ENGINEERS AUSTRALIA PRACTICE NOTE
FEA in the design of reinforced concrete concr ete buildings buildings How to make the best use of FEA packages and avoid potential pitfalls
TIM MESSER
“FEA in the design of reinforced concrete buildings”
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Author: Tim Messer Messer BEng CPEng RPEQ RPEQ NPER MIEAust,
[email protected] [email protected] With contributions from Mal Wilson, Director Advanced Structural Designs Reviewed by Dr John Mullard, Associate and Newcastle office manager at Lindsay & Dynan Consulting Engineers
Acknowledgement: Gil Brock – Australia’s Concrete Structures Code Committee BD2 and owner/developer of RAPT Software
Editor: Dr Dietrich Georg
Copyright 2014 © Engineers Australia. Endorsed by the Structural College of Engineers Australia All rights reserved reserved Published by Engineers Media Pty Ltd, Crows Nest, Sydney, www.engineersmedia.com.au, www.engineersmedia.com.au, on behalf of Engineers Australia Cataloguing-in-Publication entry is available from the national Library of Australia at http://catalogue.nla.gov.au/ ISBN 9781-922107-27-5
The material contained in this practice note is in the nature of general comment only and is not advice on any particular matter. matter. No one should act on the basis of anything anything contained in this note without taking appropriate appropriate professional advice upon the particular circumstances. circumstances. The publisher and the author do not not accept responsibility for the consequences of any action taken or omitted to be taken by any person on the basis of anything contained in or omitted from this note. Engineers Australia
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Author: Tim Messer Messer BEng CPEng RPEQ RPEQ NPER MIEAust,
[email protected] [email protected] With contributions from Mal Wilson, Director Advanced Structural Designs Reviewed by Dr John Mullard, Associate and Newcastle office manager at Lindsay & Dynan Consulting Engineers
Acknowledgement: Gil Brock – Australia’s Concrete Structures Code Committee BD2 and owner/developer of RAPT Software
Editor: Dr Dietrich Georg
Copyright 2014 © Engineers Australia. Endorsed by the Structural College of Engineers Australia All rights reserved reserved Published by Engineers Media Pty Ltd, Crows Nest, Sydney, www.engineersmedia.com.au, www.engineersmedia.com.au, on behalf of Engineers Australia Cataloguing-in-Publication entry is available from the national Library of Australia at http://catalogue.nla.gov.au/ ISBN 9781-922107-27-5
The material contained in this practice note is in the nature of general comment only and is not advice on any particular matter. matter. No one should act on the basis of anything anything contained in this note without taking appropriate appropriate professional advice upon the particular circumstances. circumstances. The publisher and the author do not not accept responsibility for the consequences of any action taken or omitted to be taken by any person on the basis of anything contained in or omitted from this note. Engineers Australia
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CONTENTS Summary
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1 Introduction
5
1.1
Scope of this practice note
5
1.2.
The aim of structural modelling
5
1.3
Preliminary design
6
1.4
Common myths about advanced analysis software
6
2 Types of FEA software
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2.1
3D analysis
6
2.2
2D analysis
6
2.3
Other programs
7
3 Modelling inputs
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3.1
Flexural tensile strength (modulus of rupture)
7
3.2
Modulus of elasticity
8
3.3
Poisson’s Ratio
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4 Long-term deection, AS3600-2009
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4.1
Creep
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4.2
Shrinkage
9
4.3
Volume change/support interaction
9
4.4
Temperature Temper ature
12
4.5
Cracking
13
4.6
Long-term deections
14
5 Modelling decisions
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5.1
Element type
15
5.2
Size
15
5.3
Meshing
16
5.4
Discontinuity areas (D-regions)
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5.5
Shape
17
5.6
Boundary conditions
17
5.7
Modelling elements
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5.8
Supports
18
5.9
Column stifness
19
5.10 Non-structural items
19
5.11 Walls
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5.12 Beams
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5.13 Foundations
22
5.14 Consideratin for interrupted supports and openings
23
5.15 Redistribution
23
5.16 Buckling
23
5.17 Loading
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5.18 Construction
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5.19 Loading sequence
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5.20 Changes in cross section
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5.21 Composite construction using concrete elements
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6 Ultimate limit state design
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6.1
Design moment distribution (not redistribution)
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6.2
Twisting moments
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6.3
Classical beam theory
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6.4
Torsion
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6.5
P-Delta
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6.6
Shear
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6.7
Vertical Ver tical load take down
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6.8
Interpreting results
31
6.9
Rationalisation
32
6.10 Additional reinforcing
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7 Serviceability limit state design
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7.1
Deection
33
7.2
Precamber
34
7.3
Vibration
34
8 Design
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8.1
New programs
36
8.2
Recommended reading
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9 Forensic engineering
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9.1
Load sequencing
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9.2
Anchorage of wall reinforcement
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9.3
Backspin stiness
41
9.4
Construction loading
41
9.5
Moments in steel support columns
41
9.6
Torsional stiness of the band beam
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9.7
Shrinkage restraint
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9.8
Trusting what can be observed
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10 Sensibility checks
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10.1 Comparison to known limits
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10.2 Heuristics (rules based on experience and intuition)
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10.3 Sensitivity analysis
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11 Validation
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12 Closing comments
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Appendix
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Case Study 1: Restraint eect on carpark structure
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Case Study 2: Deections of a concrete oor
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Case Study 3: Royal Palm Hotel, Guam
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Case Study 4: Sleipner oshore oil platform, North Sea
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Case Study 5: Koro–Babeldaob Bridge, Palau
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SUMMARY Concrete design packages with pre/post-processors based on finite element analysis (FEA) have become a popular method of analysing concrete slab structures for practising engineers. engineers. There are some interesting issues that surface in the use of these packages that could catch the uninitiated off-guard. This practice note seeks to present some common issues and potential pitfalls that arise in the modelling of concrete building structures using these packages, especially for new users not familiar with these design packages. It provides a general overview of the topics in finite element analysis. Readers are advised to seek further information in regard to their specific applicatio applications ns and circumstances. Keywords: Keywor ds: Concrete, finite element, modelling, design, computers, 3-dimensional, creep, shrinkage, Poisson’ Poisson’ss ratio,, mesh, twisting moments. ratio
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INTRODUCTION
1.1
Scope of this practice note
Advanced concrete design packages based on finite element analysis (FEA) are popular among structural engineers. Inexperienced engineers are drawn to FEA programs as they give them the feeling of freedom to design almost anything an architect can envisage envisage,, from complex floors to unusual loadings, without relying on experience. However, if the dependence on these FEA packages is such that the engineer cannot carry out simpler methods of analysis, his or her ability to perform a self-regulating check of their model is compromised. This also creates a potential problem for the checking engineer (senior engineer) as it is almost impossible to check that a complex model you have not generated yourself is correct. There are few sources of practical advice on how to model and analyse using FEA programs. This guide seeks to highlight some of the topics engineers must be aware of when utilising such programs. However, this should not be considered an in-depth resource and further reading in this field is recommended. The advantage of FEA is the ability to model complex issues such as transfer slabs, large large openings, irregular column layouts and unusual loading conditions, and to easily update calculations and adjust the structure if changes occur. For example, for circular slabs with column supports around the outside and one column in the centre, the equivalent slab frame method can be used but the design will be conservative without some adjustments. FEA can handle this type of arrangement effectively without the extra adjustments. The disadvantage of FEA in commercial finite element programs is that they require a steep learning curve and that checking the outcomes is difficult. This practice note discusses FEA design and analysis packages with an automatic pre/post processor, which require fewer fundamental modelling decisions than the general FEA packages such as Strand 7, where engineers must generate all inputs. Recently graduated engineers are normally not f ully educated in the analysis of concrete, hence errors can occur especially with modelling assumptions. Finite element design requires a “feel” for and experience with concrete behaviour. behaviour. Therefore users should not treat the software as a black box with all the answers and should seek to understand what assumptions are made by the software in all stages of the computations computations.. Most structural problems can be broken down into different classes: 1. Static analysis (linear/nonlinear stress analysis); 2. Normal modes (resonant frequencies and mode shapes); 3. Buckling beha behaviour viour (buckling coefficients and mode shapes); 4. Frequency response; 5. Random response; 6. Transient response (linear/nonlinear stress analysis). This practice note only discusses static analysis in finite element applications. applications. 1.2
The aim of structural modelling
The aim of structural modelling is to create a model that is acceptable for practical purposes. purposes. It is important to keep in mind that we analyse a representative model of the structure, not the structure itself. The behaviour of the model may or may not be close to the behaviour of the structure. To create a meaningful model an engineer must appreciate the behaviour of the components that make up Engineers Australia
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the structure and be able to transform this behaviour into an analysis model. This requires an understanding of forces and moments, plate membrane, beam and column behaviour, load transfer, construction sequencing, deformations, cracking, yield, buckling, actual and design loads and many other aspects. It then requires decisions on how to model different aspects of behaviour, given the capabilities of the available analysis methods/computer programs. 1.3
Preliminary design
For the preliminary design of simple regular structures it is recommended that heuristics and experience be relied upon until a working design is developed. A good reference is the Structural Engineer’s Pocket Book by Fiona Corb. The heuristics used for preliminary design can be helpful in evaluating designs by others. For complex or designs involving new material such as fibre composite, finite element modelling may be used to help evaluate the preliminary designs. In these cases it is recommended that the user consider more general software such as Strand 7 or similar so that any heuristics built into the FEA model are minimised. 1.4
Common myths about advanced analysis software •
Finite element analysis returns lower bending moments or deflections – This is only true if the previous
techniques were conservative. Studies have shown that the results from FEA compared to traditional techniques give similar results; • Deflections will be more accurate – Previous experience indicates the best estimate of deflection is in the range of +15% to -30% using any technique, thus FEA is only as accurate as its assumptions. (Using the multipliers such as K cs, as defined in AS3600-2009, in the FEA instead of modified stiffness methods for long-term deflection will cause the analysis of deflections to become speculative rather than calculation based); • FEA computer programs save time – This is only true if/when the in-depth checking of the results is omitted. Hand calculations can be used to check the models and overall a time saving maybe made; • Using software will give accurate results – No software is “error” free. Most programs have only limited accuracy. For example, rounding errors and modelling assumptions will have an effect on the results; • FEA will provide “correct” design results – According to Elms 1985, “all models are wrong, some are useful”. FEA should be treated as a calculation with limited accuracy as t o the ability to represent a concrete structure in a model, as the model is based on many assumptions and should not be used as the only basis of design.
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TYPES OF FEA SOFTWARE
It is prudent to take the time to understand any design software before using it. This is especially the case with FEA software. There are many different types of FEA software, from 3D whole-frame to 2D programs for each floor. The common situation for a program with a complete design component is 3D analysis used to do the load take down/lateral analysis with floors exported to a separate 2D package for reinforcement and deflection design. 2.1
3D analysis
Normally this is a linear based analysis package with global adjustments in stiffness made to columns and floors to correct the model (see Figure 1). The concrete is treated as an elastic material and an assumption is made that concrete can transfer the forces as nominated in the model. This is fine for ultimate limit state if P-Delta effects can be ignored, but for service limit state the same does not hold, hence the 2D package requirement. 2.2
2D analysis
Normally this is a nonlinear analysis package (see Figure 2). It enables the software to predict cracked concrete properties within a set accuracy. To achieve this accuracy the software needs to be able to do the processing interactively. Generally this is based on Branson or Bischoff modifications to I ef (effective second moment of area), as software with the ability to take yielding of the reinforcement into account directly is considered a rarity and generally reserved for scholarly type applications.
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Figure 1: 3D model of a building.
2.3
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Figure 2: A plan view of a slab modelled in 2D.
Other programs
There are programs similar to RAPT (reinforced and post tension analysis program) which are based on FEA but don’t create complete slab models. They should be interrogated to the same extent, but due to their simpler nature this is less difficult.
