Document [original]

Preprint No. 2019-03

Decision making in structural engineering problems under

polymorphic uncertainty

A benchmark proposal

Y. Petryna∗1, M. Drieschner1

February 22, 2019

∗Correspondence: Y. Petryna: yuriy.p[email protected]

1Technische Universität Berlin, Faculty VI Planning Building Environment, Department of

Civil Engineering, Chair of Structural Mechanics, Gustav-Meyer-Allee 25, 13355 Berlin,

Germany

Suggested Citation: Y. Petryna, M. Drieschner. Decision making in structural en-

gineering problems under polymorphic uncertainty - A bench-

mark proposal. Preprint-Reihe des Fachgebiets Statik und Dy-

namik, Technische Universität Berlin, Preprint No. 2019-03, 2019.

http://dx.doi.org/10.14279/depositonce-8240.

information see https://creativecommons.org/licenses/by/4.0/.

Preprint-Reihe des Fachgebiets Statik und Dynamik, Technische Universität Berlin auf

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1 Y. Petryna, M. Drieschner

Decision making in structural engineering problems under

polymorphic uncertainty

A benchmark proposal

Y. Petryna, M. Drieschner

Abstract

The treatment of diverse uncertainties is an important challenge in structural engineering prob-

lems, especially from the viewpoint of realistic analysis. Inaccuracy and variability are always

present and have to be quantified by either probabilistic, possibilistic, polymorphic or other

uncertainty approaches. Regardless to the applied uncertainty quantification method, the nu-

merical predictions have to be useful for structural assessment and decision making. The authors

propose in this contribution a benchmark example of a portal frame structure including various

uncertainties. The goal of this benchmark study is to compare justifications and decisions pro-

vided by probabilistic and non-probabilistic methods with respect to clear challenges of decision

making with and without measurements and data assimilation. The engineering problem itself

is simple enough to understand and complex enough not to be reduced to a simple formula with

uncertain parameters.

Keywords benchmark; polymorphic uncertainties; portal frame; decision making; data as-

similation

1 Introduction and goals

The treatment of diverse uncertainties is an important challenge in structural engineering prob-

lems, especially from the viewpoint of realistic analysis. Inaccuracy and variability are always

present and have to be quantified by either probabilistic, possibilistic, polymorphic or other un-

certainty approaches. Regardless to the applied uncertainty quantification method, the numerical

predictions have to be useful for structural assessment and decision making. Dealing with un-

certainties is since long subject of research in several fields like weather and climate predictions

[17], geosciences [8] or nuclear energy [4]. In structural engineering, uncertainty quantification is

mainly related to the design and optimization problems [6, 9, 19, 20, 10, 16], geotechnics [11] or

system and parameter identification [18].

In spite of numerous publications, there were only very few comparative studies involving

epistemic and aleatory uncertainties simultaneously, i.e. dealing with polymorphic uncertainties

[5]. Benchmark studies on special problems usually help comparing diverse methods and ap-

proaches with respect to their efficiency and accuracy and developing a standard. Some relevant

benchmarks are known in nuclear energy [2], geosciences [1], control problems [13] or structural

health monitoring [18]. However, benchmarks dealing with polymorphic uncertainties in struc-

tural engineering are very rare. The recent test bed example [14] originates from a joint activity

of the Priority Programme SPP 1886 of the German Research Foundation (DFG) entitled "Poly-

morphic uncertainty modelling for the numerical design of structures". The goal is an objective

comparison of various non-probabilistic methods with respect to decision making in engineering

problems with polymorphic uncertainties.

Preprint No. 2019-03, FG Statik und Dynamik, TU Berlin 2

The present contribution proposes a benchmark of a structural engineering problem that

pursues several typical goals: (i) encourage researchers to develop new methods for polymorphic

uncertainty modeling, (ii) provide a problem description and a measurement data set to test

such new methods under realistic conditions, and (iii) allow an objective comparison of proposed

methods based on a true data set of input and output. At that, the authors realize the need

for a benchmark that would consider interpretable uncertainty for decision making based on the

polymorphic uncertainty.

The considered engineering system is a portal frame under vertical and horizontal point loads.

It is a simple mechanical system that helps avoiding false interpretation of results with respect to

system behavior and failure states. On the other hand, it possesses two competing failure modes,

material failure and stability failure, which make the limit state function strongly nonlinear and

sensitive to uncertainties. Such a combination of failure modes is typical for many structural

systems. The benchmark should demonstrate how real decision making can work in the case

of polymorphic uncertainties. One of the crucial issues is an objective comparison of results

with different origin, namely, probabilistic and non-probabilistic ones. Perhaps, this could be

done on the basis of the decisions made. The second challenge of the benchmark deals with the

question how data assimilation can help in the decision making. Finally, a design problem under

polymorphic uncertainty is given in which more demanding operation requirements have to be

fulfilled.

