Analysis and Visualization of Equilibrium in
Masonry Structures
by
Hijung Valentina Shin
Submitted to the Department of Electrical Engineering and Computer
Science
in partial fulfillment of the requirements for the degree of
Master of Science in Computer Science and Engineering
ARCHIVES
iMASSACHUSETTS WNS MUTE
OF TECHNOLOGY
at the
APR 10 2014
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
BRAR1ES
February 2014
@ Massachusetts Institute of Technology 2014. All rights reserved.
A uthor...............................
..
Department of Electrical Engineering and Computer Science
y 27, 2014
Certified by ...................
Fredo Durand
Professor of Electrical Engineering and Computer Science
Tlgsis Skupugr
C ertified by ..............................
John A. OchKndorf
Professor of Architecture
Thesis Supervisor
A ccepted by .......................
/Leslid Klo
dziejski
Chairman, Department Committee on Graduate Students
Analysis and Visualization of Equilibrium in Masonry
Structures
by
Hijung Valentina Shin
Submitted to the Department of Electrical Engineering and Computer Science
on January 27, 2014, in partial fulfillment of the
requirements for the degree of
Master of Science in Computer Science and Engineering
Abstract
This thesis presents novel analysis and visualization methods to explore the equilibrium of masonry structures. Following a previous approach, we model the stability
problem as a quadratic program. When a structure is unstable the quadratic program
returns a measure of infeasibility. We extend this model to include tensile structures
such as cables. Then, we derive a closed-form gradient of stability with respect to
geometry modifications, and apply it to the design of structurally sound buildings.
In addition, we analyze various properties related to the equilibrium state of structures and visualize the result. We study the sensitivity of equilibrium with respect
to block weights, and from that we trace the flow of forces in the structure.
Finally, we compare the equilibrium approach to the finite element analysis (FEA)
method-the most widely used alternative. We point out the disadvantage of FEA
that comes from formulating the contact constraints and propose an improvement
based on an iterative constraint relaxation algorithm.
Masonry analysis; Statics; Equilibrium analysis; Structural optimization; Stability
visualization; Finite element analysis; Contact constraints
Thesis Supervisor: Fredo Durand
Title: Professor of Electrical Engineering and Computer Science
Thesis Supervisor: John A. Ochsendorf
Title: Professor of Architecture
Acknowledgements
First of all, I would like to thank my advisor, Professor Fr6do Durand and my co-advisor, Professor John Ochsendorf for their wisdom,
patience, enthusiasm and support through my research at MIT. I am
particularly grateful to them for allowing me to explore and develop
my ideas, not only with their patient support, but also with their clear
and to-the-point feedback.
I will always value the many hours we
spent discussing new ideas. It is a wonderful privilege to work with
both professors. I dedicate this work to them.
I am indebted to Professor Thomas Funkhouser, who introduced me
to the field of computer graphics and inspired me to pursue research
in graduate school. I am extremely grateful for his constant encouragement and friendship.
Numerous colleagues kindly helped me, and I would like to thank
Emily Whiting and Etienne Vouga in particular, both of whom I turned
to on many occasions. I am also grateful to David Levin, Sylvain Paris,
Desai Chen for their timely advice and generous assistance at various
stages of my research. I would like to thank Yichang Shih for his constant optimism and encouragement.
I am pleased to acknowledge the Samsung Scholarship Foundation for
their very generous financial support, which offered me the chance to
come to this stimulating institute and meet many amazing people.
I would like to thank all of my friends who have taught me so much
about life, and especially my chavruta, Adriana Schulz. Most of all, I
would like to thank my family, both near and far, for their unending
support and encouragement.
Contents
Introduction
Introduction . . . . . . . . . . . . . .
Problem statement and contributions
13
. . . . . . . . . . . . .
13
14
Related works
Structural analysis . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . . . .
3 D structure visualization . . . . .
15
15
16
Equilibrium Analysis
Static equilibrium analysis . . . . .
Analytic structural gradient . . . .
Extension to tension elements . . .
Application to design of structures
Future work . . . . . . . . . . . . .
17
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17
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18
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18
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19
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21
Visualization of Equilibrium
Block weights . . . . . . . . . . . . . . . . . . . . . . . . . .
23
Force flow
. . . . . . . . . . .. . . . . . . . . . . . . . . . .
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
27
28
Comparison of Equilibrium and Elastic Analysis
. . .. -.
Linear elastic formulation . . . . . . . . . . . .
. . ... .
. .. . . . .
Tension failure . . . . . . . .. . .
.. . . .
.
.
tension
of
&
interpretation
Contact constraints
. . ..
Towards a compression-only solution . . . . . .
Comparison to the equilibrium analysis . . . . . . .. . . .
Future work . . . . . . . . . . . . . . . . . . .. . . . . -.
31
31
33
34
35
36
37
Conclusions
39
Bibliography
41
List of Figures
1
Parameterization of forces for tension elements . . . . . .
19
2
Convergence of infeasibility metric . . . . . . . . . . . . .
20
3
Optimization of design using structural gradients
. . . .
21
4
Sensitivity analysis with respect to block weights . . . . .
24
5
6
Maximum weight of blocks in an arch . . . . . . . . . . .
Minimum weight of blocks in a vault . . . . . . . . . . . .
25
7
8
Thrust lines and block weights . . . . . . . . . . . . . . . .
Necessary weights for equilibrium . . . . . . . . . . . . .
25
9
Important weights within structures . . . . . . . . . . . .
Visualization of force flow through blocks . . . . . . . . .
27
Contact constraints in FEA formulation . . . . . . . . . .
Equilibrium vs. elastic analysis of thin arch . . . . . . . .
Mechanism of three hinged arch . . . . . . . . . . . . . . .
