# The Redundancy Matrix as a Performance Indicator for Structural Assessment

0000-0002-1069-7070David Forster  University of Stuttgart, Institute for
Structural Mechanics<EMAIL_ADDRESS>0000-0002-4156-2355Malte
von Scheven  University of Stuttgart, Institute for Structural Mechanics

###### Abstract

Abstract
The degree of static indeterminacy and its spatial distribution characterize
load-bearing structures independent of a specific load case. The redundancy
matrix stores the distribution of the static indeterminacy on its main
diagonal, and thereby offers the possibility to use this property for the
assessment of structures. It is especially suitable to be used in early
planning stages for design exploration. In this paper, performance indicators
with respect to robustness and assemblability are derived from the redundancy
matrix. For each of the performance indicators, a detailed matrix-based
derivation is given and the application is showcased with various truss
examples.

###### keywords:

redundancy matrix, structural assessment, robustness, assemblability,
structural optimization

##  1 Introduction

###  1.1 Motivation

In civil engineering, several requirements must be satisfied when designing a
building. Besides aesthetic and sustainability aspects, a key aspect of the
design is structural safety, meaning that the structure withstands external
forces such as wind and dead load but also temperature changes and exceptional
influences like vehicle impact. The national building codes are mainly
restricting stresses for the ultimate limit state and displacements for the
serviceability limit state, taking into account different load cases and
safety factors depending on the probability of their respective occurrence [8,
3]. Those concepts are well-defined and known to structural engineers.

In contrast, the notions of redundancy and robustness, as well as the degree
of static indeterminacy and the distribution of internal constraint are only
vaguely touched in building codes. Especially the quantification of these
structural performance indicators is not specified. The redundancy matrix and
thus the distribution of the degree of static indeterminacy in the structure
expands the possibilities for structural engineers to assess also these
aspects of structural design on a quantitative basis.

Dealing with robustness, according to the German building codes, collapse must
be prevented and the effect of damage and its cause must be somewhat
proportional [9]. This means, for example, that a small event must not lead to
an overall collapse of the structure.

Another aspect of structural assessment, which is not covered at all in
building codes, is the assembly process and the interplay between
prefabrication and on-site manufacturing. Geometric imperfections can cause an
initial stress-state in the assembled structure and by this influence the
load-bearing behavior. Adjusting the design or assembly sequence, the
assemblability of structures can be improved and the initial stresses caused
by imperfections can be kept minimal.

In this paper, quantitative design criteria for the robustness and the
assemblability of structures based on the redundancy matrix are proposed. When
dealing with robustness and structural assembly, the redundancy matrix serves
as a suitable measure, since it is a measure for the internal constraint and
independent of specific load cases. This is especially important in very early
design stages, where various design options with different topologies, cross-
sections and geometries need to be assessed without explicit knowledge about
load cases and governing load combinations.

###  1.2 State of the Art

The concept of the redundancy matrix and the related distributed static
indeterminacy, as it is used in the present paper, was proposed in the group
of Klaus Linkwitz in the context of geodesic and structural mechanics research
[29]. Based upon this work, [4] and [41] describe the redundancy matrix as an
idempotent matrix, quantifying the spatial distribution of the degree of
static indeterminacy on its main diagonal, referred to as redundancy. [42] use
this information to quantify the sensitivity towards imperfection of a
structural system. In [45], the matrix-based derivation of the redundancy
matrix is summarized for trusses and plane beams and extended to continuous
systems. Applications in the field of adaptive structures are presented in
[46] and [16], using the redundancy matrix for actuator placement to
compensate for external loads with force or displacement manipulation within
the structure. [12] give a brief overview of the concept of using the
redundancy for the design of structures and describe the calculation for
three-dimensional frame structures. In [17], the redundancy matrix is used to
assess robotically assembled carbon fiber composite structures with a focus on
capturing deviations stemming from geometric imperfections due to the
manufacturing process.

The concept of redundancy is of course closely related to the notion of
robustness. But there exists a much larger variety of definitions of
robustness in the literature. Not all of them can be mentioned here. Only few
of these definitions of robustness make use of the redundancy matrix or the
static indeterminacy. We will show later that in fact the redundancy
distribution as a measure for robustness is identical to other definitions in
literature.

[22] present the idea that the factors of safety in designing structural
elements should be adapted according to the member’s importance. Based on the
probabilistic approach of the building codes, the authors present a
reliability index, which is shifting the normal distribution curve for
resistance depending on whether the member is of high importance for the load
transfer or highly redundant.

[13] are taking into account brittle, ductile and hardening behavior of the
structural elements to assess redundancy. Amongst several definitions and
interpretations of structural redundancy, the authors define four different
criteria. Two of those are the degree of static indeterminacy [30] and the
load bearing capacity of different states of the structure, which makes this
measurement of robustness dependant of a specific load case scenario. [11] use
an index based on the material’s strength to quantify the performance and show
related optimization of structures. The above-mentioned four different
criteria are also used for structural optimization and the extension of
redundancy for continuous structures [14, 34].

The contribution by [38] distinguishes between the different causes of damage,
disregarding external influences and points out that the majority of
structural defects are due to the design, followed by wrong execution and
improper use. This underlines the fact, that early design stages are of utmost
importance when it comes to structural safety. Therein, the term redundancy is
defined as the structure’s ability to provide different load paths in order to
compensate for individual failure of members, adding safety to the structure
beyond the requirements in building codes. The contribution also rises the
question of manufacturing and therefore the assembly process as a measure for
structural performance.

Describing examples of structural collapse due to missing robustness, [18]
propose a quantification of robustness using a score which is again dependent
on a distinct external exposure. A list of measures to design robust
structures includes the structure itself but also the maintenance and the used
material. [5] present a framework based on probabilistic risk analysis,
quantifying the direct consequences of damage as well as subsequent impacts.

[20] define a so-called strong redundancy, taking into account the spatial
distribution of the static indeterminacy within the structure. Within a truss
example, this strong redundancy counts the maximum number of elements that can
be removed before the structure fails, without taking into account the order.
By this, the method identifies critical paths and non-redundant parts within a
structure. [24] present examples in the context of redundancy and robustness.
They introduce a recursive method to calculate the redundancy matrix for the
modified system after failure of a certain structural element in order to
capture progressive failure.