2 Computational model

2.1 Reference structure: portal frame

A portal frame structure consisting of two columns of height Hand one girder of length L(Fig. 1)

is considered. Such frames are typical for many technical systems, for example industrial facilities

or portal cranes. The columns are fixed in the foundation and the joints between columns and

girder are rigid. The frame is loaded by a vertical crane force FVwhich can be located arbitrarily

between two limit positions and is always directed downwards. The operational field of this force

is marked gray in Fig. 1. In addition, the crane truck can cause a horizontal brake force FBat

the same position as FV. The direction of FBcould be either to the left or to the right. Finally,

the left column is loaded horizontally by a force FHacting always from the left to the right.

The position of FHis fixed. All loads are considered as static ones and, thus, the problem to be

solved is a static problem.

Figure 1: Mechanical model of the portal frame

3 Y. Petryna, M. Drieschner

The columns and the girder are made of steel and have rectangular cross-sections of the same

width and different heights, as shown in Fig. 1. Real hollow steel profiles are not considered

here for simplicity, since the cross-sections cannot be varied continuously but taken from an

assortment list at disposal. The material behavior is assumed to be linear elastic until failure.

2.2 Potential failure modes

It is assumed that the frame suffers only deformations in the plane, i.e. no out-of-plane defor-

mations are considered. Under given loads, the frame experiences both normal forces N, lateral

forces Qand bending moments M. Typical distributions of these internal forces as well as a

typical deformation state are given exemplarily in Fig. 2 for better understanding of system be-

havior. Two failure modes with respect to ultimate limit states are considered: material failure

and stability failure.

Figure 2: Internal forces Nand Mas well as deformation state of the portal frame

Material failure (local stress problem) The material failure is always local and occurs due

to the violation of a limit stress, in this case, the yielding stress of steel σy. The maximum

acting stress can be calculated from the known normal forces and bending moments in each

cross-section as follows:

σmax =N

A±M

W(1)

with the cross-section area Aand the resistance moment Wof a rectangular cross-section ac-

cording to Fig. 1

A=bh and W=bh2

6.(2)

The limit state function for material failure is implicit since it requires each time the calcu-

lation of internal forces and corresponding maximum stresses at critical locations:

g(σmax, σy) = σy

|σmax|−1.0 = λmat

λ−1.0=0.(3)

Here, λmat and λindicate the critical load factor for material failure and the current load factor

λ= 1.0, respectively.

Stability failure (global buckling) The stability failure occurs due to the loss of the sys-

tem equilibrium and is, therefore, always global. Stability failure is typically characterized by

the buckling load λstabPand the buckling mode Φ, i.e. the deformation state appearing after

buckling. At that, the load factor λstab marks the critical value of a given load case P. A typical

buckling mode of the portal frame under consideration is shown in Fig. 3.

The corresponding limit state function is also implicit since it requires a step of the system

stability analysis and can be written in terms of the buckling load λstabPand a given load λP

as follows:

g(λ, λstab) = λstab

λ−1.0 = 0 .(4)

Preprint No. 2019-03, FG Statik und Dynamik, TU Berlin 4

Figure 3: Buckling mode of the portal frame

System failure (material or stability failure) It is assumed that any local exceedance of

yielding stress and any buckling of the frame are unacceptable limit states for the crane operation

and, thus, correspond to the system failure. The load, structural and material parameters in

this benchmark are chosen in such a way that both material and stability failure occur at similar

load levels. Therefore, the real failure mode is sensitive to uncertainties.

2.3 Structural analysis

Structural analysis of the frame is carried out by the displacement method similarly to the finite

element method. All necessary explanations are given below and provided in the appendix to

the benchmark, so that all participants can use the same computational model. Thus, the model

uncertainty can be excluded from this test aiming at the role of polymorphic data uncertainties.

The stiffness relation of the Bernoulli beam elements with the stiffness matrix Ke, load vector

Peand the nodal displacement vector Veis well known [3]:

Ke·ve=







l0 0 −EA

l0 0

12EI

l3−6EI

l20−12EI

l3−6EI

4EI

l06EI

2EI

l0 0

12EI

6EI

symm.4EI













ϕl

ϕr



















=Pe.(5)

It provides exact solution with respect to displacements:

ve=K−1

e·Pe.(6)

The portal frame is discretized by five classical Bernoulli beam elements and 12 active degrees

of freedom (nodal displacements), as shown in Fig. 4.

