Prestressed concrete as the name implies is a concrete that is subjected to prior compressive stresses before its working life. This technology is born out of the need to optimize the inherent weakness of concrete in tension, hence prestressing a concrete member seeks to eradicate or minimize the tensile stress in such a member.

The Reinforced Concrete Solution

Concrete is strong in compression to about 100N/mm² of cylindrical strength and 120N/mm² of cubic strength; it is however weak in tension such that its tensile strength is just about 10percent of its compressive strength. This inherent weakness in tensile strength is meant to be overcome by reinforcing the concrete with steel bars. A reinforced concrete is a composite material that harnesses the compressive strength of concrete and the tensile strength of steel. When the composite material is subjected to stress, the concrete resists the compressive stress while the steel resists the tensile stress. However, for this composite action to take place, the concrete section has to crack around the region subjected to tensile stress. These cracks allow the reinforcing steel to kick in and resist the tensile stress thereby complementing the compressive strength of the concrete in a composite action.

 

Disadvantages of the Reinforced concrete Solution

The development of crack which is necessary for the composite action of reinforced concrete has two main drawbacks:

1. The crack portion of the concrete becomes a burden as it does not add to the overall strength of the section. It only adds to its dead weight thereby making the section uneconomical.

 

2. The need to limit the crack width to about 0.3mm under service condition necessitates that high-strength                     steel cannot be used so as to reduce the stress around the reinforcement. This also contributes to                                     uneconomical use of steel in reinforced concrete.

 

Besides the above-mentioned main drawbacks, other disadvantages present in reinforced concrete are:

3.  Reinforcement bars in cracked concrete are more vulnerable to rust thereby affecting the durability of the            member.

4.  The strength of a cracked section is less efficient to resist shear stress.

5.  Cracked section loses considerable amount of stiffness which makes it less desirable under heavy loads                   when serviceability state of deflection is critical.

6.   Crack formation in reinforce concrete is irreversible

 

 The Prestressed concrete Solution

The mentioned disadvantages in the use of reinforced concrete are not present in prestressed concrete because:

1. Tensile stress can be completely eliminated in the member such that the whole section contributes to resist loads such that there is no cracked portion which only adds to dead weight without contributing to the overall resistance of the section.

2.  High-strength steel cannot only be used but desired so as to apply maximum prestress load on the section and          also prevent stress loss due to relaxation. Advantage can be taken of high strength to make economical use of              steel.

Method of Prestressing

There are two methods of applying prestressed concrete.

1. Pre-tensioning
2. Post-tensioning

 

Pre-tensioning

Pre-tensioning is a method where prestressed is applied prior to casting of the member. The stress is transferred to the member by means of bond between the concrete and the prestressing steel. This method is particularly suitable for producing large numbers of similar elements like rail sleepers, bridge beams, poles, deck panels etc. It is ideally carried out in a factory.

Steps of pre-tensioning

1. Prestressing tendons are installed and tensioned against the abutment
2. The member is cast around the stressed tendons and left to hardened and reach the desired strength for transfer stage.
3. The tendons are cut and set loose from the abutment so that they transfer their forces to the concrete member in the process of contracting.

 

Challenges of Pre-tensioning

As earlier established that pre-tensioning is typically carried out in factories in a long line where series of identical members on the line are stressed in a swoop. One major challenge is that the line has to remain straight. This makes it difficult to alter the eccentricity of the strands in order to control the permissible stress along the member. One of the ways to overcome this challenge is through debonding.

Debonding is the process of preventing bond from developing between concrete and prestressed wire strands by wrapping a coating around them. Since the major technique of transferring prestress force in a pre-tensioned member is through bonding between the concrete and the steel strands, debonding some of these strands will allow the control of stress along the member.

 

Post-tensioning

 Post-tensioning is a method of prestressed where prestressing force is applied on a member by means of anchors at the ends of the member after casting the member.

