,
STOCKHOLM SWEDEN 2020
Parametric optimization of
reinforced concrete slabs
subjected to punching shear
SOFIA THURESSON
KTH ROYAL INSTITUTE OF TECHNOLOGY
MASTER OF SCIENCE THESIS STOCKHOLM, SWEDEN 2020
Parametric optimization of reinforced
concrete slabs subjected to punching shear
S
OFIA
T
HURESSON
Master Thesis in Concrete Structures, May 2020 TRITA-ABE-MBT-20419
© Sofia Thuresson 2020
Royal Institute of Technology (KTH)
i
Abstract
The construction industry is currently developing and evolving towards more automated and optimized processes in the project design phase. One reason for this development is that computational power is becoming a more precise and accessible tool and its applications are multiplying daily. Complex structural engineering problems are typically time-consuming with large scale calculations, resulting in a limited number of evaluated solutions. Quality solutions are based on engineering experience, assumptions and previous knowledge of the subject.
The use of parametric design within a structural design problem is a way of coping with complex solutions. Its methodology strips down each problem to basic solvable parameters, allowing the structure to be controlled and recombined to achieve an optimal solution. This thesis introduces the concept of parametric design and optimization in structural engineering practice, explaining how the software application works and presenting a case study carried out to evaluate the result. In this thesis a parametric model was built using the Dynamo software to handle a design process involving a common structural engineering problem. The structural problem investigated is a reinforced concrete slab supported by a centre column that is exposed to punching shear failure. The results provided are used for comparisons and as indicators of whether a more effective and better design has been achieved. Such indicators included less materials and therefore less financial cost and/or fewer environmental impacts, while maintaining the structural strength. A parametric model allows the user to easily modify and adapt any type of structure modification, making it the perfect tool to apply to an optimization process.
The purpose of this thesis was to find a more effective way to solve a complex problem and to increase the number of solutions and evaluations of the problem compared to a more
conventional method. The focus was to develop a parametric model of a reinforced concrete slab subjected to punching shear, which would be able to implement optimization in terms of time spent on the project and therefore also the cost of the structure and environmental impact.
The result of this case study suggests a great potential for cost savings. The created parametric model proved in its current state to be a useful and helpful tool for the designer of reinforced concrete slab subjected to punching shear. The result showed several solutions that meet both the economical and the punching shear failure goals and which were optimized using the parametrical model. Many solutions were provided and evaluated beyond what could have been done in a project using a conventional method. For a structure of this type, a parametric strategy will help the engineer to achieve more optimal solutions.
iii
Sammanfattning
Just nu utvecklas Byggbranschen mot mer automatiserade och optimerade processer i projektdesignfasen. Denna utveckling beror till stor del på teknikutveckling i form av bättre datorprogram och tillgänglighet för dessa. Traditionellt sett löses komplexa
konstruktionsproblem med hjälp av tidskrävande och storskaliga beräkningar, vilka sedan resulterar i ett begränsat antal utvärderade lösningar. Kvalitets lösningar bygger då på teknisk erfarenhet, antaganden och tidigare kunskaper inom ämnet.
Användning av parametrisk design inom ett konstruktionsproblem är ett sätt att hantera komplexa lösningar. Dess metod avgränsar varje problem ner till ett antal lösbara parametrar, vilket gör att strukturen kan kontrolleras och rekombineras för att uppnå en optimal lösning. Denna avhandling introducerar begreppet parametrisk design och optimering i
konstruktionsteknik, den förklarar hur programvaran fungerar och presenterar en fallstudie som genomförts för att utvärdera resultatet. I denna avhandling byggdes en parametrisk modell med hjälp av programvaran Dynamo för att hantera en designprocess av ett vanligt konstruktionsproblem. Det strukturella problemet som undersökts är en armerad betongplatta som stöds av en mittpelare, utsatt för genomstansning. Resultaten används för att utvärdera om en bättre design med avseende på materialanvändning har uppnåtts. Minimering av materialanvändning anses vara en bra parameter att undersöka eftersom det ger lägre kostnader och/eller lägre miljöpåverkan, detta undersöks under förutsättning att
konstruktionens hållfasthet bibehålls. En parametrisk modell gör det möjligt för användaren att enkelt modifiera en konstruktionslösning med avseende på olika parametrar. Detta gör det till det perfekta verktyget att tillämpa en optimeringsprocess på.
Syftet med denna avhandling var att hitta ett mer effektivt sätt att lösa ett komplext problem och att multiplicera antalet lösningar och utvärderingar av problemet jämfört med en mer konventionell metod. Fokus var att utveckla en parametrisk modell av en armerad
betongplatta utsatt för genomstansning, som kommer att kunna genomföra optimering med avseende på tid som spenderas på projektet och därmed också kostnaden för konstruktionen och miljöpåverkan.
Resultatet av denna fallstudie tyder på att det finns en stor möjlighet till kostnadsbesparingar och anses därför vara ett mycket hjälpsamt verktyg för en konstruktör. Resultatet visade flera lösningar som uppfyllde de konstruktionsmässiga kraven samtidigt som de gav en lägre materialanvändning tack vare optimeringen. Många lösningar tillhandahölls och utvärderades utöver vad som kunde ha gjorts i ett projekt med en konventionell metod. En parametrisk strategi kommer att hjälpa ingenjören att optimera lösningen för en konstruktion av denna typ.
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Preface
This master thesis work was written for the Division of Concrete Structures at the Department of Civil and Architectural Engineering, KTH Royal Institute of Technology Stockholm Sweden. The thesis marks the end of the master program Civil and Architectural Engineering at the School of Architecture and the Built Environment, KTH.
This thesis was written at the structural engineering department Sweco Structure AB in Stockholm.
For the contribution to the work of this thesis, the author wishes to thank the following people:
Primarily, supervisor and examiner Adj. Prof. and Tech. Dr. Mikael Hallgren at the department of Civil and Architectural Engineering, KTH for his time and guidance. His invaluable expertise helped create this work.
Further, Model Manager and Construction Engineer, Tim Örun at Sweco Structure for his valuable time and guidance throughout this thesis.
And thanks to Sweco Structure AB for providing office space, laptops and requires software licenses.