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MODELLING INPUTS
The following discussion in this practice note assumes the use of a 2D analysis type software package, which enables the prediction of cracked concrete properties. “Garbage in equals garbage out”. Selecting appropriate inputs for modelling is crucial for success. Reinforced concrete is a material made up of reinforcing steel, aggregates, water, cementitious material (some unhydrated), admixtures and voids. Reinforced concrete has some unique features that distinguish its behaviour from other materials. At flexural failure, concrete slabs develop hinge lines, which mobilise the tension reinforcement passing through the section to resist the moment along its length. The total amount and orientation of the reinforcement along a design section governs the collapse load. Once a structure has cracked, the reinforcement determines the fashion in which the applied loads are resisted. It is the orientation and the amount of reinforcement that govern the path that the load takes to the supports. These features rely on the ability of the structure, once past the elastic limit, to redistribute forces. Adequate ductility becomes a prerequisite; generally the reinforcement requirements contained in building codes safeguard this ductility. By varying the constituents of the concrete, varied results for its structural behaviour are obtained. For example, Young’s modulus depends highly on the aggregate selected and the quantities. The properties of concrete can be externally modified as well, such as by weather, age of loadings, workmanship and curing conditions. The main concrete codes allow concrete to be modelled as an elastic isotropic material, but there are a number of assumptions that are made to enable this. These assumptions will be discussed throughout this practice note. 3.1
Flexural tensile strength (modulus of rupture)
The flexural tensile strength of the concrete is important as the concrete cracks once the tensile strength of the concrete is exceeded in the extreme fibre. In AS3600-2009 the tensile strength (modulus of rupture) is taken as 0.6√ f’c MPa (f’c is the characteristic compressive cylinder 28 day strength). The standard is silent on lightweight concrete. Engineers Australia
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“FEA in the design of reinforced concrete buildings” Table 1: Flexural tensile strengths used in some of the international codes. Code
Flexural tensile strength (MPa)
AS3600-2009
0.6√f’c
ACI
0.62√f ’c
IS456
0.7√f ck (f ck = Characteristic compressive cube strength)
CSA A23
0.6λ√f ’c λ = 1.00 for normal density concrete λ = 0.85 for semi low density concrete in which all of the ne aggregate is natural sand λ = 0.75 for semi low density concrete in which none of the ne aggregate is natural sand
Eurocode 2
0.21fck ^2/3
SABA 0100
0.5√f’c
The author recommends referring to the Canadian concrete code (CSA A23) for reference of lightweight concrete values or requesting testing from manufacture. The tensile strength has an influence on the deflection of slabs and shallow beams through tension stiffening. The value reported by various investigators for the exural tensile strength varies from 0.33 √f’c MPa to 1.0√f’c MPa. Table 1 shows the exural tensile strength in some of the international codes. The deformation values can vary by up to 40% using different expressions of exural tensile strength. The tensile strength is highly variable, therefore higher and lower bounds should be considered when evaluating critical deections.
Factors affecting modulus of elasticity of concrete
Moisture state of the concrete at loading
Elastic modulus of cement paste
Volume fraction of cement paste and porosity
Elastic modulus of the aggregate
Volume fraction of the aggregate
Figure 3: Factors aecting modulus of elasticity of concrete.
3.2
Modulus of Elasticity (Young’s Modulus)
Concrete is a composite inhomogeneous material with non-linear behaviour. Most codes allow it to be modelled as a linear isotropic material with limitations imposed. The value of elastic modulus can vary markedly depending on aggregate type, workmanship, time and curing condition to name a few (see Figure 3). Researchers have established several empirical equations for predicting the elastic modulus of concrete. AS3600-2009 gives an equation based on the mean compressive strength. It should be noted that the code points out that the E j (Young’s modulus) can vary by 20% under good conditions. This should be taken into account when assessing the deflections. A sensitivity analysis varying the time dependent parameters is recommended. 3.3
Poisson’s Ratio
Normally taken as 0 to 0.2, these values ensure the compressive stresses are overestimated, which is acceptable for concrete models and important for cracked sections. Conversely, in the primary reinforcement areas a minimum of 20% of primary reinforcement should be provided in the transverse direction to account for errors relating to the Poisson Ratio and transverse strength requirements.
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References:
• • • • •
AS5100.5-2004, Australian Standard for Bridge Design, Part 5: Concrete, Standards Australia 2004. AS3600-2009, Australian Standard for Concrete Structures, Standards Australia 2004. Lamonf, J.F. & Pielert, J.H. 2006, “Significance of tests and properties of concrete and concrete making materials”, ASTM (the US testing and materials society) International Standard Worldwide, p 169. Cobb, F. 2008, Structural Engineer’s Pocket Book, 2nd Edition. Webster, R. & Brooker, O. “How to design concrete structures using the Eurocode”, The Concrete Centre.
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LONGTERM DEFLECTIO N, AS36002009
4.1
Creep
Creep is a phenomenon whereby the compressive strain in the concrete increases over time under constant compressive stress. All building materials experience creep (plastic flow) strains. When added to the elastic strains this can increase deflection for concrete spanning members by a factor of 2 to 7. The quantity depends on many factors. Age at and duration of loading, environment, and proportioning of materials are some of the main factors. To accurately predict the creep, deflections would require a large amount of effort with regards to testing etc. The methods available to predict an upper bound deflection including the creep are the ageadjusted effective modulus method (AEMM) and Eurocode 2. These two methods would be appropriate under the AS3600-2009 code. Creep shortening is important in vertical members, especially if different materials are used for the vertical elements such as a steel truss core with concrete columns, shear walls with concrete columns or if columns have different stress levels; this results in differential deflections. This creates extra stresses in the column slab connections and, if the building is not symmetrical, will cause sway deflections (this is under vertical loading and is a permanent condition). Creep should be considered for any other permanent loading conditions, such as water, earth and equipment loads. These loads can be either vertical or horizontal; for the horizontal loading, careful consideration needs to be given to these effects to ensure the building doesn’t become unstable over time. Rule of thumb for tall buildings: A good way to mitigate for moderate differential vertical creep is to ensure t he entire vertical concrete elements have the same average stress under long-term loads with the same concrete properties. 4.2
Shrinkage
The reinforcement restraint induced curvature should be included in the calculations for deflection, whereas the supports restraint effects will be discussed further under volume change (see below). Shrinkage curvature depends on the water/cement ratio, relative humidity and the size and shape of the member. The effect of shrinkage in an asymmetrically reinforced section is to induce a curvature that can lead to significant deflection in shallow members. Gilbert and others have proposed curvature equations based on the reinforcement in the slabs/beams. This effect should be considered in the deflection calculations; it is included in the AEMM through the equation below as suggested by Gilbert et al. f cs is the maximum shrinkage induced tensile stress on the uncracked section at the extreme fibre at which cracking occurs and may be taken as
f cs
=
1.5 p 1 + 50 p
E s cs
where p is the reinforcement ratio ( A st /bd ), ε cs the nal design shrinkage strain and E s the Young modulus of steel. 4.3
Volume change/support interaction
Volume change due to thermal loads, shrinkage and creep causes forces and strains to build up in restrained concrete members; these actions should not be ignored. These strains can cause tensile stress in beams and slabs and shear/moments in columns (see Figure 4). Since the volume changes take place over a period of time, the effect of shortening on shear and moments is reduced due to creep and micro-cracking effects. This causes the estimation of restraint forces to be problematic at best, with assumptions for connections, footings etc playing a major role. For instance, if you assume fixed foundation supports, the forces will be overestimated. Conversely, if you assume pin foundations, the forces will be underestimated. Thus slab restraint Engineers Australia
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Figure 4: Forces in structures due to volume change.
cracking is the most common cause of deflection estimates being significantly different to recorded values. The question is how you determine the amount of restraint. Some programs account for shrinkage restraint caused by the reinforcing, few account for restraint forces. A published method which attempts to give a method for calculating the restraint forces is the method proposed by PCI (Precast/Prestressed Concrete Institute). The analysis method involves the use of an equivalent shortening principle. This allows you to compute a tensile force in the slab. This can then be used to adjust the expected tensile strength of the concrete to assess deflections. It can be used to work out forces imposed on columns similar to Figure 4 (see Cl 3.4 on volume change in the PCI Precast Design Handbook 6th edition). The forces for volume change are larger and real and if not detailed appropriately can cause problems (see Figure 5). James Deaton provides a method for using the temperature load T sh in the analysis package to estimate restraint forces due to shrinkage. This method is similar to M. H. Baluch, et al. T sh
=
sh
= specified shrinkage strain α = coeff. of thermal expansion ε
sh
Figure 5: A block wall subjected to volume change forces causing it to crack. Engineers Australia
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Alternative positions Figure 6a : Location of movement joints.
Preferable layout of columns and walls (low restraint)
Non-preferable layout of columns and walls (high restraint) Figure 6b: Alternative layouts of walls and columns for dierent levels of restraint.
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In an investigation of a carpark (see Figure 5) he found that if concrete joints had been provided, this would have relieved shrinkage stress by a factor of 3.5. Spacing of contraction joints at 30m, instead of 95m, would have been more appropriate. For typical concrete structures, a qualitative approach to volume-change design is recommended as opposed to explicit calculation of volume-change forces. Designers may rely on previous experience. Advice on primary joint spacing for different building types can be variable and conflicting, recommending between 25m and 60m depending on the wall layouts and pour strips. It is recommended that previous success in your local area should be reviewed for selection of joint spacing. Secondary movement joints should be considered for walls and finishes as required. For atypical structures, movement joints should be considered in locations of change in building configuration (as shown in Figure 6a). Lockable dowels are beginning to gain wide acceptance as an alternative to delayed pour strips and careful consideration needs to be given to how much force will be transferred to the dowels after they are locked and how cracking can be controlled as load transfers from the slabs and into the dowel. Careful consideration should also be paid to what areas may need to be waterproofed. Layout of walls and columns affects the build-up of strains and stresses within the structure. Careful consideration of wall layouts can reduce these forces significantly. See Figure 6b for an idealised good and poor layout of walls and columns with regards to these restraint forces. 4.4 Temperature
Temperature changes in a member cause thermal expansion and contraction. Because the heat source is generally only on one side of the member, the expansion will be asymmetric which in turn can cause tensile stress and lead to extra deflections. This asymmetric expansion occurs in addition to the overall volume change as discussed above. It is important for members in roofs, walls and in any other position where they are exposed to the weather. This effect is perhaps at its most extreme when wide precast hollow-core panels unreinforced in their transverse direction are exposed to thermal loads. A perfect storm of factors are at play here as: • the holes act to reduce heat transfer • the holes act as crack inducers • the deflected geometry of the steel support beams exacerbates the transverse arching effect • there is often no reinforcement to carry the resultant loads. The above issue was highlighted to spectacular effect at a car parking structure in Canberra in 2002 when one of the large number of longitudinal cracks intersected with a saw cut around a column and partially collapsed a section of panel. Fortunately the panel only fell 80mm before getting caught on a cleat that was a shop detailing error. Longitudinal cracking is as closely spaced as 300mm and confined almost exclusively to areas exposed to the sun and rain. Finite element modelling of the panels under thermal loads identified the pointed apex of the panel voids (top side only) as a significant contributor to the phenomenon.
Figure 7: Longitudinal cracking from thermal loads leading to a structural failure. Engineers Australia
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“FEA in the design of reinforced concrete buildings”
Because temperature does not often affect the ultimate limit state of the structure, any deflection due to temperature is sometimes not considered in design. For uncracked members, effects of temperature can be included in deflection calculations in a relatively straightforward manner. For statically indeterminate systems after cracking, the deflections, stiffness and temperature are inter-related, and an alternative procedure is required for a correct solution. ACI 435.7R-85 (1997) provides guidance for this analysis. Movement due to temperature can lead to problems with joints and sealants as it will cause tensile stress and fatigue in the sealants. This should be taken into account when selecting sealants for multistorey buildings, in situations where sealant replacement is expensive. For roof slabs a 25m maximum spacing for movement joints should be considered. 4.5
Cracking
Deflection of structure is directly related to the amount of cracking. Cracking should be analysed in all directions and not just assumed. Tension stiffening plays a major role in determining the amount of deflection for concrete slabs. It is necessary to know the time of first cracking; this is of interest if the construction loadings are higher than the service loadings, as once the slab has cracked the loss in stiffness is permanent. Figure 8 shows the tension stiffening effect on a load versus deflection curve. The concept of effective moment of inertia, I ef , to reflect the concrete cracking was conceived originally by Branson. He assumed bilinear load-deflection behaviour of a cracked section and proposed I ef as a function of the level of cracking. This concept has been developed further by others and most programs will give you a selection of different methods, through the selection of different deflection models.
Deflection assuming no cracking
Load
Tension stiffening, δΔ
E
B
D
Actual response
Pservice Pcr
C
A
Concrete carries no tension anywhere
Concrete carries no tension in the cracked regions
0
Deflection
Figure 8: Typical load versus deection relationship.
Branson’s Formula
For a given cross-section, I ef is calculated using Branson’s formula (Branson, 1963): m M m M cr I cr I ef * I g 1 cr * M M s s
I cr is the second moment of area of the fully-cracked; I g is the second moment of area of the gross concrete section about its centroidal axis; M s is the maximum bending moment at the section and M cr the cracking
moment. For AS3600-2009 a value for the index m of 3 is used, because it averages the effective section over the span of the beam. Branson recommended a value of 4 be used as discrete sections are to be utilised. Several authors have suggested a “modified Branson” model to include modifications to the formula and vary m with different reinforcement ratios. While the ACI and Australian standards use Branson, Eurocode 2 uses the Bischoff method.