 

Steps of post-tensioning

1. A duct affixed to reinforcement in the member is laid to desired profile
2. The prestressed tendons are placed inside the installed ducts
3. The member is cast and left to hardened and reach the desired strength for transfer stage.
4. The tendons are tensioned on one or both ends with the aid of a jack, held in position with wedge plate, and then anchored on the concrete using external permanent anchors
5. The protruding tendons from the anchors are cut off.
6. The duct is filled with a grout under pressure in order to establish a bond between concrete and steel and also as protection against corrosion.
7. The wedge plate and cables are covered with grout cap and other layer of protection deem necessary in the working environment.

 

Design of Prestressed concrete for Serviceability.

 Prestressed concrete is designed in reverse order as reinforced concrete design. In prestressed concrete design, the member is designed for serviceability limit state of cracking by limiting the stress to predetermined value while further checks are made for the ultimate limit state of flexure, shear etc.

Critical Stages in Prestressed concrete design for Serviceability.

When designing for the limiting permissible stress, two stages are important.

• Transfer Stage: This is the stage just after the prestress force is transferred to the member and there are only immediate losses. The loss of prestress at transfer stage is often estimated to be about 10% and the remaining prestress is denoted by Pmo
• Service Stage: This is the stage when the member is in full-service condition and there are already time-dependent losses. The loss of prestress at service stage is often estimated to be about 25% and the remaining prestress dubbed effective prestress is denoted by Pm,t

 

Permissible stress in concrete

 Eurocode 2 gives guidance on the permissible stress in prestressed concrete members as discussed below:

 Permissible stress at transfer

• The permissible compressive stress at transfer is limited to 0.6f_ck

(f_ck is the characteristics cylindrical strength of concrete at transfer – which is at the time of stressing the tendon for post-tension members and at the time of cutting the tendon from end abutment for pre-tension members)

• The permissible tensile stress is limited to f_ctm (mean tensile stress of concrete)

where:

f_ctm = 0.30f_ck^2/3   when f_ck is less than 50/60

f_ctm = 2.12In (1.8 + 0.1fck) when f_ck is greater than 50/60

NB: f_ck is the cylindrical strength of concrete at transfer

 Permissible stress in service

• The permissible compressive stress in service is limited to 0.6f_ck or 0.45f_ck under quasi permanent load to ensure linear creep deformation (f_ck is the characteristics cylindrical strength of concrete at 28^th day after casting)
• The permissible tensile stress is limited to f_ctm (mean tensile stress of concrete)

f_ctm = 0.30f_ck^2/3   when f_ck is less than 50/60

f_ctm = 2.12In (1.8 + 0.1fck) when f_ck is greater than 50/60

NB: f_ck is the cylindrical strength of concrete at service

Should Sections be designed as cracked or uncracked?

Except when the structure is such that cracking is not permitted as in the case of penetrating liquid retaining structures, it is often more economical to design members such that controlled cracking is allowed. A member designed to allow for control cracking is called partially prestressed. When cracking is not allowed, the member is said to be fully prestressed.

The prestressing force required to make a member fully prestressed can be really large vis a vis economy. There is also concern of over-cambering in the transfer stage which may lead to cracking of the top region of the member, thereby compromising the compressive strength of the fibers at the top region of the member in service stage. All these and also considering that the required serviceability conditions to make the member optimally functional can be achieved in a partially prestressed member makes fully prestressed member unnecessary unless for exceptional scenarios.

Additionally, even when a member is designed as fully prestressed, it is still advisable that the designer takes precaution to provide extra bonded tendons or bonded conventional reinforcement to limit crack width if ever cracking happens. This is because cracks are very difficult to predict in concrete due to many factors. It might even be that the concrete had – latently though – cracked during transfer stage when the concrete is immature against all precautions. Add since Eurocode ordinarily permits crack in prestressed concrete, it is just prudent that designer explore this to prevent against excessive crack even if crack is not ordinarily intended in the design.