Tension forces and contact constraints in FEA . . . . . . .
32
Comparison results from equilibrium & elastic analysis .
37
lo
11
12
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14
15
25
26
28
33
34
36
List of Tables
1
Comparison of equilibrium vs. elastic analysis . . .
...
33
Introduction
Introduction
Much of the world's most important architectural heritage consists of
masonry structures. It may seem superfluous to point out that these
structures are valuable even by the mere fact that they are ancient and
that they still exist. However, as Heyman points in his article, The
Stone Skeleton, this observation has force when placed in the structural
context [Heyman, 19661. The survival of Gothic cathedrals reflect the
consummate expertise of the medieval master builders that brought
out such extreme stability in their structures.
Strange though it may seem, despite the great advancement in building technology made since those times, the structural actions of these
historical buildings still remain in large part a mystery. So much so
that explaining even as simple a structure as an arch becomes a debate
within the theory of structures. While computational techniques and
visualization tools have considerably broadened the range of possible
analyses, when it comes to practice we are still limited to a number
of standard approaches such as finite element analysis to compute
stresses inside structures. A deeper understanding of the stability
of masonry structures would be valuable not only for the restoration
and preservation of historical buildings but also for the design of new
structures. The question remains: how do we explain the stability of a
structure?
This thesis explores that question. The thesis consists of three main
parts. The first part is a condensed summary of the equilibrium analysis method as presented in Whiting's thesis [Whiting, 2012], and an
extension of this method to structures with tension elements. This part
served as the basis for a paper presented in SIGGRAPH Asia [Whiting
et al., 2012] and provides a starting point for this research. The second
part consists of experiments in visualizing the equilibrium of 3D structures. We tried to ask different types of questions in order to illustrate
the static equilibrium of a 3D structure from various angles. In the
course of this exploration, we faced the problem of static indeterminacy of masonry structures. This led us the third part of the thesis,
14
which consists of the comparison between linear elastic finite element
analysis(FEA) and equilibrium analysis.
Problem statement and contributions
This thesis investigates computational and visualization tools to explore the equilibrium of masonry structures. In this context, we make
the following contributions: 1) We extend a previous equilibrium analysis to include structures with tensile elements and apply it to the
design of stable structures. 2) We present novel visualizations which
bring new insights about structural equilibrium. 3) Finally, we demonstrate the differences between the two most widely used methods for
masonry analysis: the equilibrium analysis and the elastic analysis.
Related works
Structuralanalysis
Computational analysis of structural soundness of mechanical parts or
buildings has been widely applied. In the context of computer graphics, stability analysis has been useful especially for animation and recently also for 3D modeling and printing. For example, [Shi et al.,
2007] uses static equilibrium to determine realistic character poses.
[Umetani et al., 2012] performs stability analysis in an interactive tool
to design physically durable plank-based furniture. [Zhou et al., 2013]
presents a method to identify structural problems for a broad range
of objects designed for 3D printing. This thesis focuses on masonry
structures and their equilibrium.
Approaches to masonry analysis can be broadly grouped into two categories. Strength methods represented by finite element analysis (FEA)
assess structures by computing its stress and taking into consideration
the limits of material strength. FEA applied to masonry structures
has been studied extensively. The reader is referred to [Tzamtzis and
Asteris, 2003] for a comprehensive summary. A different approach
proposed by Jacques Heyman [Heyman, 1966] is commonly referred
to as limit state analysis. Based on three main assumptions 1) that
masonry has no tensile strength, 2) that masonry can resist infinite
compression, and 3) that no sliding will occur within the masonry,
these methods assess stability only from the geometry of the structure.
These methods assume that geometrical stability rather than material
strength compared to stresses is key to structural soundness, and focus
on attaining force equilibrium. For an extensive historical overview of
equilibrium methods the reader is referred to [Huerta, 2008]. Here, we
focus on several most relevant works.
Traditionally, equilibrium methods were applied to masonry using
graphical techniques called graphic statics. Recently, [Block et al., 2006]
created interactive graphic statics tools which suggested the possibility
of using computerized graphical methods for the analysis of masonry
structures. [Block and Ochsendorf, 2007] extended graphic statics to
Network Analysis (TNA)
3 D masonry shells by introducing the Thrust
16
method. Using projective geometry and dual force diagrams they find
a compression-only structure from an input mesh and boundary constraints. In the computer graphics community, in particular [Vouga
et al., 2012] and [Panozzo et al., 2013] applied TNA to design and optimize unreinforced masonry structures.
This work is most closely related to and builds upon an alternative
equilibrium method introduced by R.K. Livesley [Livesley, 1978], and
further developed by Emily Whiting in her thesis [Whiting, 2012].
While TNA is specific to shell structures with topologies that can be
projected onto a 2D plane, Whiting's method is applicable to arbitrary
topologies. This thesis 1) extends the method to include tension-only
elements, 2) applies it to develop visualizations of structural equilibrium, and 3) compares it to the finite element analysis method.
3D structure visualization
Architects and engineers rely heavily on visual tools to illustrate 3D
objects or environments in order to explain their parts, functions, spaces
and structures. A good visualization is key to both understanding an
existing structure and designing new structures.
In computer graphics and visualization, researchers have developed
various techniques to explore different objects, each focusing on different aspects. Some of them are concerned with illustrating the geometry of complex objects. For example, [Niederauer et al., 2003] present
a system for interactively producing exploded views of 3D architectural environments such as multi-story buildings, and [Karpenko et al.,
2010] visualize complicated mathematical surfaces using techniques
inspired by hand-designed topological illustrations. Other works attempt to demonstrate the function of an object. For instance, [Mitra
et al., 2010] introduce an automated approach for generating visualizations that depict the operation of complex mechanical assemblies
starting from their 3D models. [Shao et al., 20131 develop a system to
facilitate exploration of product functions from their concept sketches.