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Bischoff’s (2005) Formula
I ef
I cr m I M cr cr 1 1 I gt M s*
For Eurocode 2 a value for m of 2 is adopted. I cr is the second moment of area of the fully-cracked section; M s is the maximum bending moment at the section. In Brischoff’s Formula Gilbert has recommended the use of the transformed second moment of area, I gt . β is a factor that allows for loss of tension stiffening. It is important to understand that these theories are all based on different tests upon which they were developed. Each equation has a set validity range. Branson’s formula/theory was developed from medium reinforcement ratios, while Bischoff’s was based on lighter reinforcement ratios. Knowing the validity range for the methods used within the software is important. For slabs with lighter reinforcement ratios Bischoff’s Formula is recommended. 4.6
Long-term deections
AS3600-2009 allows a multiplier K cs for long-term deflections which includes an allowance for compression steel. This has created confusion as often the steel in the top of slabs was considered to be compression reinforcing. For compression reinforcing to have an effect on long-term creep, the reinforcement must at least be in the top half of the compression zone γkud, not just at the compression face of the member where it may be in tension or low compression stress and will have no effect on creep. This issue has been clarified in the latest edition of the code. A general rule of thumb is for a slab thickness of less than 250mm, compression reinforcement will not affect long-term deflection significantly other than to provide symmetric reinforcement which in turn reduces shrinkage curvature. The AEMM or Eurocode methods are far superior methods for estimating the deflection. They should be used for long-term deflection estimates. The Kcs method is not usable for PT (post-tensioned) slabs or beams. The Kcs multiplier fails to take into account reinforcement induced shrinkage and also ignores shrinkage and creep features of concrete. The author’s opinion is that the K cs multiplier should be removed from AS3600-2009. At best it will give a “ballpark” on the predicted deflections and at worst could lead to serviceability problems for the structure in question. References:
• • • • • • • • • • • • •
Shen, P., Fang, H. & Xia, X. 2009, “Effect of concrete creep and shrinkage on tall hybrid-structures and its countermeasures”, Frontiers of Architecture and Civil Engineering in China, pp. 234-239. Gilbert, R.I. & Ranzi, G. 2011, Time-dependent behaviour of concrete structures, Taylor & Francis, UK. Gilbert, R.I. 2008, “Calculation of long-term deflection”, paper presented at the CIA Seminar of Control of Long-term Deflection, Brisbane. Doug J, 2009, “Predicting the deflection of concrete structures in practice”, paper presented at the Concrete Solutions 09 conference, Sydney. Gilbert, R.I. 2001, “Shrinkage, cracking and deflection – the serviceability of concrete structures”, Electronic Journal of Structural Engineering, pp. 2-14. Gilbert, R.I. 1988, Time effects in concrete structures, Elsevier Science Publishers , Amsterdam, p 321. Gilbert, R.I. 1992, “Shrinkage cracking in fully restrained concrete members”, ACI Structural Journal, Vol. 89, No. 2, March-April 1992, pp. 141-149. Gilbert, R.I. 1999, “Deflection Calculations for reinforced concrete structures – why we sometimes get it wrong” ACI structural journal 96 (6), pp. 1027-1032. Scanlon, A. & Bischoff, P.H. 2008, “Shrinkage restraint and loading history effects on deflections of flexural members”, ACI Structural Journal, Vol 105, issue 4, pp. 498-506. Gilbert, R.I. 2007, “Tension stiffening in lightly reinforced concrete slabs”, Journal of Structural Engineering American Society of Civil Engineers (ASCE), Vol. 133, No. 6, pp 899-903 ACI Committee 435, 1985, “Observed deections of reinforced concrete – slab systems, and causes of large deections”, SP 86-2 ACI Journal , US. Gilbert, R.I. & Kilpatrick, A. 2001, “Improved prediction of the long-term deections of reinforced concrete exural members”, Proceedings b Symposium, Prague. Klein, G.J. & Lindenberg, R.E. 2009, “Volume-change response of precast concrete buildings”, PCI Journal
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• • • • • • • • •
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Fall 2009 , pp 112 -131. Deaton, J.B. & Kahn, L.F. 2010, “Lessons learned from forensic FEA of failed RC structures”, paper presented at the ACI Fall 2010 Convention. Iqbal, M. 2010, “Design of expansion joints in parking structures”, Structure Magazine, Oct 2010, pp. 12-14. DeSerio, J.N. 1971, “Thermal and shrinkage stresses – they damage structures!”, American Concrete Institute Special Publication pp. 43-49. ACI Committee 435, 1985, “State-of-the-art report on temperature-induced deections of reinforced concrete members”, SP 86, ACI Manual of Concrete Practice, Part 3. Jenkins, D. 2006, “Prediction of cracking and deections, international code provisions and recent research”, paper presented at the Concrete Institute of Australia seminar, Sydney, 2006. Liao, S., Klein, G., Mikhlin, Y. & Grossman, J.S. 2010, “Vertical structural deformation estimation and control for a deformation-sensitive building”, Structure Magazine, Sept, 2010, pp. 34-36. Pfeiffer, M.J. & Darwin, D. 1987, “Expansion joints in buildings”, Federal Construction Council Technical Report No 65. Baluch, M.H. Rahman, M.K. & Mahmoud, I.A. 2008, “Calculating drying- shrinkage stresses”, Concrete International, Vol 30, No 07, pp. 37-41. ACI Committee 435, 1985 (reapproved 1997), “State of the art report on temperature induced deections of reinforced concrete members”, SP 86 ACI Journal , US
5
MODELLING DECISIONS
5.1
Element type
Commercial concrete design software allows limited specific types of elements for building design. These are generally plate and beam elements for reinforced concrete, with shell elements used for PT slabs. The plates can be in the form of triangles and squares with either nodes at corners, or at corners and mid-edges (see Figure 9). It should be noted that these types of elements are only of use for flexure design and should not be used for bearing or shear applications (span on depth <10). For the latter applications brick elements should be considered, which are not available in most commercial concrete FEA programs discussed in this practice note. Accordingly, reference to other methods is required to supplement the finite element models. 5.2
Size
The selection of the size of elements is paramount to the accuracy of the design. Since the only place where forces are calculated is at the nodes, the number of these is important for the accuracy of the models. For example, for a 100m long beam modelled with three nodes, mistakes are certain and the model is unacceptable. Alternatively, by using the same beam and providing nodes at 1m centres the model is more likely to be acceptable. Rule of thumb: The size of the elements for slabs and beam modelling mesh size should be no greater than 1m, span/10 or half width of load patch.
4 Nodes
8 nodes
3 nodes
6 nodes
Figure 9: Typical element types. Engineers Australia
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5.3
Meshing
Most FEA programs do the meshing automatically; the larger plates that have been entered by the engineer get turned into a smaller matrix. The engineer must assess the finesse of the mesh; generally the automated meshing options will give choices of density. Figure 10 shows a typical surface member before and after meshing. When a coarse mesh is selected, the results will not give an accurate representation of the structure, particularly near supports, openings and under load points. Conversely, if too fine a mesh is selected, excessive time to compute can be a problem. The maximum hogging moments the FEA shows will be affected by the size of the mesh. The finer the mesh generally the more intense the support moment. Given the software will do this automatically, it is advisable during the modelling stages to use a coarse mesh to refine the model and to ensure it is error free, and to use the finer mesh for design. This reduces the time for modelling and increases the accuracy for design.
a) Surface member
b) Surface member divided into mesh
Figure 10: Typical mesh.
Items to look for when assessing the fitness of the automatically generated mesh include: • Mesh near re-entrant corners or sharply curved edges; • Mesh in the vicinity of concentrated (point) loads, concentrated reactions, cracks and cut-outs; • Changes in mesh in the interior of structures with abrupt changes in thickness, material properties or cross sectional sizes; • When stress maximums are of interest, check whether the stress contours are smooth in the highly stressed areas and whether the stress changes across an element are appropriate. 5.4
Discontinuity areas (D-regions)
Discontinuity areas or D-regions are parts of a structure with a complex variation in strain. They include portions near abrupt changes in geometry (geometrical discontinuities) or concentrated forces (static discontinuity). Based on St Venant’s principle, a D-region spans about one section depth on either side of the discontinuity. Beams, flat plates and shells cannot be used to model D-regions. Finite element models are seldom capable of reproducing the complexities of boundary conditions and related stress disturbances in the beam column joints. Therefore commercial FEA programs should be limited to applications were the Bernoulli principle would be applicable (see Figure 11).
Figure 11: Example of the division between B-regions and D-regions (with B-regions being parts of a structure in which Bernoulli’s hypothesis of straight-line strain proles applies).
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In regard to D-regions it is recommended that further analysis is undertaken to ensure adequate reinforcing is provided. Good complementary analysis methods for this are strut-tie analysis or specialist non-linear finite element applications. 5.5
Shape
Meshing is normally carried out by the computer. However, the user needs to ensure a well-conditioned model is created. To ensure the model is acceptable the ratio of shapes should not exceed 1:4 (the minimum length to maximum length). It is important to ensure that in the areas in the model where forces change rapidly more nodes are present to ensure accurate results are obtained. Figure 12 shows examples of good and bad shapes for meshing.
Good Shapes
Bad Shapes
Figure 12: Element shapes.
5.6
Boundary conditions
Design packages often require restraint conditions to be specified at the boundary edges. The forces generated need to be transferred to boundary elements. The design package will not check the validity of these boundary conditions and the engineer must do so. For example, a slab or beam framing into a thin wall may be closer to pinned than fixed. It is important that boundary conditions are reviewed for different loading stages, as the boundary conditions may change significantly from lower to higher loads. Boundary conditions will vary from full 3D to 2D models and a continual review of the boundary conditions at these different analysis points is required. For 2D packages the support conditions at columns and walls need to be consistent with the stiffness of the columns and walls (see the discussion on supports below). In 2D the best results are given when column walls are modelled representative of their stiffness. 5.7
Modelling elements
FEA programs require faithful modelling of the geometry accompanied by engineering judgement. Most software packages offer a limited number of modelling elements – plates (shell for PT) and beams. Plate elements are generally triangular or quadrilateral, with nodes at corners and sometimes include additional nodes on the sides (see Figure 9). Beam elements are used to model narrow beams, while plate elements are used to model wider beams. This is due to the accuracy of the slab bending moment when beam width increases for beam elements. AS3600-2009 allows moments to be taken at an offset distance from the face of the supports such as beams for slabs and columns for beams (Clause 6.2.3 AS3600-2009). Engineers must keep in mind when modelling beams that torsional stiffness is important. While most programs allow the torsional stiffness to be ignored when modelling beams, for plates this isn’t possible. This must be taken into account for deflections where the reduction in stiffness due to cracking in torsion can be in the vicinity of 90%. In programs where plate elements are used to design deep beams torsional stiffness may be based on the cube of the depth rather than the cube of the width. When a structure is not dependent on torsional resistance for equilibrium, most codes say that torsion can be ignored. Nevertheless, if torsional stiffness is present in a computer model, torsion and the torsional stresses developed must be included in the design. The author has found a varying array of ways that torsion is handled
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by programs; the engineer must understand the assumptions implicit in their design and the computer package being used. Curves and circles are only able to be modelled by straight edge shapes. This should be kept in mind when modelling, as a large mesh will give inaccurate answers. Figure 13 gives an example of this.
Poor modelling
Good modelling
Figure 13: Circular meshing.
5.8
Supports
Finite element analysis programs presume that the bending of the beams and columns continues to the centre of the beam/column joint. While this is fine for small structures with slight column and beam sizes, this is not satisfactory for structures where the beam/column joints have larger dimensions (ie 1m x 1m) and where the concrete is shearing – not bending. There are ways to fool FEA software into thinking that the bending stops at the face of the beam/column joint, but then the shear calculations may not be properly analysed. Some software programs perform this trick well, while others do it poorly. Rigid offsets and thicker “dummy” areas are two possible methods as shown in Figure 14. Reference should be made to the concrete code as to the location of the bending moment for design.
Infinitely sff link (rigid offset) thicker area
rigid offset
Figure 14: Alternative methods for modelling the area of the column.
It is important to model supports in concrete slabs as accurately as possible. Supports modelled correctly enable bending moments for punching shear calculations to be appropriate (important for edge and corner columns in flat slabs). If the corner and edge columns are modelled as pin-roller supports, bending moment inaccuracies in the forces around the support will be present. This could cause punching shear problems at loads less than the ultimate design load. The ways in which the columns are modelled can vary drastically, from the most inaccurate way at a single node to the more appropriate ways using rigid offsets or modelling a thicker area over the column. Neither of these latter methods is perfect, but they will provide more correct deflection than a single node support. Plastic assumptions are not possible in FEA. Slabs to walls can be modelled in different ways. Some finite element packages use zipper elements to allow the walls to be meshed independently from the slabs. These zipper elements can allow larger moments to be transferred than preferred by the designer. So it is important to check the transfer of moments between elements.
