Reliability Assessment of Structural Concrete with Special Reference to

Shear Resistance

By

Kenneth Kwesi Mensah

PhD Research Proposal

Promoters: Dr. C. Barnardo-Viljoen & Prof. J.V. Retief

March 2012

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Table of Contents

1. INTRODUCTORY COMMENTS 1

2. INTRODUCTION 2

2.1 Concept of basis of design and codification 2

2.2 Uncertainties in structural design and structural reliability 2

2.2.1 Aleatoric and Epistemic uncertainties 4

2.2.2 Quality control 4

2.3 Research motivation and significance 5

2.4 Research objectives 6

2.5 Work done in M-thesis 8

2.6 Expected outcomes of the research (Hypothesis) 10

2.7 Research contribution 12

3. LITERATURE OVERVIEW OF THE STUDY 14

3.1 Background of the reliability basis for structural design 14

3.2 The Problem of shear and its reliability basis 18

3.3 EC 2’s variable strut inclination design method for shear 20

3.4 The MCFT 21

3.5 Reliability analysis of EC 2’s variable strut inclination method for shear 22

3.6 Parametric analyses 23

ii

4. RESEARCH METHODOLOGY 24

4.1 General Methodology 24

4.2 Methodology for reliability analysis and calibration of variable strut inclination

method 26

5. PROPOSED RESEARCH PROGRAM 28

REFERENCES 30

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1. INTRODUCTORY COMMENTS

This research represents a continuation and extension of reliability-based investigations

assessing the appropriate application of modern basis of design formats in deriving design

guidelines for structural concrete resistance. The study began at master’s level, from which a

masters dissertation was submitted in December 2011 and has been recommended by the

reviewers for an upgrade to a PhD at the University of Stellenbosch. For all purposes in this

proposal, the masters dissertation will be referred to as the M-thesis.

The objectives of the PhD research are an extension and continuation of those considered in

the M-thesis to further harmonise the application of reliability principles in deriving design

guidelines for structural concrete, particularly in South Africa. However, following

preliminary reliability analyses conducted in the M-thesis, the specific objective for this

research of calibrating the EC 2 design method for the shear of members requiring stirrups

arises. Assessments of the applications of the principles of structural reliability continue in

an attempt to identify ways in which the process can be advanced, not only for South Africa

but on an international platform as well.

In the M-thesis, the terms EC 0 and EC 2 were established as convenient short forms for the

Eurocode Basis of Structural Design Standard EN 1990 (EN 1990, 2002) and the Eurocode

Standard for the Design of Concrete Structures EN 1992-1-1 (EN 1992-1-1, 2004),

respectively. These conventions are maintained for use in this proposal.

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2. INTRODUCTION

2.1 Concept of basis of design and codification

The structural engineering fraternity has the social responsibility to ensure that all structures

designed and constructed are safe and, further, perform as expected during service. To be

able to achieve safe and durable structures, design and construction professionals require a

system of verifying adequate structural performance of buildings and structures by applying

rational and safe procedures through the different stages of a project: planning, design,

analysis, detailing, construction, and maintenance of structures. This is typically done

through a code of practice with a well-established design basis specifying procedures and

guidelines that enable assessments of structural performance and safety. Guidelines are

therein also given to achieve certain levels of performance through detailing rules and quality

management provisions for design and construction.

A design code, or design provisions in general, should represent sound and well-established

methods of engineering practice that have been thoroughly researched and validated by

relevant experience (Ellingwood, 1994). In deriving code provisions, they should be

calibrated extensively to validate their use across the field of application in practice.

Provisions should however not be too complex to use by design engineers in practice who do

not always have time to study innovative trends in research, and are often under time

demands of projects. A code is therefore a platform of disseminating efficient and current

methods of design and construction between research and practice.

2.2 Uncertainties in structural design and structural reliability

In general, problems of structural design must be resolved in the face of various uncertainties.

Uncertainties arise not only in the assessment of actions which the structure has to sustain,

and from the occasional lack of control during the production processes of the materials and

components required, but also from incomplete knowledge about the mechanical

formulations describing the response of the structure and its capacity to sustain those actions.

Structural reliability techniques, compared to other basis of design formats, are aimed at

rationally quantifying and assessing the effects of uncertainties associated with all aspects of

3

structural design. The uncertainties in the design and construction process are represented by

way of mathematical statistics and the assessment of structural performance is conducted

through probabilistic concepts and analyses. Such treatment of uncertainties gives a rational

scientific (decision tool) approach to the calibration of structural design provisions.

Modern and technologically advanced design codes adopt the Partial Factor Limit States

Design Method as their basis for design. This method applies partial factors, the vector ���, to increase action values as well as reduce material property and resistance values to

generate their design values for use in a limit state assessment. Characteristic values, the

vector ����, are also introduced into limit state functions where partial factors are applied to

make an economically but safe assessment of structural performance. The governing

condition of a limit state assessment is that the action effects should be less than the available

resistance. In this method, dimensions are generally implemented at nominal values, but in

some cases (second-order effects, geometrical imperfections, buckling) can assume design

values by applying some tolerance limit. This method can account for the variability of

materials by applying partial safety factors to the material properties. Further, it can also be

used for safety verification of cross-sections and members as well, since the action effects

and resistance force of cross-sections are calculated for use in the limit state verification.

Until recently, partial safety factors used in limit state design verifications were derived

mainly by expert judgement and by reference to sound traditional designs, thereby lacking the

appropriate rational and scientific treatment they require. Structural reliability techniques

arise as an attempt or method to represent variability and performance of physical models of

structural systems by taking account of the distributions of the basic variables in mechanical

formulations used for limit state verifications. Basic variables are the most fundamental

quantities the designer has to consider in mechanical formulations.