Still other works focus on depicting properties inside 3 D objects. [Dick
et al., 2009] visualize stresses inside bones under load to facilitate orthopedic surgery, and [Laramee et al., 2005] visualize fluid flow inside
cooling jackets in order to investigate and evaluate the design of an
automotive engine.
Most visualizations are application specific and require some knowledge about the object or environment. To a certain extent, this is inevitable since both what and how to visualize are determined by the
application. The second part of this thesis focuses on visualizing different aspects of equilibrium in masonry structures with the goal of
gaining a better understanding about how structures stand.
EquilibriumAnalysis
Static equilibrium analysis
In her thesis, Whiting develops a structural analysis method by extending an approach introduced by [?]. This method formulates the
static equilibrium problem-i.e. whether a structure is in equilibrium
or not-as an optimization problem under linear constraints. Here, I
provide a condensed summary of the method and deal with its extension to tension elements in more detail. For a thorough explanation of
the method, I refer the readers to [Whiting, 2012].
Whiting models structures as assemblage of rigid blocks, and discretizes the force distributions at the interfaces between these blocks.
We place a 3D force fi at each vertex of the interface. Each force fi
is decomposed into three components with respect to the local coordinate system of the interface: a normal component, fn and two in-plane
friction components, orthogonal to each other, fi" and f]. The normal
and friction forces on a shared interface between two blocks have opposite orientation for each block.
The stability analysis, also commonly referred to as limit analysis, lies
on three basic assumptions of masonry behavior introduced by [Heyman, 19661. They are: (i) Masonry has no tensile strength; (2) stresses
are so low compared to material strength, that masonry has effectively
unlimited compressive strength; and (3) sliding failure does not occur.
Finally, the safe theorem of the plasticity theory states that if one possible, lower bound, solution can be found that satisfies equilibrium,
the structure will be safe, i.e. it will stand. Under these assumptions
the static equilibrium can be expressed as a linear system as follows.
minimize
f
g(f) =- fTHf
subject to
Aeq - f = -w
2
f > 0
A, - f = 0
(1)
18
The first equality constraint enforces that the net force and net
torque for each block is equal to zero. w is a vector containing the
weights of each block, f is the vector of interface forces and Aeq is the
sparse matrix of coefficients for the equilibrium equations. The second
inequality constraints apply to normal forces and enforce the forces
to be compressive. Finally, the equality constraint Afr - f = 0 approximates the friction cone constraints. If 1, 1f I < af , Vi E interface
vertices.
If a solution f exists that satisfies all the constraints, the structure is
in equilibrium. Otherwise, the structure is unstable. Furthermore, by
softening the compression-only constraint and allowing tension forces,
it is possible to solve for the forces of infeasible structures. In this
case, H is a diagonal weighting matrix of forces, which places a large
penalty on the tension forces and low penalty on the remaining forces.
[Whiting, 2012] uses the function value ifTHf as a measure of infeasibility of a structure.
Analytic structural gradient
After formulating the static equilibrium problem as a linear optimization and defining a measure of infeasibility of a structure, [Whiting,
2012] derives its closed-form derivative with respect to the displacements of the vertices describing the structure geometry. The gradient
expresses how the infeasibility of a structure changes with respect to
changes in the structure geometry.
Extension to tension elements
Whiting's approach can handle other types of constraints such as those
corresponding to tension-only elements. I extended the method to include cables in structures. Whereas the same principles of static analysis and gradient computation apply, cables are different from rigid
blocks in several ways:
1. Cables can resist tension, but fail under compression. I model cables
with infinite tensile capacity, and penalize compression forces.
2.
There are no friction constraints. The cables are modeled as infinitely thin elements that are firmly attached to each other or to a
block.
3. Cables can apply forces only along their axis. This means that a
structure can be infeasible not only due to the sign of the required
forces (i.e. compression inside cables or tension in between blocks),
19
but because of their limited degrees of freedom. For example, Figure 1 shows a case where there is no configuration of forces along
the cable axis that achieves static equilibrium. In order to compute a
measure of infeasibility for such structures, I define virtual torques
around the centroid of each element and include them in the penalty
function.
cable
tA,
t
ty
The geometry of each cable is defined by its two end points {Po, P1}
which may be attached to an adjacent cable or to a point on a block
surface. The direction of the tension force, ft, is along its axis. Similar
to blocks, f' is split into positive and negative components, f+1, and
ft. Compression, f+f is penalized for cables. It is possible to place an
upper bound on ft with inequality constraints, which reflect the scale
of the cable since strength is relative to cross-sectional area.
Cable elements are assigned a weight, w = pLg, where L is the length
of the cable and p is the mass per unit length. For gradient computation, we parameterize cables using the x, y, z coordinates of their
end points. In order to compute the structural gradient, we need the
derivative of the weight vector, and the derivative of the cable tension
direction (it) with respect to the position of its end points (w). The
derivative of the weight vector aw/w for cable is given by:
aw
L
U
(2)
1(po- pi) - O
aw
2
(3)
|po-pil
where coordinates po, p1 are two ends of the cable.
The derivative of the cable tension direction is
a
Ca
w
PO - Pi
(4)
a w ||,p '- p 1|||HPO
-
P1||H(
- (PO- P1) - (
flpo - pi2
g
P ll)
(5)
For details on how to derive the analytic structural gradient please
see [Whiting, 2012].