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5.9
Column stiness
Saint-Venant’s theory for torsion is used for equivalent column stiffness when calculating the bending moments using the strip method. The equivalent columns theory has been shown to be a good theory for ultimate strength design. It does have its shortcomings for deflection design. Care is required for larger spans or significant difference in length. This problem doesn’t exist with FEA programs as all parts of the building are modelled. However, the column stiffness still needs to be modelled. Some programs will set the column stiffness to 0.7I g for design, but this is only true for a particular load and not correct for heavily loaded columns or lightly loaded columns. Column stiffness can vary from 0.4I g to 1.2Ig. Therefore it needs to be taken into account in the model, as explained by Elwood, K. J. and Eberhard, M. O.: “For example, a low estimate of the effective stiffness of columns in a moment-resisting frame usually leads to a conservative (high) estimate of the displacement demands. In contrast, a low estimate of the effective stiffness for columns in a shear-wall building would lead the designer to conservatively underestimate the elastic shear demands on the columns.” Edge and corner columns are often reduced in stiffness due to cracking. Care should be taken in reducing the column stiffness especially with punching shear. Care should also be taken in over-estimating the stiffness and attracting more moment to the column than should be. The detailing of the joint should match or exceed this assumption. Elwood and Eberhard proposed the following values for column effective stiffness: EI ef =
P
0.2
A g f c '
EI g
EI ef EI g
5 P =
EI ef =
3 A g f c
4
−
0.2 <
30
0.7
0.5 <
EI g
≤
0.2
P A g f c '
≤
0.5
P A g f c '
P is the axial load in the column and A g is the gross area of the column. More advanced formulas for effective stiffness have been developed by researchers in recent times. These should be considered if better effective stiffness models are required in complex or seismic applications. Figure 15 shows the basic stiffness equations for columns for checking of models.
4EI 1
3EI1
L1
L1
L1
L1
4EI 2
L2
3EI2
L2
L2
L2
Figure 15: Modelling column stiness.
5.10 Non-structural items
Non-structural items can affect buildings in many different ways. They can cause significant changes to the behaviour of the structure. For instance, the framing of full-height interior walls in buildings can affect damping. Masonry infill walls can cause short columns, which will attract extra loads due to stiffness being proportionate Engineers Australia
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to roughly a cube of their length. Such higher loads, especially lateral loads due to earthquakes or wind, must be taken into account. Figure 16 shows some common situations that create short columns. These should be avoided if possible. However, if short columns are required, attention must be paid to detailing to ensure that the column can handle the forces and displacements until the full moment or brace frame are fully activated. It is recommended that non-structural items that can or will affect the structure be included in the models.
Building shape before loading
Load Load
Load Load
Load Load
Building shape after loading
Short column created by Short column created masonry inll walls. by masonry infill walls
Short Shortcolumn columncreated created by bysloping slopingground. ground
Short created Short column created by by structural structural layout. layout
The a column column isisapproximately approximatelya acube cube length. Hence, short column Thestiffness stiffness of of a ofof it'sits length. Hence the the short can take up to 8x or more the load of the longer columns due to equal displacements. column take can take u to 8x or more the load of the lon er lon er columns due
Figure 16: Multistorey frames with short columns in unloaded versus loaded conditions.
5.11 Walls
Wall elements are normally modelled as vertical plates. The engineer needs to decide if this is suitable. There are many possible support conditions, such as knife edge, with walls free t o uplift or not. Slabs with corners can develop corner levers as shown in Figure 17. The Figure shows the bending moment distribution that results from the corner restrained for uplift and the yield line pattern. These bending moments can be seen in finite element models. They can become large in skew slabs with acute angles. In this application extra reinforcement above the code minimums might be required. Depending on the design it can be important to account for corner levers in the modelling. This corner condition produces M xy moments which will be discussed later. Cl 9.1.3.3 (e) AS3600-2009 has a detailing requirement for these areas to ensure that, if corner levers develop, cracking doesn’t become uncontrolled. Rombach suggests that approximately 20% of a slab edge will attempt to uplift with a 13% increase in bending moment and an increase in support reactions in the walls if the wall is able to uplift. Figure 18 shows this phenomenon. 5.12 Beams
FEA programs presume that everything is co-linear and co-planar. In reference to a concrete structure, to design a T-beam properly, the slab should be modelled at a different elevation to the beam and integral as with the beam.
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Mxy twisng moments develop
Secon A
M*1
M*1
upli f orce
M*2 Secon A M*1 = Corner lever Moment
yieldline
two way slab supported on four side yi eld l ine paern
Figure 17: Corner levers.
Figure 18: A double spanning plate lifting up at the corner.
Depending on the computer package the beams can be modelled using different techniques, the most common being with horizontal level of centroids for the beam and slab matching. When the centroids match the beam, effective depth needs to be increased for the eccentricity (for PT slabs this is not recommended and the model should correctly represent the depth and centroid of the beam). Figure 19 shows that the equivalent beam can be of substantial depth compared to the depth of the T-beam web for the example provided. To ensure all the moments within the T-beam it is also important that effective flange width is used to size the reinforcing in the beam.
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Figure 19: Depth of equivalent beam h w. Extract from G.A. Rombach
Where beams are loaded by slabs at the bottom of the beam rather than at the top, additional ‘hanging reinforcement’ is required to supplement the normal shear reinforcement. Many programs will not supplement the reinforcement as required so these situations may need to be independently checked. This problem is most significant in short heavily loaded beams. The above should not be confused with suspension zone reinforcement which again may not be adequately catered for within a given design program. 5.13 Foundations
Generally foundations are between fully fixed and pinned and should be modelled as partially fixed or by way of another conservative assumption depending on the action being considered. If you are limited to fixed and pinned foundations it is recommended you model both cases to ensure the worst case effect is computed. Most FEA packages allow the user to enter soil properties at different locations and estimate settlements. This is useful on buildings with mat foundations that might be subject to differential settlement. Differential settlement can be caused by many situations: varying loads, varying strata, clay shrink/swell, water table variations and varying properties of the soils. FEA programs will vary in how they model the soil conditions, with different programs using different approaches. Some of the approaches support settlement input directly, spring supports, equivalent elements and volume 3D elements. Use of this feature to estimate settlements should be approached with consultation of the geotechnical engineer on the project. Most good geotechnical engineers will be able to provide the properties required for the different methods such that the settlements estimated by the software are similar to the settlements estimated by the geotechnical engineer. Acceptable settlements given by different published sources are listed in Table 2. Table 2: Acceptable settlements oered by various sources. Published source
Acceptable settlement isolated (exible) foundations
Acceptable differential settlement
McDonald and Skemption (1955–56)
75mm clays 50mm sands
40mm clays 25mm sands Maximum angular distortion L/300
Sowers (1962)
50–100mm
L/300
Bjerrum (1963)
Not total maximum recommended
L/150
European Committee for
50mm
20mm (L/300)
Standardization of Differential Settlements parameters
Differential settlement rather than settlement itself is the main concern with concrete buildings. Buildings can accept large settlements providing the differential settlements are within tolerable limits. Differential settlement can lead to unexpected cracking and tilting of the building (see Figure 20).
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Figure 20: Graphical representation of varying geological conditions under a building.
Properties for use in your FEA package for the soils should be sought from the geotechnical engineer on the project or other reliable sources. It is unlikely that the FEA program will be able to help with estimates of differential settlement due to clay shrink/swell. Other references and programs such as SLOGS, AS2870 and RAFTS are available to assist in this. Soil reports only provide the designer with total settlement results. A general rule of thumb is that even with ostensibly homogeneous soils, differential settlement should be assumed to be half the total settlement. It is recommended that you consult with the geotechnical engineer on the project for specific recommendations. 5.14 Considerations for interrupted supports and openings
Infinite stresses, both shear and bending, can be developed at the edge of line and point supports due to numerical modelling. These peaks are not actual stresses and are created from the modelling process. If the discontinuity in the line support is smaller than 15 times the depth, Rombach suggests that you can ignore these in your bending analysis. Alternatively, if your opening is greater than 15 times the depth, it is suggested that engineering judgment be used to decide on the redistribution required for the theoretical bending moment versus the numerical results. Support problems can occur for other reasons such as closely spaced walls. In these situations results will show sharp peaks in the bending moments, shear and support reactions. This is due to singularities similar to columns being modelled on a single node. These peaks should be distributed across sections due to cracking and yielding. One method to handle this is to use spring supports to spread the peak moments to the surrounding nodes. Discussion with the software developer will help to understand which of the available approaches have been used and what the ramifications of these decisions may be. For large openings, consideration should be given to stiffening the slab by using beams around the opening, and torsion should be considered in beams surrounding openings. 5.15 Redistribution
How do you handle redistribution of moments with FEA? This is easy for the equivalent frame method, but when you have moments in contours, do you redistribute the maximum moment? The average? It is common for FEA programs to redistribute the moments from the columns due to singularities. It is not intended for redistribution to reduce the moment taken by the columns (when modelled correctly). This is due to punching shear being a brittle (non-ductile) failure mode and therefore redistribution of the design moment transferred from the slab to the support is not allowed. In the author’s experience, redistribution of moments in beams is of limited benefit because moment redistribution cannot be used for the service moments. For design it is recommended that all actions be redistributed after the actions have been distributed into strips. Nonlinear analysis will automatically allow some redistribution, due to cracking. The author recommends that no redistribution of moments be undertaken if a nonlinear analysis is adopted. It is worth noting here that the degree to which moment distribution is permitted under AS3600-2009 is also a function of the ductility of the reinforcement. 5.16 Buckling
Buckling in concrete buildings can be a governing design consideration for slender elements. Slender columns, slabs with large openings, slender inverted T-beams, and walls need to be taken into account during analysis. Depending on the FEA method the program can help with this analysis. Care should be taken when using the software as there will be strict limits to its capabilities. Selecting the correct effective length is the normal problem for programs, with engineering judgment required to ensure realistic response. Generally it is advisable that the engineer check any slender elements for buckling by recognised methods, using the factored up loads as appropriate for sway and non-sway condition. In the absence of a detailed buckling analysis section 8.9 of AS3600-2009 does of course nominate slanderous limits that must be adhered to. Engineers Australia
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5. 17 Loading
The engineer needs to determine how many load cases are required. AS3600-2009 requires pattern loading to be included for situations where the live load is over 75% of the dead load. The author prefers to do pattern loading for all slabs in accordance with AS1170 because even though we treat the load as uniform for design, loads are never uniform. This can increase the number of load combinations substantially and a checker pattern is not a “one fits all” solution. Figure 21 provides a number of possible configurations of loadings that should be considered for patterning of slab loadings. It is up to engineering judgement, if point loads from the slabs above should be patterned for transfer slabs and columns. Punching shear associated with point loading will often not be checked by the software and hand calculations may need to be carried out to ensure code compliance. Point loads if modelled as a single point will cause singularities; they should be distributed over an area of the actual loading. Generally codes have foot prints for consideration of point loads; it is recommended that these be incorporated using a high pressure load over a square rather than a point load, if possible. 5.18 Construction
For high-rise buildings commercial time pressures often lead t o a requirement to strike the formwork as soon as possible and move on to subsequent floors, with a minimum of propping. Tests on flat slabs have demonstrated that as much as 70% of the loads from a newly cast floor (formwork, wet concrete and construction loads) may be carried by the suspended floor directly below. This early high loading has the potential to cause deflections and cracking of the concrete. It is essential that all members of the project design and construction team understand the implications of this load and make adequate allowances to accommodate it. It can generally be assumed that early striking of formwork will not greatly affect the deflection after installing the cladding and/or partitions. This is because the deflection affecting partitions will be smaller if the slab becomes “cracked” before, rather than after, the installation of the cladding and/or partitions. Construction loads should not be ignored. Writing that the structure should be fully propped by the contactor until the structure is fully stable is NOT engineering. On most projects the arrangement for back propping is to back prop for three floors. AS3610 outlines the minimum requirements in terms of project documentation of formwork and only the building designer is in a position to provide a good deal of this information. 5.19 Loading sequence
The loading sequence and timing is critical in determining the deflections, because it will influence the point at which the slab cracks. A loading sequence from the St George Wharf study shows the relatively high loads applied during casting of the floor above (see Figure 22). If an earlier stage proves critical, the crack depth at that stage should be carried forward to all subsequent stages, as once a slab is cracked it remains cracked and the stiffness reduction is permanent. The timing of formwork removal is particularly crucial where walls over act to support structure under and this needs to be highlighted on drawings as necessary. Staged stressing can also be a critical requirement of construction that needs to be highlighted on the drawings. When software is used to check prestress transfer stresses, some software for example may assume that all self-weight is present at transfer. But this is of course only true if it has been constructed. If some loads from floors over are categorised as self-weight, it may for example be necessary to change them to dead loads. This last point is a trap in both 3D and 2D structural programs.
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Load Array 1
Load Array 2
Load Array 3
Load Array 4
Load Array 5
Load Array 6
Load Array 7
Load Array 8
Load Array 9
Load Array 10
Load Array 11
Load Array 13
Load Array 12
Load Array 14
Figure 21: Suggested loading arrangements for pattern live loading broken into an idealised strip layout – yellow represents imposed plus permanent, blue represents permanent only.