Another important aspect for the assessment of structures is the assembly
process and the induced stress-states during on-site assembly. In construction
industry, tasks on site are mainly performed manually by skilled workers,
offering the opportunity to account for dynamic environmental changes and
uncertainties, while at the same time assistance through automatic control to
execute repetitive tasks increases [19]. With an increasing digitization and
automation in construction industry, as described e.g. by [23], effects from
predefined assembly sequences and manufacturing imperfections on the
performance of the structural system need to be addressed.

Many publications deal with assembly planning to reduce the amount of formwork
or even achieve self-supporting structures. [21] use a method based on so-
called backward assembly planning [28] to assemble shell structures with a
minimum amount of formwork. Imperfections in the manufacturing process, which
can impact the initial stress-state of the assembled structure, are not taken
into account. Also, recent publications in the field of robotically assembled
structures mainly deal with self-supporting structures that avoid scaffolding,
without referring to stress-states or imperfection sensitivity of the assembly
process [35, 6]. In the context of robotically aided on-site assembly, [26]
present an automated process for timber cassettes that is showcased on a real
construction site. [27] show the automated assembly of spatial timber
structures using single-axis robots and standardized timber struts.

Manufacturing imperfections and initial stresses induced during the assembly
procedure are not considered in most of these publications. Since
manufacturing imperfections introduce states of stress in a structure, the
ultimate load-bearing capacity can be reduced by these initial stresses.
Therefore, from a structural engineering point of view, it is important to
either minimize the imperfections or to decrease their negative effect by a
customized assembly sequence. Within this paper, the influence of
manufacturing imperfections on the strain distribution of a structure is
presented. Subsequently, the effect of different assembly sequences, which
lead to different structural configurations, on intermediate strain
distributions is shown.

###  1.3 Outline

The paper is structured as follows. Section 2 briefly introduces the
theoretical fundamentals of structural mechanics, including matrix structural
analysis, the definition of the redundancy matrix and its properties. In
Section 3, a measure for robustness based on the redundancy matrix is derived
and showcased with a 3D truss structure. Section 4 shows the assessment of a
structure in regard to the assembly and the respective derivation of a
quantitative measure. Section 5 summarizes the work and gives an outlook on
future research.

##  2 Fundamentals of Structural Mechanics

###  2.1 Matrix Structural Analysis

In this section, relevant quantities and equations of matrix structural
analysis for linear static analysis of discrete models of spatial truss and
frame structures are summarized. The formulation is based on the natural mode
formulation originally presented by [1] and [2]. This formulation describes
the deformation of an element by decoupled strain inducing modes and rigid
body modes.

Given is a discrete model consisting of $n$ degrees of freedom,
$n_{\mathrm{n}}$ nodes, and $n_{\mathrm{e}}$ elements, each of which carries
loads via $n_{\mathrm{m}}$ load-carrying modes. The number of load-carrying
modes is equal to the number of generalized stress resultants or generalized
elastic deformations in this element and is $n_{\mathrm{m}}=1$ for plane or
spatial truss elements, $n_{\mathrm{m}}=3$ for plane beam elements and
$n_{\mathrm{m}}=6$ for spatial beam elements. In general, models can consist
of a combination of truss and beam elements, i.e., $n_{\mathrm{m}}$ can vary
between the elements. Therefore, the total number of load-carrying modes of
all elements is introduced as $n_{\mathrm{q}}$. For models consisting of only
one element type $n_{\mathrm{q}}=n_{\mathrm{m}}n_{\mathrm{e}}$.

The relation between the external loads ${\mathbf{f}}\in\mathbb{R}^{n}$ and
the generalized displacements ${\mathbf{d}}\in\mathbb{R}^{n}$ is described by
the three field equations static equilibrium, elastic material law and
compatibility:

$\displaystyle{\mathbf{A}}^{\mathrm{T}}{\mathbf{s}}$
$\displaystyle={\mathbf{f}}$ $\displaystyle{\mathbf{s}}$
$\displaystyle={\mathbf{C}}{\mathbf{e}}_{\mathrm{el}}$
$\displaystyle-{\mathbf{e}}_{\mathrm{el}}$
$\displaystyle=-{\mathbf{A}}{\mathbf{d}}+{\mathbf{e}}_{0}.$ (1)

${\mathbf{A}}^{\mathrm{T}}\in\mathbb{R}^{n\times n_{\mathrm{q}}}$ is the
equilibrium matrix, ${\mathbf{A}}\in\mathbb{R}^{n_{\mathrm{q}}\times n}$ is
the compatibility matrix, and ${\mathbf{C}}\in\mathbb{R}^{n_{\mathrm{q}}\times
n_{\mathrm{q}}}$ is the material matrix, which is a diagonal matrix with
positive entries. The vector ${\mathbf{s}}\in\mathbb{R}^{n_{\mathrm{q}}}$
represents the generalized stress resultants of all elements,
${\mathbf{e}}_{\mathrm{el}}\in\mathbb{R}^{n_{\mathrm{q}}}$ represents the
corresponding generalized elastic deformations and
${\mathbf{e}}_{0}\in\mathbb{R}^{n_{\mathrm{q}}}$ represents the generalized
pre-deformations.

generalized
---
elastic deformations
---
${\mathbf{e}}_{\mathrm{el}}\in\mathbb{R}^{n_{\mathrm{q}}}$
---
generalized
---
displacements
---
${\mathbf{d}}\in\mathbb{R}^{n}$
---
external loads
---
${\mathbf{f}}\in\mathbb{R}^{n}$
---
generalized
---
stress resultants
---
${\mathbf{s}}\in\mathbb{R}^{n_{\mathrm{q}}}$
---
elastic material law
---
${\mathbf{s}}={\mathbf{C}}{\mathbf{e}}_{\mathrm{el}}$
---
compatibility
---
$-{\mathbf{e}}_{\mathrm{el}}=-{\mathbf{A}}{\mathbf{d}}+{\mathbf{e}}_{0}$
---
static equilibrium
---
${\mathbf{A}}^{\mathrm{T}}{\mathbf{s}}={\mathbf{f}}$
---
${\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{A}}{\mathbf{d}}={\mathbf{f}}+{\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{e}}_{0}$
---
Figure 1: Overview of relevant equations and quantities in matrix structural
analysis for linear elastostatics (inspired by Tonti’s diagram for
elastostatic problems [44] and by [40]).