Structural reliability techniques are consistent with the Partial Factor Limit States Design

format in the sense that partial factors can be derived from reliability analyses and calibration

exercises and then applied in limit state verifications. The application of structural reliability

as the theoretical basis for limit states design ensures that improved economic performance is

achieved together with improved safety performance across a wide range of practical design

situations. The design provisions of the suite of structural Eurocodes are formulated on

reliability principles.

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2.2.1 Aleatoric and Epistemic uncertainties

Other uncertainties, apart from those associated with the prediction of action effects and

resistance, affect structural performance. Aleatoric uncertainties arise due to the natural or

inherent variability in a physical process which may never be determined with accuracy. It is

simply a random uncertainty we may have to deal with but try to control through efficient

design practice. Epistemic uncertainties are more systematic. They are due to one of two

reasons:

1. Either due to insufficient knowledge or lack of understanding that causes some aspect

to be constantly overlooked,

2. Or due to conservative assumptions and simplifications made, which are derived from

extensive research to make final design equations manageable and relatively quick to

manipulate for safe and efficient design practice.

Regardless of any of the sources of epistemic uncertainties bulleted above, they can be

quantified and subsequently calibrated against to build sufficient conservatism into design

procedures. Model uncertainty is an epistemic uncertainty.

2.2.2 Quality control

Structural failures are not only caused by the unfavourable uncertainties that affect a limit

state assessment. Gross errors are, in fact, found to be the major cause of structural failures.

Table 2.1 shows the origins and causes of structural failures.

Table 2.1. Origin and causes of structural failure (ISO workshop, 2011)

Origin Design Execution Use Others

20% 50% 15% 15%

Causes Gross errors Adv. cond.

80% 20%

Gross errors can be limited by quality control during design and construction, as well as

through routine maintenance. An important part of assuring reliability is to give guidelines

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on quality management that aim primarily to reduce gross errors in design and increase the

quality and integrity of constructions.

2.3 Research motivation and significance

The European Commission’s initiative to harmonise technical barriers between EU member

states to allow exchange of information and intensify trade relations has caused each member

states’ national structural design standards to be replaced by a unified set of Eurocodes. In

this transformation, The British Standard The Structural Use of Concrete BS 8110-1 (BS

8110-1, 1997) on which the currently operational South African standard The Structural Use

of Concrete SABS 0100-1 (SABS 0100-1, 2000) is based, is being withdrawn and replaced

by a new operational Eurocode Standard for Design of Concrete Structures EN 1992-1-1 (EN

1992-1-1, 2004). For the on-going revision of South Africa’s standard for the design of

concrete structures, which will be newly referred to as SANS 10100-1, the South African

Concrete Code Committee has chosen to adopt EN 1992-1-1 as reference.

The application of the principles of structural reliability to establish a standardised basis for

structural design using partial factor limit states design procedures is done in the European

Standard for the Basis of Structural Design EN 1990 from which it is adapted to the South

African Basis of Design Standard for Building and Industrial Structures SANS 10160-1

(SANS 10160-1, 2010). The basis of design requirements stipulated in EN 1990 and SANS

10160-1 apply to all aspects of structural design: This includes reliability levels of structural

performance and their differentiation and management; identification of various limit states

and design situations; the specification of all the basic variables; separate treatment of actions

and material-based resistance. However, application of these requirements is then primarily

focused on actions whilst the provision for structural concrete is then left to the materials

based design standards.

The Eurocodes can be viewed as a general set of reference standards which need to be made

operational as national standards through the selection of Nationally Determined Parameters

in National Annexes. A key parameter for which national choice is allowed and has grave

effect on matters concerning reliability is the selection of the target level of reliability, of

which the Eurocode recommends a value of � � 3.8 and South Africa uses a value of

� � 3.0. Retief and Dunaiski (2009) propose that the reliability assessment of a future South

6

African concrete standard could therefore consist firstly of reviewing the degree to which EN

1992 complies with and applies reliability principles as set out in EN 1990; and secondly to

calibrate it in accordance with SANS 10160-1 requirements, including required levels and

classes of reliability for the restricted scope of building structures.

2.4 Research objectives

The general objective of this research, advancing from the masters, is to systematically study

and trace the extent that EN 1990 and EN 1992-1-1 are harmonised in terms of the reliability

based framework. To continuously and systematically achieve such objective, the reliability

framework and its requirements are first identified in EN 1990 and other basis of design

documents such as the JCSS Probabilistic Model Code (JCSS, 2001) and the Draft 2010 fib

Model Code (fib, 2010). Thereafter, the nature and extent of the implementation of the

reliability framework for structural concrete resistance is traced by studying the provisions of

EN 1992-1-1 as well as relevant background documentation. The background documentation

concerning the reliability basis of EC 2’s design provisions may be found in documents such

as the Eurocode 2 Commentary and Worked Examples (European Concrete Platform, 2008)

and various fib and CEB-bulletins.

During the harmonisation process, it is imperative that an assessment is made of the

implications for South Africa where national choice is allowed. Further, where abstraction or

incompleteness in the implementation of the reliability framework is identified in structural

concrete provisions, improvements or suggested actions to harmonise design practice are

recommended. Such efforts were made in the M-thesis concerning some quality aspect of

reliability management. In the M-thesis, requirements of the framework were also exercised

through extensive assessment of the model factor and reliability performance of the

provisions for members requiring design shear reinforcement. Therefore, the specific

objectives of the PhD research can be outlined as:

1. To continually map out and study the reliability framework and requirements as

presented in EN 1990.

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2. To continually trace the extent to which the reliability framework is implemented in

deriving the EN 1992-1-1 design provisions by use of relevant references and

background documents.

3. To extend the reliability framework where abstraction or incompleteness is found in

provisions for structural concrete resistance.