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1st slab above cast d 2nd slab above cast
c
e
) m / N k ( d a o l l a c i t e r o e h T
2
a b
f
Slab struck
3rd slab above cast
Time from casting (days)
Figure 22: St George Wharf study loading of sla b during construction. Extract from the concrete centre study of the St George project.
5.20 Changes in cross section
If the section properties (the shape) of a beam or slab change as you cross the span, accurately modelling the shape of the changing section is virtually impossible in FEA packages. For most situations the shape can be split into equivalent sections. However, careful consideration is required to ensure that the approximations made are appropriate. Generally a minimum of three sections is required to model an appropriate approximation of a haunched area, this could be used as a guide to more complex arrangements. 5.21 Composite construction using concrete elements by Mal Wilson
Most of the discussion presented here to this point has assumed that slabs and beams are constructed in a single pour but there is a significant segment of the residential and commercial market that uses reinforced or prestressed precast concrete elements to support cast in situ toppings. Such systems vary, from precast prestressed joists with infill panels to hollow core floor panels or Beamshells. All of these systems are increasingly being designed using concrete finite element packages. While some suppliers may indicate that these elements can be designed as if the concrete were monolithic, this is not strictly correct for a number of reasons and great care needs to be taken in modelling their strength as well as their serviceability performance. These precast concrete members are often constructed using concrete many times stronger than the cast in situ concrete surrounding them. If plate elements are used in the modelling, only one set of strength and stiffness parameters can be set, and which is most appropriate will depend on the moment direction and the situation being considered. The concrete in the precast element is in a very different place in time with respect to its shrinkage and creep history when compared to the concrete poured on site so the construction is fundamentally composite in nature. How old the precast unit may be when it becomes part of the composite element is something outside the control of the designer. It is important to check that allowable longitudinal shear stresses are not exceeded at the interface between the existing and new concrete. Consideration needs to be given not only to the stresses induced by the shear from the ultimate load condition but also any stress that may be present due to differential creep and shrinkage. When joists or slabs are subjected to very high shear loads longitudinal shear stresses can quickly become a critical consideration especially in prestressed joists constructed of very high strength concrete. In some instances in the past at least one Australian manufacturer has recommended details that do not comply with AS3600–2009 reinforcement anchorage requirements. It is the author’s view that AS3600–2009 detailing requirements must be complied with unless it can be clearly shown (preferably through testing programs) that any alternate detail offers the same degree of strength and ductility as well as the equivalent robustness in the event of unexpected loading. Precast suppliers have in the past provided advice to suggest that codified longitudinal shear stress limitations are unduly conservative. It is the author’s experience that claims of vastly improved longitudinal shear capacity
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gained through one-off testing do not stand up to rigorous scrutiny. So where such claims are made, test data, coefficients of variation used and general test methodology should be checked first hand prior to any conclusions being drawn. If the precast manufacturer claims the raw data is confidential, the advice should simply be ignored as sound engineering design should be based on peer reviewed fully transparent and independently verified findings. Certainly finite element analysis can be used to assist in understanding the behaviour of these composite systems but no FEA programs commonly used are set up to model the behaviour directly, and considerable engineering judgement is required to establish what modifications to the normal design parameters are appropriate and what conclusions may be drawn from such modelling. Particular attention needs to be paid to anchorage of prestressed strands as many commonly used FEA programs will assume all strands are anchored by live and dead ends which is clearly not the case in precast construction. This point alone can affect designed reinforcement layout, shear capacity and even deflections. Once the effects of elastic shortening, creep and shrinkage have been accounted for in the precast elements themselves it is not necessary to fully account for them again in the same way that would be required in posttensioned construction. Further creep and shrinkage does inevitably occur but the amount and effect need to be carefully considered. References:
• • • • • • • • • •
6
Vollum, R. 2004, “Backprop forces and deflections in flat slabs: construction at St George Wharf”, BRE Report, No. 463, UK. Enochsson, O. & Dufvenberg, P. 2001, “Concrete slabs designed with finite element methods”, Lulea University of Technology Thesis p 146. Elwood, K.J. & Eberhard, M.O. 2006, “Effective stiffness of reinforced columns”, Research Digest No. 2006-1, Pacific Earthquake Engineering Research Center pp. 1-5. The Concrete Society 2005, TR43: Post-tensioned Concrete Floors Design Handbook , Concrete Society Technical Teport, UK, p 160. Mota M.C. & KaMara, M. 2006, “Floor openings in two-way slabs”, Concrete International Magazine, July 2006, pp. 33-36. Rombach, G.A. 2004, Finite element design of concrete structures, Thomas Telford Publishing, UK. Skemption, A.W. & MacDonald, H.H. 1956, “The allowable settlement of buildings”, Proceedings of Civil Engineers, Part 3, Vol. 5, pp. 727-784. Sowers, G.F. 1962, Shallow foundations, foundation engineering , G.A. Leonards (ed.), McGraw-Hill Book Co, New York, pp. 525-632. Bjerrum, L. 1963, “Allowable settlement of structures”, Proceedings of the 3rd European Conference on Soil Mechanics and Foundation Engineering, Wiesbaden, 2, Brighton, England, pp. 135-137. European Committee for Standardization, 1994, “Geotechnical design, general rules – Part I” , Eurocode 7, Brussels, Belgium.
ULTIMATE LIMIT STATE DESIGN
Ultimate limit state design replaces working stress design. Other than this minor change structural engineering hasn’t changed much in the past few decades. 6.1
Design moment distribution (not redistribution)
FEA programs generally report moments and reinforcement in contours. It is recommended that moments be distributed across the column and middle strips as appropriate, keeping in mind all detailing requirements of the code (see Figure 23). This is due to the micro-cracking relieving the slab at the support locals to the surrounding areas. There is a temptation to provide reinforcement to resist the peak moments; this should be avoided as this is too conservative.
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column/middle strip design 60 50 t n e 40 m o m30 g n i d n 20 e B
Moments from FEA average bending moments
10 0
column strip Middle 0 1 strip
2 Distance
3
Middle 4 strip
Figure 23: Column/middle strip distribution.
Design strips and sections must be defined for the serviceability and strength checks. The advantage of FEA is that design strips can be defined after the slab has been modelled. Design strips can be defined by code definitions or points of zero shear. Engineering judgment should ensure that the design strip is designed for the load acting on it (see Figure 24). The point of zero shear is especially useful for complex geometries. Defining the strips based on FEA results should lead to more economical reinforcement for complex support situations (refer to Concrete Society report TR43 (2)). It is important for strips with moments of different signs to be integrated separately as they produce top and bottom reinforcement and do not cancel each other out. Lines of zero shear
code defined strip
Figure 24: Dening the column strips.
A useful rule of thumb for verifying the results is that top reinforcement in the column strip be in about twice the area of the bottom reinforcement. While distributing the steel the engineer needs to keep in mind the requirements for shear. 6.2
Twisting moments
Modelling slabs as plate elements can lead to interpretation problems for bending moments. FEA gives bending moments in the Mx and My directions, but due to the modelling used it will give M xy moments (see Figure 25). This moment should be included in the design of reinforcement as it can be significant. The most common methods for including this in the reinforcement design are proposed by Wood Armer or Denton and Burgoyne. These methods are slightly conservative and your FEA program may use more complex methods. Most FEA packages allow you to include M xy in the outputs for M x and My; this should be selected by the user. The Standards Australia committee has recently reinforced this position.
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Figure 25: Slab twisting shear.
The term Mxy, the twisting moment, represents the twist, that is the rate of change of slope in the x-direction as one moves in the y-direction or vice versa. The twisting moment results in shear stress parallel to the plate surface except near the ends. Because of this shear flow difference, the reinforcement to prevent torsional beam failure should not be confused with the reinforcement to prevent twisting plate failure. (see Figure 26). My
Mxy
Mx Mxy
Mx My
Mxy
Mxy Figure 26: FE bending moment output.
Many people think the plate nite element is just a “smaller” plate and that the nodal reactive moments M’ x and M’y are the same as M x and M y in classical plate theory. Well, they are not! Let’s take a look at Lagrange’s equation: 2
2
M x 2
x
+
2
M xy xy
2
+
M y y
2
=
q
(where – q is the load).
This equation for plates shows that Mx, My and Mxy are coupled. Therefore, according to the Lower Bound Theory, this allows the apportioning of Mxy moments to the Mx and M y moments. For slabs, this is natural and can be easily achieved by increasing the orthogonal M x and My reinforcement using the Wood Armer equations or similar. 6.3
Classical beam theory
With the plane-remains-plane assumption and shear deformation excluded, the beam theory equations are simple and will not be repeated here. However, the torsion equation will be discussed. The torsion as shown in Figure 27 in beams is not related directly to the twisting moment M xy. These are two different actions and should be treated as such. If anything, torsion in beams should be related to M x and My. It is important to realise that: • the placements of longitudinal reinforcement and torsional stirrups are coupled; • beam torsion results in circular shear stress. Engineers Australia
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Beam Beam Figure 27: Basic torsion diagram for a solid section.
6.4 Torsion
Cracking of concrete can reduce the strength of concrete in torsion significantly as discussed earlier. It is important that any torsional steel designed by the program is reviewed for accuracy, and that it is known what assumptions the program is making when designing the amount of torsion steel required. Most programs design the beam as single beams without taking into account the extra torsional strength provided by the internal slab and beam arrangement. Research by Warner & Ragan found during tests on beams integral with slabs that slab restraint increased the shear/torsion capacity by a factor of 4 to 6. These tests were carried out on torsion beams with same depth as the slab. Yew-Chaye Loo et al showed that the increased resistance depended on the depth of the beam relative to the slab. Engineering judgement should always be used when selecting the option for design with or without torsion and when the compatibility torsion design is selected. This is particularly important to torsion members which, if they were to fail in torsion, would compromise the structure. Some programs use plate elements to model beam behaviour and, while these may perform satisfactorily for shallow beams, care must be taken where beams are deeper than they are wide. Torsional properties may be linked to the cube of the depth which will vastly overestimate the torsional stiffness of a deep beam. 6.5 P-Delta
P-Delta is a non-linear action occurring in all structures with axial loads both vertical and horizontal. The effect is a change in structure with possible changes in deflection and moments. These second order effects are relative to the magnitude of the applied axial force, displacement and slenderness of the elements making up the structure. These effects can generally be classified as: • P-“BIG” Delta (P-∆) – a structure effect; • P-“little” Delta (P-δ) – a member effect. It should be ensured that, if required, P-Delta effects (see Figure 28) are taken into account in the analysis. Care should be taken to understand and work within the limitations of the software. FEA programs generally don’t allow for the reduction in load-bearing capacity from slenderness, so users must review slender columns for load distribution.
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Figure 28: P-Delta eects.
6.6 Shear
FEA models produce shear stress results; these results generally are unable to help in punching shear checks. It is recommended that punching shear be checked using the requirements of AS3600-2009, and that column stiffness included in the model be correct or upper and lower bounds be used. If you prefer to use the software to carry out these checks, all openings (even small) must be modelled correctly especially if near the shear perimeter. Note: Punching shear is required to be calculated at the edges of drop panels and similar. Most software will not carry out this check; the designer should complete this external to the program. The requirements of the code should be included in designs including minimum transfer of moments between columns and slabs for punching shear and detailing requirements for reo near columns. The 25% detailing rule Cl 9.2.2 AS3600-2009 is often not well understood; the detailing is required for punching shear. The reinforcing forms the tie in the crude strut tie over the columns. The author hopes that in the future the % steel over the support will be included in the punching shear calculations to ensure this is clear to the design engineer. 6.7
Vertical load take down
An FEA model based on the static reaction for vertical loads can be un-conservative, as elements can be designed to take tension from above the floor under construction, t hus creating an unstable situation during construction or a different load path. An FEA model based on the area method for vertical loads generally is conservative as no elements are designed to take tension. The area method or construction sequence is recommended for vertical loads and reactions; the construction sequence method will simulate the gradual increase of the loading as the building is constructed. This is especially important if hanging beams or outriggers are present in the building structure. 6.8 Interpreting results
Before the invention of complete building modelling software, engineers had to analyse each design strip for moment compression/tension, shear and torsion, combined. This gave them a good understanding of the building and response to loadings. Engineers should carry out this in-depth analysis of critical points using engineering judgement. If you don’t have the experience to decide the critical points, review every point. However, because of the huge volume of results produced by the models a single engineer will find it hard to review all results. Therefore it is recommended that the model be reduced to more simple strips for analysis purposes, with results and calculations being recorded. Rationalisation of the design results is an important step in the design process. Often with highly advanced analysis methods you can get highly varying reinforcement results which, if provided in the design drawings, would allow for less reinforcement overall but would not represent the best cost outcome (as shown Engineers Australia
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in Figure 29). Time
Slow
Fast
High Minimum overall cost
t s o C
Finance
Labour, plant & preliminaries
Low
Material Highly detailed
Usual
Rationalised
Highly Rationalised
Level of rationalisation of reinforcement Figure 29: Relationship between cost of building, rationalisation of reinforcement and time.