Application to design of structures
In the design process of a structure, it is customary to perform structural analysis only after the aesthetic design has been determined. An
architect designs the shape and then passes it to structural engineers
who try to make the structure stand using appropriate material and
reinforcement. In contrast to this type of approach, [Whiting, 2012]
Wbock
Figure 1: The tension force ft applied
by the cable balances the block weight,
but creates unbalanced torque. Virtual
torques are added in directions ix, ty,
and iz around the centroid of the block
to solve for static equilibrium.
20
suggests using the structural gradient to directly suggest ways to improve the geometry in order to enhance structural soundness in the
design process.
Given an infeasible structure, the structural gradient informs how
to modify the geometry (i.e. move the block or cable vertices) in ways
to improve feasibility. The following figures show examples of infeasible input structures and their modifications to a stable configuration.
In modifying the structure, only geometric changes are considered.
Topological changes such as attaching or detaching a cable are not
considered. Most results are achieved after a number of iterations.
Again, Whiting's thesis provides details of the implementation concerning geometry modification.
Figure 2: Progress of the infeasibility
metric as the unstable arch is modified
into a stable structure. At each iteration
the value of the infeasibility decreases
and converges toward 0 after a small
number of gradient steps. The performance is improved further by L-BFGS.
stable output
unstable input & gradients
700
-gradient descent
LBFGS
0
5s00
~400-300
200
100
0
1
2
3
4
5
Iteration
In addition to computing the first-order gradient, I implemented
the limited-memory Broyden-Fletcher-Goldfarb-Shanno method (L-BFGS)
for non-linear optimization [Fletcher, 1987]. L-BFGS searches for the
minimum by approximating the Hessian matrix, using the first-order
gradient. Experience suggests that L-BFGS converges to a stable solution with fewer iterations than using only the first-order gradient
directly (Figure 2), particularly for models with higher complexity.
Figure 3 demonstrates an example of using the structural gradient to
optimize a cable-stayed bridge design. The input bridge is unstable
because the horizontal walkway is too heavy for the vertical arch to
hold up. In the first case, we fix the horizontal walkway and the position of the cables, and only modify the vertical arch. After several
21
iterations we arrive at a stable solution, where the arch has thickened
and leaned backwards to balance the torque. In the second case, we
allow the horizontal walkway to deform, as well as changing the position of the cables. As the function of the bridge requires, we constrain
the top surface of the walkway to remain horizontal. Now, the bridge
is optimized to a different stable structure, where the vertical arch has
significantly thickened, the base arch has become thinner in the middle
and thicker at the supports.
Figure 3: An example of optimizing a
structure to improve stability
(a) A
cable-supported bridge structure originally infeasible. (b) Constraint and corresponding feasible output. The horizontal arched walkway and cables are
fixed and only the vertical arch is mod-
ified. (c) An alternative feasible output,
where only the normal of the arch walkway is fixed to remain horizontal.
(a)
(b)
(c)
Future work
The feasibility or infeasibility of the linear program in equation (1)
gives us a binary answer to whether a structure is stable or not. When
the structure is unstable, the sum of tensile forces gives a continuous
metric of infeasibility. It would be useful to develop similar metrics to
indicate the degree of stability for stable structures.
In computing the structural gradient, we considered only geometric
changes to improve stability. However, increasing stability in this way
does not guarantee a feasible solution and topological changes may be
necessary. It would be interesting to consider, for example, optimizing
for the number of cables or blocks.
Along with structural gradients, user constraints and user-defined objectives presented in [Whiting et al., 2012] provide a useful tool to
22
guide structurally stable designs. However, there is still much room
for improvement in the area of interactive design and optimization.
For example, it would be valuable to combine editing and visualization algorithms presented in 3D modeling tools such as [Igarashi and
Hughes, 2001] and [Gal et al., 2009] with analysis and optimization
algorithms.
Visualization of Equilibrium
According to Heyman's safe theorem, the existence of an equilibrium
solution for a structure indicates that the structure can indeed stand.
Yet, this is just one answer that leads to many more questions. For
example, how stable is the stable structure? Can we quantify this stability? How is the equilibrium affected by perturbations? What does
the equilibrium state consist of? In other words, we are interested in
not only whether the structure is standing or not, but how it is standing.
The challenge here lies as much in formalizing what we mean by how
as in obtaining and visualizing the result. This section takes a first step
in exploring this topic.
Block weights
In the previous chapter, the equilibrium of a structure was expressed
as a series of equality constraints enforcing the equilibrium of individual blocks. In the absence of external forces, the right hand side of
Aeq - f = -w becomes a vector containing the weight of each block.
One approach to studying the equilibrium is to do a sensitivity analysis with respect to these block weights. We ask how the force solution,
f, changes with respect to variations in w.
Computing the analytic gradient Vfwi is similar to computing the analytic structural gradient. We use the closed form expression of the force
solution derived in [Whiting et al., 20121. The closed form expression
is derived for a given geometry 0, by fixing the active constraints and
transforming them into equalities at the local solution:
fn = H-1CT(CH-
CT)-lb
(6)
C is a concatenation of the static equilibrium constraints and the
active inequality constraints, and b is the concatenation off -w and
o for the active inequality constraints. Then the expression for the
derivative of fn with respect to the i-th block's weight wi is:
n =H-1 CT (CH-CT)-lbi
aw;
(7)
24
where bi is a vector containing all zeroes except 1 for the i-th
block's weight.
Figure 4: Force change with respect to
increase in block weights for a dome
structure. (a) Making the bottom block
heavier does not significantly affect the
rest of the structure. (b) The block in the
middle, on the other hand, affects the
forces on the blocks below and around
it.
