Prestressed Concrete Design Lecture Notes PDF

Preview

Summary

PRESTRESSED CONCRETE DESIGN Lecture 1 – Introduction to Prestressed Concrete Structures Principles of Prestressed Concrete  In a reinforced concrete beam subject to bending, the tensile zone cracks and all the tensile resistance is provided by the reinforcement.  Stress that may be permitted in th...

Description

PRESTRESSED CONCRETE DESIGN Lecture 1 – Introduction to Prestressed Concrete Structures Principles of Prestressed Concrete 

In a reinforced concrete beam subject to bending, the tensile zone cracks and all the tensile resistance is provided by the reinforcement.



Stress that may be permitted in the reinforcement is limited only by the need to keep the cracks in concrete to acceptable widths under working conditions (thus no need for use of very high strength steels which are available)



Design is therefore uneconomic in 2 respects: ◊ Dead weight includes ‘useless’ concrete in the tensile zone ◊ Economic use of steel resources is not possible ‘Prestressing’ means artificial creation of stresses in the structure before loading, so that the stresses which then exist under loading are more favourable/efficient. As concrete is strong in compression the material in a beam will be used most efficiently if it can be maintained in a state of compression throughout.

    

Presence of a longitudinal compressive force acting on a concrete beam may therefore be overcome the disadvantage of reinforced concrete listed above. Not only is the concrete fully utilised, but the need for conventional tension reinforcement is removed. The compressive strength is usually provided by tensioned steel wires or strands which are anchored against the concrete and, since the stress in this steel is not an important factor in the behaviour of the beam but merely a means of applying the appropriate force, full advantage may be made of very high strength steels.





The way in which the stresses due to bending and an applied compressive source may be combined is demonstrated in pic. In the case of an axially applied force acting over the length of a beam, the stress distribution at any section will be equal to sum of the compression and bending stresses. It is therefore possible to determine the applied force so that the combined stresses are always compressive.

     

If the compressive force is applied eccentrically, a further stress distribution – due to the bending effects of the couple thus created – is added. This offers further advantages when attempting to produce working stresses within required limits. Methods of prestressing There are two basic techniques commonly employed in the construction of prestressed concrete. The difference between the two is primarily about whether the steel tensioning is carried about before or after the hardening of the concrete. The choice of method is governed by the type and size of the member coupled with the specification of either a precast element or an element constructed in situ. PRETENSIONING ◊

◊

◊

◊ ◊ ◊

◊

In this method the steel wires or strands are stretched to the required tension and anchored to the ends of the moulds for the concrete The concrete is cast around the tensioned steel, and when it has reached sufficient strength, the anchors are released and the force in the steel is transferred to the concrete by bond, There is an immediate drop in prestress force due to elastic shortening of the concrete as well as occurrence of long-term losses due to creep, shrinkage and relaxation. Due to the dependence on bond, the tendons for this form of construction generally consist of small dimeter wires or small strands which have good bond characteristics Anchorage near the ends of these wires is often enhanced by the provision of small indentations in the surface of the wire This method is ideally suited for factory production where many identical units economically made under controlled conditions – e.g. long line system where several units can be cast at once, end-toend, and the tendons merely cut between each member. An advantage of factory production: specialised curing techniques such as steam curing can be employed to increase the rate of hardening of the concrete and allow earlier transfer of prestress.



POST-TENSIONING ◊ This method is the most suitable for in situ construction ◊ It involves the stressing against the hardened concrete of tendons or steel bars which are not bonded to the concrete. ◊ ◊ ◊ ◊ ◊

◊

◊ ◊ ◊

The tendons are passed through a flexible sheathing which is cast into the concrete into the correct position. They are tensioned by jacking against the concrete and anchored mechanically by means of steel thrust plates or anchorage blocks at each end of the member. Alternatively, steel bars threaded at their ends may be tensioned against bearing plates by means of tightening nuts. It is necessary to wait for the concrete to become sufficiently strong under in situ conditions before stressing. The use of tendons consisting of a number of strands passing through flexible sheathing offers considerable upside in that curved tendon profiles can be used.