There is lack of evidence that the variable strut inclination method for the shear

design of members requiring stirrups adopted in EC 2 is properly calibrated. A

preliminary reliability investigation for shear was conducted in the M-thesis. It was

motivated by the fact that the modelling factor associated with the shear prediction

model is excessively conservative, coupled with some very inconsistent behaviour at

varied amounts of shear reinforcement provided in design. The results of the

preliminary reliability investigation indicate that further characterisation and

subsequent calibration of EC 2’s shear design method is necessary. The primary

objective of the PhD research is to fully calibrate the variable strut inclination method

to both SANS and EC 0 reliability requirements.

The Partial factor modification scheme, particularly reduction, prescribed for use in

an EC 2 Annex, has not been harmonised with the defined differentiation scheme

warranting such action as set out in EC 0. In the M-thesis, a link was established

between the two and a reduction scheme has been proposed for use in materials codes

reflecting the requirements set out in EC 0. Further developments and examples of

how this framework could be applied in practice are warranted.

4. To present and publish the research innovations and findings at conferences and in

journals. This will serve to disseminate and impart advancements and motivate

similar such action for other modes of resistance and materials, particularly in South

Africa. Table 2.2 below shows a list of papers that are currently being conceptualised

and planned for submission as Journal publications. Sufficient research has thus far

been undertaken to publish most of the papers outlined in Table 2.2. These papers

are therein indicated to be based on the M-thesis, whilst the fourth paper will require

completion of the PhD study.

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Table 2.2. List of possible Journal publications

No. Title of paper Based on:

1 Review of Reliability Basis of Structural Design (RBoSD) and its

application in South Africa

M-thesis

2 Model uncertainties: Characterisation and implications for reliability

modelling

M-thesis

3 Reliability analysis EC 2's variable strut inclination design method for

shear of members requiring stirrups

M-thesis

4 Reliability calibration of EC 2's design method for members with

stirrups

PhD

2.5 Work done in M-thesis

The thesis studied the principles of reliability presented in EC 0 and more importantly,

reviewed their level of implementation in deriving EC 2’s guidelines. The investigation

identified that:

1. Model uncertainties are important and deserve proper treatment and characterisation

in reliability modeling

2. Annex A in EC 2 which allows for partial factor reduction given certain quality

requirements is consistent with the reliability differentiation framework in EC 0 that

allows and guides such reduction.

Action was taken in the thesis to:

1. First determine the model factor of the variable strut inclination method for shear to a

compiled database of 222 tests. The statistics of the model factor were then

determined to ready its use for reliability modeling. The model factor associated

with the Modified Compression Field Theory’s (MCFT) prediction of shear

resistance for members requiring design shear reinforcement was also determined by

comparison to a subset of 116 tests. The prediction quality of MCFT was not a

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primary objective, but was applied in independent reliability modelling of design

situations as step of validating previous obtained results.

2. To generate the common reliability model where the design method is converted for

use as the general probabilistic model (gpm). In reliability modeling, the gpm serves

as the true descriptor of shear resistance where any conservative bias incorporated for

use in design are omitted such as the use of partial factors, characteristic values and

some simplifications or modifications. Mean values of the basic variables are used in

gpms.

An effective tool of comprehensive calculation steps set to characterise the reliability of EC

2’s calibrated variable strut inclination design method were established. First, extensive

reliability modelling of EC 2’s design method for shear is considered paying attention to all

basic variables that contribute to shear reliability performance. Further analysis then

condensed the process to highlight the most dominant, hence important, basic variables

affecting reliability performance. Reliability models provide resourceful insights that aid

decision making and ultimately calibration. However, European models of basic variables

published in literature (JCSS, 2001; Holický, 2009) are mostly used in the thesis due to lack

of availability of similar models based on South African practice and standards of

workmanship and quality.

Due to time constraint, reliability modelling was conducted for only two important and

critical design situations. In one instance, a section representative of a design situation with

low amounts of shear reinforcement (Test Case 1) was investigated whilst, conversely, the

other contained a relatively high amount of shear reinforcement (Test Case 2). It was found

that:

1. In general, the reliability of the design situations considered was

acceptable/satisfactory, particularly according to SANS 10160-1 requirements

2. The reliability of test case 2 did not meet EC 0’s reliability requirements although it

did not fall alarmingly below the threshold value of � � 3.04

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3. In order to validate the initial reliability model, the more accurate and rational-

scientific MCFT was applied in reliability modelling. Using the program Response-

2000, the MF associated with the MCFT was determined by comparing its

predictions to a subset of 116 tests.

4. It was found that a more severe assessment of reliability results from the

conventional process of using the design prediction model as gpm as opposed to

using the more accurate MCFT.

2.6 Expected outcomes of the research (Hypothesis)

The expected outcomes of this research are discussed as bulleted arguments below.

1. More extensive reliability investigation for shear considering more design situations is

warranted for PhD study. For calibration purposes, more insight is required into shear

reliability performance over a wide range of design situations.

2. To fully calibrate the model for shear. At current, tools for analysis have been

developed and have merely been used to determine the reliability of two test cases in

the M-thesis. This is not nearly enough to consider the use of partial factors �� � 1.5

and �� � 1.15 as an economical set of factors used over a wide range of design

situations. Already for Eurocode requirements, the aforementioned partial factor set

fails to meet the basic requirement of safety; failing to achieve target reliability for

test case 2. Full calibration of partial factors for design resulting from extensive

parametric analyses are required, to find an economic and safe combination for EC 2

requirements and to optimise the combinations according to SANS requirements.

There is need for comprehensive parametric analyses that explore reliability trends

across a wide range of design situations and across various plausible partial factor

schemes. The partial factor scheme that leads to the most economic reliability

performance as long as minimum reliability requirements are satisfied across a wide

range of design situations should be an outcome of this study.