Method of Prestressed Concrete design for serviceability

• Combined Stress Method
• Load balancing method

The Combined Stress Equation for Serviceability Limit state

The combined stress method assumes elastic properties for concrete and superimposes the stresses acting on the member to get the final stress. The mid span and end support are the two most critical sections that are isolated to evaluate the final stress in the member.

The possible stresses acting on a prestressed member are always three:

1. The axial stress due to the prestress force (P/A).
2. The stress due to the eccentricity of the prestress force (Pe/Z).
3. The bending stress due to self-weight of the member and external load (M/Z).

 

 

These three stresses have to be considered to evaluate the final stress on the member. The stress due to the eccentricity of the prestress force (Pe/Z) will be zero if the prestress force is applied concentrically, while the bending stress due to self-weight of the member and external load (M/Z) will be zero at the support of the member.

 

The stress equation for serviceability limit state given a prestress force P and eccentricity e is given as:

At transfer

Top

$
\sigma \,\,=\,\,\left( \frac{-P}{A}\,\,+\,\,\frac{P\,\,e}{Z_t} \right) \gamma _{sup}\,\,-\,\,\frac{M_{\min}}{Z_t}
$

Bottom

$
\sigma \,\,=\,\,\left( \frac{-P}{A}\,\,-\,\,\frac{P\,\,e}{Z_b} \right) \gamma _{sup}\,\,+\,\,\frac{M_{\min}}{Z_b}
$

At Service

Top

$
\sigma \,\,=\,\,\left( \frac{-P}{A}\,\,+\,\,\frac{P\,\,e}{Z_t} \right) \gamma _{inf}\,\,-\,\,\frac{M_{\max}}{Z_t}
$

Bottom

$
\sigma \,\,=\,\,\left( \frac{-P}{A}\,\,-\,\,\frac{P\,\,e}{Z_b} \right) \gamma _{inf}\,\,+\,\,\frac{M_{\max}}{Z_b}
$

Click here to study a worked example on how these equations are applied to verify the stresses on a prestressed beam

The Magnel diagram

Determining the prestress force (P) and eccentricity (e) that serves both transfer stage and service stage can be very tedious using the combined stress equation, hence magnel diagram provides a graphical representation of the feasible region for prestress force and eccentricity that will satisfy the stress constraints. Click here to study more on magnel diagram.

Design of Prestressed Concrete for ultimate limit State

Often times when a member is designed for serviceability, it passes the ultimate strength check. This, however, is not always true. Hence the need for independent strength check is important.

Ultimate Limit State of Bending

The ultimate strength of a member in bending is one of the important criteria to check after the design of a prestressed member for serviceability. Analysis of prestressed section under ultimate load is similar to that of reinforced concrete, only that the initial strain due to prestress has to be considered.

Stress Block in Bending for Concrete

The stress distribution in a concrete section in compression can be simplified from rectangular parabolic to a uniform one of breath ηfcd over a depth of the neutral axis times λ as shown in the rectangular stress block below.

 

Stress and Strain in concrete

If Fck < 50 MPa,  λ = 0.8

If 50 < fck < 90MPa,  λ = 0.8 – ((fck – 50)/400)

 

If Fck < 50 MPa,  η = 0.8

If 50 < fck < 90MPa,  η = 0.8 – ((fck – 50)/200)

 

Fcd = fck/1.5

 

ε_cu3 = 3.5 x 10^-3   if σc ≤ 50MPa

ε_cu3 = (2.6 + 35 x ((90 – fck)/100)⁴ ) x 10^-3    if σc > 50MPa

 

Stress and Strain in Steel

The stress and strain in steel in prestressed concrete is due to prestress and due to bending

Stress and Stain due to prestress

σ_pe  = Ps/As

ε_pe  = σ_pe/Es

Stress and Stain due to bending

ε_b = ε_cu3(d – x)/X

Total Stress and Stain due to bending and prestress

ε_t = ε_b + ε_pe

σ_t = ε_t x Es

Steps in calculating the ultimate strength of bending

• Assume a depth for the neutral axis
• Calculate the value of tensile force and compressive force for the assumed neutral axis and check for equilibrium. If there is no equilibrium, start from step 1 again until equilibrium is achieved.
• Haven found equilibrium for a particular neutral axis, take the moment of the tensile and compressive forces about the soffit or any common point within the member to get the ultimate bending strength.