The diagram in Figure 1 summarizes the relevant equations and quantities in
matrix structural analysis for linear elastostatics and states the equation to
compute the generalized displacements ${\mathbf{d}}$ from the external loads
${\mathbf{f}}$:

$\displaystyle{\mathbf{K}}{\mathbf{d}}$
$\displaystyle={\mathbf{f}}+{\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{e}}_{0}$
with $\displaystyle{\mathbf{K}}$
$\displaystyle={\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{A}}.$ (2)

${\mathbf{K}}\in\mathbb{R}^{n\times n}$ is called the elastic stiffness
matrix. It is symmetric by definition due to the diagonality of
${\mathbf{C}}$.

It is assumed throughout the paper that the structures are statically
indeterminate with a degree of static indeterminacy
$n_{\mathrm{s}}=n_{\mathrm{q}}-\text{rank}({\mathbf{A}}^{\mathrm{T}})$.
Furthermore, it is assumed that the structures are kinematically determinate,
i.e., $\text{rank}({\mathbf{A}})=n$ [37, 36], which is equivalent to
${\mathbf{K}}$ being regular. The latter assumption can be satisfied by
properly choosing structural topology and boundary conditions. It ensures that
the structures are able to equilibrate loads without pre-stress (and thus
geometric stiffness effects) such that linear structural theory is applicable.

###  2.2 Definition of the Redundancy Matrix

Based on the quantities and equations of matrix structural analysis defined in
the previous subsection, the concept of the redundancy matrix [29, 4, 41, 45]
is recapitulated in the following. As state-of-the-art, the redundancy matrix
is only defined for the linear setting.

The redundancy matrix is a measure of the internal constraint in a structure
and is therefore independent of the external loads. Thus,
${\mathbf{f}}={\mbox{$\mathbf{0}$}}$ is assumed. Solving Equation 2 for the
generalized displacements ${\mathbf{d}}$ and inserting those into the
compatibility Equation Equation (1c) yields a relation between the negative
generalized elastic deformations $-{\mathbf{e}}_{{\mathrm{el}}}$ and the
generalized pre-deformations ${\mathbf{e}}_{0}$:

$\displaystyle-{\mathbf{e}}_{{\mathrm{el}}}=({\mathbf{I}}-{\mathbf{A}}{\mathbf{K}}^{-1}{\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}){\mathbf{e}}_{0}={\mathbf{R}}{\mathbf{e}}_{0},$
(3)

with the redundancy matrix ${\mathbf{R}}\in\mathbb{R}^{n_{\mathrm{q}}\times
n_{\mathrm{q}}}$

$\displaystyle{\mathbf{R}}={\mathbf{I}}-{\mathbf{A}}{\mathbf{K}}^{-1}{\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}$
(4)

and the identity matrix ${\mathbf{I}}\in\mathbb{R}^{n_{\mathrm{q}}\times
n_{\mathrm{q}}}$.

Considering Equation 3, the redundancy matrix component $R_{ik}$ maps the
initial elongations imposed in element $k$ onto the negative elastic
elongations in element $i$. Therefore, the redundancy matrix contains column-
wise the negative generalized elastic deformations caused by a unit
generalized pre-deformation in the respective element $k$. For a truss system,
this corresponds to removing element $k$ from the structure and reassembling
it after assigning a unit elongation. Squeezing this imperfect element into
the structure will cause elastic deformations in other elements (column $k$ of
the redundancy matrix). The amount by which the initial elongation in element
$k$ is reduced by the surrounding structure is a measure of the constraint
imposed on the element and also its redundancy in the structure. For a very
high constraint, the resulting total deformation in element $k$ will be close
to zero, the elastic deformation close to one and also the redundancy $R_{kk}$
will be close to one. On the contrary, an element with little constraint from
the surrounding structure will yield a large total deformation and a small
elastic deformation and therefore a small diagonal entry and redundancy.

This definition of the redundancy can be applied to all discrete structural
system, like truss systems in 2D and 3D as well as frame systems in 2D [45]
and 3D [12, 43].

###  2.3 Properties of the Redundancy Matrix

The redundancy matrix ${\mathbf{R}}$ describes a parallel projection of
initial elongations into the subspace of elastic elongations
($\operatorname{im}({\mathbf{R}})$) parallel to the subspace of compatible
elongations ($\ker({\mathbf{R}})$).

The matrix ${\mathbf{R}}$ is idempotent and its trace is equal to
$n_{\mathrm{e}}-n_{\mathrm{d}}=n_{\mathrm{s}}$, $n_{\mathrm{s}}$ being the
total degree of static indeterminacy in the structure [45]. As
$\operatorname{tr}({\mathbf{R}})=n_{\mathrm{s}}$, the diagonal entries
$R_{kk}$ of the redundancy matrix can be interpreted as the contributions of
the individual elements to the total degree of static indeterminacy
$n_{\mathrm{s}}$ [4, 41]. Therefore, the diagonal entries are also called
distributed static indeterminacy [10, 48, 7]. This allows to distribute the
total degree of static indeterminacy $n_{\mathrm{s}}$ amongst the elements of
the structure. Properties known for statically determinate or indeterminate
structures can be transferred to the element level: Constraint load cases will
not yield internal forces in an element with zero redundancy as this element
is statically determinate and removing a statically determinate element, i.e.
an element with zero redundancy, will lead to (partial) failure of the
structure. This makes the redundancy matrix very useful for the assessment of
structures with respect to robustness (reducing the impact of element failure)
and assemblability (avoiding stresses due to geometrical imperfections).

The redundancy matrix ${\mathbf{R}}$ can also be interpreted as an influence
matrix. It describes the influence of initial deformations on the elastic
deformations in the structure. In some cases, it is not the influence on
deformations that is important, but the influence on stresses or stress
resultants. Then the elastic deformations can be directly converted into the
stress resultants using the material matrix ${\mathbf{C}}$. The influence
matrix for the stress resultants is $-{\mathbf{C}}{\mathbf{R}}$.

