design freedom?
Discretize
the design domain
Determine whether each element
is a solid or void
Truss Topology
Optimization
Advantages
Computationally simple
Convex formulations attainable
Straight members
Disadvantages
Restricted design freedom
24
Problem Formulation
min
d
,

F
T
d
V
such
that
1
2
d
e
T
K
0
e
d
e
v
e
for
all
elements
Ensures uniform strain energy
densities*
Maximize external work while minimizing strain energy
Minimum Compliance (maximum stiffness) convex formulation
After a series of substitutions, we are able to remove areas (
e
) and
create a
mathematically equivalent
convex problem
“Design”
variables
25
Problem Formulation
design variable bounds
min
e
F
T
d
d
T
K
(
e
)
d
such
that
K
(
e
)
d
F
e
v
e
e
V
0
e
max
e
equilibrium
volume (mass) constraint
External work (= Internal strain energy)
Minimum Compliance (maximum stiffness) formulation
Design variables
26
Problem Formulation
min
d
,

F
T
d
V
such
that
1
2
d
e
T
K
0
e
d
e
v
e
for
all
elements
Ensures uniform strain energy
densities*
Maximize external work while minimizing strain energy
Minimum Compliance (maximum stiffness) convex formulation
After a series of substitutions, we are able to remove areas (
e
) and
create a
mathematically equivalent
convex problem
“Design”
variables
Truss Solution Strategy
•
Optimizer
: Interior point method or other gradient

based approach
•
Design
: Areas extracted from Lagrange Multipliers:
e
=
λ
e
/
v
e
(KKT conditions)
•
Properties
: Uniform strain energy densities (uniformly stressed design)
•
Stabilization
: Joint instabilities are not an issue in strut

and

tie models
Benefits of imposing a minimum length scale
•
Eliminates checkerboards and mesh dependency
•
Influences sensitivity to flaws
•
Provides a means for imposing minimum size design specifications and
fabrication constraints
•
Provides a means for influencing design complexity (fabrication cost)
27
Original implementation
•
Introduces a length scale
r
min
(
d
min
)
into the
problem, defined as the minimum allowable radius
(diameter) of a structural member
Length scale control via Heaviside Projection Method (HPM)
d
min,solid
d
min,void
Algorithm Details
(Guest, Prevost, and
Belytschko
, 2004; Guest 2009)
•
Introduces an auxiliary independent design variable that is projected onto element
space to obtain topology.
Length scale control is implicit.
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