Stockholm, June 2020
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Content
Abstract ... i Sammanfattning ... iii Preface ...v Symbols ... ix 1. Introduction ... 11.1 Optimization in civil engineering ... 1
1.2 Parametric design ... 2
1.3 Optimization- Concrete flat slab ... 3
1.4 Aim and objective ... 3
1.5 Scope and limitations ... 4
Theoretical background ... 5
2.1 Optimization Method ... 5
2.2 Objective functions, search space and design parameters ... 5
2.3 Algorithms... 6
Mathematical-based algorithms ... 7
Stochastic methods ... 7
2.4 Punching shear failure ... 8
2.5 Mechanical models of Punching shear failure ... 10
Kinnunen and Nylander ... 10
Hallgren ... 11
Muttoni ... 13
2.6 Eurocode 2 Section 6.4 ... 16
2.7 Shear Reinforcement ... 23
Shear reinforcement Eurocode 2 ... 23
2.8 Design for bending moment ... 28
2.9 Elasticity theory of concrete slabs ... 30
Strip Method... 32
Case study ... 34
3.1 Sollentuna Municipal building ... 34
Load conditions ... 35
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Method parametric design of the concrete columns ... 37
4.1 Dynamo as a parametric design approach ... 37
Revit ... 38
Generative Design for Revit and Dynamo ... 38
Python... 40 4.2 Parametric model ... 40 Geometric model ... 40 Visualization of Reinforcement ... 41 Parameters ... 42 Loads ... 44 Bending Moment ... 46
Anchoring and shortening ... 47
Volume and Price calculations ... 49
Punching shear failure ... 49
Optimization ... 53
Generative design ... 55
Evaluation of the program ... 56
5.1 Iteration ... 57
5.2 Visualization ... 57
5.3 Optimization ... 58
Minimization of reinforcement ... 59
Minimization of concrete ... 61
Minimization of reinforcement and concrete... 63
Minimization of reinforcement and concrete with shear reinforcement . 69 Discussion and conclusions ... 72
6.1 Visualization ... 72
6.2 Optimization ... 72
6.3 Case study comparison ... 72
6.4 Economic Evaluation ... 73
6.5 Conclusions ... 73
6.6 Future research studies ... 74
References ... 76
List of figures and tables ... 78
ix
Symbols
Latin upper-case letters
𝐴𝑐 Area of concrete [𝑚𝑚2]
𝐴𝑐𝑦 Area of concrete in y-direction [𝑚𝑚2]
𝐴𝑐𝑧 Area of concrete in z-direction [𝑚𝑚2]
𝐴𝑠𝑤 Area of one perimeter of shear reinforcement around the column [𝑚𝑚2]
𝐴𝑠𝑤,𝑚𝑖𝑛 Minimum required area [𝑚𝑚2]
𝐴𝑐𝑜𝑛𝑡 Basic control area [𝑚𝑚2]
𝐴𝑙𝑜𝑎𝑑 Loaded area [𝑚𝑚2]
𝐵 Column width [𝑚𝑚]
𝐶𝑅𝑑,𝑐 Parameter/Factor [−]
𝐷 Diameter of the critical column [𝑚𝑚]
𝐸𝑠 Modulus of elasticity of flexural reinforcement [𝐺𝑃𝑎]
𝐺𝐹∞ Fracture energy of concrete [𝑀𝑃𝑎]
𝑀 Bending moment [𝑁𝑚]
𝑀𝐸𝑑 Design value of the applied internal bending moment [𝑁𝑚]
𝑁 Number of shear reinforcement [−]
𝑁𝐸𝑑 Normal force [𝑁]
𝑁𝐸𝑑,𝑦 Normal forces in y-direction [𝑁]
𝑁𝐸𝑑,𝑧 Normal forces in z-direction [𝑁]
𝑃 Load [𝑁]
𝑇 Compression force [𝑁]
𝑉 Shear force [𝑁]
𝑉𝐸𝑑 Design value of the applied shear force [𝑁] 𝑉𝑓𝑙𝑒𝑥 Force associated to flexural capacity [𝑁]
𝑉𝑅𝑐 Punching shear resistance [𝑁]
𝑉𝑅𝑑,𝑐 Shear force resistance [𝑁]
𝑉𝑅𝑑,𝑚𝑎𝑥 Maximum punching shear resistance [𝑁]
𝑊1 Distribution of shear [𝑚𝑚]
Latin lower-case letters
𝑎𝐼 Distance supplement for moving the moment curve [𝑚𝑚] 𝑏𝑧 Length of the control perimeter in z-direction [𝑚𝑚] 𝑏𝑦 Length of the control perimeter in y-direction [𝑚𝑚]
𝑐 Diameter of a circular column [𝑚𝑚]
𝑐𝑑 Parameter/factor [𝑚𝑚]
𝑐1 Column length parallel to the eccentricity of the load [𝑚𝑚] 𝑐2 Column length perpendicular to the eccentricity of the load [𝑚𝑚]
𝑐𝑥1.. 𝑐𝑥4 Dividing lines x- direction [𝑚𝑚]
𝑐𝑦1.. 𝑐𝑦4 Dividing lines y- direction [𝑚𝑚]
𝑑 Depth [𝑚𝑚]
𝑑𝑎 Maximum aggregate size [𝑚𝑚]
𝑑𝑒𝑓𝑓 Effective depth of a concrete slab [𝑚𝑚]
𝑑g0 Length of the control perimeter [𝑚𝑚]
𝑑𝑦 Effective depth in y-direction [𝑚𝑚]
𝑑𝑧 Effective depth in z-direction [𝑚𝑚]
𝑒 Eccentricity [𝑚𝑚]
𝑒𝑦 Eccentricity along y [𝑚𝑚]
x
𝑒𝑝𝑎𝑟 Eccentricity parallel to the slab edge [𝑚𝑚]
𝑓𝑐 Cylinder compressive strength for concrete [𝑀𝑃𝑎] 𝑓𝑐𝑑 Design value of concrete compressive strength [𝑀𝑃𝑎] 𝑓𝑐𝑘 Characteristic compressive strength of concrete [𝑀𝑃𝑎]
𝑓𝑐𝑡 Tensile strength [𝑀𝑃𝑎]
𝑓𝑐𝑡𝑑 Design value of the ultimate bound stress [𝑀𝑃𝑎]
𝑓𝑦 Yielding strength [𝑀𝑃𝑎]
𝑓𝑦𝑑 Design yield strength of steel [𝑀𝑃𝑎]
𝑓𝑦𝑤𝑑 Tensile strength of the shear reinforcement [𝑀𝑃𝑎] 𝑓𝑦𝑤𝑑,𝑒𝑓 Effective design strength of the punching shear reinforcement [𝑀𝑃𝑎]
ℎ Effective depth [𝑚𝑚]
𝑘 Parameter/factor [−]
𝑘1 Parameter/factor [−]
𝑘𝑚𝑎𝑥 Parameter emendation [−]
𝑙𝑏𝑑 Anchorage length [𝑚𝑚]
𝑙𝑏,𝑚𝑖𝑛 Minimum anchorage length [𝑚𝑚]
𝑙𝑏,𝑟𝑞𝑑 Basic anchorage length [𝑚𝑚]
𝑚𝑓 Field moment [𝑁𝑚]
𝑚𝑠 Support moment [𝑁𝑚]
𝑚𝑥 Moment in x- direction [𝑁𝑚]
𝑚𝑦 Moment in y- direction [𝑁𝑚]
𝑚𝑥𝑦 Torsional moment [𝑁𝑚]
𝑞𝑑 Design area load [𝑁/𝑚2]
𝑟𝑐𝑜𝑛𝑡 Further control perimeter [𝑚𝑚]
𝑟𝑠 Distance between the column axis and the line of contra flexure bending moment
[𝑚𝑚]
𝑠𝑘 Characteristic snow load [𝑁/𝑚2]
𝑠𝑟 Radial spacing of the perimeter of shear reinforcement [𝑚𝑚] 𝑠𝑡 Average tangential spacing of perimeters of shear reinforcement
measured at the control perimeter
[𝑚𝑚]
𝑢0 Column perimeter [𝑚𝑚]
𝑢1 Basic control perimeter [𝑚𝑚]
𝑢1∗ Reduced control perimeter [𝑚𝑚]
𝑢𝑖 Length of control perimeter [𝑚𝑚]
𝑢𝑜𝑢𝑡 Control parameter outside shear reinforcement area [𝑚𝑚] 𝑢𝑜𝑢𝑡,𝑒𝑓 Effective length of the control perimeter outside the shear
reinforcement area.
[𝑚𝑚]
𝑣 Vertical load [𝑁]
𝑣𝐸𝑑 Maximum shear stress [𝑃𝑎]
𝑣𝑐𝑠 Design value of the punching shear resistance of slab with shear reinforcement
[𝑃𝑎]
𝑣𝑚𝑖𝑛 Minimum punching shear resistance [𝑃𝑎]
𝑣𝑅𝑑,𝑐 Design value of the punching shear resistance of slab without shear reinforcement
[𝑃𝑎] 𝑣𝑅𝑑,𝑐𝑠 Design value of the punching shear resistance of slab without shear
reinforcement
[𝑃𝑎] 𝑣𝑅𝑑,𝑚𝑎𝑥 Maximum punching shear resistance [𝑃𝑎]
𝑥 Depth of compression zone [𝑚𝑚]
xi
Greek lower-case letters
α Ratio of modulus of elasticity [−]
𝛼 Angle between the shear reinforcement and a beam axle [−] 𝛼 Ratio of the average stress to the maximum stress, 𝛼 = 0.80 [−] 𝛼1 Coefficient, for the effect of the form of the bars assuming adequate
cover
[−] 𝛼2 Coefficient, for the effect of concrete minimum cover [−] 𝛼3 Coefficient, for the effect of confinement by transverse
reinforcement
[−] 𝛼4 Coefficient, for the influence of one or more welded transverse bar [−] 𝛼5 Coefficient, for the effect of the pressure transverse to the plan of
splitting along the design anchorage length.