3. Once sufficient calibration has been achieved, and giving the results obtained,

judgement based approaches should be exercised in specifying how effective shear

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design using the variable strut inclination method should be applied in practice. The

results may comprise either of or any combination of the following:

a. Set of partial factors to be used

b. Range of design situations over which design method can be applied

effectively or limit of its use

c. Aspects of quality management and perhaps detailing can be considered over

general design scenarios as well as critical design situations, to specify

effective use of these measures to ensure that acceptable reliability is achieved

in practice. Also, poor or highly uncertain quality control expected in practice

could warrant slightly conservative partial factors to be prescribed for design

as compared to those deemed sufficient by calibration studies. This may be

the case when calibrating the design method according to SANS requirements,

considering the fact that at current reliability modelling has been based on

some European models of basic variables. European levels of quality control

in production and level of workmanship are generally perceived to be higher

or stricter than those in Africa. This gives rise to the next expected outcome

of PhD research.

4. To consider, in so far as is possible, how best SA models of basic variables can be

derived and used in actual reliability modelling. This would better reflect the

reliability performance of the variable strut inclination design method in an

assessment of South African design requirements, conditions and practice.

5. An important step, already partially achieved in the M-thesis, is to validate the

reliability models used in the thesis. This has been achieved through the use of the

MCFT in independent reliability analyses to check the validity of the results when the

design prediction model is converted for use as the gpm in reliability modelling.

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2.7 Research contribution

This research adds to existing studies that have dealt, in various ways, with the effective

application of reliability techniques and principles in deriving structural design guidelines.

However, the study of applying these principles to properly calibrate the EC 2 variable strut

inclination method for shear is not only unique but important as well. This design method is

used in practice to provide shear reinforcement for reinforced concrete members. Most

beams in practice are provided with shear reinforcement, save for those of minor structural

importance. Shear reinforcement is provided in design to avoid and limit cracks due to shear,

thereby avoiding sudden and brittle shear failures. More importantly, effective control and

avoidance of shear failures allows members to reach their full flexural capacity which, unlike

shear failure, provides enough warning about impending failure through excessive cracking

and deflections. Proper calibration of the variable strut inclination method is therefore

essential to achieve economic and sufficiently safe designs over a specified and suitable

range of design situations.

Comparison of the variable strut inclination prediction model to an experimental database of

lab tests indicates that the unbiased model is generally very conservative, coupled with some

unconservative predictions when relatively high amounts of shear reinforcement are provided

in design. The object of good design practice is to ensure that economic designs are

implemented (therefore not excessively conservative) and that they meet set safety

requirements, currently reflected internationally by � in modern calibration procedures. MS

Excel tools that enable the reliability analysis, based on the FORM method, for design

situations for shear according to the EC 2 design method were developed in the M-thesis.

During the PhD research, these tools will be used to conduct the full parametric investigation

to determine the combination of partial safety factors that best achieve economic and safe

performance for shear. To achieve such objective, plausible partial factor combinations will

be investigated across a wide range of design situations, particularly at varied amounts of

shear reinforcement, in which the code may be applied. The variable strut inclination

prediction model for shear will therefore be calibrated in this PhD research, in an attempt to

assess and characterise the reliability performance of members designed for shear against

Eurocode and South African requirements.

The use of the MCFT in the process of reliability validation also presents a unique and

thorough approach to the investigation of shear reliability performance, giving credibility to

13

obtained results. The explicit derivation of the statistical properties of the modelling factors

from a compiled database for the two shear prediction models applied in reliability modelling

presents a thorough treatment of the basic variables of concern and is also a unique research

contribution. Efficient representation of the South African quality requirements and practice

will be implemented, specifically in terms of promoting the use of South African models of

basic variables used in reliability modelling. This would imply that a thorough assessment of

the performance and applicability of the model for shear for South African design practice

will be obtained.

Annex A in EC 2 allows partial factor reduction based on the control of geometry of critical

sections and concrete strength. However, no clear guidance is given in EC 2 on how partial

factors can be reduced for a specified reliability class, whereas the principle for such action is

given in EC 0. A detailed assessment of how the reliability differentiation framework can be

used to effect partial factor reductions for specified reliability classes was performed in the

M-thesis. A more detailed assessment of how such procedures could be practically applied in

design, particularly for shear, should be demonstrated and would provide insight and

initiative in applying it efficiently in design; across all other materials and modes of

resistance. The key is to emphasise that quality control is essential to reliability management.

Prescribing reliability levels would, in effect, give the designer an incentive and opportunity

to demand certain quality requirements, motivated by his use of reduced partial factors in

design.

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3. LITERATURE OVERVIEW OF THE STUDY

3.1 Background of the reliability basis for structural design

This research is mainly centred on the effective application of the more advanced calibration

methods described in Annex C of EC 0: Basis for Partial Factor Design and Reliability

Analysis. Particular interest is taken in the application of these methods to assess the

reliability performance and calibrate the variable strut inclination method for members

requiring stirrups. The recommendations for management of structural reliability for

construction works in Annex B of EC 0 have been implemented in the M-thesis to

complement partial factor reduction. Reduction is allowed for a given reliability class

provided levels of standard design supervision and site inspection are increased. Applying

the differentiation format from EC 0 gives a formal approach, consistent with basis of design

requirements, that introduces the need and benefit of better quality at all phases and levels of

construction.

Annex C in EC 0 gives recommendations on code calibration methods for structural models

used in design, highlighting methods of partial factor calibration. Probabilistic methods that

incorporate levels of structural performance ��� in partial factor determination are outlined.

Figure 3.1 below depicts the different methods of partial factor calibration.

Figure 3.1. Overview of Reliability Methods (EN 1990, 2002)

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Annex C in EC 0 states that most partial factors and action combination factors proposed in

the currently available Eurocodes are derived through Method a; that is, on the basis of

judgement calibrated to a long experience of building tradition. This Research is aimed at the

effective application of Method c in calibrating the partial factors used in the variable strut

inclination method, according to the requirements of both EC 0 and SANS 10160-1. In the

First Order Reliability Method (FORM), � is the measure of structural performance whose

value can be adjusted in the process to reflect different levels of safety calibration of partial

factors. The probability of failure ��, is related to the reference level of reliability �, by:

�� � ���� [3.1]

Where Φ is the cumulative distribution function of the standardised Normal distribution.