Interactive design methods require fast feedback to inform designers and
assist them in their decision-making process. Direct feedback on the
redundancy distribution in a structure is particularly useful for topology
exploration with respect to assemblability and robustness [12]. But due to the
inverse of the stiffness matrix and the matrix-matrix multiplications, the
computational complexity for the calculation of the redundancy matrix is given
by $\mathcal{O}(n\cdot n_{\mathrm{q}}^{2})$. Since $n$ is typically
proportional to $n_{\mathrm{q}}$, the complexity scales cubically with the
problem size. A more efficient computation of the redundancy matrix is
proposed by [43]. The closed-form expression is derived via a factorization of
the redundancy matrix that is based on singular value decomposition.

If in a design or optimization process, a structure is iteratively examined
with the help of slight adjustments, the resulting changes to the redundancy
matrix can be computed via a rank one update. A generic algebraic formulation
for efficiently updating the redundancy matrix (and related matrices) is
presented by [25]. The formulations based on the Woodbury formula include
various modifications like adding, removing, and exchanging elements and are
applicable to truss and frame structures.

##  3 Robustness

The redundancy distribution within a structure can be used to quantify its
robustness. We assume a system to be robust if the change in elastic
deformations due to a given load is minimized in the event that an element
fails and is therefore removed. A detailed derivation is conducted, starting
from the change in stiffness due to the removal of an element up to a compact
form of the calculation of the effect on the elastic deformations, see also
[4]. The details and the notation for the matrix calculation are based on
[25]. Thereby, the compatibility matrix ${\mathbf{A}}$, the material matrix
${\mathbf{C}}$, and the stiffness matrix
${\mathbf{K}}={\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{A}}$ refer to the
initial system. The following derivation is based on one load-carrying mode
$n_{\mathrm{m}}$. For beam structures, the modes can be evaluated separately.
The removed element is denoted as $r$, thus, the element’s redundancy of the
removed element is given by $R_{rr}$. The row of the compatibility matrix
related to the element to be removed is described by
${\mathbf{a}}_{r}\in\mathbb{R}^{1\times n}$ and its stiffness by
${\mathbf{C}}_{rr}=c_{r}$.

With this at hand, we can write the flexibility matrix of the modified system
as the inverse of the stiffness matrix of the modified system
$\tilde{\mathbf{K}}$ as

$\displaystyle\tilde{{\mathbf{K}}}^{-1}$
$\displaystyle=\left({\mathbf{A}}^{\mathrm{T}}{\mathbf{C}}{\mathbf{A}}-{\mathbf{a}}_{r}^{\mathrm{T}}c_{r}{\mathbf{a}}_{r}\right)^{-1}.$
(5)

Using the Woodbury formula [47], the above equation can be rewritten as

$\displaystyle\tilde{{\mathbf{K}}}^{-1}$
$\displaystyle={\mathbf{K}}^{-1}+{\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}c_{r}\left(1-{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}c_{r}\right)^{-1}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}.$
(6)

Since we want to examine the change of flexibility if an element is removed,
we define

$\displaystyle{\bm{\Delta}}{\mathbf{K}}^{-1}$
$\displaystyle=\tilde{{\mathbf{K}}}^{-1}-{\mathbf{K}}^{-1}={\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\left(c_{r}^{-1}-{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\right)^{-1}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}$
(7)

as the change of flexibility matrix when removing element $r$.

According to [43], the main-diagonal entry of the redundancy matrix for the
element to be removed, $R_{rr}$, can be computed as

$\displaystyle R_{rr}$
$\displaystyle=1-{\mathbf{a}}_{r}{\mathbf{K}}^{-1}c_{r}{\mathbf{a}}_{r}^{{\mathrm{T}}}.$
(8)

This expression can be re-written as the ratio of the redundancy of the
element and the stiffness of the element as

$\displaystyle\frac{R_{rr}}{c_{r}}$
$\displaystyle=\left(c_{r}^{-1}-{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\right).$
(9)

Inserting Equation 9 into Equation 7, the change in flexibility can be
expressed as

$\displaystyle{\bm{\Delta}}{\mathbf{K}}^{-1}$
$\displaystyle={\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\frac{c_{r}}{R_{rr}}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}.$
(10)

The change in displacements due to the removal of element $r$ under an
arbitrary load ${\mathbf{f}}$ can be calculated using the change in the
flexibility matrix:

$\displaystyle{\bm{\Delta}}{\mathbf{d}}={\bm{\Delta}}{\mathbf{K}}^{-1}{\mathbf{f}}$
$\displaystyle={\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\frac{c_{r}}{R_{rr}}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{f}}.$
(11)

To further simplify this expression, it can be multiplied by the row of the
compatibility matrix related to the removed element ${\mathbf{a}}_{r}$. This
yields the change of elongation ${\Delta}e_{r}$ of the removed element, or as
the element is removed, the change in distance between the corresponding nodes
considering linear kinematics.

$\displaystyle{\Delta}e_{r}$
$\displaystyle={\mathbf{a}}_{r}{\bm{\Delta}}{\mathbf{d}}={\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}}\frac{c_{r}}{R_{rr}}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{f}}.$
(12)

Although this is only a local criterion, it describes the effect on the load-
bearing behavior at the location and in the direction of the structural
modification. Using Equation 8, rewritten as
$1-R_{rr}={\mathbf{a}}_{r}{\mathbf{K}}^{-1}c_{r}{\mathbf{a}}_{r}^{{\mathrm{T}}}$,
we can formulate the above equation as

$\displaystyle{\Delta}e_{r}$
$\displaystyle={\mathbf{a}}_{r}{\bm{\Delta}}{\mathbf{d}}=\frac{1-R_{rr}}{R_{rr}}{\mathbf{a}}_{r}{\mathbf{K}}^{-1}{\mathbf{f}}=\frac{1-R_{rr}}{R_{rr}}{\mathbf{a}}_{r}{\mathbf{d}}.$
(13)

Equation 13 shows that the change in element elongation ${\Delta}e_{r}$ caused
by removing the element $r$ depends on the factor $\frac{1-R_{rr}}{R_{rr}}$
and therefore on the redundancy of the removed element $R_{rr}$. The larger
the redundancy of the removed element $R_{rr}$, the smaller the effect on the
load-bearing behavior of the structure. This means that for a robust behavior,
the redundancy of the removed element should be as large as possible.

Since robust behavior of a structure is associated with being independent of
the element to fail, the redundancies of all elements need to be as large as
possible. As the sum of all redundancies equals the degree of static
indeterminacy and is independent of the element to be removed, a homogeneous
distribution maximizes all redundancies, and thus can be used as an objective
to design robust structures.