A post-tensioned structural member may be constructed from an assembly of pre-cast units which are constrained to act together by a means of tensioned cables which are often curved as illustrated above. Alternatively, the member may be cast as one unit in the normal way but a light case of untensioned reinforcing steel is necessary to hold the ducts in position during concreting. After casting, the remaining space in ducts may be filled via high pressure with grout (‘bonded’) or left empty (‘unbonded’) Although the grouting assists in transferring forces between the steel and concrete under live loads – and improves the ultimate strength of the member – the principal use is to protect steel from

corrosion The bonding of the highly stressed steel with the surrounding concrete beam also greatly assists with demolition as the beam can be safely chopped up w/out release of stored energy (could be dangerous). Prestressing Materials ◊ Prestressing Wire  Types: smooth, indented, Crimped ◊



◊

 Diameter: 4-7mm dia.  Characteristic Strength: 1510-1770MPa  Force: up to 60kN  Elastic Modulus: 205GPa Prestressing Strands  Types: Standard, Super, Dyform    

All sizes: 8-18mm dia. Characteristic Strength: 1670-1860MPa Force: up to 380kN Elastic modulus: 195GPa

◊

  

Prestressing Bars  Types: smooth, ribbed  All sizes: 20-75 mm dia.  Characteristic strength: 1000-1100 MPa

 Elastic Modulus: 170 & 205Spa Standard Bridge beams include pre-cast Y-beams, pre-cast Y-beam deck, precast U-Beams, Precast Ubeam deck, Precast Double Tee Beams, Precast Floor Slabs (Hollowcore) Standard pre-tensioned precast beams such as the inverted T, I, M, Y and U beams are widely used in the UK Hollowcore – very common for residential buildings. Precast floor slab called hollowcore. Optimised cross section: number of different types of cores available.

Lecture 2 – Serviceability General Equations and Transfer Conditions 

Aims of Prestressing ◊ To make the concrete section fully elastic ◊ More efficient use of the concrete ◊ To span greater distances ◊ ◊ ◊

◊ ◊ ◊

◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊ ◊

To balance the applied loads This graph shows difference in behaviour between an RC beam and a PSC beam When a sagging bending is applied to a beam, the top half is under compression and the bottom half is under tension

K3 K2

K1

Concrete does not resist tension well – has a low tensile strength – we neglect it Under tension, cracks will form at the bottom of the beam in such a scenario: it’s why reinforcement is used in this region. The beam starts with an initial stiffness where the load is proportional to the deflection. But the beam loses stiffness when the cracks occur, changing the gradient of line/relationship between the two variables. This is what happens at the ‘RC cracking’ point on the graph: the RC member starts to crack and the beam loses stiffness. The next phase is when the beam reaches the ultimate limit state, which corresponds to the ultimate moment. This is idealised behaviour – we like linear behaviour – we assume when the beam reaches yielding, the system responds with no stiffness, i.e. a flat response. K3 = 0. This is a typical response of reinforced concrete structures. We don’t like this kind of response because cracks may be detrimental for the performant of the structure, particularly in harsh environments. This is a benefit of using prestressed concrete. The formation of cracks can be avoided until greater loads. In terms of response, we do not start from 0, because the system already has an initial stress: gives us an initial deflection. As we have compressed the section – centrally or w/ eccentricity – the system has the same stiffness compared to the RC beam, but start from a non-zero deflection. There is a translation of the response to the left of the graph. When the force is increased, the cracks occur, but at much larger forces. Delays the formation of cracks.

◊ ◊ ◊ ◊ 

Again, there is a reduction in stiffness, but a smaller zone/range as ~same ultimate moment. The important part is the initial stages. The concrete is used more efficiently and is more durable. It is more expensive to produce this type of component. Can have longer span with same height of beam.