Table 3.1 shows some numerical representation of the relation between � and ��.

Table 3.1. Relation between � and ��.

�� 10-1 10-2 10-3 10-4 10-5 10-6 10-7 � 1.28 2.32 3.09 3.72 4.27 4.75 5.20

The objective of calibration for concrete resistance is to determine a set of partial factors

��� , ��� for use in design that attain acceptable reliability levels through a number of

analytical situations that are representative of all practical scenarios that the code may

perceivably be applied in. Full probabilistic methods are possible but are rarely used in code

calibration due to the frequent lack of statistical data. The FORM method is a convenient

tool for calibration. It is a first order method because � is evaluated at a linearised plane on

the failure surface. The method is compatible with the use of the first (mean) and second

(variance) moments of basic variables. Full distribution statistics of basic variables are not

necessarily required. Where distributions are available, the first and second moments of

nonnormally distributed variables are expressed in the equivalent normal representation (Ang

& Tang, 1984). This transformation is illustrated in Figure 3.2.

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Figure 3.2. Schematic representation of the standard normal transformation process (Taken

from Dithinde, 2007)

The FORM method gives insight into which basic variables mostly affect reliability

performance, by determining each of their direction cosines or sensitivity factors. This makes

it an effective decision and calibration tool, particularly regarding critical design situations

where marginal reliability performance is prevalent.

Design values should be based on the values of the basic variables at the FORM design point,

which can be defined as the point on the failure surface closest to the average design point in

the space of normalised variables. EC 0 allows separate calibration of action and resistance

standards. This principle was and will continue to be used in the reliability analysis and

calibration of the EC 2 design method for shear. The separation is achieved by the use of

FORM sensitivity factors, �� and �� as shown in the following Equations:

��� � ��� � � ���� [3.2]

��! " !�� � ������ [3.3]

Where �� is negative for unfavourable actions and �� is positive for resistance or resistance

variables.

The design resistance !� is expressed in the following form:

17

!� � #$%&

!'(�,); +�, � #$%&

! -.)/0,1$2,1

; +�3 4 5 1 [3.4]

Where ��� is the partial factor covering uncertainty in the resistance model, plus geometric

deviations if these are not modelled explicitly. (�,) is the design value of material property 4.

.) is a conversion factor taking scale effects into account. �6,) is the partial factor taking

unfavourable deviations of material properties into account. +� refers to the design value of

geometrical quantities. Consistent with Equation [3.4] above, Figure 3.3 gives a schematic

diagram that shows the elements to be calibrated for use as operational partial factors.

Figure 3.3. Relation between individual partial factors (EN 1990, 2002)

Taerwe (1993) states that special calibration of the model uncertainty as part of the global

resistance factor is warranted for coefficients of variation of 20 % and above. Model

uncertainties should be taken into account. They are, however, usually treated nominally by

use of recommended models from literature or through subjective professional judgement.

Models found in literature are usually derived from and representative of European levels of

quality control and workmanship, thus incorporating further uncertainty due to lack of

knowledge of the influence of South African specific conditions on reliability performance.

South African models of basic variables should be made available and be more readily

18

applied in reliability modelling for an assessment against South African performance

requirements.

3.2 The Problem of shear and its reliability basis

Literature and published research (Cladera & Mari, 2007; Huber, 2005) have made it evident

that the variable strut inclination prediction model, in general or across board, yields a rather

conservative estimate of shear resistance. The model factor (9:), or the ratio of the

experimentally determined shear resistance to the predicted shear resistance, was used to

describe model uncertainty.

An independent investigation was carried out in the M-thesis, finding that the unbiased

prediction model has mean model factor (9:) ;<= � 1.65 when compared to a carefully

compiled experimental database of 222 tests, as well as a large scatter associated with this

result with a standard deviation ? � 0.51. Further, inconsistent predictions were realised

with varying amounts of shear reinforcement (@ABCDB EBF⁄ � provided in design as shown in

Figure 3.4 below. Figure 3.4 is taken from Chapter 7 in the M-thesis, where the unbiased

variable strut inclination method’s ability to predict true shear resistance was investigated.

Figure 3.4. Logarithmic regression trendline fit to the scatter plot of the EC 2 model factor

against the amount of shear reinforcement (taken from M-thesis)

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From Figure 3.4, it can be observed that the 9: can be as high as 2.5 at @ABCDB EBF⁄ H0.21 9�+, progressively decreasing logarithmically to values as low as 0.8 at

@ABCDB EBF⁄ H 2.6 9�+. Further, the 9: equals 1 at about 1.9 9�+, and progressively

falls below 1 with increasing amounts of shear reinforcement. Design situations with

relatively high amounts of shear reinforcement are clearly the more critical region of shear

performance as conservatism in the EC 2’s shear predictions reduces with increasing amounts

of shear reinforcement.

With these uncertainties in shear prediction, the question now is how to proceed to design for

shear as it clearly is a phenomenon affecting structural performance? The answer is twofold:

1. By the use of structural reliability techniques, to appropriately calibrate shear models

with their inherent uncertainties. This would build sufficient conservatism into the

procedures by the use of partial factors and characteristic values to ensure that safe

designs are achieved. Effort must be directed to achieve as uniform reliability

performance as possible across different design situations, to avoid excessively

conservative and therefore expensive designs at lower amounts of shear

reinforcement provided in design.

2. To apply rational scientific methods such as non-linear analyses through the use of

finite elements and use of the MCFT to improve on the model uncertainty inherent in

the model itself. In any case, as a requirement of design bases, the model would still

have to be calibrated to achieve sufficient conservatism that accounts for other

uncertainties.