This definition of a robust structure having a homogeneous distribution of
redundancy is in fact identical to other definitions in literature. The
determinant of the global stiffness matrix $\det({\mathbf{K}})$ is widely used
to quantify robustness such that the ratio of the determinant of the modified
stiffness matrix and the determinant of the initial stiffness matrix is used
as a measure and maximized. [32] denotes this ratio as the member consequence
factor, used to quantify structural integrity and [39] use this ratio to
define a stiffness-based measure of robustness.

With the help of a rank one update [31], the determinant of the modified
stiffness matrix can be written as

$\displaystyle\det(\tilde{\mathbf{K}})=\det({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}}-{\mathbf{a}}_{r}^{{\mathrm{T}}}c_{r}{\mathbf{a}}_{r})$
$\displaystyle=\det({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}})(1-c_{r}{\mathbf{a}}_{r}({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}})^{-1}{\mathbf{a}}_{r}^{{\mathrm{T}}})$
$\displaystyle=\det({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}})R_{rr}.$
(14)

Thus, the ratio of the determinant of the stiffness matrix of the modified and
the initial system is identical to the redundancy of the element to be
removed:

$\displaystyle\frac{\det(\tilde{\mathbf{K}})}{\det({\mathbf{K}})}$
$\displaystyle=\frac{\det({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}}-{\mathbf{a}}_{r}^{{\mathrm{T}}}c_{r}{\mathbf{a}}_{r})}{\det({\mathbf{A}}^{{\mathrm{T}}}{\mathbf{C}}{\mathbf{A}})}=R_{rr}.$
(15)

Equations (3) and (15) as well as the relation to the stiffness based
robustness index proposed by [39] was communicated by [15]. It underlines the
applicability of our approach of distributing redundancies homogeneously and
by this maximizing the redundancy to achieve a robust structural design. The
calculation procedure of the redundancy of an element can be made fast,
offering an advantage regarding computational time compared to the procedure
using the determinant [43].

To showcase the above-mentioned approach of distributing the redundancy
homogeneously within a structure, an optimization scheme using this objective
is described in detail.

1
---
(b) Top view, element numbering
---
(a) Isometric view, node numbering
---
1
---
2
---
3
---
4
---
5
---
6
---
7
---
8
---
9
---
10
---
11
---
2
---
3
---
4
---
5
---
6
---
7
---
8
---
9
---
10
---
11
---
12
---
13
---
14
---
$y$
---
$x$
---
$z$
---
$1.00\,\mathrm{m}$
---
$1.00\,\mathrm{m}$
---
$1.00\,\mathrm{m}$
---
$1.00\,\mathrm{m}$
---
$1.00\,\mathrm{m}$
---
$1.00\,\mathrm{m}$
---

$1.00\,\mathrm{m}$
---

$1.00\,\mathrm{m}$
---

$EA=\text{const.}=1000\,\mathrm{kN}$
---

$1.00\,\mathrm{m}$
---

Figure 2: Initial configuration of a 3D truss structure; Isometric view, node
numbering and coordinate system shown in (a); Top view and element numbering
shown in (b).

Figure 2(a) shows the initial configuration of a 3D truss structure in the
isometric view with node numbering and the coordinate system. The top view and
the element numbering can be seen in Figure 2(b). The structure consists of 14
truss elements with a constant element stiffness $EA=1000\,\mathrm{kN}$ and
has a degree of static indeterminacy of $n_{\mathrm{s}}=5$. The spatial
distribution of the redundancy is shown in color scheme in Figure 3(a and b).

${\mathbf{R}}_{kk}$
---
$0.08$
---
$0.58$
---
(b) Top view initial configuration
---
(a) Isometric view initial configuration
---
(c) Isometric view robust configuration
---
(d) Top view robust configuration
---
Figure 3: Optimization of a 3D truss structure to obtain a robust design.
Isometric view (a) and top view (b) of initial configuration shown on the
left, colors indicating the redundancies according to the colorbar. Isometric
view (c) and top view (d) of robust configuration shown on the right.

As it can be seen in the color scheme, the redundancy of the elements is
varying between 0.08 and 0.58. The individual redundancies of the elements are
additionally shown in Table 1 in line $R_{kk}$.

Element $k$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 | 13 | 14
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---
$R_{kk}$ | 0.08 | 0.08 | 0.35 | 0.35 | 0.58 | 0.58 | 0.58 | 0.58 | 0.35 | 0.35 | 0.08 | 0.08 | 0.49 | 0.49
$R_{kk,{\mathrm{opt}}}$ | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36 | 0.36
Table 1: Redundancies for initial configuration and optimized configuration
per element.

The four elements at both ends of the structure drawn in dark blue have a very
low redundancy and are of high importance for the load transfer. In case these
elements fail, little possibilities for the redistribution of forces are
given, thus these elements are very relevant for structural integrity.

In order to obtain a homogeneous distribution of the redundancies, the spatial
location of the nodal points are chosen as the design variables within the
optimization. The optimization problem can then be formulated as follows:

$\displaystyle\min_{{\mathbf{s}}}f({\mathbf{s}}),$ $\displaystyle
f({\mathbf{s}})$ $\displaystyle=R_{{\mathrm{max}}}-R_{{\mathrm{min}}},$
$\displaystyle{\mathbf{s}}^{{\mathrm{T}}}$
$\displaystyle=\begin{bmatrix}x_{1}&x_{2}&x_{9}&y_{1}&y_{2}&z_{9}&z_{10}\end{bmatrix}.$
(16)

$R_{\mathrm{max}}$ and $R_{\mathrm{min}}$ denote the maximum and minimum
redundancy of the structure, respectively. The remaining locations of the
nodal points are chosen, such that the structure remains symmetric and the
support points do not move in $z$-direction, compared to the initial
configuration. Therefore, only the seven values in Equation 16 are to be used
as design variables within the optimization.

The optimization is performed with the commercial software Matlab, using the
sequential quadratic programming algorithm, as described in detail by [33].
Figure 3(c) shows the optimized configuration in isometric view with the
homogeneous redundancy distribution, as can be seen by the equal color of all
elements. The top view of the optimized configuration is shown in Figure 3(d),
clearly indicating the symmetry of the structure. The redundancies of the
elements of the optimized configuration are shown in Table 1 in the row
$R_{kk,{\mathrm{opt}}}$.