[−] 𝛽 Parameter accounting to the concentration of the shear forces [−] 𝛽 Position of the center of gravity of the compression, 𝛽 = 0.40 [−] 𝜂1 Coefficient related to the quality of the bond [−]
𝜂2 Coefficient related to rebar diameter [−]
𝜀𝑐 Concrete stain [−]
𝜀𝑐𝑅 Radial concrete strain [−]
𝜀𝑐𝑇 Tangential concrete strain [−]
𝜀𝑐𝑡𝑢 Ultimate tangential concrete strain [−]
𝜀𝑐𝑢 Crossing strain [−]
𝜀𝑦𝑑 Design yielding strain for steel [−]
𝜀𝑐𝑍𝑢 Ultimate compressive concrete strain [−]
𝜀𝑠 Strain in the reinforcement [−]
𝜌 Average ratio of flexural reinforcement [−]
𝜌𝑥 The bonded reinforcement ratio in x-direction [−] 𝜌𝑦 The bonded reinforcement ratio in y-direction [−]
𝜎𝑐𝑝 Normal concrete stresses [𝑀𝑃𝑎]
𝜎𝑐𝑇 Tangential concrete stress [𝑀𝑃𝑎]
𝜎𝑐,𝑦 Normal concrete stresses in y-direction [𝑀𝑃𝑎]
𝜎𝑐𝑧 Normal concrete stresses in z-direction [𝑀𝑃𝑎]
𝜎𝑥 Normal stresses in x-direction [𝑀𝑃𝑎]
𝜎𝑦 Normal stresses in y-direction [𝑀𝑃𝑎]
𝜎𝑧 Normal stresses in z-direction [𝑀𝑃𝑎]
𝛾𝑐 Partial factor for concrete [−]
𝛾𝑠 Partial factor for steel [−]
ψ Angle of rotation of the slab portion outside the shear crack [−]
Ψ Rotation of the slab [−]
∝ Angle between the shear reinforcement and the bending reinforcement
[−] Δ𝜙 Central angle between radial- and shear crack [−]
∆𝐹𝑡𝑑 Traction supplement [𝑁]
𝜃 Angle between concrete pressure and beam axle [−]
𝜏𝑥𝑧 Shear stresses in x- and z-direction [𝑃𝑎]
𝜏𝑥𝑦 Shear stresses in x- and y-direction [𝑃𝑎]
1
1. Introduction
The construction industry is currently developing and evolving towards more automated and optimized processes in the project design phase. One reason for this development is that computational power is becoming a more precise and accessible tool and its applications are multiplying daily. This evolution is taking place in conjunction with growing environmental awareness about the limitation of nature’s resources, which creates a significant selling point in a competitive market. This evolving situation results in new challenges for engineers who must direct their efforts to achieve optimal performance in terms of cost and environmental impacts. A more automated and optimized project design phase results in savings for both the construction firm and consumer.
Structural engineers strive to find the most practical and suitable solution for a given building problem. Engineers strive for an optimized solution that addresses the needs of the occupants of a building, the load bearing capacity, architectural aspects, and environmental impacts. Balancing these factors drives the solution to be a more complex one.
The use of parametric design within a structural design problem is a way of coping with complex solutions. Its methodology strips down each problem to basic solvable parameters, allowing the structure to be controlled and recombined so as to achieve an optimal solution. A flat slab subjected to punching shear is an example of such a complex structural design
problem where parametric design should be considered as the design method.
Traditionally, complex structural engineering problems are time- consuming with large scale calculations, resulting in a limited number of evaluated solutions. This master thesis utilises the application of parametric design to optimize a structural engineering problem, with the goal of shortening the design phase, thereby reducing project costs.
Optimization in civil engineering
Although there are many both effective and well-worked algorithms today, the design process still relies on structural engineering experience (Solat Yavari, 2017). The structural design process is currently based on a trial and error method. With this method, in practice, the engineer adopts an initial design and adjusts it if it does not fulfil the requirements of the design standard. This adjustment continues until the design meets all listed requirements. However, this method causes some problems, one being that it is very time-consuming and might not result in providing the optimal solution for the intended project. Solat Yavari (2017) states that although structural optimization can lead to major savings as well as shortening the design process, there is a gap between the practical applications in technology and theoretical applications of optimization. He states that this is particularly the case in the application of structural optimizations to complex structures. Improving this process is particularly worthwhile as significant savings can be achieved in cost and design time when structural optimization of complex structures is carried out. Though, the term optimization can be defined in various ways, Solat Yavari says that optimization is finding the best solution to a specific problem, while addressing all of the involved constraints.
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research and practical applications are due to the lack of satisfaction of the user´s needs. He lists requirements to strive for, including the following conditions:
1. The designer should be assisted in performing the optimization. 2. The computer program should be user friendly.
3. The design process should be faster.
4. Practical design problems should be addressed and produce practical results. 5. The results should be verifiable.
The first one of these requirements is the most important from a designer’s point of view given the designer’s responsibility for all aspects of design and need for control over the process. Templeman (1983) argues that a redefinition of “optimum design” required and he suggested “optimum design aid” instead. This new term, unlike the first one, reflects an intention to help the designer speed up the process and therefore have a better grasp of the role of the optimization tool. From a cost perspective, the increase in speed is a major advantage since a significant cost lies in the design phase of a project.
Parametric design
Parametric design is progressively becoming recognized as a powerful tool for engineers in all aspects of design. Solutions to complex problems that were previously difficult are now more accessible. With the use of parametric design complex problems can be simplified and navigable as models. Parametric design dissects the problem into solvable variables which when integrated create a full simplified model of the problem. By controlling the partial parameters and approaching the problem with algorithmic thinking, the designer can access the problem and find various alternative solutions. Armed with this information, the designer can choose the optimal solution for the specified problem (Park, et al., 2010).
For several years, architects have used parametric tools that provide a deeper understanding for the design and the given target. These tools have guided the user’s decisions and created new solutions. Schnabel et al. (2008) states that parametric design enables designers to deal with one problem at a time and a more complex problems can be divided into separate problems that may solved individually. Parametric design provides an undefined set of solutions and opportunities that give the designer obvious advantages for manufacturing processes.
3
As an open source plug in to Revit, Dynamo, a script interface, is currently under
development and acts as a parametric modeling engine. Kensek (2014) defines Dynamo as a visual scripting language and this is explained further in section 4.1.
Optimization- Concrete flat slab
A parametric model allows the user to easily modify and adapt any type of structure modification. This makes it the perfect tool to apply to an optimization process. It can be useful to reduce the time spent on the design and helping the engineer find the best solution. The solution of flat slabs directly supported by columns is often used as a construction solution in various buildings such as offices and car parks as well as bridges (Ansell, et al., 2014). These structures are usually made of reinforced concrete and are subjected to many parameters that can vary and affect the result of the construction and load bearing capacity. Some of these varying parameters are the thickness, amount of reinforcement, column cross-section and height, and concrete and reinforcement quality, etc. These flat slab structures have a common type of failure to be considered, punching shear failure. Punching shear failure in concrete occurs due to highly concentrated stresses at the connection between columns and the concrete slab which causes the columns to punch through the slab. This very brittle failure can cause severe damage.
Punching shear failure and its outcome depend on the calculations and the engineering assumptions that have been made. All structural engineers in Sweden (and most other European countries) must adhere to Eurocode 2 (see section 2.6) and its presented equations and design conditions.
A flat slab structure subject to punching shear failure is a good candidate for an optimization process given the varying parameters and the serious damage that results if failure occurs. Finding what the engineer believes to be the best solution is an iterative and time-consuming process with the conventional methods used today. Moreover, the process may not even result in identification of the optimized solution. Parametric models can provide solutions to the structure that the engineer would never have found using a conventional method.
Aim and objective
The focus of this thesis is to apply parametric design to a structural engineering problem and to develop a software application to automize and optimize the design process. The structural engineering problem investigated is this case study is Sollentuna Municipal Building. In this thesis the actual structure in the 1970s is compared with how it would have been designed today with modern technology. In this case, a better solution is expected using parametric design compared to a more conventional design. The parameters included in the optimization were a combination of column cross section, reinforcement, shear reinforcement, strength and deflection requirements.