Considering the fact that the EC 2 design method for shear for members with stirrups is

currently applied in most countries where the Eurocodes are currently operational, and will

most likely be used in South Africa’s revised SANS 10100-1, proper calibration of the

method is warranted. The European Concrete Platform (2008) presents some reliability

based verification of the shear procedures for members not requiring stirrups but with no

similar justification conducted for members subjected to shear that require design shear

reinforcement. Most beams in practice are designed promoting ductile failure as extensive

warning (cracks and deflections) is given before failure. Shear failures are brittle and failure

occurs suddenly. The provision of shear reinforcement is therefore an important situation

that is conducted to limit shear failures and allow members to reach their full flexural

20

capacity. Extensive reliability assessments that properly calibrate the partial factor

requirements for the EC 2 variable strut inclination design method are essential.

3.3 EC 2’s variable strut inclination design method for shear

In the variable strut inclination method all the shear force will be resisted by the provision of

stirrups with no direct contribution from the shear capacity of the concrete itself. Crushing of

the inclined concrete struts is checked to avoid situations where premature web crushing may

occur. In design situations where the web crushing strength is predicted to be lower than the

yield strength of the stirrups, the width of the section is normally increased to an extent that

the web crushing strength exceeds or at least equals the yield strength of the stirrups. The

performance function for the reliability analysis is, therefore, based on the steel contribution

of the stirrups provided during the design of a section or member. The shear resistance

provided by the stirrups is determined by:

J��,A � KLMA N CDB� cot R [3.5]

Where J��,A is the design resistance force provided by the stirrups, @AB is the cross-sectional

area of 2-legs of the links, F is the spacing of the links, z is the internal lever arm, CDB� is the

design yield strength of the links and R is the angle of inclination of the concrete struts.

The angle R increases with the magnitude of the maximum shear force on the beam and

hence the compression forces in the diagonal concrete members. EC 2 limits R to occur

between 21.8° �cot R � 2.5� and 45° �cot R � 1�. For most cases of predominately

uniformly distributed loading the angle R will be 21.8° but for heavy and concentrated loads

it can be higher in order to resist crushing of the concrete diagonal members (Mosley et al.,

2007). The limits placed on R, which affect the quality and performance of the model’s

predictions, are set from applying the plasticity theory to the truss model.

EC 2 provides an upper limit, J��,6TU, on design shear force that is limited by the ultimate

crushing strength of the diagonal concrete strut in the analogous truss, where its vertical

component is given by:

21

J��,6TU � �VBCV�EBNW# �cot R tan R�⁄ [3.6]

Where W is a concrete effectiveness factor and �VB is a coefficient due to prestress.

The minimum amount of shear reinforcement, ZB,6)[, is given in EC 2 as:

ZB,6)[ � �0.08\CV] � CD]⁄ [3.7]

An additional requirement for links, as set by EC 2, is that the stirrup spacing must not

exceed, in any direction, the lesser of 75 % of the effective member depth, ^, and 600 __.

3.4 The MCFT

The Modified Compression Field Theory (MCFT) has been adopted in this Research to use as

a validation reliability model that aims to check the results obtained by the conventional or

more routine method used in reliability analyses. The MCFT has an extended rational base

and has been shown in a wealth of literature and research to make better predictions of shear

resistance than most prediction methods available and in use today. The MCFT, unlike

conventional truss models, does not just consider equilibrium, but additionally treats

compatability as well as more general stress-strain relationships of the steel and concrete, all

of which are formulated in terms of average stresses and average strains. The angle of

inclination of the compressive struts, R, is determined by considering the cross-sectional

dimensions of a member and its deformations, caused by bending moments concomitant with

shear at the studied section, of the transverse reinforcement, the longitudinal reinforcement

and the diagonally stressed concrete (Cladera & Mari, 2007). With these methods alongside

equilibrium conditions, compatability conditions, and stress-strain relationships for both the

reinforcement and the diagonally cracked concrete, the load deformation response of a

member subjected to shear can be determined. The MCFT may be explained as a truss model

in which the shear strength is the sum of the steel and concrete contribution. As such, it

provides itself as a general model for the load-deformation behaviour of two-dimensional

cracked reinforced concrete subjected to shear.

In this Research, the MCFT is implemented by the use of Response-2000. Response-2000 is

a non-linear sectional analysis program for the analysis of reinforced concrete elements

subjected to shear according to the MCFT. The Program was developed at the University of

22

Toronto by Evan Bentz in 2000 and is available for free download at:

http://www.ecf.utoronto.ca/~bentz/r2k.htm

3.5 Reliability analysis of EC 2’s variable strut inclination method for shear

Consistent with the FORM method, the performance function for shear is described by:

`��� � Jab6��� � J��c&dL1ef���, �� [3.8]

Where Jab6��� represents the distribution of ‘true’ shear resistance, based on EC 2’s

variable strut inclination method for shear or the MCFT, determined using unbiased values of

the basic deterministic and random variables and neglecting the use of partial factors in the

resistance model. J��c&dL1ef���, �� is the single deterministic value of shear resistance as

would be determined for a practical design situation in accordance with the stipulations in EC

2. The vectors �� and � imply that the single deterministic value of shear resistance is

calculated using appropriate characteristic representative values of all the basic variables,

which are all treated deterministically when the code method is applied for design. Figure

3.5 shows schematically the probabilistic representation of the performance function.

Figure 3.5. Probabilistic representation of the performance function for shear

23

The FORM method is used to evaluate � for a given design situation. The goal is to use

FORM to conduct parametric analyses to assess partial factor requirements that achieve

acceptable reliability performance according to EC 0 and SANS 10160-1 requirements.