This example shows, that repositioning of nodes can be used to generate a
structure with a homogeneous redundancy distribution. Finally, an exemplary
study is performed to show that this structure is also more robust, i.e.
yields smaller changes in element elongations due to a given arbitrary load
and that the change in the determinant of the stiffness matrix is independent
of the element to fail. The initial and robust configurations shown in Figure
3 are compared.

Table 2 shows in the second and third columns the changes in element
elongations due to a load of $100\,\mathrm{kN}$ in vertical direction on nodes
9, 10 and 11. Each line refers to the structural system with one element
removed, which is indicated in the first column. For most of these cases the
change in element elongation for the robust system is significantly smaller
compared to the inital system. But for certain elements, the increase in
element elongation is smaller for the initial configuration, for example if
element 5 is removed. This is in good accordance with the values of the
redundancies, since for these elements the redundancy is large in the initial
configuration and becomes smaller in the robust configuration. However, for
the robust configuration, the changes in element elongation vary on a smaller
scale and the arithmetic mean of the changes
$\overline{|{\Delta}{\mathbf{e}}|}$ is also smaller in comparison to the
initial configuration. The same analysis could be done for any other given
load or displacement and a similar result could be seen according to Equation
13.

In columns four and five, Table 2 shows the determinant of the stiffness
matrix of the system with one element removed. It can be seen, that the
determinant is independent of element to fail for the robust configuration,
for which the redundancies are distributed homogeneously. For the initial
configuration, the changes can be compared to Equation 15 and the redundancy
values in Table 1.

Additionally, the last two columns of Table 2 compare the initial and robust
configuration with respect to the change in the Euclidean norm of the complete
displacement vector. For both configurations, the displacements due to the
aforementioned vertical load of $100\,\mathrm{kN}$ on the three top nodes are
calculated for the intact system ${\mathbf{d}}$ and the system with one
element removed ${\mathbf{d}}_{r}$. Each line in the table shows the relative
change ${\beta}_{r}$ in the Euclidean norm of the displacement vectors for the
case that one element is removed. For the removal of certain elements, the
relative change ${\beta}_{r}$ is slightly larger for the robust configuration.
But the arithmetic mean of all configurations shows that the robust
configuration leads to less change in displacements in case of an element
failure.

removed |  $|{\Delta}e_{r}|$ $\text{in}\leavevmode\nobreak\ \,\mathrm{m}$ | $\det(\tilde{\mathbf{K}})/10^{23}\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \left(\frac{\,\mathrm{kN}}{\,\mathrm{m}}\right)^{9}$ | ${\beta}_{r}=\frac{||{\mathbf{d}}_{r}||_{2}-||{\mathbf{d}}||_{2}}{||{\mathbf{d}}||_{2}}\leavevmode\nobreak\ \text{in}\leavevmode\nobreak\ \%$
---|---|---|---
element r | init. config. | rob. config. | init. config. | rob. config. | init. config. | rob. config.
1 | 1.34 | 0.15 | 0.95 | 3.89 | 96.83 | 11.06
2 | 1.34 | 0.15 | 0.95 | 3.89 | 96.83 | 11.06
3 | 0.52 | 0.31 | 3.97 | 3.89 | 78.84 | 116.12
4 | 0.52 | 0.31 | 3.97 | 3.89 | 78.84 | 116.12
5 | 0.08 | 0.19 | 6.64 | 3.89 | 2.29 | 22.99
6 | 0.08 | 0.19 | 6.64 | 3.89 | 2.29 | 22.99
7 | 0.08 | 0.19 | 6.64 | 3.89 | 2.29 | 22.99
8 | 0.08 | 0.19 | 6.64 | 3.89 | 2.29 | 22.99
9 | 0.52 | 0.31 | 3.97 | 3.89 | 78.84 | 116.12
10 | 0.52 | 0.31 | 3.97 | 3.89 | 78.84 | 116.12
11 | 1.34 | 0.15 | 0.95 | 3.89 | 96.83 | 11.06
12 | 1.34 | 0.15 | 0.95 | 3.89 | 96.83 | 11.06
13 | 0.04 | 0.01 | 5.64 | 3.89 | 0.60 | 0.18
14 | 0.04 | 0.01 | 5.64 | 3.89 | 0.60 | 0.18
| $\overline{|{\Delta}{\mathbf{e}}|}=0.56$ | $\overline{|{\Delta}{\mathbf{e}}|}=0.18$ |  |  | $\overline{{\bm{\beta}}}=50.93$ | $\overline{{\bm{\beta}}}=42.93$
Table 2: Changes in element elongation due to a prescribed load of
$100\,\mathrm{kN}$ on nodes 9, 10 and 11 in vertical direction, determinant of
modified stiffness matrix and relative change of the norm of the displacements
due to the prescribed load. Different structural configurations representing
initial configuration and robust configuration for an individual element’s
removal.

The assumptions of the optimized configuration being symmetric can of course
also be neglected and various different solutions exist, that satisfy the
homogeneous redundancy distribution. Another approach to achieve this goal
would be to use the cross-sections as design variables. In case of adjustments
of the cross-sectional thickness in hollow sections, this makes the
geometrical appearance independent of the optimization.

##  4 Assemblability

###  4.1 Imperfection Induced Strains

From a structural engineering point of view, one goal is to avoid large
stresses induced during on-site assembly due to manufacturing imperfections of
certain elements. Since the stresses are proportional to the strains for a
constant Young’s Modulus, strains will be used here to assess the structure
with regard to the imperfection sensitivity.