4
conventional method; (2) to develop a parametric model of a reinforced concrete slab subjected to punching shear, which we will be able to implement optimization in terms of time spent on the project and therefore also the cost of the structure and environmental impact; and (3) to find a better way to simulate this complex problem and optimize the varying parameter and compare this model with a real problem, such as in a case study. The definition of a “more effective” and “better” design is vague and may vary depending on the user. Evaluating this term in a scientific and practical way is not easily done, especially in a certifiable way. Nevertheless, the results provided can be used for comparisons and as indicators of whether a more effective and better design has been achieved. Such indicators may be, for example, less materials and therefore less financial cost and/or fewer
environmental impacts, while maintaining the structural strength.
The research question is does applying a parametric approach in the early stage of the design result in several evaluated solutions that the designer can choose from and proceed in the direction which is most optimal and suitable for the given task? From this approach, a fully functional parametric model will be developed and used for future projects, serving as a powerful tool for construction technology development for more optimal solutions.
Scope and limitations
Although the tools for optimization of a project exist, it is not used in practice. Research must focus on helping to develop methods that can be used on real-world projects to close this gap that exist between theory and practice. The power and advantages of optimization is clearly showed, specifically in larger project and should be included in the daily work for the engineers and designers. Many projects for an engineer and designer meet this requirement and with parametric design the benefits of time spent on repetitive work can be reduced. This main body of this thesis consists of a theory part, optimization part and a result part. The theory section provides a description of the phenomenon of punching shear failure,
optimization methods and algorithm. After the theory part, a case study and an optimization are given a thorough parametric design approach. The case study considers the slab over a single column in view of punching shear capacity. The thesis body then consists of a results section. The thesis then ends with a discussion and conclusion section.
Many aspects of design are examined in this thesis, but some limitations have been made and are listed below.
This thesis only investigates flat slabs.
This thesis only considered EN 1992-1-1 (2004).
The case study only investigated one floor of the building and no complete analysis of the entire building was made.
The study did not investigate different concrete types, only cast in place concrete. The load combination only included an uniform distributed load and no point loads. No eccentricity was considered and no deeper investigation of the eccentricity factor β
were done.
5
Theoretical background
Research has been done on optimization for many years, providing different built-in
toolboxes in numerical software programs such as Grasshopper or Dynamo. This chapter will mainly present the basic theories of optimization and punching shear failure.
Optimization Method
Depending on the user, an optimization problem is about finding a set of design parameters. The user is either looking to the maximize or minimize output value due to phenomena of interest. One example is the total volume or weight of a structure. Woerli (2019) states that the space in which the design parameters evolve in is called search space and the output value is calculated throughout the usage of the function and called objective functions. He also states that the problem is usually subjected to constraints which limit the range of possible solutions.
Templeman (1983) states this problem, mathematically, in the following simplified model. 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓𝑖(𝑥), 𝑖 = 1,2, … , 𝑁 𝑁 ∈ ℕ
Subject to the constraints:
𝑔𝑗(𝑥) ≤ 0, 𝑗 = 1, … , 𝐽 𝐽 ∈ ℕ ℎ𝑘(𝑥) = 0, 𝑘 = 1, … , 𝑀 𝑀 ∈ ℕ Where 𝑓𝑖, 𝑔𝑗 and ℎ𝑘 are functions of the design vector 𝑥:
𝑥 = (𝑥1, 𝑥2, … , 𝑥𝑑) ∈ ℝ𝑑
Objective functions, search space and design parameters
Woerli (2019) states that objective function is a numerical representation of the aspect to be optimized by the designer. Depending on the situation, the applied design parameters of the function result in either a maximization or minimization of quantity.Optimization problems can be divided into several categories (Solat Yavari, 2017). These categories are as follows:
Direct or continuous optimization.
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Solat Yavari explains direct and continuous optimization, noting that design variables can only be discrete, like the number of diameters for reinforcement or the thickness of different parts, and can take any value. Optimization problems can be categorized as constrained or unconstrained, depending on the existence of limits on the design value. He states that constrained optimization can be explained from linear, nonlinear, etc. categories. Lastly, he explains, single-or objective optimization to refer to, as it sounds, only single or multi-objective problems depending on the number of quantities the problem of interest has to be optimized.
The typical structure optimization problem deals with minimization of a parameter such as, for instance, environmental impacts. Depending on the situation, choice of design parameters that will be investigated varies. Each of these parameters is limited to the search space and where their number defines the dimensions.
Woerli (2019) visualized this in a 3D graph and states that this is possible merely if two design variables belonging to the problem of interest and the objective function are known. Such a 3D plot is illustrated in Figure 2.1 where the objective functions are called
landscapes. Normally, as in this example, the objective function consists of several local extrema (minima or maxima) and only one global minimum and maximum. The search space aims to limit the possibilities of the design variables with respect to constraints of the
problem.
Figure 2.1 Search space and objective function, showing the extrema (Woerli, 2019).
Algorithms
Many methods solve optimization problems and choosing a suitable optimization algorithm is considered an important step for this. There are several guidelines for choosing the right optimization methods to generate the best result for a given problem. The first step,
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existing method in an optimization problem, algorithms present the same variation. Solat Yavari (2017) divides optimization algorithms into two major groups, mathematical-based and stochastic methods, each containing subgroups and some of these are described in the following chapter.
Mathematical-based algorithms
Traditional gradient-based methods are mathematical-based algorithms that are a fast methods to use. The methods involves the gradient of the objective function to determine its extrema and the gradient is used as a measurement of the variation of the objective function. Newton’s method is one example of a mathematical-based algorithm. Newton’s method uses Tyler’s series to linearize a function to be studied. However, this method requires studied functions that can be differential at least one or two times. The result depends on the input and even though this method is fast, it is easily trapped in a local extrema and therefore not that common anymore in optimization (Solat Yavari, 2017).
One other example of a mathematical-based method is Optimality Criterion Method. This method optimizes the features of the problem and is problem dependent. Stankovic et. al. (2015) states this method as an iterative process that applied on some physical characteristics of the problem. A disadvantage of this method is its lack of flexibility and the fact that it can only be developed for one problem at time.
Stochastic methods
Stochastic methods are usually non-deterministic and approximate, the gradient is not of usage, instead probabilistic calculations are used. Solat Yavari (2017) states that these methods as slow but more successful of finding the global minimum. Stochastic methods are inspired by nature and Genetic algorithms is an example of such a method.
Genetic algorithm
As early as the 1960s, Genetic Algorithm (GA) were first developed and many people since then have worked on the development of the algorithm. Today, GA can be used to solve complex problems due to it usage of values of objective function where no requirement of knowledge about the function or the mathematical gradient itself is necessary. Solat Yavari (2017) states that GA are evolving and based on genetic principles. Some of these principles are inheritance, selection, mutation and crossover. An advantage of GA is that it solves problems that are not yet mathematically defined. Being able to solve continuous and discrete variables, stochastic and non-smooth optimization problems are another advantage.
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process goes on and on until the stopping criteria are met. The stopping criteria in
optimization can be defined as the point at which the calculation is stopped, concluding the process of finding the optimum value. Solat Yavari (2017) explain that the overall goal for GA is to find the optimal solution with the given inputs to the function of interest. With these already explained generations, is the purpose to affect the functions so the output converges toward the stated expectation.
No dominated sorting genetic algorithm II (NSGA-II) is a famous multi-objective
optimization algorithm adopted for any optimization problem (Deb, et al., 2002). Population size and generations are working as input arguments for the functions. Multi-objective functions involve multi objectives and results in a trade-off between the resulting objectives rather than finding one single solution. This algorithm therefore differs from single objective functions where you only have one objective function to study that provides one single optimal solution. The author’s states that NSGA-II provides a better spread of solution and better convergence near the true Pareto-optimal front. The rise to a set of optimal solutions with multi objectives in a problem is known as the Pareto-optimal solution.
Punching shear failure
At the beginning of the 20th century, reinforced concrete slabs on columns were first
developed with a design that included large mushroom-shaped columns. Later, the flat slabs without capitals emerged and enabled a better usage for the room’s height and installations (Muttoni, 2008).