Sign Convention ◊ Stresses (/s)  Compression: +ve  Tension: -ve ◊ Moment (M):  Positive if t> b, i.e. a sagging moment Eccentricity of prestress (e):  +ve downwards Calculation of the 2nd Moment of area: ◊



 

Use a tabular format: Section Modulus ◊ Defined previously the section modulus, which is very important because it relates to the engineers bending equation ◊ The Engineers bending equation: ◊

 

 



= =

 

y is the elevation of the fibre considered with respect to the centroid R is the curvature 

◊

From the first 2 terms, this gives 𝑀 =

◊

Easy derivation of the bending moment



◊

This is why the section modulus is important, because can be defined as 𝑍 =

◊

This a property of the member

 





◊ Therefore, 𝑀 = 𝜎𝑍 or 𝜎 =  ◊ Allows designer to easily determine stress in a given section or ultimate moment Beam with axial prestress.

◊ ◊

When a beam with a simple axial prestress is under a sagging bending moment, the stress distribution shown is found. There is compression in the top and tension in the bottom The stress in the top and bottom may be related to the bending moment via the section modulus Elastic modulus different at top and bottom

◊

The notation dictates tensile stress signified by negative magnitude, hence minus sign

◊

When the force is introduced, centrally, the prestress is introduced as compressive stress in all fibres. A uniform compressive stress. The end result is found by summing stress in each fibre. At the bottom, the compression from prestress and tension from bending strain distribution is found. Results – in this case – with small net compressive force. This is the aim of prestressed

◊

◊ ◊

◊ ◊ ◊

concrete design: want to have compression to avoid formation of cracks Means we can derive an expression for stress at top and bottom by summing contributions at top and bottom. Second term negative as a tensile stress Under maximum loading conditions (Mmax)  

◊



(1) (2)

 

𝜎 = − 



 

(4)

For MAXIMUM loading condition, guaranteeing no tension in the beam, equation 2 becomes 

◊



𝜎 =  −

  

Under minimum loading conditions   (3)  𝜎 = +   

◊



𝜎 =  +



𝜎 =  −

 

=0

This can be rearranged to find the minimum value of prestress 

 

=

 , 

or 𝑃 =

  

◊

Therefore, the maximum stress in the top of the section can be found via substitution of Mmax to be  



We need to check the stress at the top in order to avoid crushing of the concrete Conclusions » The top is always in considerable compression if prestress is applied axially w/out eccentricity » More efficient use of concrete if the stress range at the top is similar to stress range at bottom. This can be achieved with the use of eccentric prestress force Eccentric Prestress ◊ ◊





𝜎 = +       𝜎 = 󰇡 󰇢  

◊

In this case we are interested in load applied to structure/section with a given eccentricity

◊

It can be decomposed as the summation of 2 contributions » The first, p, is similar to what has been discussed so far » The second is a bending moment, M, which is equal to the P – the applied force – times the eccentricity

◊

In this case, following the same steps as before, we have the axial prestress – the first contribution – and the additional strain – the second contribution – which is caused by the eccentricity of prestress This additional prestress is in the form of tension at the top and compression at the bottom. As a final stress distribution, we have compression on top and at base New stresses at top and bottom given by

◊ ◊ ◊

» »













𝜎 = + −    

𝜎 =  −  + 







Outer Fibre stresses with eccentric prestress ◊ Under maximum loading conditions (Mmax) » » ◊

»

(6)

    −   



𝜎 =  +

𝜎 =

   + 

−

(7) (8)

Concrete stresses with eccentric prestress ◊ Critical condition for no tension in the bottom of the beam, equation 6 becomes » ◊

So »

◊

Or »

◊



𝜎 =  − 𝑃=

  +  

=0

  



𝑀 = 𝑃 󰇡

 

+ 𝑒󰇢

In the axial prestress scenario, Mmax was found to be equal to 𝑀 = 𝑃 󰇡





󰇢, so the increase in

◊

moment capacity is Pe. Therefore the value you obtain now is larger than you could previously: more efficient

◊

However, the max stress at the top – given equation 5 – is found to be »



𝜎 =

(5)



   −  +   

Under minimum loading conditions »





𝜎 = +  −    

   

𝜎 = 󰇡 

󰇢

◊ This is the same as for the axial prestress case Comparing axial and eccentric prestress ◊ The increase in moment capacity of the beam due to the presence of eccentricity of prestress = Pe ◊ The top fibre stress are the same axial and eccentric prestress cases – do not vary ◊ Eccentric prestress allows the moment capacity you increase for the same extreme fibre stresses.