3.6 Parametric analyses

Parametric investigations should be conducted across a range of factors that are known to

affect shear resistance and its performance. Preliminary considerations from the M-thesis

indicate that these should be:

1. The concrete strength, CV

2. The size of the cross section, EB & ^

3. The amount of longitudinal tension reinforcement, Zg 4. Maximum moment to shear ratio divided by the effective depth �9/J^� or

alternatively the +/^ ratio

5. Amount of shear reinforcement, @ABCDB EB⁄ F

6. Model Factor 9:

The preliminary reliability analysis presented in the M-thesis focused on isolating the main

basic variables that affect shear reliability performance. The model factor was found to

dominate. Therefore, any control or judgement concerning shear should be applied to it’s

modelling ability, particularly at high amounts of shear reinforcement where its predictions

are known to be unconservative. Either, larger partial factors are used to achieve acceptable

reliability performance or a better model predicting shear should be used in such situations,

thereby limiting the use of the conventional method.

24

4. PROPOSED RESEARCH METHODOLOGY

4.1 General Methodology

The proposed research methodology is based on a continuation and extension of the approach

presented in the M-thesis. The approach taken to conduct the research is very much in line

with the research objectives. First, a general survey of reliability principles governing basis of

structural design are continually studied and reviewed. The main focus is, however, to

determine the extent that EC 0 reliability principles have been applied in deriving EC 2

provisions, particularly for shear resistance and in terms of regulated quality control. EC 0

presents mature reliability concepts that should be effectively applied in achieving the

guidelines for resistance. In order to achieve more innovative and unique application of

reliability techniques as basis of design, general and modern documents as the JCSS

Probabilistic Model Code (2001) and the Draft fib 2010 Model Code (fib, 2010) are reviewed.

It has been found that Annex B and Annex C from EC 0 are not fully implemented as and

where necessary in EC 2. Action is therefore stimulated to carry out reliability investigations

and complementary assessments. Various motivations stimulate the application of Annex C

to calibrate EC 2’s variable strut inclination method for members requiring stirrups.

Some contradiction exists between the reductions of partial factors allowed in EC 2 under

conditions of increased quality control, particularly of deviations of geometry of critical

sections as well as increased quality control of concrete production. The fib Model Code

seems to prefer that partial factors are not adjusted given stricter quality control of concrete

strength for a given reliability class whilst this is done in EC 2. An independent investigation

in the M-thesis justifies that partial factor reduction for resistance is feasible and applies the

reliability framework in EC 2 Annex B to partial factor modification in Annex A of EC 2.

The general methodology is summarised schematically in Figure 4.1.

25

3. Identified issues warranting treatment/ proper implementation

Figure 4.1. General research methodology

1. Map out Reliability Basis of Structural Concrete Design (RBoSD) – done in accordance with modern

international standards e.g. EC 0, JCSS PMC, CEB documents, CEB-FIP and fib Model Codes

2. Identify deficient application of reliability principles in Structural Concrete Provisions - done using EC 2 and

relevant background documentation (EC 2 Commentary & Worked Examples, Papers (Cladera & Mari), Published papers and research

applying reliability principles. Attention given to South African requirements and conditions

3a. Partial Factor reduction in EC 2 not synchronised with reliability differentiation

principles warranting such reduction. Principles given in Annex B of EC 0. Action taken to formalise this procedure in M-thesis.

Guidance given to European and South African requirements.

3b. Reliability Based Calibration in Annex C of EC 0 to be properly applied to

EC 2 shear design method for members with stirrups. Explicit representation of

modelling factor considered. Design method calibrated to both European and

South African requirements

26

4.2 Methodology for reliability analysis and calibration of variable strut inclination

method

The calibration of the variable strut inclination method for shear is central to this research.

The manner in which the model is calibrated is therefore important and should be in line with

the basic provisions given in EC 0 Annex C and other relevant documents where this issue is

dealt with. Figure 4.2 (shown overleaf) shows a detailed breakdown of reliability analysis,

up to the next step of parametric analysis, that was conducted in an effort to calibrate the EC

2 design method for members with stirrups. The extension of the parametric range of the

representative cases is an important element of the extended investigation. From the

parametric analysis, regions can be identified with sufficient reliability to excessive

conservatism; transitional conditions with marginal reliability; conditions of insufficient

reliability requiring modification in the design procedures.

27

Figure 4.2. Flowchart outlining procedure for reliability analysis and calibration of EC 2’s design method for shear

2. BASIC VARIABLES

1a. GENERAL PROBABILISTIC MODEL (GPM) FOR SHEAR, 2 cases:

1. EC 2 Strut inclination method converted to gpm

2. MCFT used as gpm – more rational case

1b. EC 2 DETERMINISTIC VALUE FOR SHEAR RESISTANCE, 3 analyses:

1. Design values 2. Characteristic values only 3. Partial Factor only (PFs applied to

mean values of basic variables)

2a. Allowance for Basic Variables to be randomly distributed:

1. MF and other basic random variables e.g. @AB , F, CDB etc.

2. Quantities can also be deterministic where applicable or when justified

2b. Deterministic design (characteristic) input variables applied with partial factors;

1. Vector (��,�) 2. Appropriate bias also expressed to

obtain mean values of basic variables

3. LIMIT STATE FUNCTION (LSF)

1. Analytical partial differentiation applied to LSF when EC 2 used as gpm 2. When MCFT is used as gpm, numerical differentiation applied using Response-2000 + explicit analytical

differentiation for MF only

4. REPRESENTATIVE CASES (Limited Parametric study)

@ABCDB EB⁄ F � 0.45 9�+

4a. CASE 1: Low shear rnft

@ABCDB EB⁄ F � 1.80 9�+

4b. CASE 1: High shear rnft

5. RELIABILITY ANALYSES

5a. Full prob. reliability model

All basic variables considered as random variables for representative cases 1 & 2

5b. Simplified reliability model

Only MF considered as basic random variable for representative cases 1 & 2

6. PARAMETRIC ANALYSES

1. Assessment of reliability given different design situations and partial factor schemes 2. Final critical judgements and decisions based on trends of parametric analyses 3. Formalised design rules for application of EC 2 shear design method in SA; assessment also done of

performance to Eurocode requirements and conditions

VaP check

VaP check

28

5. PROPOSED RESEARCH PROGRAM

A breakdown of the research program is outlined below.