The solution of slabs directly supported by columns is often used as a construction solution in various buildings such as offices and car parks as well as bridges. These flat slab structures have one common failure type to be considered, namely punching shear failure. Punching shear failure in concrete happens due to highly concentrated stresses at the connection
between columns and the concrete slab which lead to the columns punching through the slab. This type of failure is a very brittle failure and occurs very suddenly and therefore is the cause of severe and dangerous consequences (Ansell, et al., 2014). A typical problem that causes this type of failure is the resistance forces of a column acting against a slab, as the shear force exceeds the shear resistance of the slab punches through the slab (see Figure 2.2). Hallgren (1996) states that the first signs are cracks that can be observed above the columns on the upper side of the slab surface. This appears when the concrete tensile strength is reached. The cracking starts with radial cracks extending outward from the column and then shortly
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Figure 2.2 Punching shear failure (Hallgren, 1996)
There are several different methods of increasing the punching shear capacity. In Figure 2.3 four different strategies are shown, including a column head that increases the width at the top of the column, a drop panel that locally increases the height of the slab, and two types of shear reinforcement around the failure zone (Ansell, et al., 2014). A car park in Wolverhampton is known to have one flat slab that partly collapsed due to punching shear failure. According to studies several factors were involved, including deterioration of the top floor slab. The
increased load on the columns when an adjacent column failed led to a progressive collapse of the car park slab, fortunately without injuries. Enough punching shear resistance is therefore necessary when designing flat slabs (Wood, 1997).
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Mechanical models of Punching shear failure
Different theories about punching shear failure have been developed through time as the knowledge of the subject and technology have developed. These theories generally do not agree on the mechanisms behind the failure mode, but most originate from what has been observed and measured in experiments. This thesis presents some of the different models and theories in the following section.
Kinnunen and Nylander
While performing a series of punching shear tests on circular concrete slabs without shear reinforcement, Kinnunen and Nylander (1960) were able to develop a mechanical model to explain the punching shear failure and calculate the ultimate limit state. This mechanical model is based on axisymmetric in a flat slab around a column. The slab is loaded with a distributed load along the circumference and the slab is symmetrically supported on a column. They observed from the test that the conical shear crack and the radial cracks formed
concrete segments that rotate like rigid bodies, looking like a piece of a cake (Figure 2.4). From these observations, the mechanical model was created. The model comprises,
geometrically, one of the slab segments bounded by the radial crack and the shear crack with the centre angle Δ𝜙 in-between.
Kinnunen and Nylander based their model on equilibrium and the segment is subjected to part of the external loads and to all the internal forces resulting in static equilibrium. The fraction of the total load P applied along the slab, denoted 𝑃∆𝜙
2𝜋, act upon the slab segment. The
resultant force from the force in the tangential reinforcement is denoted R1 and denoted R2 in the radial reinforcement. The centre of the rotation coincides with the neutral plane and underneath this plane, concrete is being exposed to compressive strains. In this compressive zone are the resultant to the tangential force denoted R4. The slab position outside the shear cracks is assumed in the model to be carried by a compressed conical shell, developed from the column to the root of the shear crack, with the resultant 𝑇Δ𝜑
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Figure 2.4 Mechanical model (Kinunnen, et al., 1960).
Kinnunen and Nylander assume that the segment is in equilibrium and base the model’s failure criterion on that failure occurs when tangential concrete strain in the compression side reaches a characteristic value. The tangential concrete strain is proportional to the critical value at the distance y from the column to the distance from the centre of the slab. Kinnunen and Nylander therefore formulate the tangential concrete strain as
𝜀𝑐𝑇 = 0,0035(1 − 0,22ℎ𝐵) (2.1)
where B is the width of the column and h is the effective depth.
Hallgren
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Kinnunen and Nylander in mind, the Hallgren (1996) model, is based on the equilibrium of the force acting upon a slab segment. In addition to what is mentioned in the previous section, Hallgren introduced a dowel force D in the reinforcement. The compressed conical shell that was presented in the model by Kinnunen and Nylander (1960) is now modified as a truncated wedge that carries the slab segment. Along the radius r of the slab, tangential stresses in the compressive zone vary as presented in Figure 2.5.
Figure 2.5 Variation of tangential stresses (Hallgren, 1996).
At the distance y from the column face, along the bottom outer border of the truncated wedge, Hallgren assumes that the concrete is in a bi-axial compression stat 𝜀𝑐𝑇 = 𝜀𝑐𝑅, see Figure 2.6. It is assumed that the concrete is in a three-axial compressive state, along the column
perimeter. The stated conclusion is that the macro-cracks start to develop at this level as a consequence of tensile stress acting perpendicular to the compressive stresses. It also shows that at this point, tensile strains and compressive strains are almost in equilibrium. This implies that the horizontal crack opens in the tangential compressive zone immediately before failure and the concrete strains can therefore be written as 𝜀𝑐𝑇 = −𝜀𝑐𝑍𝑢.
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Figure 2.6 Bi-axial and three-axial stresses (Hallgren, 1996).
Compared to Kinnunen and Nylander (1960), mentioned in section 2.5.1, Hallgren formulates the ultimate tangential strain as
ℰ𝑐𝑡𝑢 =3,6 ∗ 𝐺𝐹 ∞ 𝑥 ∗ 𝑓𝑐𝑡 (1 + 13 ∗ 𝑑𝑎 𝑥 ) −1/2 (2.2) where 𝑓𝑐𝑡 and 𝐺𝐹∞ are the tensile strength and the fracture energy of concrete, respectively. The maximum aggregate size expressed as 𝑑𝑎 and x is the depth of the compression zone. Hence, the failure criterion considers both the size effect and the brittleness of the concrete.
Muttoni
According to Muttoni, et al. (1991) the failure load is dependent on the crack pattern. The crack pattern in turn depends on the initial state of stresses and load history. An example of parameters that impact the position of the crack and the load bearing capacity is the
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Figure 2.7 Slab elements (Muttoni, et al., 1991)
Muttoni (2008) bases his failure criterion on the critical shear crack theory (CSCT). This criterion describes the relationship between punching shear strength of a slab and its rotation at failure. The previous rational models and design formula are based on the result of
experimental tests performed on thin slabs (slab diameter=0.1 to 0.2m). The few tests that were performed with thick slabs (slab diameter= 0.4 m) present a notable size effect. Therefore, there is a need for rational model correctly describing punching shear while accounting for this size effect. Muttoni continued with the work of Kinnunen and Nylander (1960) and accounted for the size effect that Kinnunen and Nylander (1960) did not consider. From tests with the punching shear strength as function of the amount of reinforcement and slab thickness, Muttoni and Schwartz (1991) first presented evidence that with increased slab thickness a considerable reduction in punching shear resistance may occur. Hence, the geometry of the size effect is of great importance.
Muttoni (2008) bases CSCT on the event where a diagonally flexural crack, a critical shear crack, arises and interferes with the shear transfer action for concrete members without transvers reinforcement. The punching shear and linear shear strength is dependent on the roughness and width of the crack. This critical shear crack disturbs the shear transfer action that leads to reduction of strength and will result in punching shear failure.
From the assumption, the critical shear crack width is proportional to the product of the rotation of the slab outside the shear crack, ψ, and the effective depth of the slab. Muttoni’s failure criterion for the punching shear load is formed. He determined the punching shear load by finding the intersection of the load-rotation curve and the failure criterion. This
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Figure 2.8 Punching shear resistance and rotation capacity (Muttoni, 2008). The tangential cracks and the radial curvature are concentrated in the vicinity of the column leading to a single failure criterion expressed in the following equation:
VRc b 0· d · √fc= 3 4 1 + 15 · ψ · d dg0 + dg (2.3)
The various components of the equation are defined as follows: VRc is the punching shear recistance, 𝑑g0 is the length of the control perimeter which is d/2 from the edge of the column, and d is the effective depth. The cylinder compressive strength for concrete is
expected as 𝑓𝑐, dg0 is a reference size equal to 16mm, dg is the maximum aggregate size, and the rotation of the slab is ψ.