Click here to study a worked example on design of prestressed concrete beam for bending capacity

Ultimate Limit State of Shear

Shear force is an internal force that tends to make the part of a structural member slide against another. It is often significant close to supports and under point loads. BS EN 1992-1-1: 2004 adopts the approach of variable strut inclination method for shear reinforcement design; this method allows for flexibility in the angle of inclination of the compressive strut between 22 and 45.

Shear resistance of section without Shear reinforcement

Before shear reinforcement are designed, it is important that the concrete section is checked whether it has sufficient shear capacity (V_Rdc ) to resist the maximum design shear force (V_Ed) that act on it without shear reinforcement. This is particularly important in prestressed member as the initial compression from prestress enhances the shear resistance capacity of the section without shear reinforcement and this is duly considered by the code.

Another thing to consider is whether the section to be designed for shear is cracked or uncracked in bending. The section tends to be cracked where three is high internal moment (M) and low shear force (V), and remains uncracked where large shear fore (V) shear force and low bending moment.

Shear resistance of section uncracked in bending

The shear resistance of section uncracked in bending is given in clause 6.2.2 (2) as

$
V_{Rdc\,\,}=\,\,\frac{I.b_w}{S}\,\,\sqrt{\left( f_{ctd} \right) ^2+\,\,\propto _1\sigma _{cp}f_{ctd}}
$

S is the first moment area of part of the sction above the centroid and about the centroid

Shear resistance of section cracked in bending

For member which shear reinforcement is not required, the shear capacity of cracked section is given in section 6.2.2 (1) as:

The shear resistance of section uncracked in bending is given in clause 6.2.2 (2) as

V_Rdc = (C_RdcK(100_ρLf_ck)^1/3 + K1σ_cp) b_wd  ≥ (Vmin + K1σ_cp )b_wd

where:  σ_cp = N_Ed/A_c

C_{Rdc =}  0.18/ϒ

K  = (1 +√200/d) ≤  2.0

ρL   = Asl/bwd  ≤ 0.02

K1  = 0.5

 

When the value of C_Rdc and K1 is substituted into the equation it becomes:

(0.12K(100_ρLf_ck)^1/3 + 0.5σ_cp) b_wd

 

Location of cracked region within a member

Before designing a region of a beam as cracked or uncracked in bending, it is important to be able to decide at what point does crack start and end in such a member. A conservative approach can be taken by designing the sections of interest as both cracked and uncracked and then the least resistance is adopted.

Alternatively, since a section is considered as cracked if the flexural tensile stress exceeds fctd, a more flexible route is to equate the combined stress equation at the service stage to fctd and solve for x as shown in the below equation.

$
\frac{-P}{A}\,\,-\,\,\frac{P\,\,e}{Z_b}\,\,+\,\,\frac{0.5q\left( Lx\,\,-\,\,x^2 \right)}{Z_b}\,\,=\,\,f_{ctd}
$

The upper and lower limit of x provides the range of cracked region within the member, other sections outside this region can as well be designed as uncracked.