The magnitude of the imperfections in length are assumed to be relative to the
length of the respective element. As described in Section 2.2, for truss
structures, each column $k$ of the redundancy matrix represents the negative
elastic elongations in all elements that occur, if a prescribed unit
elongation is applied on element $k$. Therefore, we can use the redundancy
matrix scaled column-wise by the lengths of the elements to evaluate the
strains induced by the imperfections. For this, we introduce the diagonal
matrix ${\mathbf{L}}\in\mathbb{R}^{n_{\mathrm{e}}\times n_{\mathrm{e}}}$ that
contains the lengths of the individual elements on the main diagonal:

$\displaystyle{\mathbf{L}}=\left[\begin{array}[]{cccc}L_{1}&0&\cdots&0\\\
0&L_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&L_{n_{e}}\end{array}\right].$ (21)

Furthermore, the matrix ${\bm{\alpha}}\in\mathbb{R}^{n_{\mathrm{e}}\times
n_{\mathrm{e}}}$ is introduced to specify the magnitude of the imperfection as
the percentage of the original length for each element individually:

$\displaystyle{\bm{\alpha}}=\left[\begin{array}[]{cccc}{\alpha}_{1}&0&\cdots&0\\\
0&{\alpha}_{2}&\cdots&0\\\ \vdots&\vdots&\ddots&\vdots\\\
0&0&\cdots&{\alpha}_{n_{e}}\end{array}\right].$ (26)

${\mathbf{E}}_{\text{ass}}\in\mathbb{R}^{n_{\mathrm{e}}\times n_{\mathrm{e}}}$
expresses now column-wise the elastic elongations in all members caused by
imperfections:

$\displaystyle{\mathbf{E}}_{\textnormal{ass}}=-{\mathbf{R}}{\bm{\alpha}}{\mathbf{L}}.$
(27)

To obtain the strains in each element from these elongations, the entries of
${\mathbf{E}}_{\text{ass}}$ need to be divided row-wise by the original length
of the respective element:

$\displaystyle{\bm{\varepsilon}}_{\text{ass}}={\mathbf{L}}^{-1}{\mathbf{E}}_{\text{ass}}=-{\mathbf{L}}^{-1}{\mathbf{R}}{\bm{\alpha}}{\mathbf{L}}.$
(28)

The matrix
${\bm{\varepsilon}}_{\textnormal{ass}}\in\mathbb{R}^{n_{\mathrm{e}}\times
n_{\mathrm{e}}}$ contains column-wise the distribution of strains in the
structure due to a length imperfection relative to the original length in one
element. Compared to a standard finite element calculation of imperfection-
induced strains, the above proposed procedure offers a compact matrix-based
calculation that avoids repetitive analysis of the full structure. Different
norms can now be applied to the columns $k$ to define a measure that can be
compared easily. While the maximum norm
$\max_{i}({\bm{\varepsilon}}_{\text{ass},ik})$ concentrates on the largest
value of strain induced by an imperfection, the Euclidean norm
$||{\bm{\varepsilon}}_{\text{ass},ik}||_{2}$ takes into account the effect on
all members of the structure. The effect of imperfections in the members of
the structure can now be compared and the design and/or assembly sequence
adapted accordingly. In order to evaluate the effect of all imperfections, a
corresponding matrix norm can be applied to the complete matrix
${\bm{\varepsilon}}_{\textnormal{ass}}$.

In the following, we will showcase the influence of manufacturing
imperfections and how the influences can be altered within an optimization
scheme. In a second example, different assembly sequences are compared with
regard to intermediate strain states showing that the sequence itself is
largely influencing the maximum strain throughout the construction process.

###  4.2 Influence of Geometric Imperfections

$2.00\,\mathrm{m}$
---
$2.00\,\mathrm{m}$
---
$2.00\,\mathrm{m}$
---

$2.00\,\mathrm{m}$
---

$0.0$
---
$0.5$
---
${\mathbf{R}}_{kk}$
---
(a) Structural system
---
(b) Redundancy distribution
---
13
---
14
---
15
---
16
---
1
---
2
---
3
---
4
---
5
---
6
---
7
---
8
---
Figure 4: Truss structure with two prefabricated modules (grey) and four
elements for final assembly (black); Element 14 with 100 times stiffness
compared to all other elements (a). Redundancy distribution of the structure
(b) in colorscheme.

Figure 4 shows a simple 2D truss structure with node and element numbering on
the left and the redundancy distribution in color scheme on the right. The
stiffness of element 14 is 100 times higher than the constant stiffness of all
other elements, and therefore the element is drawn thicker. This leads to a
very low redundancy for element 14. The total degree of static indeterminacy
is $n_{\mathrm{s}}=4$. In this scenario, the imperfection in length is defined
as 10 % of the perfect length of the members, i.e.
${\bm{\alpha}}=0.1*{\bm{1}}_{n_{e}}$.

The grey elements on either side of the structure are assumed to be pre-
fabricated and thus no geometric imperfections are assumed for them. The black
elements 13 to 16 are used for final assembly on site. In this study we are
interested in the influence of manufacturing imperfections for different
elements. Element 14 is the one with the lowest redundancy, meaning that
according to the interpretation of the redundancy matrix, it is the least
constraint by the surrounding. Nevertheless, the maximum strain and the
Euclidean norm of the strain is larger in comparison to the elements 13 and
16, see Table 3. This means that for the scenario that one element is
imperfectly manufactured, element 13 or 16 would influence the strain
distribution on a smaller scale in comparison to elements 14 and 15.

Element $k$ | 13 | 14 | 15 | 16
---|---|---|---|---
${\mathbf{R}}_{kk}$ | 0.1504 | 0.0043 | 0.4254 | 0.1504
$\max_{i}({\bm{\varepsilon}}_{\text{ass},ik})$ | 0.0213 | 0.0425 | 0.0425 | 0.0213
$||{\bm{\varepsilon}}_{\text{ass},ik}||_{2}$ | 0.0361 | 0.0722 | 0.0722 | 0.0361
Table 3: Assessment of assembly parameters for initial truss structure (Figure
4).

In a second scenario, where element 14 is said to be imperfectly manufactured,
the strains that are induced by a length imperfection should be minimized.
This can be done by a shape optimization using the nodal positions as design
variables. It is prescribed that the supports remain at their original
position, the lower chord of the truss remains straight and the system stays
symmetric. Therefore, only 5 design variables are used and the locations of
the remaining nodes are derived from these design variables. During the
optimization, the Young’s modulus and the cross-sections are kept constant.
The optimization problem can be defined as follows:

$\displaystyle\min_{\mathbf{s}}f({\mathbf{s}}),$ $\displaystyle
f({\mathbf{s}})$ $\displaystyle=||{\bm{\varepsilon}}_{\text{ass},i14}||_{2},$
$\displaystyle{\mathbf{s}}^{\mathrm{T}}$
$\displaystyle=\begin{bmatrix}x_{2}&x_{5}&x_{6}&y_{5}&y_{6}\end{bmatrix}$ (29)

The optimization was again performed with Matlab using sequential quadratic
programming. Figure 5 shows the original configuration of the truss on the
left and the optimized geometry according to Equation 29 on the right. Table 4
shows the resulting values for the redundancy, the maximum strain and the
Euclidean norm of the strain. One can see that the Euclidean norm of the
strain was reduced by 12 % from 0.0722 to 0.0632. One can also see that the
difference in the Euclidean norm between element 13 and 14 decreased
drastically, meaning that the impact of the change in length regarding the
strains decreased from 100 % difference in the initial configuration to 13 %
in the optimized configuration.