For practical reasons, the load-rotation relationship has been simplified to the following equation: ψ = 1.5 ·𝑟𝑠 𝑑· 𝑓𝑦 𝐸𝑠 · ( 𝑉 𝑉𝑓𝑙𝑒𝑥 )32 (2.4)
In this case, 𝑟𝑠 is the distance between the column axis and the line of contra flexure bending moment and 𝑓𝑦 is the yielding strength of the flexural reinforcement. The modulus of
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Eurocode 2 Section 6.4
This section presents the current Eurocode 2 for punching shear resistance, obtained from section 6.4 in EN 1992-1-1 (2004).
The phenomena, punching shear failure, may occur when a concentrated load that affects a slab of a small area, so called load area, is present. A suitable model that verifies these phenomena, in its ultimate limit state, is presented in Figure 2.9.
Figure 2.9 Model verifying punching shear failure (EN 1992-1-1, 2004).
When a control for punching shear failure is implemented, the control is performed at the face of the column and at the basic control perimeter u1. When shear reinforcement is required, a future perimeter, uout,ef ,can be found where shear reinforcement no longer is required.
DESIGN CONDITIONS
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𝑣𝐸𝑑 ≤ 𝑣𝑅𝑑,𝑚𝑎𝑥 (2.5)
Shear reinforcement is not necessary if the following conditions are satisfied.
𝑣𝐸𝑑 ≤ 𝑣𝑅𝑑,𝑐 (2.6)
where 𝑣𝐸𝑑 is the maximum shear stress, 𝑣𝑅𝑑,𝑚𝑎𝑥 is the maximum punching shear resistance and 𝑣𝑅𝑑,𝑐 is the punching shear resistance of a slab without shear reinforcement. 𝑣𝑐𝑠 is the punching shear resistance of a slab with shear reinforcement. If the condition is not met and shear reinforcement is necessary, the flat slab should be designed in accordance with section 6.4.5 in EN 1992-2-2 (2005).
LOAD DISTRIBUTION AND BASIC CONTROL PERIMETER
Effective depth, 𝑑𝑒𝑓𝑓, of a slab is assumed constant and normally determined by the following equation:
𝑑𝑒𝑓𝑓 =
𝑑𝑦+ 𝑑𝑧
2 (2.7)
where 𝑑𝑦 and 𝑑𝑧 is the effective depth for the reinforcement in the respective y- and z-direction.
(1) Normally the basic control perimeter, 𝑢1, is placed at the distance of 2,0d from the loaded area and should be drawn in a way that minimizes its length, see Figure 2.10.
Figure 2.10 Typical illustration of the basic control perimeter (EN 1992-1-1, 2004).
(2) If the concentrated force is opposed by a higher pressure, or by the effect of a load or reaction within a distance 2d from the load area, the control perimeter should be considered at a shorter distance than 2d.
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Figure 2.11 Basic control perimeter near an opening (EN 1992-1-1, 2004).
For loads near or at edges or corners should the basic control perimeter be defined as shown in Figure 2.12. This is only of use when the control perimeter is smaller than the control perimeter derived from (1) and (2). For load areas less than d from a corner or edge, special reinforcement must be added to the slab, see section 9.3.1.4 in EN 1992-1-1 (2004).
Figure 2.12 Basic control perimeter for loaded areas located near corners and edges (EN 1992-1-1, 2004).
ECCENTRICITY
If the support reaction is eccentric with respect to the control perimeter, the maximum shear stress is
𝑣𝐸𝑑 = 𝛽 𝑉𝐸𝑑
𝑢𝑖∙ 𝑑 (2.8)
Where 𝑉𝐸𝑑 is the design shear force and 𝑑 are the effective depth. Parameter 𝛽 is obtained from
𝛽 = 1 + 𝑘𝑀𝐸𝑑 𝑉𝐸𝑑 ∙
𝑢1
𝑊1 (2.9)
19 𝛽 = 1 + 0.6 ∙ 𝜋 ∙ 𝑒
𝐷 + 4 ∙ 𝑑 (2.10)
where 𝑒 =𝑀𝐸𝑑
𝑉𝐸𝑑 which is the eccentricity of the load and 𝐷 is the diameter of the column.
Parameter 𝛽 is obtained from the approximated equation (2.11), when there is an inner rectangular column where the loading is eccentric in both axes
𝛽 = 1 + 1.8 ∙ √(𝑒𝑦 𝑏𝑧 ) 2 + (𝑒𝑧 𝑏𝑦 ) 2 (2.11)
where 𝑒𝑦 is the eccentricity along the y-axial, 𝑏𝑧 is the length of the control perimeter in z-direction, 𝑒𝑧 is the eccentricity along the z-axial and 𝑏𝑦 is the length of the control perimeter in y-direction.
A moment, parallel to the slab edge, is formed for edge column with eccentricity only towards the interior. The punching force is then considered to be uniformly distributed along the control perimeter u1*, showed in Figure 2.13.
Figure 2.13 Reduced control section with the length 𝑢1∗ for edge and corner columns (EN 1992-1-1, 2004).
Parameter 𝛽 derives from the equation: 𝛽 = 𝑢1
𝑢1∗+ 𝑘 𝑢1
𝑊1𝑒𝑝𝑎𝑟 (2.12)
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When the eccentricity is loading toward the interior of the slab for a corner column, and the punching force is assumed to be uniformly distributer along the reduced control parameter, are the parameter 𝛽 calculated as
𝛽 = 𝑢1 𝑢1∗
(2.13) If the lateral stability of the structure does not depend on frame action between the slab and the column, and where the length of the adjacent span does not differ more than 25%, approximate values for 𝛽 may be used according to Figure 2.14
Figure 2.14 Recommended values for 𝛽 (EN 1992-1-1, 2004). Values of k for rectangular loads areas are showed in Table 2.1
Table 2.1 Values of k for rectangular loaded areas.
Where and 𝑐1 and𝑐2 are dimensions seen in Figure 2.15
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SHEAR DISTRIBUTION Wi
The distribution of shear can be expressed as 𝑊𝑖 = ∫ |𝑒|𝑑𝑙
𝑢𝑖
0
(2.14)
The parameter 𝑑𝑙 is a length increment of the perimeter and 𝑒 is the distance of the 𝑑𝑙 from axial about which the moment 𝑀𝐸𝑑 acts.
The value of 𝑊1 depends on different cases. For a column with rectangular shape, is 𝑊1
𝑊1 = 𝑐1 2
2 + 𝑐1∙ 𝑐2+ 4 ∙ 𝑐2∙ 𝑑 + 16 ∙ 𝑑
2+ 2 ∙ 𝜋 ∙ 𝑑 ∙ 𝑐
1 (2.15)
For an edge column with rectangular shape, is 𝑊1 𝑊1 =
𝑐12
4 + 𝑐1∙ 𝑐2+ 4 ∙ 𝑐1∙ 𝑑 + 8 ∙ 𝑑
2+ 2 ∙ 𝜋 ∙ 𝑑 ∙ 𝑐
2 (2.16)
If the eccentricity is perpendicular to the slab edge towards the exterior, is the expression according to equation (2.9).
PUNCHING SHEAR RESISTANCE OF SLAB WITHOUT SHEAR REINFORCEMENT. The punching shear resistance is calculated as
𝑣𝑅𝑑,𝑐 = 𝐶𝑅𝑑,𝑐∙ 𝑘(100𝜌1 ∙ 𝑓𝑐𝑘)1⁄3+ 𝑘1∙ 𝜎𝑐𝑝≥ (𝑣𝑚𝑖𝑛+ 𝑘1∙ 𝜎𝑐𝑝) (2.17)
where 𝑘 = 1 + √200
𝑑 ≤ 2.0 and 𝑑 are expressed in mm.
𝜌𝑙 = √𝜌𝑙𝑦∙ 𝜌𝑙𝑧 ≤ 0.02 where 𝜌𝑙𝑦 and 𝜌𝑙𝑧 represent the bonded reinforcement ratio in the respective y- and z-direction. This parameter should be calculated as a mean over the slab width equal to the column width plus 3d on each side.
𝜎𝑐𝑝 =𝜎𝑐𝑦+𝜎𝑐𝑧
22 𝜎𝑐,𝑦=
𝑁𝐸𝑑,𝑦
𝐴𝑐𝑦 and 𝜎𝑐,𝑧 =
𝑁𝐸𝑑,𝑧
𝐴𝑐𝑧 where 𝑁𝐸𝑑,𝑦 and 𝑁𝐸𝑑,𝑧 are the longitudinal forces action over a
defined slab section for the internal columns and the control section for an edge column. This load derives from pre-stressing actions. Ac is the section area as defined by 𝑁𝐸𝑑.