6.9 Rationalisation
Rationalisation is the process of eliminating unnecessary variation by reducing complexity so that provision of materials is easier and efficient. Finite element reinforcement results generally do not undertake the rationalisation process to an appropriate level, other than to select a common reinforcement bar size. The engineer needs to review the results from the finite element program such that overly complex details are not provided. Simple philosophies that can help with this objective are: • Rather than detailing each element separately, the engineer can try to identify typical reinforcement arrangements that will be suitable for common elements; • Rather than having varying reinforcement lengths along supports, try to keep the changes in length to a suitable number; • Rather than providing small bars sizing at close centres, try larger bar sizes at more appropriate centres. This may result in some elements being ‘over designed’, but there will be subsequent cost saving because of the reduction in the time taken to provide alternative arrangement onsite, hence this will be faster. 6.10 Additional reinforcing
While the analysis provides the majority of reinforcing for the building, there is additional reinforcement required for serviceability and detailing. While serviceability considerations will be discussed later, this section concentrates on reinforcement due to sound engineering judgement: • Additional steel may be required around openings (possible recesses) to prevent shrinkage cracking or similar. Extra steel is also recommended at re-entrant corners; • The Australian code has a requirement to offset the bending moment at a distance d (depth to tension reinforcing centroid) in each direction. Most computer programs don’t include detail for development length and offset d. This is important near edges of slabs and transferring moments. The Sleipner accident should serve as a warning in regard to the importance of anchorage and detailing. While that accident cannot be attributed to anchorage and detailing alone, detailing of the joints was inadequate and the structure did fail more radically than it would have with correct detailing. All anchorage requirements and lapping should be checked, most programs do not provide extra steel required for this purpose. Over-reinforcing of slabs and thin elements can lead to restraint deflection from steel. Designers need to be careful that all the reinforcing is included in deflection analysis of shallow members.
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References:
• • • • • •
7
Jakobsen, B. & Rosendahl, F. 1994, “The Sleipner accident”, Structural Engineering International, IABSE, Vol. 3, pp. 190-193. Wood, R.H. 1968, “The reinforcement of slabs in accordance with a predetermined field of moments”, Concrete 2 No 2, pp. 69-76. Li, J. 2002, Reinforced plate design for M xy twisting moment, Solutions Research Centre, Hong Kong, p 7. Timoshenko, S.P and Woinowsky-Krieger, S. 1959, Theory of plates and shells, McGraw-Hill International Editions, p 580. Morris, A. 2008, A practical guide to reliable finite element modelling, John Wiley & Sons, p 380. FIB Bulletin 45, 2008, Practitioners guide to finite element modelling of reinforced concrete structures – state of the art report, p 337.
SERVICEABILITY LIMIT STATE DESIGN
Hardy Cross once wrote: “Strength is essential but otherwise not important”. Modelling of the structure must reflect the required serviceability performance of the building with regards to cracking, deflection, crack and stress limitation. Serviceability dominates the design of most structures, with deflection being influenced by concrete strength (both compression and tension), creep, shrinkage, elastic modulus, restraint, loading/time of loading/duration of loading, ambient conditions, formwork stripping procedures and durability. 7.1
Deection
Deflection design has many influences, none of which can be predicted accurately. Thus deflection predictions are best estimates, and the estimates you make should be the upper bound for deflection, not the lower bound. While there are minimum deflections quoted in the code, there are a lot of instances where deflection is critically important. The designer needs to decide which of these apply to an individual project. Often the load which affects the critical deflection (eg deflection affecting cladding) is not applied at the same time as the initial loading. Some critical situations for deflections: • Cladding walls can only handle a finite amount of deflection, ranging from 1/250 to 1/2000, and some cladding manufacturers state the systems can only handle 5mm of deflection. The design is best carried out in coordination with the design of the walls. Edge beams can be used to control deflections of the external facade. Failure to account for deflection under walls reduces the expected life of seals and joints. • Ceiling and light-weight partition walls need to be considered for visual deflections, and if the edge of the slab is visible, this should be considered. • In light-weight slabs vibration needs to be checked as well. This is extremely important in mixed-use areas such as a gym in an office building or hospitals. • Glass walls are sensitive to deflections. • Operable walls have stringent requirements for deflection and manufacturers’ input should be sought early for each project. • Roof structures with membranes need extra care to ensure that accelerated wear of the membrane is not a problem. Membrane problems have been experienced where deflection reduces the drainage of the roof or where there is cracking of the supporting slab. • Coordination with the architect or hydraulic consultant is essential to ensure that outlets are located in appropriate locations such that minimum fall will be achieved throughout the life of the structure. To check the deflections in complex slab systems, the author recommends the procedure shown in Figure 30. The span is defined in any direction.
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Note: a
a
The maximum deection at any point can be dened as 2a/n (n = span to deection ratio as dened by AS3600-2009, eg span/250 maximum deection may be critical on grid lines).
Figure 30: Interpreting deection.
Actual concrete deections are inuenced by many factors which cannot be fully taken into account. They include: • Tensile strength of concrete (a change in strength from 2.7 to 2.1 can increase deflections by 50%) • Modulus of concrete +/- 20% • Early construction loading • Shrinkage warping. Always remember loads can only be estimated and even dead loads cannot usually be calculated to within 5% accuracy. It is advisable to give a suitable range/warning with any estimate of deflection that others are relying on. Possible methods for calculating deflections using FEA: • Deemed to comply span on depth ratio as per Cl 9.6.3 AS3600-2009 (the author questions the advantage of finite element modelling) • Linear analysis with section properties adjusted for cracking factored up deflections using K cs (the author suggests that this is guess work) • Nonlinear analysis with adjusted elastic modulus or similar advanced modelling (the author recommends the use of non-linear analysis with adjusted elastic modulus in accordance with AEMM or Eurocode). With any of the above methods it is recommended that a sensitivity analysis on deflections be conducted on the vital input parameters. This will ensure that any deflections reports are reported with the appropriate range. This is especially important if providing deflections for the use of others. 7.2
Precamber
Modelling of precamber in slabs in FEA programs is difficult. Slab and beams can be precambered to reduce the effect of deflection. In practice, if precamber is utilised too much it is generally estimated and the slab remains permanently cambered. This is because of the difficulty in accurately calculating deflection and also representing it in models to see the effect across all areas. It is recommended that precamber be set at a conservative value. However, the author is of the opinion that it is better to design for the deflection requirements of the AS3600-2009 rather than precamber. It is important to remember precamber does not reduce the deflections affecting partitions or cladding. A positive development in the industry has been that with the near universal move to levelling with laser levels rather than taking heights off the deck there has been a major move away from precambering. 7.3
Vibration
More efficient design utilising stronger materials can lead to lighter structures with lower natural frequencies which makes them potentially more vulnerable to vibration problems. The most common causes of vibrations are human activities (walking, running, dancing, jumping or gymnastics). Conventional methods of predicting floor accelerations are only suitable for a narrow range of floor layouts and materials. FEA models allow any type of layout of walls and floors to be modelled for deflection. Using finite element methods vibration sources can be modelled and the effects on other areas evaluated due to the Engineers Australia
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decreased accuracy required when dealing with small loads and strains. FEA will allow the user to include the effects of the rotational stiffness of columns, and the effects of partition walls, stairwells, services and finishes. The author has found Steel Construction Institute publication P354 entitled Design of Floors for Vibration: A New Approach describes a reasonable method for modelling in FEA which is applicable to any type of suspended floor. A realistic allowance for non-structural mass, due to services, fit-out finishes etc should be included. It is normally possible to add non-structural mass to an element, and some programs also allow static load cases to be converted to additional mass. If neither of these options is available, then the density of the material representing the slab must be modified. In general, only include additional mass that is likely to be there in practice. Over-estimating the mass can be non-conservative for footfall response. The movement of some offices towards a paperless environment which has been muted for 20 years is finally gaining significant traction. This is a situation which increasingly needs to be considered in design. There are several parameters that can influence vibration. They include the steel, the modulus of elasticity, damping and the extent of cracking. In regard to steel beam connections, for strength or serviceability design structural engineers often assume pinned end connections. For the very small strains associated with footfall-induced vibration, it has been found from tests that connections will normally act as if they are fixed rather than pinned, and so can be modelled without releases. The modulus of elasticity for vibration analysis is larger than the static values, in particular when high strength concrete is used. Damping has an inherently high variability that is difficult to determine before a floor system is placed in service. The recommended values from reference [Allen, D.E., and Murray, T. M., 1993] vary from 2% – 3% for bare concrete floors to 5% – 8% with full height partitions. Cracking reduces floor stiffness and consequently lowers its natural frequency. For conventionally reinforced concrete it is important to allow for cracking. When evaluating the above parameters, reference to the base method and the assumptions used both in development and testing should be reviewed. References: • Buettner, D.R. & Ghosh, S. K., ACI Committee 438.8R-8, 1997, “Observed deections of reinforced concrete slab systems, and causes of large deections”, SP 86-2 ACI Journal, p 47. • • •
• • • • • •
8
Thomas, M.M., Allen, D.E. & Ungar, E.E 2003, “Floor vibrations due to human activity”, Steel Design Guide Series 11, American Institute of Steel Construction, p 69. Allen, D.E. & Murray, T.M. 1993, “Design criterion for vibrations due to walking”, AISC Engineering Journal, 4th Qtr, pp.117-129. Smith, A.L., Hicks, S.J. & Devine, P.J. 2009, “Design of oors for vibration: A new approach”, SCI P354 , The Steel Construction Institute, p 114. Willford, M.R. & Young, P. 2006, A design guide for footfall induced vibration of Structures, the Concrete Centre, Gillingham House, 38-44 Gillingham Street, London, p 79. AS3600-2009, Australian Standard for Concrete Structures, Standards Australia 2009. Elwood, K.J. & Eberhard, M.O. 2006, “Effective stiffness of reinforced columns”, Research Digest No. 2006-1, Pacific Earthquake Engineering Research Center, pp. 1-5. Cobb, F. 2004, Structural engineer’s pocket book , Elsevier, UK, p 354. Morrison, J. & Jones, T. 2003, “Use of computers in the design of concrete structures”, Concrete Magazine, May, 2003, pp. 40-42. Brooker, O. 2006, How to design reinforced concrete flat slabs using finite element analysis, The Concrete Centre, London, p 16.
DESIGN
Many FEA programs handle the reinforcement and bending moment calculations for the design of the structure
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to the relevant codes. Thus the engineer needs to have a good method for checking to ensure the assumptions made by the software designer are in accordance with the design being undertaken by the engineer. Simple or alternative calculations are very important for this, some possible checks are: • Calculate wl2/8 (a basic bending moment formula) for a span and check the FEA model gives the same value between the positive and negative moments (10% difference could be considered a pass, anything greater would need further investigation). • Compare the total slab weight against the total reactions under dead load. • Span on depth ratios: again if you are well above normal limits then it would be worth checking again. • Use alternative analysis program (like RAPT) to do a few lines up and down the building and compare. • Is the span/depth or height/depth ratio in line with standard practice, if not why? • Simple hand bending and shear diagrams. • Using the direct methods from the code and compare, if these vary why? • Are supports modelled; how are they going to really behave? Check walls to slab connections as these are difficult to reinforce for full moment transfer. • Do the contour plots have similar results as the Pucher influence charts (Pucher charts are a series of contour plots of influence surfaces for various plate and loading geometries). • Static equilibrium; compare total loads to total reactions. • Check the load increase (and face shear) in a column at any given floor is approximately equal to the load on the floor area notionally supported by the column. • One of the commonly used methods for estimating the fundamental frequency is known as the selfweight method. If the maximum gravity deflection of a single bay structure (under self-weight plus service super-imposed load) is δ, then the fundamental frequency (Hz) can be estimated using f = 18/√ δ, where δ is the maximum deflection in millimetres. This equation works reasonably well for most singlespan beam or floor plates. Some items to be considered in design but not discussed in this practice note are: • How much does the slab contribute to the beam load? • Properties of concrete flat slabs, one-way slabs, waffle slabs, and slabs acting as diaphragms supported on steel joists. • Torsional and flexural effects of such systems on the actual stiffness of beams. • Interaction of shear walls and beams. • Shear lag effects on interconnecting concrete walls (in elevator and stair shafts). • Skewed slabs – in skew slabs infinite stress will be caused in the corners and special consideration is required. Refer to “Finite element design of concrete structures” by G.A. Rombach for further information on modelling possibilities. • Most software assumes the centre of elements with different thickness will be aligned in the vertical plane, so the offset of the drop or beam should be defined in the model. • The output is usually in the form of contour plots, and interpretation is required at the interface of elements with different thicknesses. • The discussions in this practice note are not for the design of post-tensioned/pre-stressed flat slabs. Most importantly, the discussions in this practice note are not intended to be a substitute for engineering judgement.