(a) Initial configuration
---
(b) Optimized configuration
---
14
---
Figure 5: Truss structure from introductory example with stiffer diagonal element 14 (a). Structure with optimized nodal positions to minimize $||{\bm{\varepsilon}}_{\text{ass},i14}||_{2}$ (b). Element $k$ | 13 | 14 | 15 | 16
---|---|---|---|---
${\mathbf{R}}_{kk}$ | 0.3001 | 0.0037 | 0.3682 | 0.3001
$\max_{i}({\bm{\varepsilon}}_{\text{ass},ik})$ | 0.0321 | 0.0368 | 0.0368 | 0.321
$||{\bm{\varepsilon}}_{\text{ass},ik}||_{2}$ | 0.0551 | 0.0632 | 0.0632 | 0.0551
Table 4: Assessment of assembly parameters for optimized truss structure
(Figure 5).

###  4.3 Assembly Sequence

The following case study aims to understand the influence of different
assembly sequences on the strain state within a structure. For different
states of the assembly $l$, reaching from the first assembled element $a$ to
the last element $f_{l}$ assembled in this step, the strain distribution can
be calculated in vector format as follows:

$\displaystyle{\bm{\varepsilon}}_{\textnormal{seq}}^{l}=\sum_{k=a}^{f_{l}}{\bm{\varepsilon}}_{\textnormal{ass},ik}^{l}.$
(30)

The matrix ${\bm{\varepsilon}}_{\textnormal{ass}}^{l}$ describes the state $l$
within the assembly sequence. Since there exist many possibilities with
various intermediate structural configurations for the assembly sequence, an
efficient update can drastically decrease the computational effort [25].

Figure 6 shows the structural system of a plane truss on the left. The
statically determinate part of the structure is shown in light grey and is
said to be constructed without any geometric imperfections. The elements
labelled 9 to 12, drawn in solid black, are the ones hat will be assembled
with given imperfections of ${\alpha}_{9}={\alpha}_{10}=0.1$,
${\alpha}_{11}=0.3$ and ${\alpha}_{12}=-0.3$. On the right of Figure 6, the
maximum absolute strain of three exemplary construction sequences is shown
with different colors. The x-axis represents the assembly steps, starting from
the initial step 0 to the final assembly step 4. On the y-axis, the maximum
absolute strain value of all elements of the structure is given, according to
Equation 30. In the initial state, the strain is zero in the whole structure.
In the final state, the maximum absolute value is similar for all assembly
sequences. Since we are dealing with linear static analyses, the theorem of
Betti-Maxwell applies and the final distribution of strains is independent of
the assembly sequence.

One can obtain that different assembly sequences yield different maximum
strains and thus stresses throughout the process. The sequence that is drawn
in red, where element 11 is assembled last, yields a higher maximum strain at
step 3 than in the final state. This means that sequence 1 should be avoided,
otherwise the intermediate maximum strain outreaches the one that is
unavoidable in the final assembly. One could of course also track the strain
of individual elements throughout the assembly process to choose a sequence
that is defined as optimal for any given scenario. This can be especially
useful if a specific element is very sensitive to initial strains or if for
example sensors are placed and initial strain deviations should therefore be
avoided.

$1.00\,\mathrm{m}$
---

$1.00\,\mathrm{m}$
---

$1.00\,\mathrm{m}$
---

$EA=\text{const.}=1000\,\mathrm{kN}$
---
9
---
10
---
11
---
12
---

$0$$0.05$$0.1$$0.15$$0.2$$0$$1$$2$$3$$4$$\textnormal{max}|{\bm{\varepsilon}}_{\textnormal{seq}}|$Assembly
step$\textnormal{Sequence 1:}\leavevmode\nobreak\
9-10-12-11$$\textnormal{Sequence 2:}\leavevmode\nobreak\
11-9-12-10$$\textnormal{Sequence 3:}\leavevmode\nobreak\ 10-12-11-9$

Figure 6: Truss structure of already assembled elements in grey and four
elements for final assembly in solid black (left). Maximum strain of all
assembled elements throughout different assembly sequences (right). Sequence 1
outreaches the final maximum strain and proves therefore unfeasible.

##  5 Conclusion

The paper addresses the assessment of structures using the redundancy matrix
and by this, using the distribution of static indeterminacy. On this basis,
quantitative performance indicators for the robustness and assemblability are
presented. These additional measures for structural assessment enlarge the
possibilities for design exploration in very early design stages.

A detailed derivation of the matrix calculations for these two structural
performance indicators was given and showcased with various examples. It was
shown that the design of robust structures can be achieved by distributing the
redundancy homogeneously within the structure. Different measures for the
structural performance were used to compare the robustness of an initial
configuration and an optimized configuration with a homogeneous redundancy
distribution. In the context of the construction process, the influence of
geometric imperfections and the assembly sequence on initial strains was
predicted using the redundancy matrix. By optimizing the assembly sequence the
maximum initial strain could be reduced.

The presented methods are applicable to truss and frame structures and can be
especially useful in building systems that are sensitive towards
imperfections. The extension of the notion of the redundancy matrix to plates
and shells is ongoing work. In addition, the extension of the redundancy
matrix to the non-linear setting is work in progress and will allow a
straightforward transfer of the proposed indicators to non-linear problems as
well.

The application of the presented methods to the behavior of a structure during
progressive collapse is still an open question. It this situation, the
redundancy matrix changes constantly after damage started. In each damage
state the presented methods can be applied, but a repeated update of the
redundancy matrix is necessary.

## Acknowledgments

This work is partly funded by “Deutsche Forschungsgemeinschaft” (DFG, German
Research Foundation) under Germany’s Excellence Strategy – EXC 2120/1 –
390831618.

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