Its linear response to static loads can be calculated by use of the system stiffness relation,

i.e. system of algebraic equations:

K·V=λP with λ= 1.(7)

The elastic system stiffness matrix Kij, i, j = 1,...,12 and the system load vector

Pi, i = 1,...,12 are derived analytically according to the system discretization in Fig. 4 and

stay explicitly for all participants at disposal. A MATLAB R

file executing an exemplary deter-

ministic calculation is available under [15] in order to minimize mistakes and misunderstanding

with respect to the structural analysis.

5 Y. Petryna, M. Drieschner

Figure 4: Discretized mechanical model with five beam elements and 12 nodal degrees-of-freedom

At that, the following input parameters are defined: material properties E,σy; geometry L,

H; element cross-sections bi, hi, i = 1,...,5; load magnitudes FV,FB,FHand load positions LV,

HH.

Material failure (local stress problem) The analysis steps for evaluating the limit state

function according to Eq. (3) are shortly described in the following:

1. calculate the system nodal displacements: V=K−1·P

2. transform the system displacements Vto the local element ones ve,i

3. calculate the internal forces of each element: se,i according to Eq. (5)

4. calculate the maximum stress in Eq. (1) of each element: σmax,i

5. calculate the maximum stress in portal frame: σmax = max(σmax,i)

6. calculate the critical load factor λmat =σy

|σmax|

7. calculate the limit state function value according to Eq. (3): g(σmax, σy)

If the frame losses its stability and buckles, it takes a new deformed state called buckling

mode. The critical load at which it happens is called buckling load. The buckling load factor

λand the buckling mode Φcan be calculated according to the classical stability theory from

the generalized eigenvalue problem for the elastic (K)and the geometric (Kg)system stiffness

matrices as follows:

(K−λKg)Φ = 0 (8)

The geometric system stiffness matrix Kgof the frame accounts for the deformed state and de-

pends on the acting normal forces Niand, thus, on the load level λP. The linearized geometric

stiffness matrix Kg,efor classical beam elements is well known [3]. The geometric system stiffness

matrix Kgof the frame with respect to 12 active nodal degrees-of-freedom (Fig. 4) is derived

analytically by the authors from the element matrices Kg,eand stays at disposal for all partic-

ipants. This kind of stability analysis provides only a linearized prediction of the buckling load

and does not account for imperfections. However, it is considered as sufficient for the purpose of

the present benchmark.

Preprint No. 2019-03, FG Statik und Dynamik, TU Berlin 6

Stability failure (global buckling) The analysis steps for evaluating the stability limit state

function according to Eq. (4) are shortly described in the following:

1. calculate normal forces from the linear elastic analysis at λ= 1.0:Ni

2. calculate the geometric system stiffness matrix: Kg

3. solve the eigenvalue problem in Eq. (8) and determine the minimum eigenvalue, i.e. load

factor: λstab = min(λi)

4. calculate the limit state function g(λ, λstab)according to Eq. (4).

2.4 Result interpretation

Material failure occurs if g(σmax, σy)≤0, stability failure takes place if g(λ, λstab)≤0. The

failure mode with the smallest load factor λstab or λmat occurs first.

3 Available data

3.1 Input data

The geometry, load, material and cross-sectional parameters are defined with uncertainties as

described below.

Geometry and cross-sections The lengths of the columns and the girder are explicitly mea-

sured and considered as deterministic: L= 10m and H= 8m (see Fig. 1). The quality man-

agement of the manufacturer gives a usual fabrication tolerance of ±2mm for the cross-sections

of the frame. According to the original design, the cross-sections of the members should be as

follows:

•width: b1=b2=b3=b4=b5= 0.20m

•height: h1=h2=h5= 0.14m (column), h3=h4= 0.85m (girder).

Loads The operation field of the vertical load FVis defined deterministically within two limit

positions from the left edge LV∈[2.0m,8.0m] (Fig. 1). The position LVof the vertical load

within this interval can be arbitrary. It is known that usual live loads of the crane vary between

1000kN and 2000kN. A special sensor prevents the crane operation if the vertical load is larger

than 3000kN.

The crane also causes brake loads when moving freight. Depending on the movement velocity

and type of braking, the brake force can be determined with regard to the vertical force as follows:

|FB|=αFVwith α∈[0.0001,0.001] .(9)

The brake force FBcan be directed both to left or right, while the vertical load FVis directed

always downwards.