(a)
(b)
Figure 4 shows examples from the sensitivity analysis on different
blocks of a dome. The force gradient is computed with respect to the
weight of a selected block. The results show blocks color coded by the
total change of forces acting on their interface. The colored area can be
interpreted as the sphere of influence of the selected block. As expected,
in the case of the bottom block, changing its weight does not significantly affect the forces in the rest of the structure. The small force
change in the blocks above it indicated by the slight red is a side effect
of using a L2-norm for the objective function. That is, the objective in
(1) tries to minimize the sum squared forces by distributing the forces
equally as much as possible. Therefore, when we increase the weight
of the bottom block, the weights of the blocks above it are distributed
more to the sides in order to balance the forces at the ground interfaces. On the other hand, when we increase the weight of the block in
the middle of the dome the added weight is transmitted to the blocks
below it and around it, increasing both the longitudinal arching forces
and the latitudinal hoop forces.
A related question is: how much can we increase or decrease the
weight of a block while preserving equilibrium? We approach this
problem by a brute-force method. For each block, we decrease or increase its weight and solve the optimization in (1) until the problem
becomes infeasible. Figure 5 shows the maximum load of each block
in an arch. Not surprisingly, the blocks at the supports of the arch
can carry large weight. More interesting is the crown of the arch. It
has slightly larger load capacity than the blocks to its side. Figure 6(a)
shows the minimum load of each block in a vault supported by flying
buttresses. Only the outer buttresses need weight in order to resist
the outward thrust of the vault. However, as Figure 6(b) illustrates
our analysis is too local and the results depend on the discretization
of the structure. If the buttress is subdivided into smaller blocks, only
25
the top blocks need weight. If we subdivide these blocks even further,
no individual blocks would need weight. As long as the other blocks
have weight, the buttress as a whole can stand. A different type of
analysis would be necessary to probe the nature of the structure that
is independent of discretization.
Figure 5: Maximum weight of blocks
of an arch. The supports near the bottom have large load capacity. The crown
block also has slightly greater load capacity than the blocks to its side.
41
41
I
(a)
1
-j
-L
\..
Figure 6: Minimum weight of blocks in
a vault supported by flying buttresses.
(a) Weight is necessary only at the outer
buttresses. (b) However, our analysis is
sensitive to discretization. If we subdivide the outer buttress into smaller
blocks, only the top block needs weight.
~,
(b)
The investigation about the minimum necessary weight of blocks leads
to the following insight. There are at least two different roles that a
block contributes to equilibrium. A block can provide a path for the
transmission of forces through a structure, and/or it can deflect the
force path by its weight. Figure 7 illustrates this idea using thrust lines.
A thrust line represents the locus of internal reaction points found
by determining the location of the interface force reactions required
to maintain equilibrium [Ochsendorf, 2002]. Comparison of the two
buttresses shows that the added weight on the top of the right buttress
pushes the thrust line away from the outer edge of the wall, adding
stability to the structure.
There are many ways to approach this question of distinguishing the
different roles of each block in the equilibrium state. One way is to
fix the block weight to be 0 and check if a feasible equilibrium solution exists. If a solution exists, the block is acting primarily as a path
to transmit forces, and its weight has an insignificant effect. Another
approach asks a slightly different question. For a single block with a
fixed non-zero weight, we can ask what are the minimum necessary
Figure 7: Flying buttresses with the lines
of thrust. Blocks transmit forces through
the structure. (Right) The extra block
weight deflects the thrust line closer to
the center of blocks.
26
weights of the rest of the blocks that achieve equilibrium. The equilibrium formulation becomes:
minimize
h(x) = xTpIX
2
subject to
Aeq -x = -W
(8)
Afr f = 0
f
> 0, wJ > 0
where the variable x is a concatenation of the interface forces and the
unknown block weights wj's, and * is a vector of all zeros except for
the fixed block weight. Extra inequality constraints are added to ensure that block weights must be nonnegative.
Figure 8 illustrates two examples. In the case of the stacked
Figure 8: Minimum necessary weights
for equilibrium. In each case, only
the highlighted block is assigned fixed
weight. Red blocks need weight whereas
white blocks only act as path to transmit forces. Red intensity indicates magnitude of necessary weight.
(a)
(b)
blocks, the middle block has fixed weight. The top block must have
positive weight in order to exert a downward force that balances the
torque in the middle block. On the other hand, the bottom block need
only transmit the forces downward to the ground. The case of the
arch is more complex. In order to balance the torque of the highlighted block, blocks on the opposite side need weight. This in turn
creates thrust that is balanced by placing weights on the bottom right
side. Similarly, in most structures there is a chain of dependency between blocks, where in order to support one block another block needs
weight, and in order to support that weight, yet another block needs
weight and so on.
Finally, by adding the minimum necessary weights computed with respect to each block in the structure, we could visualize which parts
of the structure have weights that are significant for the equilibrium.
Figure 9 presents the results for an arch and a dome. Weights are most
important at the haunches of an arch. In real arches, this is sometimes handled by placing concrete or rubble fillings on the haunch
[Jagadeesh and Jayaram, 2000]. As expected, block weights do not
have a significant role at the bottom and top of a dome.
27
Figure 9: Sum of the minimum necessary weights solved with respected every other block (normalized by block
volume). Redder color indicates blocks
that need weight in order to support
other blocks.
Forceflow
Underlying the previous considerations-about a block's weight deflecting the thrust line or the weight of a single block being transmitted through other blocks-is the idea of the flow of forces within the
structure. Force flow or load paths is a concept frequently used in the
structural analysis of masonry. For example, [O'Dwyer, 1999] tries to
identify the principal load path in order to compute a force network
that represents the surface of thrust in a masonry vault. Similarly,
in his thesis about Thrust Network Analysis, Block discusses the importance of choosing a network topology that can represent the force
flows [Block, 2009].