THE VARIABLE STRUT INCLINATION METHOD FOR SHEAR REINFORCEMENT DESIGN

As for section which has the member shear capacity (V_Rdc) less than the design shear force (V_Rdc) the shear reinforcement is required and designed using the variable strut inclination method. This method is based on an imaginary truss model where concrete acts as the top chord and also acts as the diagonal strut members inclined at an angle ϴ, the bottom chords are the main tensile reinforcement while the designed links serve as the transverse tension members. The angle ϴ changes in value proportionately to the shear force acting on the member. The angle ranges between the lower and upper limit of 22 and 45 degrees respectively. It should be noted that in this imaginary truss, the contribution of the concrete to shear resistance is ignored.

Before the vertical stirrup is designed, it is important to check whether the design shear force is not too large enough to cause the crushing of the inclined compressive strut. To ensure this does not happen, the maximum resistance of the section must be greater than the design shear force.

ie: V_Rdmax    > V_Ed

The formula for V_Rdmax is given in equation (6.9) of the code as:

V_Rdmax     =   ∝_cwb_wZv₁f_cd/(cotϴ + tanϴ)

where ∝_{cw  is:}

_{1 +  σcp/fcd  for 0 ≤ σcp/fcd ≤ 0.25}

1.25 for 0.25 ≤ σ_{cp/fcd  ≤ 0. 5}

_{2.5 (1 – σcp/fcd) for 0.5 ≤ σcp/fcd  ≤ 1}

v₁ according to equation 6.6N is:

v_{1 = 0.6(1 – fck/250)}

z = 0.95d

_{fcd = fck/1.5}

When the value of v_1, fcd, and Z are entered into the formula, it can be further simplified as:

V_{Rdmax       = (}0.36∝_cwbwd(1-fck/250)fck)/(cotϴ + tanϴ)

The value of ϴ to be used to check whether the section is adequate such that the compressive strut is not crushed is 45 degrees as this is the highest value of ϴ allowed by EC2. When 45 is substituted for ϴ then the equation becomes:

V_Rdmax(45)   =   0.81∝_cwbwd(1 – fck/250)fck

Should V_Rdmax be less than  V_Ed, the compressive strut will fail by crushing so the section is inadequate and have to be resized. However, if V_Rdmax is greater than V_Ed then we can proceed to designing the vertical links that will be required to resist the shear in the member. Different values of ϴ can be tried to achieve a shear resistance greater than the design shear force. The first value to be tried is 22 degrees as this angle gives the least resistance. If substituting 22 for ϴ does not give sufficient resistances, then a higher value of ϴ will be tried.

As an alternative and easier workaround to trying different values for ϴ until a suitable one is found, the design shear force should be equated to V_Rdmax so that ϴ is made the subject of the formular and a suitable value of  ϴ  is obtained in one swoop. Thus:

V_Ed = (0.36∝_cwb_wd(1 – f_ck/250)f_ck)/(cotϴ + tanϴ)

Once the perfect value of ϴ is obtained having modified the above equation to make ϴ the subject of the formular, this value can be plugged into the equation below: To get the shear resistance of a vertical stirrup of area A_sw and spacing s, the code gives the equation to be used as:

V_Rd,s = (A_sw/s)Zf_ywdCotϴ 

Where; f_ywd = design yield strength of shear reinforcement.

Substituting for Z = 0.95d and f_ywd = f_ywk/1.15, then the equation becomes

The ratio of the Area and spacing of the link should not however be less than the minimum ratio as given in equation (9.5N) thus:

Asw/s = 0.08b_w(√f_ck/f_ywk)

Additional Longitudinal tension reinforcement

There’s always need to provide additional longitudinal tension reinforcement at the bottom face of the section to resist the horizontal component of the force in the inclined strut. The required force to be resisted is given in equation (6.18) of EC2 which goes:

F_td:  0.5V_Ed(Cotϴ – Cotα)

For all practical purpose, when curtailment length of longitudinal bars is increased beyond the position where they are not needed, then required force to counteract the longitudinal component of the force on the strut is unwittingly provided, hence separate calculation for additional longitudinal steel is often superfluous.

Click here to Study a Worked Example on design of Prestressed Concrete Beam for Shear Strength