Lecture 3 – Section Sizing and Prestress Design    



Transfer: how the forces are introduced in the beam considered Principle of Limit State Design – a general approach found in codes that considers 2 limit states: ultimate limit state & serviceability limit state 2 different conditions/ types of checks you have to deal with when designing a structure Ultimate limit states (ULS) are those associated with structural failure are concern ◊ Safety of people - priority ◊ Safety of the structure Serviceability limit sate (SLS) correspond to conditions beyond which specified service requirements are no longer met and concern: ◊ The functioning of the structure (durability) ◊ The comfort of the people; and appearance » In some cases need to define with the client Key serviceability limit states for structural concrete are ◊ Cracking ◊ Deflection ◊ Vibration – generally neglected for concrete, but relevant for long spans ◊



 



The design of a prestressed concrete member is based on maintaining the concrete stresses within specified limits at all stages of the life of the members Hence, the primary design is based on the serviceability limit state, with the concrete stress limits based on the acceptable degree of flexural cracking, the necessity to prevent excessive creep and the need to ensure that excessive compression does not result in longitudinal and micro cracking SLS equations with stress limits ◊ Under maximum loading conditions (Mmax) » » ◊

»

 

𝜎 =

(9) (10)

    −  



𝜎 =  + 𝜎 =

  

−

+



≥ 𝑓

≤ 𝑓

(11) (12)

What are we doing when we are checking these equations? fmax corresponds to the value at which concrete crushes, fmin the minimum value of resistance, i.e. tensile strength. Need to avoid exceeding capacity of concrete. ◊ We aim to avoid cracks at the bottom. The allowable concrete compressive tress in bending is given in EC2 as limited to ◊ 0.6 fck under the action of characteristic loads ◊ 0.45 fck under the action of quasi-permanent loads ◊

 

  −  ≤ 𝑓      −  +  ≥ 𝑓   

Under minimum loading conditions (Mmin) »





𝜎 = + 

Transfer At initial transfer of prestress to the concrete, the prestress force will be considerably higher than the long-term value as a result of subsequent loses which are due to various causes including elastic shortening, creep and shrinkage of the concrete member. Since these loses start immediately, the condition at transfer represents a transitionary stage in the life of a member – must consider limiting both the compressive and tensile forces at this stage. The concrete at this stage is also usually relatively immature and not at full strength, hence transfer is a critical stage and should be considered carefully ◊ Prestress force (at transfer): P0 ◊ Permissible stresses: f’max & f’min ◊ Loss factor: K (Long term P=kP0) = approx. 0.8 ◊ ◊ ◊ ◊ ◊ ◊

When forces are applied using either post or pre-tensioning, there is a loss of stress The properties of concrete at initial application of force are not the same through life – properties change with time. The max/min stresses change – not constant but vary with time. Start from a smaller value of max/min stress when concrete is cast Conventional time at which concrete is cast s The compressive stress at transfer should be limited to 0.6 fck where fck is based on the strength of the concrete at transfer. The tensile strength should be limited to 1 MPa for sections designed not to be in tension.



Transfer equations

◊

◊

Common critical cases (condition you have to consider when designing): » Minimum moment at transfer » Maximum moment at transfer At transfer » »

◊

In service » »

    

  +      𝜎′ = −  

𝜎′ =

𝜎 =

𝜎 =

   

+

−

   

−

+

 ≥ 𝑓′  ...