We approach the problem of identifying force flows by considering the
change in the force solution f in response to external forces. This idea
was employed in section to access the sensitivity of the structure to
block weights. This was useful, for example, to visualize the sphere of
influence of a particular block within a structure. Here, the goal is more
demanding. We would like to trace the weight of a single block (i.e.
the force that the block exerts on the rest of the structure) through consecutive adjacent blocks and eventually to the ground. This becomes
a more local and sequential process. At each step, we are only considering the flow of forces to adjacent blocks (or interfaces since that
is where we represent the discrete forces in our model). If we have
a method to trace forces from one interface of a block to the other,
we can apply it recursively until the forces reach the interface at the
ground.
We visualize the load path of a single block(B) as follows:
1. Start traces pointing downward from the center of mass of B. The
number of traces is proportional to the weight of B. Each trace represents a force with a location (center of mass) and direction (downward).
2. Compute the change in force solution df at B's interface with respect
to the force represented by each trace.
3. Define a probability distribution on the interfaces of B according to
df. Larger change in force means higher probability. The probability
28
is interpolated on the surface of the interface and normalized to add
up to 1.
4. Trace each path to a point on an interface according to the probability distribution.
5. Each trace again represents a force with a location and direction
defined at an interface. Repeat steps 2-4 for each trace recursively
until it reaches the ground.
Figure lo presents two examples. The load traces illustrate that the
weight of the middle block in the H structure is supported equally
by the two columns, whereas the weight of the top block in the right
structure is supported mostly by the left column. In the same manner,
we could trace the weight of every block in a structure and superimpose the traces. The density of the traces would show principal load
paths where forces tend to concentrate inside the structure.
Figure 10: The weight of the selected
block (highlighted) is transmitted through
adjacent blocks into the ground. (a)
Weight trace starting from the center of
mass, and (b) starting from random samples inside the block.
(a)
(b)
Future work
The approaches suggested in this chapter are a first step towards better analysis and visualization of equilibrium in structures. There are
many interesting and challenging aspects that are waiting to be developed, some of which we hope to investigate in a continuation of this
research.
Visualizing properties on 3D structures is in itself a challenging subject. As we consider structures with increasing complexity and number of blocks (e.g. a cathedral), the problems of occlusion, navigation
and interactivity become more important. The choice of both what
and how to visualize would affect the effectiveness of the result. For
example, exploring volumetric or texture-based representation of force
flows inside a structure, or a hierarchical representation depending on
the required level of detail leaves much room for future investigation.
29
Developing computable metrics or definitions for the various qualities to visualize is also an important area. For instance, stability, force
paths or interdependence between blocks are useful concepts, yet too
vaguely used. The primary goal of visualization is to aid understanding about the state of the structure. Therefore, one starting point may
be to consider applications and what information is useful for potential users. For instance, a commonly used measure of stability is the
tilt test, which measures the maximum angle of tilt before collapse
([Ochsendorf, 2002] and [Zessin, 2012]. Depending on the application,
other measures of stability would also be useful.
The equilibrium method presented in [Whiting, 2012] provides a useful way to do limit state analysis for masonry structures. However, it
gives little clue about the actual force state. First, the objective function
that is used to find the optimal force solution is not based on physical
principles. It attempts to minimize total forces while penalizing tension, but this is not an accurate description of how a structure finds
equilibrium. It is not a question of finding a magical objective function
either. Rigid body physics by itself does not specify enough information about the structure to determine one actual state for a given
structure. In most cases, masonry structure is hyper-static or statically
indeterminate, meaning an infinite number of equilibrium solutions
exist. So, the question arises about how to deal with this inherent indeterminacy. Finite element analysis is one way, and this is where we
turn to next.
Comparison of Equilibrium and Elastic Analysis
Finite element analysis (FEA) represents the most widely used method
for structural analysis in many fields of engineering, including masonry. However, recently, several studies pointed to the shortcomings
of FEA and suggested that the problem is not only in the level of accuracy it attains but that the method is inherently inappropriate for
masonry structural analysis ([Whiting, 2012], [Block, 2005]). Instead
they propose equilibrium analysis such as the one presented in the
previous chapters. Still, other than few example cases where FEA predictions fail, to our knowledge, no attempt has been made at a clear
and thorough explanation about why equilibrium analysis is more appropriate than FEA. This chapter examines the problem of FEA closely
in order to clarify why it is not suitable for masonry. Based on our
observation, we also propose an improved algorithm for FEA.
FEA divides a structure into many small elements and prescribes material properties to them in terms of stiffness matrices. It specifies
boundary conditions on the displacements of the elements. Finally,
based on variational principles, it computes the displacement of the
elements, which are used to derive stresses and strains in the structure. Masonry structures have several characteristics that make FEA
challenging. For instance, [Whiting, 2012] calls attention to the fact
that FEA is designed to model a continuum rather than a discontinuous set of blocks, that it does not consider the indeterminate property
of masonry structures, and that stresses in masonry are low compared
to material stiffness so elastic deformations are almost negligible.
In what follows, we identify characteristic cases where FEA outputs
incorrect results and present a clear interpretation for the cause of its
failure. These cases also shed light on a possible step towards unified
comparison of the FEA with the equilibrium analysis which we plan
to follow up in future research.
Linear elasticformulation
First, in order to compare with the equilibrium method presented in
previous chapters we formulate and implement a simple finite element
32
analysis as follows:
i. Each block constitutes a hexahedral element with eight nodes, one
at each corner of the block. The stiffness matrix Ki is computed
for each element. We use Ki = f_ BTEBdV, assuming that the
stress-strain matrix E is constant over the element. B is the straindisplacement matrix. The integral over the volume V is replaced
by a numerical integration at the Gauss quadrature points. [Felippa
and Clough, 1970]
2.
The global stiffness matrix K is computed for the entire structure by
assembling the Ki element matrices.