𝐶𝑅𝑑,𝑐, 𝑣𝑚𝑖𝑛 and 𝑘1 can be found in the national annex, recommended values for 𝐶𝑅𝑑,𝑐 = 0.18 𝛾𝑐 , 𝛾𝑐 = 1.5, 𝑣𝑚𝑖𝑛 = 0.035 ∙ 𝑘 3 2 ⁄ ∙ 𝑓 𝑐𝑘 1 2 ⁄ and 𝑘1 = 0.1.
PUNCHING SHEAR RESISTANCE OF SLAB WITH SHEAR REINFORCEMENT If shear reinforcement is needed, punching shear resistance is now calculated as
𝑣𝑅𝑑,𝑐𝑠 = 0.75 ∙ 𝑣𝑅𝑑,𝑐+ 1.5 ∙ 𝑑
𝑠𝑟∙ 𝐴𝑠𝑤∙ 𝑓𝑦𝑤𝑑,𝑒𝑓∙ ( 1
𝑢𝑖 ∙ 𝑑) sin 𝛼 ≤ 𝑘𝑚𝑎𝑥∙ 𝑣𝑅𝑑,𝑐 (2.18) where 𝐴𝑠𝑤 is the section area of the shear reinforcement along one perimeter around the column. 𝑠𝑟 is the radial distance between perimeters of shear reinforcement and 𝛼 is the angle between the radial shear reinforcement and the plane of the slab. 𝑘𝑚𝑎𝑥 is an emendation in EN 1992-1-1 and set to 1.6 according to the national annex. 𝑓𝑦𝑤𝑑,𝑒𝑓 is the effective design strength of punching shear reinforcement that’s derived from 𝑓𝑦𝑤𝑑,𝑒𝑓 = 250 + 0.25 ∙ 𝑑 ≤ 𝑓𝑦𝑤𝑑 where 𝑓𝑦𝑤𝑑 is the tensile strength of the shear reinforcement.
The punching shear resistance should not outdo the design value of the maximum punching shear resistance:
𝑣𝐸𝑑 =𝛽 ∙ 𝑉𝐸𝑑
𝑢0∙ 𝑑 ≤ 𝑣𝑅𝑑,𝑚𝑎𝑥 (2.19)
where 𝑢0 varies depending on column.
For center column, 𝑢0= the perimeter of the cross section of the column [mm] For edge column, 𝑢0= 𝑐2+ 3 ∙ 𝑑 ≤ 𝑐2+ 2 ∙ 𝑐1 [mm]
For corner column, 𝑢0= 3 ∙ 𝑑 ≤ 𝑐1+ 𝑐2 [mm]
The control section outside which shear reinforcement is not required are the control perimeter, 𝑢𝑜𝑢𝑡, calculated as
𝑢𝑜𝑢𝑡 = 𝛽 𝑉𝐸𝑑
𝑣𝑅𝑑,𝑐𝑑 (2.20)
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Figure 2.16 A and B shows the parameter, 𝑢𝑜𝑢𝑡 (EN 1992-1-1, 2004).
Shear Reinforcement
As already mentioned in section 2.4 Punching shear failure, shear reinforcement is a way of increasing the punching shear capacity. There are different types of shear reinforcement, one of those investigated in this thesis is vertical shear reinforcement. One type of vertical shear reinforcement is double-headed studs. According to Brooks Forgings (2020) the typical diameter of the stud’s head can vary from 30-75 mm, the stud diameter from 10-25 and the length of the stud depends on the customers’ requirements. It has strength of 500 N/mm2 and meets the request specified in Eurocode 2. Vertical shear reinforcement, named HALFEN HDB is a type of shear reinforcement that is strengthening at the transition between the concrete and the bending reinforcement. HALFEN (2020) highlights the user friendly installation of the reinforcement and notes that it contains a higher strengthening capacity than the original shear reinforcement. Another type of shear reinforcement that can be used to increase the punching shear capacity in the structure is bend up/down-bars and stirrup
reinforcement. These types of reinforcement will not be examined further in this thesis.
Shear reinforcement Eurocod e 2
In section 9.4.3 Eurocode 2, EN 1992-1-1 (2004) a more detailed description about the punching shear reinforcement is presented. Placement of stirrups is presented in Figure 2.17
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As illustrated in Figure 2.17, the distance between the shear reinforcement should not extend 0.75d and if shear reinforcement is necessary, there should be at least two lines of stirrups. When bent up- and down-bars bars are used, a single row may be considered enough. The required area, 𝐴𝑠𝑤,𝑚𝑖𝑛 is determined based on the difference given in equation (2.21):
𝐴𝑠𝑤,𝑚𝑖𝑛 = (1.5 ∙ sin ∝ + cos ∝)/(𝑠𝑟∙ 𝑠𝑡) ≥ 0.08 ∙ √(𝑓𝑐𝑘)/𝑓𝑦𝑘 (2.21)
where ∝ is the angle between the shear reinforcement and the bending reinforcement. The distance between the radial direction and the leg are named 𝑠𝑟 and the distance between the tangential direction and the leg are named 𝑠𝑡. The distance between the support or the edge of the loaded area closest to the shear reinforcement, shall not exceed d/2. This distance needs to be defined by means of tensile reinforcement. Bent-down bars pass throughout the loaded area or within 0.25d from the same area may be used as shear reinforcement.
Figure 2.18 Visualization of placement for bent-down bars (EN 1992-1-1, 2004). Section 8.4.4, Eurocode 2, EN 1992-1-1 (2004) provides a further explanation regarding determination of anchorage length for reinforcement. The anchorage length, 𝑙𝑏𝑑 is determined as
𝑙𝑏𝑑= 𝛼1𝛼2𝛼3𝛼4𝛼5𝑙𝑏,𝑟𝑞𝑑 ≥ 𝑙𝑏,𝑚𝑖𝑛 (2.22)
where 𝛼1, 𝛼2, 𝛼3, 𝛼4 and 𝛼5 are coefficients determined from Figure 2.19. 𝛼1 is for the effect of the form of the bars assuming adequate cover 𝛼2 is for the effect of concrete minimum cover
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𝛼5 is for the effect of the pressure transverse to the plan of splitting along the design anchorage length
Given the product 𝛼2𝛼3 𝛼5 ≥ 0.7
𝑙𝑏,𝑟𝑞𝑑 = (𝜙/4)(𝜎𝑠𝑑/𝑓𝑏𝑑) (2.23)
where 𝜎𝑠𝑑 is the design stress of the bar at the position from where the anchorage is measures from and 𝑓𝑏𝑑 is the design value of the ultimate bound stress and calculated as 𝑓𝑏𝑑 =
2.25𝜂1𝜂2𝑓𝑐𝑡𝑑.
The value for 𝜂1is coffecient related to the quality of the bond and set equal to 1 when “good” conditions prevail. 𝜂2 is set to 1.0 when use of rebar diameter less than 32mmm occur. 𝑓𝑐𝑡𝑑 is the design value of concrete tensile strength.
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Figure 2.20 Value of K for beams and slabs (EN 1992-1-1, 2004).
Figure 2.21 Values of the dimension 𝑐𝑑in different cases (EN 1992-1-1, 2004). Section 8.5, Eurocode 2, EN 1992-1-1 (2004) provides an additional explanation regarding link and shear reinforcement anchorage. The anchorage links and shear reinforcement should normally be affected by bends and hocks, or welded transvers reinforcement. The anchorage must be in accordance with Figure 2.22.
Figure 2.22 Anchorage of shear reinforcement (EN 1992-1-1, 2004). Section 9.2.1.3 Eurocode 2, EN 1992-1-1 (2004), provides a supplementary explanation regarding shortening of longitudinal tensile reinforcement.
For structures with shear reinforcement shall a traction supplement, ∆𝐹𝑡𝑑 is calculated according to equation (2.24). If the structure contains no shear reinforcement the supplement can be estimated by moving the moment curve a distance 𝑎𝐼 = 𝑑.