8.1
New programs
New programs are being created all the time. These can increase design speed, with some programs developed to analyse design detailing and drawing from one package. The engineer using such programs must understand the software, understand the limitations and things it doesn’t do. The possible time saved by using such programs should be spent on checking to ensure a safe and durable structure. 8.2
Recommended reading
This practice note has only scratched the surface of finite element modelling for reinforced concrete structures. Further reading is recommended to fully understand the more complex issues of finite element modelling. The author recommends: Engineers Australia
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• • •
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Finite element design of concrete structures, G.A. Rombach. The Sleipner Accident, Jakobsen and Rosendahl. Theory of plates and shells, Timoshenko S.P and Woinowsky-Krieger S, McGraw-Hill International Editions (this is an old classic book, which presents a number of solutions for elastic plates, which may be helpful for some simple cases). A Practical Guide to Reliable Finite Element Modelling, Alan Morris, Emeritus Professor of Computational Structural Analysis, Cranfield University, UK. Practitioners guide to finite element modelling of reinforced concrete structures – state of the art report – FIB bulletin 45. Influence surfaces of elastic plates, Pucher, A, 5th revised edition, Springer Verlag, New York, 1977.
FORENSIC ENGINEERING
by Mal Wilson
In forensic engineering we look to discover why a structural element has collapsed, cracked or deformed in a way that is unexpected and FEA can provide a very useful tool in discovering what may have led to the situation being explored. FEA may also contribute to an understanding as to what the response of a given element may be under additional loading. In this form of endeavour much of the advice given in the preceding chapters needs to be reconsidered and in certain circumstances ignored completely for a number of important reasons which will be considered below. It is often the case that when a structure is deflecting or cracking unexpectedly, detailing rules set out in the Australian Standards have not have been followed. It must be remembered that design details such as where and how positive and negative reinforcement must be curtailed can affect the ability of a structure to cope with unexpected moment changes along a given member. Limits to the amount of flexural reinforcement may also affect the capacity of a section to redistribute moments prior to a section failing. Much of the advice given previously is based on the assumption that the AS3600–2009 or other code requirements have been fully implemented. When such detailing has occurred the structure is ‘guaranteed’ a certain amount of ductility as well as a capacity for the expected moment envelope to move left or right without catastrophic consequences. It is this attention to detailing that means that ‘close enough really is good enough’ when it comes to FEA. Detailing rules are in place for a number of purposes one of which is to enable the structure to more reliably deal with unexpected loads and form an important role in the overall safety of the structure. These detailing requirements are therefore not conditions to be lightly traded off when considering whether a structure is safe to remain in service. Essentially it needs to be shown that the building is not only safe under a given load condition but it should also be as safe as it would have been had all code detailing provisions been complied with in the first instance. This requirement can be considered a measure of the structure’s robustness. When code detailing rules are ignored it fundamentally changes the guidance on FEA given so far as suddenly any modelling assumption we make can be extremely critical depending on the circumstances. Effectively the safety net in terms of the structure’s ductility and its capacity to redistribute moments or carry shear loadsmay have been compromised and a failure to model a behaviour precisely may prove critical or even catastrophic. Under such circumstances nothing should be taken for granted and parametric studies are often required for a full understanding of the structure in its current form as well as the potential risks under additional loading. Such studies are a form of sensitivity analysis that lead to an understanding of how robust the current situation is. The other important consideration in forensic work is that we are no longer necessarily working within a carefully controlled set of construction parameters. In design work we can, theoretically at least, control the construction and loading process through careful documentation and prudent supervision. Our documents can and should for example specify the: • maximum amount and distribution of loading during construction • sequence and method of stripping of the formwork • prestress sequencing and staging (as necessary) • concrete strength to be achieved before loading or stripping of form work • maximum amount and distribution of in-service loading • weld sizes, types and categories Engineers Australia
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• lap lengths for reinforcement • code requirements to be adhered to • chemical anchor depths and type. In design we are operating in a ‘perfect world’ where every parameter is theoretically within our c ontrol. When carrying out forensic investigative work we too often find that some or all of the parameters listed above are unspecified, unmonitored and unrecorded. Even the raw data gathered on site is sometimes ‘polluted’ by construction inaccuracies (initial out of plumb or level etc) which can make interpretation challenging. In many instances this situation can devalue any attempt to model the behaviour of the structural system with FEA as the models become a series of hypotheses, a number of which may fit the data gathered on site equally well. When this situation occurs the physical data will quite naturally take precedence over the FEA model and residual strength considerations may need to be based more heavily upon what can be observed on site rather than what can be modelled on a computer. The important point to take from this is that it is often what is not specified within our documents or recorded on site that renders our FEA questionable. It may be difficult for some readers to grasp the significance of all of this. The following example illustrates how failure to comply with code detailing provisions or to adequately document or record the construction process can lead to unexpected problems within the resultant structures. Moreover, this example highlights that FEA assumptions that may be perfectly reasonable in general design may be dangerously un-conservative when applied to existing poorly detailed structures. Figure 31 shows a view of a suspended slab where cracks exist at the extremity of some precast joists which carry a cast-in-situ slab and are supported by a cast-in-situ band beam on a line between X and Y. The extent of these cracks is indicated as a red line at the underside of the band beam (see Figure 32).
Figure 31: Isometric view of a suspended concrete oor.
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Figure 32: View of the underside of a suspended concrete oor. The red line depicts the extent of cracking.
Figure 33 is a typical cross section through the band which depicts the crack at the point where the precast beam terminates.
Figure 33: Section through slab showing location of crack.
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Figure 34: Cracks at end of Ultraoor beams.
From Figure 33 it can be seen that the bottom tendons in the precast beam do not continue to the support. They do not for example continue past and above the bottom reinforcement of the band beam. Most structural engineers will be well aware AS3600 cl 8.1.8.4 c) requires at least one quarter of the bottom reinforcement continue past the near face of support. To a prospective FEA modeller the designer’s failure to comply with this basic design rule should ring alarm bells as a positive moment at the slab band junction could significantly compromise the capacity of the floor system to carry shear loads. For many structural engineers the visible cracking at bottom of the interface and the associated face step in some places would provide more than enough evidence to suggest a shear problem exists already at serviceability loads (see Figure 34) but let us consider for a moment what the modelling challenges might be for this and uncracked areas with similar geometry. Under the circumstances we can only imagine that the original designer assumed that the walls crossing the band beam (which incidentally are three stories tall) would act to stop any beam rotation and induce a negative moment in the adjacent slab. While this situation would not be code compliant in terms of bottom reinforcement anchorage it would significantly enhance the shear capacity as the d o (see AS3600 cl 8.7.2.1) would jump from around 35mm to around 160mm (negative V s positive bending). When modelling the structure using finite elements the following matters need to be carefully considered. 9.1
Load sequencing
To obtain the maximum benefit from the walls over it would be important that the slabs loading the bands remain propped until the walls over are in place. In this instance no such requirement has been noted on the drawings and the precast supplier’s web advice suggests that props can be removed 7 days after the slab over is poured. When site records are not available this adds greatly to the complexity of the FEA task as walls coming on after the props are removed may still act to reduce creep deflections and also to eventually carry an amount of load, but the quantum will be difficult to predict. 9.2
Anchorage of wall reinforcement
The N16 vertical bars in the block walls over (see Figures 31 and 32) are called up at 600c/c but there is no indication of what epoxy is to be used or whether the starter bars are required to finish at the ends of the walls. The hole size nominated for the N16 bars is 18mm which is certainly not enough to correctly epoxy the bar and we would expect that the bar may need to be hammered into its 125mm deep hole. When the performance of the wall anchorage reinforcement is very poorly specified FEA becomes more challenging and it may be instructive to search for gaps appearing between the blockwork and the slab, or indeed prudent to ignore the contribution of the bars entirely. Certainly any attempt to model load stiffness behaviour of the anchor would seem futile given the state of the specification.
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9.3
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Backspan stiness
The FEA shows the short backspan to the left of point A in Figure 35 in significant negative bending. In this instance the backspan is passively reinforced with low ductility mesh and the longer span between B and D is primarily in positive bending and prestressed (pre-tensioned). Clearly it will be important in any analysis to calculate and use the correct cracked section moduli as the back span will crack far more which increases the likelihood of a positive moment at B. Much of the FEA software available for this type of modelling is set up for post-tensioned situations, so great care needs to be taken in using such software to model pre-tensioned beams.
Figure 35: Diagram showing deected shape (deections factored for clarity).
It is important in this case to fully understand the concept of plastic and elastic moment redistribution and the effects that this may have on resultant moment at point B. For a fuller understanding we recommend Scott, R.H. & Whittle, R.T., University of Durham, Arup Research and Development, 2005, “Moment redistribution effects in beams”, Magazine of Concrete Research 2005, Vol. 57, Issue 1, February, pp. 9-20. 9.4
Construction loading
As noted earlier construction loading can have a significant effect on how much cracking occurs in the slabs and band beams which can also effect the effective section moduli and in turn the moment distribution. 9.5
Moments in steel support columns
Clearly any moment that can be delivered to the column will act to increase the likelihood of a negative moment at B. In this instance N16 bars on top of the column (see Figure 33) are specified but the number is ambiguous, the weld procedure is unspecified and the hooks are not anchored over any bars. Reliably modelling the moment in the column may therefore prove difficult but the moment resulting from edge loading of the column could be considered a likely minimum. 9.6 Torsional stiness of the band beam
Figure 33 shows that despite the large amount of torsion on the band beam the reinforcement documented does not include any closed ties so hairline torsional cracks may act to significantly reduce torsional rigidity of the beam. 9.7
Shrinkage restraint
With all FEA there are practical limits to the size of model that can be run and for this reason it would be quite normal (given the complexity of the structural form) to run the size of model similar to what we have indicated in Figure 31. One problem that can result from such an approach is that we have neglected to include the restraining effect of the retaining walls that the car park slab has been rigidly connected to. These retaining walls act to restrain the slab shrinkage which can also have a significant effect on the shear capacity at the critical section (see AS3600 cl 8.2.7.1 β2). What this code rule is effectively suggesting is that if the critical shear zone is in 3.5MPa of tension across the gross area then Vuc = 0. To complicate matters further the extent of stress from constrained shrinkage can be influenced by the age and construction of the retaining walls the pour sequencing of the slab the type of curing the constituents of the concrete as well as many other factors. The assumption of full restraint may be a prudent approach when it comes to shear capacity of a critical element with no shear reinforcement and no anchorage of critical reinforcement. Shrinkage stresses can also adversely affect flexural and torsional st iffness of the various elements.
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9.8 Trusting what can be observed
When so little can be determined with confidence it is often useful to go back to the basics and measure what you can see. In this case if we look at Figure 35 the deflection between A and B varies from 4mm to 18mm and a line drawn between B, C and D indicates minimal deflection which all points to a positive moment at point B. More compelling than this is the fact that there is a crack at B and the compressive zone of a negative moment does not by definition include any cracks as fresh air is free to move in and out of the crack and cannot be compressed. If there remained any doubt the face step in the cracks in some areas should be enough to highlight a very significant problem that may well represent the onset of shear failure. In other cases where these ‘smoking guns’ are not present FEA is an excellent tool but it must be used with great care as it is not the structure itself but rather our best guess at the structure. Parametric studies (or sensitivity analysis) hold the answer to a realistic analysis of the risks we face in any given circumstance and we encourage engineers working in this field to see FEA as a tool to explore options rather than some type of absolute model of structural behaviour.
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SENSIBILITY CHECKS
10.1 Comparison to known limits
FEA models are difficult to check. It is recommended that simple sensibility checks be developed to easily establish if the structure is proportioned appropriately. Table 3 lists a number of preferred stress ranges which the author nds helpful in determining areas within a design that need further review. Table 3: Preferred stress ranges for concrete design elements.
Element
Action
Preferred stress range (MPa)
RC slabs
M°/bd2 V°/bd M°/bd 2 V°/bd
0.5 ~ 2.5
RC beams/bands RC columns PT slabs PT beams/bands
N°/A M°/bd2 P/A P/A
0.31 ~ 1.5 0.9 ~ 3.0
0..80 ~ 2.0 0.2 ~ 0.51f’c 0.5 ~ 3.0
1.0 ~ 1.8 1.5 ~ 2.5
(Note: These are not maximums or minimums for stress ranges). Mο is the design moment, V ο is the design shear, P is the design axial force, b is the element effective width, d is the element effective depth, A is the area. 10.2 Heuristics (rules based on experience and intuition)
Every engineer is going to have heuristics that they have collected or established over time, for instance span/ depth ratios. These are good tools for reviewing outputs from computer programs. 10.3 Sensitivity analysis
For complex models a sensitivity analysis is a useful method. Sensitivity analysis can provide the following: • Increased appreciation of relationships between input and output variables in a model • Recognising model inputs that cause uncertainty in the output and should therefore be the focus of attention • Establishing the robustness of a model • Finding for errors in the model. By understanding these above points a confidence level can be established in the model.