3. The principle of minimum total potential energy states that the
structure will deform to a position that minimizes the total potential
energy. From this principle, the FEA is formulated as an optimization problem:
minimize
U
subject to
E (u) =
2
An - u = 0
TKu + eu
(9)
We are solving for the nodal displacements u. We use a linear elastic
model. 1uTKu is the elastic potential energy, and eT u is the gravitational potential energy due to the weight of the blocks distributed
to the nodes. Finally, the linear equality constraint An - u = 0 states
the boundary conditions, described next.
4. The boundary conditions limit the space of allowable nodal displacements. In the physical world, blocks that are in contact cannot
interpenetrate. The constraints ensure that adjacent nodes at an interface do not pass through each other. We express this by requiring
their displacements to be the same. That is, for every pair of nodes
pi and p1 that are adjacent at an interface, ui = up. (See Figure 11.)
5. Once we find the optimal solution u*, the elastic force is computed
as fK = -Ku*. Similarly, the contact forces between elements is
computed from the contact constraints and their Lagrangian multiApliers so that fc =A
There are many ways to refine the method, for example by using
smaller elements or a nonlinear stress-strain relationship. Our aim in
formulating the FEA was to capture the essential characteristic of elastic analysis-elastic deformation of the material according to its property and the principle of minimum potential energy-in the simplest
possible model.
Figure 11: Each pair of adjacent nodes
pi and pj at the interface is constrained
to remain in contact and not penetrate
through the other block. This is done by
enforcing their displacements to be the
same (ui = uj).
33
Equilibrium
Elastic (FEA)
Assumptions
rigid elements
safe theorem
elastic elements
min total potential energy
Constraints
YZ(force per block) = 0
E(torque per block) = 0
ui = uj for nodes i, j
in contact
ff >
Table i: Comparison of equilibrium and
elastic analysis
0
Friction
Coulomb friction cone
oo
Output
forces
displacements
for contact nodes
0 if contact is released
Tension failure
Table i presents a summary of the two methods. Unlike the equilibrium formulation, the elastic formulation does not give a direct answer
as to whether the structure is stable or not. However, each contact constraint (A', the i-th row of An) comes from an interface j between two
blocks, and it gives rise to contact forces (f = Ai - (A') T) at the nodes.
By comparing these contact forces with the interface normal (f - ni),
we can check whether the force is compressive or tensile.
In pointing out the shortcomings of FEA, [Block, 2005] refers to the
case of two arches, one thicker and the other thinner. Equilibrium
analysis shows that the thicker arch has a compression-only equilibrium solution, whereas the thinner arch is too thin to contain a thrust
line and therefore would not stand under its own weight without tensile forces. However, FEA cannot reveal this major difference and outputs very similar results for the two arches. In fact, Whiting points
out that linear elastic FEA predicts tension for both arches despite the
thicker arch having a compression-only equilibrium solution.
Figure 12: An example often pointed out
as a failure case of FEA. (a) Elastic analysis predicts tension forces (highlighted) at
the crown where (b) equilibrium analysis finds a compression-only solution.
(a) Tension in elastic solution (FEA)
(b) Compression-only equilibrium solution
We verify this result for an arch with thickness to radius ratio (t/R)
This is sufficiently thicker than the theoretical minimum required for a stable arch, which is t/R = 0.1075 [Ochsendorf, .2006].
Figure 12 presents the result. Although the equilibrium analysis finds
a compression-only solution, where the thrust line is completely conof
0.12.
34
tained within the arch blocks (12b), FEA outputs a solution with tension forces at the crown of the arch (12a).
Contact constraints & interpretationof tension
Now, let us take a closer look at the tension forces in FEA. We observe that tensile forces in FEA should not be interpreted simply as
a failure to find a compression-only solution. Rather, tensile forces in
FEA indicate that the boundary condition does not correspond to the
physical reality of the structure we are trying to model. We will see
that, in fact, we must change the boundary condition to correspond to
the structural reality, and then search for a compressive equilibrium
solution.
As explained earlier, the constraint confines the space of allowable
nodal displacements. Specifically, when adjacent blocks in compression try to interpenetrate each other, the boundary condition prevents
this from happening. This applies for nodes that are in compression,
but the same is not true for nodes in tension.
Physically, blocks cannot interpenetrate, but they can separate from
each other. This is an important difference between masonry structures and other continuum structures modeled in traditional FEA. The
element boundaries in continuum structures are artificial boundaries
that have both compressive and tensile capacity. On the other hand,
block interfaces are physical boundaries that have zero tensile capacity. It is perfectly possible for blocks to separate from each other. In
fact, the artificial tensile forces appear precisely because blocks try to
move away from each other, but false constraints do not represent this
reality and incorrectly prevent nodes from separating.
The arch in Figure 13 illustrates this case. At the top part of the crown
interface (t), the blocks are pushing toward each other, so the nonpenetration constraint applies. On the other hand, since the abutments
of the arch give way with time, at the bottom part of the same interface
(b), the blocks try to separate away. If the boundary condition does not
express this reality, tensile forces appear. This is the case, even though
the three-hinged arch is a perfectly stable configuration that is not at
the point of collapse.
Previous works have alluded to this as a failure of FEA to find a
compression-only equilibrium solution. However, this is because the
boundary condition represents an incorrect physical model. We need
to modify the boundary constraint to include hinging behavior in its
feasible region. This is what we discuss next.
t
b
Figure 13: The arch is supported by an
abutment, which will give way slightly.
The lower figure, greatly exaggerated
shows how the arch accommodates it
self by developing three hinges. The arch
is stable.