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where 𝛼 is the angle between the shear reinforcement and a beam axle, perpendicular to the transverse force. 𝜃 is the angle between concrete pressure and the same beam axle. The value of cot𝜃 should be limited to, 1 ≤ cot𝜃 ≤ 2.5. Figure 2.23 illustrates a shortening of
longitudinal tensile reinforcement.
Figure 2.23 Illustration of shortening of longitudinal tensile reinforcement (EN 1992-1-1, 2004).
This “shift rule” may also be used as an alternative for members with shear reinforcement, where 𝑎𝐼 is calculated from
𝑎𝐼= 𝑧(cot 𝜃 − 𝑐𝑜𝑡𝛼) /2 (2.25)
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Figure 2.24 Notations for shear reinforcement (EN 1992-1-1, 2004).
Section 9.4.1 (3) Eurocode 2, EN 1992-1-1 (2004), highlights the importance of a lower edge reinforcement (≥ 2 rebar), in two perpendicular directions, that is inserted and passed through the centre of the inner columns.
Design for bending moment
The theory of bending moment presented in the following section derived from (Ansell, et al., 2014) textbook, Concrete structures.
After the concrete has cracked, tension reinforcement has the task of absorbing the tensile force of the concrete. The reinforcement in the compression zone has the task of strengthening compressed concrete by absorbing a certain amount of the compressive force. Reinforced concrete structure should therefore be equipped with both tensile and compressive
reinforcement and should be designed so that the serviceability of the structure is maintained even when cracks occur. Bending reinforcement is reinforcement whose task is to take up forces caused by bending moment.
When the load increases, the concrete passes through three different stages. The first one, Stage I, is characterized by an un-cracked concrete. Due to the low tensile strength that concrete can manage, this stage is only valid under small loads. The next stage is Stage II and that begins when the concrete in the tensile zone first begins to crack. Generally, this stage corresponded to normal reinforced structure under service conditions. The third stage, Stage III is the final stage. It describes the conditions immediately before a bending fracture occurs. An increasing load results in bending cracks that propagate until the ultimate limit load is reached.
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Normally reinforced (under reinforced) cross- section:
The yield point, 𝑓𝑦𝑑 is reached in the tension reinforcement before the crossing strain 𝜀𝑐𝑢 is reached in the concrete.
The strain in the reinforcement 𝜀𝑠 reaches the value 𝜀𝑦𝑑 = 𝑓𝑦𝑑/𝐸𝑠, where 𝐸𝑠 is the modulus of elasticity of steel. The concrete stain 𝜀𝑐 is less than the ultimate strain 𝜀𝑐𝑢.
Calculation of the height, x of the compression zone can be done from the following equations.
𝜀𝑠 = 1 −𝑥 𝑑 𝑥 𝑑 ∙ 𝜀𝑐𝑢 ≥ 𝑓𝑦𝑑 𝐸𝑠 (2.26) 𝑥 𝑑 = 𝑓𝑦𝑑 ∙ 𝐴𝑠 𝛼 ∙ 𝑏 ∙ 𝑑 ∙ 𝑓𝑐𝑑 (2.27)
If condition (2.26) is met, it can be concluded that the cross-section is normally reinforced. The ultimate limit stage moment can therefore be calculated:
𝑀 = 𝛼𝑓𝑐𝑑𝑏𝑑2𝑥 𝑑(1 −
𝛽𝑥
𝑑 ) (2.28)
where 𝛼 is the ratio of the average stress to the maximum stress and set to 𝛼 = 0.80 and 𝛽 indicates the position of the center of gravity of the compression zone and 𝛽 = 0.40 according to BBK 94 (1994), b is the thickness of the structure and d is the effective depth and 𝐴𝑠 is the crosss-area of the reinforcement. In future similar work, these values should be selected from Eurocode 2 to be completely consistent throughout the study.
Over- reinforced cross- section
In these cases, the reinforcement remains elastic resulting in 𝜀𝑠 < 𝜀𝑦𝑑. The calculation of height x is in this case is therefore determined from the following equation:
𝜀𝑠 = 1 − 𝑥 𝑑 𝑥 𝑑 ∙ 𝜀𝑐𝑢 ≤ 𝑓𝑦𝑑 𝐸𝑠 (2.29)
If the reinforcement is larger than the reinforcement balance, it is possible to draw the conclusion that the cross-section is over-reinforced. The height of the compressive zone, x should then be calculated from:
𝐸𝑠𝜀𝑐𝑢𝐴𝑠 1 − 𝑥 𝑑 𝑥 𝑑 − 𝛼𝑓𝑦𝑑𝑏𝑑𝑥 𝑑 = 0 (2.30)
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Elasticity theory of concrete slabs
The theory regarding elasticity of concrete slabs presented in the following section is taken from (Nilsson, et al., 2012) the book, Concrete Slabs: Theory and Design Methods.
The thickness of the plate and the shortest side of the plate should not be greater than about 1/5. In this context plate and slabs are both synonymous and the later will be used here. When no external loads are acting on the surface area, and a normal pressure is present on the slab, 𝜎𝑧, 𝜏𝑥𝑧, 𝜏𝑦𝑧 will have a small impact. The shear stresses 𝜏𝑥𝑧, 𝜏𝑦𝑧 are small compared to the normal stresses 𝜎𝑥, 𝜎𝑦 and the shear stress 𝜏𝑥𝑦 which create the flexural and torsional moments in the slab. Figure 2.25 shows an illustration of the varying stress in slab. The following constitutive relations are valid, given by Hooke´s law (see equation (2.31)).
{ 𝜎𝑥 = 𝐸 1 − 𝑣2(𝜀𝑥+𝑣𝜀𝑦) 𝜎𝑦 = 𝐸 1 − 𝑣2(𝜀𝑦+𝑣𝜀𝑥) 𝜏𝑥𝑦 = 𝐸 2(1 + 𝑣)𝛾𝑥𝑦 (2.31)
Figure 2.25 Stress variation (Nilsson, et al., 2012)
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Figure 2.26 Bending of torsion moment. (Nilsson, et al., 2012)
By using Mohr´s circle, the principal moment can be found by examining where the torsion moment is equal to zero. This is presented in Figure 2.27 together with equation (2.32) and (2.33).
Figure 2.27 Principal moment (Nilsson, et al., 2012)
𝑚1.2 = 1 2(𝑚𝑥+ 𝑚𝑦) ± √ 1 4(𝑚𝑥− 𝑚𝑦) 2 + 𝑚𝑥𝑦2 (2.32) 𝜙 =1 2arctan ( 2𝑚𝑥𝑦 𝑚𝑥− 𝑚𝑦) (2.33)
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Figure 2.28 Shear force in a slab. (Nilsson, et al., 2012) Equilibrium conditions of the slab can be expressed by the following equation:
𝑞 = − (𝜕 2𝑚 𝑥 𝜕𝑥2 + 𝜕2𝑚 𝑦 𝜕𝑦2 + 2 𝜕2𝑚 𝑥𝑦 𝜕𝑥𝜕𝑦) (2.34) Strip Method
Nilsson, et al. (2012) stated that the first time the strip method was presented was in 1956 by Arne Hillerborg. The basic principle for strip method is that the torsional moment is chosen to be zero with respect to the directions of the reinforcement. This results in a reduced
equilibrium equation from the already presented equation (2.34). 𝑞 = − (𝜕 2𝑚 𝑥 𝜕𝑥2 + 𝜕2𝑚𝑦 𝜕𝑦2 ) (2.35)
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Case study
Sollentuna Municipal building
Figure 3.1 Sollentuna Municipal building (Boberg, et al., 2019)
The Municipal building located in Sollentuna, Sweden, needs renovation. The municipal building was built in the middle of the 1970s and no major renovation has been done since then. The municipal building contains seventeen floors and every floor has an area of approximately 19x65 meters. The framework is made of cast concrete columns and floors. The columns at the front of the facade are placed with a distance of 3,6 m and the two centre rows columns are placed with a distance of double that, 7,2 m. Stairwells are located at both ends of the building and two ventilation shafts are centrally located in the building. The different floors are more or less identical to each other (Boberg, et al., 2019).