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VALIDATION
With any analysis it is important to validate the software; you should request from the s oftware developer designs/ tests/comparisons that have been used to validate the software. Often the company will have comparisons that have been published. A plea by the author is for software developers to produce detailed documentation on the technical assumptions made for the design analysis of the software. Table 4 provides a checklist for evaluating the software you are using.
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“FEA in the design of reinforced concrete buildings” Table 4: Checklist for evaluating the software you are using.
Software Query
Discussion
Critical for
Does the program allow for Analysis of vertical loads using a Construction Sequence?
An FE analysis based on static reaction for vertical loads can be un-conservative. An FE analysis based on area method for vertical loads generally is conservative. The area method or construction sequence is recommended for vertical loads and reactions.
Transfer slabs/beams, columns,
Ability to reduce torsion stiffness?
This is extremely important for beams in torsion. Can the beam generate the nominated stiffness required to take the torsion loading, if not can you reduce the stiffness.
Transfer Edge beams, equilibrium torsion, service deflections
Partially cracked section properties calculated and recalculated for subsequent iterations for every element, in all directions.
Reduction in stiffness due to cracking is important. Cracked section properties vary throughout the slab and in both x and y directions.
Deflections
What column/wall stiffness does the program assume?
Column stiffness is hard to calculate due to the large interaction of P/A and bending. Extremely important for flat slabs as punching shear calculations depend on the moment in the columns.
Column to slab and column moments
What does the program take into account when working out bending moments and reinforcement.
The bending moments in orthogonal directions mxy need to be taken into account for reinforcement and deflection design (e.g. are Wood Armer or Denton and Burgoyne methods used for steel design?)
Reinforcement
Does the program generate bending moments Mx, My and M xy or converted moments Mux and M uy.
The unconverted moments reported by FEA (M x, My, Mxy ) are not the same as moments reported by simple analysis (M ux, Muy ). The moments reported by FEA need to be converted to design moments either using Wood Armer or Denton and Burgoyne so that checking can be completed.
Comparing/checking moments
Automatically apply load patterning to determine worst case design forces.
Ensures “worst credible” design forces obtained. AS1170 requires that patterning be taken into account in design.
Moment and shear forces
Does Software analyse in-plane forces ie variations in centroid elevation?
Allows realistic analysis of structure with varying thicknesses containing beams etc.
Beam stiffness and step in the slabs
Incorporate curvature due to free shrinkage strain.
Required for determining deflections accurately.
Deflections
Partially cracked properties are calculated.
Tensioning stiffening will prevent a fully cracked situation in thin slabs.
Deflections for slabs
Separate analysis used for ULS and SLS.
Less cracking occurs at the SLS, so the slab is more stiff.
Deflections
Software calculates creep coefficients, tensile strength for each change in loading throughout the life of the slab.
Important for the long term deflection calculations.
Long term deflections
What creep or shrinkage properties are assumed for the vertical elements?
This is especially important if you have a different material used for vertical elements (eg a steel core with concrete column, as the columns will creep and shrink and the core will not).
Column deflections/ slab slopes
Areas of required reinforcement can be averaged over a specified width.
This automation saves time for distributing over the strips.
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CLOSING COMMENTS
The ultimate end game of any design/analysis should be proportionate to the design requirements. This practice note explores the world of modelling in finite element programs. Nonetheless, alternative models should still be used to verify the model produced. Most programs have a good solver, thus the results for bending moments etc are dependent on inputs by the user. The same cannot be said for the post processor for designing reinforcement. These post-processors are less tried and tested. The engineer needs to know how to interpret their results. Now that you have read this practice note you should be able to evaluate your program and understand the full implications of the models you create, validating and interpreting the results given by your FEA software. Understand that software is a utensil to do this in a faster manner, not a substitute for engineering knowledge or experience. “As a rule, a program should be used only if engineers can predict the general deflection and distribution of moments in the structure prior to obtaining a solution. The computed solution is used to verify the results previously predicted by the engineers. If the solution is significantly different from the prediction, engineers should use the results only if they can satisfactorily explain the reason for the discrepancy and find it acceptable.” (ACI President’s Memo José M. Izquierdo-Encarnación, 2003.)
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APPENDIX: CASE STUDIES Following are five case studies – the first two illustrate failure of serviceability and the other three deal with collapsed structures due to inexpert application of FEA. Even though case studies 4 and 5 are non-building structures they are included here because they illustrate in a dramatic way the potentially very serious consequences of faulty use of FEA. The lessons learned could be put to good use in avoiding the recurrence of similar failures in the future. These case studies are a summary of the findings from larger reports and investigations. They are the author’s opinions and may not match the conclusions from the official investigations. It is recommended that for further information on these case studies the references be accessed.
CASE STUDY 1: Restraint eects on carpark structure Introduction: • 108m × 40m, two storey parking deck, unknown location • Extensive early-age cracking of slabs reported • Case study used the approach by James Deaton for the analysis; however, plate analysis is used rather than solid elements • FEA model that was created to represent the structure is simplistic including assumptions such as fixed foundations, reduction in stiffness, ramps ignored, thermal effects ignored. Findings: • No expansion joints in the structure • Construction sequence had no visible pour breaks in the slab • Parking structure serviceability failure – the concrete floor had extensive cracks visible and greater than 1mm wide.
Figure 1: 3D view of forensic model.
Derivation of temperature load for shrinkage modelling for use in the finite element models for Figure 2 and 3: ∈SH = 0.00085, α = 9.9× 10-6/°C ∆Tsh = -85 ° C (see chapter 4.3) sigmax kPa -6592.0 -487.0 5618.1 11723.2 17828.2 23933.3 30038.4 36143.5 42248.5 48353.6
Figure 2: Top oor In-plane stress; average stress 6.5MPa.
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sigmax kPa -3713.5 -1763.3 186.9 2137.1 4087.3 6037.5 7987.7 9937.9 11888.1 13838.3 Figure 3: Top oor In-plane stress if contraction slab joints were provided at 36m crts; Average stress 1.1MPa.
Lessons learnt: • Design of contraction joints at 36m versus 108m would be more suitable. • Shrinkage performance criteria in mix design are important if joints are to be spaced above normal recommended practices. • Force from shrinkage can induce significant loads in stiff elements such as walls. Reference: • Deaton, J.B. & Kahn, L.F. 2010, “Lessons learned from forensic FEA of failed RC structures”, paper presented at the ACI Fall 2010 Convention.
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CASE STUDY 2: Deections of a concrete oor Introduction • Office building, unknown location • Third storey slab deflection concerns, report by Peter Taylor • Peter Taylor reported a building slab had significant deflections and used Rapt program as the analysis method. This showed that Rapt could provide good correlation between measured and calculated deflections • The slab span on depth ratio is greater than recommended by rules of thumb • FEA model that was created to represent the structure for the purpose of this investigation is simplistic. • The AAEM method was used for deflection estimations.
Figure 4: 3D view of Forensic Model.
Figure 5: Contour deection results from non-linear model.
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Figure 6: Measured deection. EXTRACT FROM PETER TAYLOR’S REPORT
Findings: • Measured deflections are greater than predicted by both Rapt and FEA models when live load is assumed to be 0kPa. • Correctly constructed FEA models can provide guidance on maximum deflection results. • Precamber was used on the slab with limited to no success. Lessons learnt: • AAEM method in FEA can be used to estimate maximum expected deflections • Rule of thumb for slabs provides a good indication. • Precamber in slabs can be problematic in practice. Reference: • Taylor, P.J., ‘The Initial and Long-Term Deflections of Normally Reinforced Concrete Flat Slabs and Plates’, a special projects report for the ACSE, June, 1997. • Taylor, P.J., “Initial and Long-Term Deflections of a Reinforced Concrete Flat • Plate Structure”, Civil Engineering Transactions (Sydney), V. CE12, No. 1, Apr. • 1970, pp. 14-20
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CASE STUDY 3: Royal Palm Hotel, Tumon Beach, Guam Introduction: • 12-storey reinforced concrete moment resisting frames • Construction of the building was completed in 1993 • On 8 August 1993 a powerful earthquake shook the island, causing partial collapse of the structure.
Figure 7: the Royal Palm Hotel building after the earthquake. PHOTO EARTHQUAKE ENGINEERING RESEARCH INSTITUTE
Figure 8: Damage to columns on the third oor. PHOTO EARTHQUAKE ENGINEERING RESEARCH INSTITUTE
Findings: • The analytical model used to design the structure had numerous errors, including several columns that were rotated 90° from their actual orientation. • Masonry infill walls created short-column conditions throughout the structure, these infill walls were not included in the models. • Strong-column weak beam principles were not applied. • Additional confinement hoops required around column splices were not specified on the drawings. • The contractor substituted “U” shaped stirrups for the closed ties required in the joints of the moment resisting frame. • The contractor omitted closed hoops in many joints of the concrete moment frame. Lessons learnt: • It is hard for supervising engineers to check large models. Engineers Australia
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• Detailing of connections in structures is important and ensuring that these are followed onsite is paramount. • Consideration of all structural and non-structural elements is important in modelling. Ignoring masonry infill walls in the analysis of the structure allowed the short column conditions to go undetected at the modelling stage. If infill walls are to be ignored, detailing of these walls is important to ensure that short column condition cannot develop. References: • Guam Earthquake Reconnaissance Report, Earthquake Engineering Research Institute, Oakland California, April 1995. • Hamburger, R. O., “Supplemental Report: Failure Investigation, Beach Wing, Royal Palm Resort, Tumon, Guam, EQE International Report, June 2004. • Moehle, J. P., “Royal Palm Resort, Guam – An evaluation of the causes of the failure in the earthquake of 8 August 1993,” Engineering Report, March 1997.
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CASE STUDY 4: Sleipner A oshore oil platform, North Sea Introduction: • Sleipner A is a condeep gravity-based offshore oil and gas drilling platform about midway between Scotland and Norway in the North Sea. • The original concrete support structure for Sleipner A sank in the Gandfjord near Stavanger, during a controlled ballast test. • The hydraulic pressure acted within the hollows formed by the intersections of the tricell joints. The leak occurred at one of these tricell joints adjacent to shaft D3 (see Figure 9). • The SINTEF (Stiftelsen for industriell og teknisk forskning, Norway) undertook an extensive investigation into the failure of this platform to establish the failure cause. Findings: • The investigations showed that during the design of the structure theoretical fundamentals were overlooked in several instances. The first instance required an advanced knowledge of finite element modelling to appreciate. The second however was a disturbing omission of essential engineering knowledge. 1. The engineer did not understand the consequences of using distorted elements in their finite element model. 2. The engineer forgot the basic mechanics of materials which would require a linear modelling of the shear stresses rather than a parabolic one. The resulting shear stress from the parabolic modelling was 45% different to beam analysis results. 3. The tie reinforcement as shown in Figure 10 was too short. Increasing the length of this bar would have increase the strength of this connection by 50%. Lessons learnt: • Simple verification using alternative method of analysis can be useful in evaluating the model’s accuracy. Differences in results shouldn’t be greater than 10%, and if so this difference should be examined. • Detailing of joints is important and strut tie analysis is helpful in these D regions. • Experienced engineers need to supervise the design process and provide direction on critical modelling decisions for complex elements.
Figure 9: Sleipner A: Water Levels at the time of failure and location of failure. GRAPH BY SINTEF
Figure 10: Sleipner A Tricell Strut-and-Tie Model. ILLUSTRATION BY UNIVERSITY OF WISCONSIN Engineers Australia
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References: • The Sleipner Platform Accident, by B. Jakobsen and F. Rosendahl, Structural Engineering International 4(3), August 1994, pp. 190-193. • The Failure of an Offshore Platform, by R. G. Selby, F. J. Vecchio, and M. P. Collins, Concrete International 19(8), August 1997, pp. 28-35. • Rettedal, W. (1993) “Design of concrete platforms after Sleipner A-1 sinking,” Proceedings of the International Conference on Offshore Mechanics and Arctic Engineering – OMAE, pp. 309-316
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CASE STUDY 5:
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KORORBABELDAO B BRIDGE, PALAU
Introduction: • The Koror–Babeldaob Bridge was a balanced cantilever prestressed concrete box girder bridge with a main span of 240.8m and total length of 385.6m. • It was the world’s largest bridge of its type at the time of construction. • On 26 September 1996 the bridge suddenly collapsed.
Figure 11: The Koror-Babeldaob Bridge before collapse.
Figure 12: The bridge after collapse.
Findings: • Creep had caused the midline of the bridge to sag 1.2m, causing discomfort to drivers and concern for officials. • No final cause has ever been definitively published. Lessons learnt: • Oversimplification of creep structural analysis using one-dimensional beam-type analysis leads to errors in deflections including prestress loss for box girders. To capture shear lags in slabs and webs box girders should be analysed in three-dimensional models. • The effects of the differences in slab thicknesses within the cross sections on the shrinkage and drying creep rates must be considered. Engineers Australia