35
Towards a compression-only solution
The difficulty in forming the correct boundary condition comes from
determining which nodes are in contact and therefore where the constraints apply. If the nodes are trying to move toward each other,
constraints must ensure that they remain in contact and not interpenetrate. Otherwise, if they are trying to move apart from each other,
the constraint must be released. However, it is impossible to know a
priori which nodes should be constrained. To resolve this problem, we
propose an iterative constraint relaxation algorithm inspired by similar approaches developed for physical simulations.
Contact resolution is recognized as a fundamental challenge in multibody dynamics. The problem comes from the coupling between friction impulses and contact impulses. In order to compute the friction
impulses, contact points must be resolved. However, in order to determine the contact points accurately friction impulses must be known. A
recent approach proposed by [Kaufman et al., 2008] solves this problem using a predictor-corrector method. First the predictor computes
the contact points, and then the corrector uses this result to solve for
friction. The two steps are interlaced until the system converges. Their
algorithm is able to produce robust and accurate simulations of rigid
and deformable bodies.
Following the same principle, we propose the following approach, in
which we iteratively relax the contact constraints for nodes that develop tension forces.
1. First, FEA is solved with all the contact constraints present, requiring adjacent blocks to remain together.
2.
The contact force f* is computed from the displacement solution u*,
its corresponding active constraints and their Lagrange multipliers
(f* = AT -A).
3. If a contact force on a node is tensile, we release the constraint on
that node by deleting the corresponding rows from An.
4. We solve for the modified FEA with fewer contact constraints.
5. Steps 2 to 4 are repeated until a compression-only solution is found,
or until a maximum number of iterations is reached.
In several cases, we were able to find a compression-only FEA solution where the initial solution had predicted tension. For example, the
arch in Figure 12 converged to a compression-only solution after two
iterations. Figure 14 shows an example of a horizontal block supported
by three columns. This is a stable configuration, where the weight of
36
the horizontal bar is distributed to the three columns. In the elastic solution, however, the leftmost column is thin and it shrinks more than
the thicker middle column. This causes the horizontal bar to lean leftward and the initial solution predicts a tensile force at the interface
of the rightmost column. After releasing the contact constraint at this
interface, FEA finds a compression-only solution. If we scale up the
displacements of this solution, we observe a hinge where the initial
solution predicted tensile forces.
Figure 14: (a) FEA predicts tension at
the interface of the rightmost column.
(b) The different thicknesses of the three
supports cause the top block to lean and
form a hinge. This is more clearly seen
after releasing the contact constraints
where tension was predicted.
(a) Tension in FEA solution
(b) Exaggerated displacement
after releasing contact constraint
Comparison to the equilibrium analysis
An alternative and more direct way to solve the contact resolution
problem would be to specify the boundary condition such that it corresponds exactly to the non-penetration principle. That is, we could
directly express the condition that the nodes are free to slide on or
separate from the interface, but not interpenetrate. However, this constraint is nonlinear and non-convex, making the problem difficult to
formulate and to solve.
The equilibrium analysis presented in previous chapters avoids this
difficulty by (1) assuming perfect rigidity, and (2) formulating the
problem in terms of forces. Since the blocks do not deform, there is
no nodal displacement and hence no interpenetration. The no tensile
requirement can be expressed as a simple inequality constraint on the
normal forces.
The equilibrium and elastic analysis use different physical models
based on different approximations. Moreover, the objective function
used in the equilibrium analysis (minimizing the sum squared forces)
is rather arbitrary and not physically justified. Naturally, the solution
found by the iterative FEA and the equilibrium analysis are usually
different. Figure 15 compares the forces computed by the two methods
in a dome structure. The equilibrium analysis outputs predominantly
vertical forces at the bottom and hoop forces at the top. On the other
37
hand, the elastic analysis outputs larger hoop forces in all levels of the
dome.
Figure 15: Both equilibrium and elastic
methods find a compression-only solution indicating that the structure is stable. However, the two solutions are
different. (a) The equilibrium analysis
outputs predominantly vertical forces in
the bottom of the structure, whereas (b)
the elastic analysis outputs larger hoop
forces.
(a) Equilibrium analysis
(b) Elastic analysis (iterative FEA)
Futurework
Resolving contact nodes do not solve the problem of indeterminacy
in masonry. Consider a four-legged stool. It could be balanced by
only two opposite legs, or equally by the four legs. In fact, there are
infinitely many possible force configurations to make the stool stand
and all of them are physically valid. Developing an efficient method
to analyze hyper-static structures is an interesting challenge.
Further investigation is necessary to delineate the relationship between
rigid body equilibrium analysis and elastic finite element analysis. The
force formulation of equilibrium analysis and the displacement formulation of FEA resemble a primal-dual relationship. A unified framework to compare the two methods along these lines would be helpful.
Efficient algorithms to handle contact and friction would significantly
improve both methods. For instance, the safe theorem which states
that if a compression-only equilibrium solution exists the structure is
stable applies only to hinging failure under tension. For both equilibrium and elastic analysis, how to efficiently compute friction and how
to handle sliding are open questions.
Conclusions
This thesis explored computational and visualization tools to study
the equilibrium of masonry structures. The following contributions
are presented:
1. We extend the equilibrium formulation to determine the stability of
masonry structures to include tensile elements such as cables. The
quadratic formulation provides a measure of infeasibility, and the
gradient of feasibility with respect to geometric modifications.
2.
We apply the gradient of feasibility in the design of stable structures. Starting from an unstable input model, we modify the geometry to increase its stability while satisfying user-defined constraints.
3. We investigate various properties related to the equilibrium of a
structure and visualize the results.
4. We compare the equilibrium method to the finite element analysis
(FEA). We point out the correct way to interpret tension in FEA,
and explain the drawback of FEA that comes from formulating the
contact constraints. We propose an iterative constraint relaxation
approach to address this problem.
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