The horizontal load FHresults from the wind and attacks the frame always at the fixed

position HH= 4.0m from left to right. During two separate measurement campaigns, each of

five months duration, extreme values of the wind load FHper month have been measured, see

Table 1. The dead load of the portal frame is ignored for simplicity.

7 Y. Petryna, M. Drieschner

Table 1: Measured extreme loads FH

Measurement 1 7.0kN 2.8kN 5.8kN 8.3kN 10.3kN

Measurement 2 10.1kN 4.6kN 12.4kN 8.2kN 15.7kN

Material It is known that the frame is built of steel S 355 [7] with the characteristic value of the

yield strength of 275MPa, that is the 5%-quantile value of the statistical distribution. The mean

value of the Young’s modulus Ecan be taken as 210000MPa. According to the Probabilistic

Model Code [12], log-normal distributions are recommended to be applied with a coefficient of

variation of 7% for the yield strength σyand with a coefficient of variation of 3% for the Young’s

modulus E.

4 Challenges

4.1 Challenge 1: Decision making

The engineering decision to be taken sounds: Is the operation of the frame under conditions

described above allowed or not? The participants are expected to give the answer "yes" or "no".

The answer "yes" is correct, if the failure rate remains smaller than 10−3, i.e. 1failure per 103

load operations, that is the requirement of the operator. The answer "no" is correct, if the failure

rate is larger than 10−3. Besides the answer "yes" or "no", participants are expected to explain

their classification of uncertainties, suitable approaches to handle them and the background of

the decision. An independent check of the decision taken by every participant can be done by

uncertain simulations of the frame with "true" structural and load parameters, which are known

to the authors. The true value of the parameters and the resulting failure rate will be declared

and explained.

4.2 Challenge 2: Decision making by data assimilation

The engineering decision to be taken sounds: Is the operation of the frame under conditions

described above allowed, if the measurement data available from the operation is taken into

account? The participants are expected to give the answer "yes" or "no". The answer "yes"

is correct, if the failure rate remains smaller than 10−3, i.e. 1failure per 103load operation.

The answer "no" is correct, if the failure rate is larger than 10−3. The focus of this challenge is

directed toward the data assimilation and its contribution to the uncertainty quantification. The

true material and cross-sectional parameters are expected to be identified and specified. Again,

an independent check of the decision will be done, and the true values of the parameters and the

resulting failure rate will be declared and explained.

Operation measurement data During operation, that means without occurrence of failure,

the response parameters given in Table 2 have been measured. There is a text file containing

N= 5000 data sets of the measured values V4,V8and LV. These data sets have been gener-

ated numerically, by calculating the frame with the true structural parameters and varying the

load parameters within the conditions defined above. The wind load is taken into account as

well. However, no information on load magnitudes is recorded. Several measurement errors and

measurement noise are already incorporated into the data sets.

Remarks to the measurement data Due to the stiffness relations in the frame, it can be

assumed that the whole girder experiences the same horizontal displacement. The vertical dis-

placement V8is measured at the measured load position LV, so both values belong together in

each measurement. All measurements have been performed by a laser distance meter, which pro-

vides a nominal accuracy of ±1mm. It is also known that some measurements could be disturbed

Preprint No. 2019-03, FG Statik und Dynamik, TU Berlin 8

Table 2: Measurements during operation

Measured value Denomination

horizontal displacement [m] V4

vertical displacement [m] V8

position of the vertical load [m] LV

by foreign objects breaking the laser rays during measurements. In this case, measurements con-

tain wrong values.

4.3 Challenge 3: Design under uncertainties

The final challenge is to design the frame in such a way that the failure rate becomes smaller than

10−4. The only design parameters are the cross-section heights of the two columns h1=h2=h5

and that of the girder h3=h4. All other parameters are the same as in challenge 1. The best

design is that one with the minimum weight of the frame and the target failure rate fulfilled. An

independent check of each design can be done individually by uncertain simulations of the frame

with "true" structural and load parameters and the design values of the cross-section height

provided by each participant.

5 Final remarks

Researches dealing with uncertainty quantification are invited to participate in this benchmark

study and submit their solutions to the corresponding author via e-mail

(yuriy.petryna@tu-berlin.de). Each solution will be registered and independently checked as

described above. A joint publication including comparison of all submitted results will be pre-

pared and discussed with the participants.

The information on the benchmark as well as all necessary data are available under [15].

Acknowledgements

The authors gratefully acknowledge the financial support of the German Research Foundation

(DFG) within the Priority Programme "Polymorphic uncertainty modelling for the numerical

design of structures – SPP 1886".

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