Reinforced Concrete Analysis and Design with Truss Topology Optimization

ovariesracialUrban and Civil

Nov 25, 2013 (3 years and 11 months ago)

89 views

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.