FIRST SEMESTER 2012 (Period mid-March to early July)

1. Extend the reviewing of literature and planning of investigative techniques for parametric

analysis - towards full calibration

2. Compile present results on reliability performance of EC 2 and provisions for shear

resistance design as background material to the adoption of EC 2 as South African standard.

3. Publish journal papers based on findings from M-thesis

4. Planning of conference papers, submission of abstracts, followed by preparation of full

papers (based on acceptance)

5. Begin Parametric investigation. {Graphs, trends, design situations, analysis}

SECOND SEMESTER 2012 (Period early August to mid-December)

6. Continue Parametric analyses {bear in mind that this is done concurrently with results analysis

and validation of reliability modelling using MCFT}

7. Collection and review information or data of available models for basic variables produced

from surveys of local practice. Bayesian updating may be possible (additional process).

8. Incorporate South African variables into analyses for South African requirements

9. Complete analyses

10. Continued assessment of review of reliability basis of design and its possible applications.

The relevance of the results to the issue of punching shear is considered with the objective of

giving guidance.

FIRST SEMESTER 2013 (Period mid-January to end of July)

11. Consideration and critical appraisal, including judgement-based arguments, of the results

yielded by parametric analyses

12. Final recommendations on calibrated elements {partial factors, reliability performance

levels, design rules including NDPs for SA practice of all basic variables describing shear}

13. Final thesis write-up and compilation of research (mid-Feb to end of July)

29

14. Possible conference attendance (ACCTA 2013)

15. Submission of PhD dissertation

16. Continued assessment of review of reliability basis of design and its possible applications.

SECOND SEMESTER 2013 (Period mid-August to mid-Dec)

17. Publication of paper on the calibration process of the variable strut inclination method for

shear according to European and South African requirements.

18. Possible conference attendance (SEMC 2013)

19. Graduate Dec 2013

Table 5.1. Dates of some conferences for possible attendance

Conference/ symposium Deadlines Conference

date

City

Abstract Full Paper

The International Conference on

Advances in cement and concrete in

Africa, ACCTA 2013

31 Mar 2012

15 Aug 2012

28 - 30 Jan

2013

Johannes

burg

The Fifth International Conference

on Structural Engineering,

Mechanics and Computation, SEMC

2013

30 Sept 2012

01 Mar 2013

2 - 4 Sept

2013

Cape

Town

2013 fib symposium 2 Apr 2012 ???? 22-24 Apr

2013

Tel-Aviv

30

REFERENCES

Ang, A.H-S., & Tang, W.H. (1984). Probability Concepts in Engineering Planning and

design. Volume II - Decision, Risk and Reliability. Wiley, New York.

BS 8110-1. (1997). Structural use of concrete - Part 1: Code of practice for design and

construction. British Standards Institution, UK.

Cladera, A., & Mari, A.R. (2007). Shear strength in the new Eurocode 2: A step forward?.

Structural concrete, 8 (2), 57-66.

Dithinde, M. (2007). Characterisation of Model Uncertainties for Reliaility Based Design of

Pile Foundations. PhD Thesis, Department of Civil Engineering, 327 pp.

Ellingwood, B.R. (1994). Probability-based codified design: past accomplishments and

future challenges. Structural safety, 13, 159-176.

EN 1990. (2002). Eurocode – Basis of structural design. European Committee for

Standardisation (CEN), Brussels.

EN 1992-1-1. (2004). Eurocode 2: Design of concrete structures – Part 1-1: General rules

and rules for buildings. European Committee for Standardisation (CEN), Brussels.

European Concrete Platform. (2008). Eurocode 2 Commentary. Retrieved Dec 2008, from:

http://www.ding.unisannio.it/ricerca/gruppi/ingciv/ceroni/commentario_EC2_2004.pdf

European Concrete Platform. (2008). Eurocode 2 Worked Examples. Retrieved Dec 2008, from: http://www.europeanconcrete.eu/index.php?option=com_docman&task=doc_view&gid=28

fédération internationale du béton (fib). (2010). fib bulletin 56, Model Code 2010 – First

complete draft, Volume 1. fib, Lausanne, Switzerland.

fédération internationale du béton (fib). (2010). fib bulletin 56, Model Code 2010 – First

complete draft, Volume 2. fib,Lausanne, Switzerland.

Holický, M. (2009). Reliability analysis for structural design. ISBN 978-1-920338-11-4,

SUN MeDIA, Stellenbosch, South Africa.

31

Huber, U.A. (2005). Reliability of Reinforced Concrete Shear Resistance. Masters Thesis,

Department of Civil Engineering, University of Stellenbosch, 188 pp.

ISO Workshop. (2011). Harmonised Structural Standards Workshop held at the University

of Stellenbosch.

JCSS. (2001). Probabilistic Model Code, Part 1 to 4, Basis of design, Load and resistance

models, Examples. Joint Committee on Structural Safety. http://www.jcss.ethz.ch/

Mosley, B., Bungey, J., & Hulse, R. (2007). Reinforced Concrete Design to Eurocode 2 (6th

ed.). Palgrave Macmillan, New York, USA.

Retief, J.V., & Dunaiski P.E. (2009). The limit states basis of structural design for SANS

10160-1. In Retief, J.V., & Dunaiski, P.E. (Editors) Background to SANS 10160. SUN

MeDIA, ISBN 978-1920338-10-7.

SABS 0100-1. (2000). The structural use of concrete – Part 1: Design. South African

Bureau of Standards, Pretoria.

SANS 10160-1. (2010). Basis of structural design and actions for buildings and industrial

structures. Part 1: Basis of structural design. South African Bureau of Standards, Pretoria.

Taerwe, L. (1993). Towards a Consistent Treatment of model uncertainties in reliability

formats for concrete structures. CEB Bulletin no. 219: Safety and Performance Concepts.

Comité Euro-International du Béton (CEB), Switzerland.