Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

Page 1 of 11

CHAPTER 7

CONTINUOUS PRESTRESSED CONCRETE

BEAMS

7.1 INTRODUCTION

Advantages of Continuous Prestressed Concrete Structures

- Smaller bending moments indeterminate structures results in smaller deflections

which in turn, may justify shallower sections or larger spans.

- In prestressed continuous structures the continuous tendons placed provide up to

twice as much upward force as the tendon in a simply supported member (Fig. 1).

- At ultimate a continuous cable provides flexural resistance in the positive moment

region as well as the negative region resulting in a much higher flexural resistance

(Fig.2).

- The elimination of expansion joints at intermediate supports and a reduction of the

number of prestressing anchors.

- Redundant structures are capable of carrying more overload than simply supported

structures.

Disadvantages of Continuous Prestressed Concrete Structures

- Higher friction losses

- Shortening of the member can induce forces in supporting members

- In the presence of high live loads the design of continuous prestressed members

may be problematic. A high live load to dead load ratio will result in moment

reversals in the span(s) adjacent to the loaded span. Moment reversal requires that

the tendon eccentricity in the span has to be small which, in turn, requires an

increase in prestressing force. In such structures as crane girders or short railway

bridges it is actually better to use simply supported girders.

- Differential settlement, if any, may introduce substantial moments

- Simultaneous occurrence of large M + V more difficult to analyze and design

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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7.2 FORCES DUE TO PRESTRESSING IN STATICALLY

INDETERMINATE MEMBERS

The design of continuous structures requires a good understanding of the internal forces

generated by prestressing. In statically indeterminate structures so-called secondary

moments are introduced in addition to the primary moments. The total moments due to

prestressing, Mp, are the sum of the primary moments, Mp', and the secondary moments,

Mp":

''' ppp MMM = +

The primary internal forces can be calculated directly from the prestressing force, its

eccentricity and its slope. The forces exerted by the prestressing tendons on a statically

determinate concrete members are:

Normal force Np' = P cos α ≅ P

Bending moment Mp' = Ne ≅ Pe

Shear force Vp' = P sin α

The moment Mp is combined with the moments due to (unfactored) dead load MD, and

(unfactored) live load ML for stress analysis. When checking flexural strength, the

secondary moment Mp'' is added to the load side of the equation.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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7.3 METHODS OF ANALYSIS FOR STATICALLY

INDETERMINATE STRUCTURES

The redundant secondary moments due to prestressing can be found by any method of

analysis. However, for the case of parabolic cables in a structures subjected to uniformly

distributed gravity loads the so-called load-balancing method introduced by T.Y. Lin may

be used advantageously.

The most frequently used methods to determine the effects of prestressing are:

- load balancing method

- force method

- equivalent load method

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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7.3.1 Load Balancing Method (parabolic cables only)

A parabolic cable produces a uniform load of the following intensity:

2

8l

sPwp =

In this equation s is the cable sag, l is the span and P is the prestressing force. P being a

negative force when acting on the concrete means that in a simple span member wp is a

negative (upward) load per unit length. This upward load counteracts a portion of the

gravity load w. The gravity load is now split into two parts:

(1) The balanced load: wbal = -wp

(2) The unbalanced (net) load: wnet = w – wbal = w + wp

Since wbal + w = 0, no bending moments, or shear forces, will be present under the

balanced load and the only force to act on the beam is the axial force P. Thus, under the

balanced load the stress in the member (statically determinate or indeterminate) is the

normal stress due to axial load P:

APfa =

The unbalanced load wnet produces moments, and stresses, as in any statically determinate

or indeterminate, member.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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The flexural stresses due to the unbalanced load moment Mnet is calculated in the usual

way:

yI

Mf netb =

The total stress is the sum of fa and fb.

In statically determinate structures the splitting of the load into balanced and unbalanced

components does not offer any advantage in the analysis. In continuous structures,

however, the load balancing concept simplifies the analysis in that no secondary moments

have to be determined.

The load balancing concept also offers the advantage that the effect of prestressing can be

easily visualized and understood. It also allows the engineer to establish quickly how

much of the dead load, wD, or of the dead plus live load, wD+L , should be balanced by

prestressing in common types of structures.

For normal two-way flat slabs for example 75 to 85% of the dead load is normally

balanced by the effective prestressing force, Pe. In medium span bridges, balancing 80 to

100% of the dead loads will lead to satisfactory designs. The exact percentage of the dead

load to be balanced by prestressing, depends, of course, also on the live load, but the

percentages mentioned will allow a first estimate of the prestressing required for a

structure.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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The load balancing method is, however, applicable for analysis purposes only if parabolic

cables are used over the full length of the members. Any intermediate cable anchor as

they are used frequently in bridge design, or any straight cable makes the load balancing

method unworkable. Also, the reversed curvature which is always present in the region of

the intermediate supports makes the method somewhat inaccurate (Fig. 3). The magnitude

of the error introduced depends on the length (relative to the span) over which the

reversed curvature occurs and on the type of curve selected for the reversed curvature. If

second order parabolas are selected for the zones of positive and negative curvature of the

interior span of a continuous beam (see Fig. 3) with the lengths of the zones, of reversed

curvature a = αl, the error introduced when neglecting the curvature reversal in the

calculation of the moment at the supports is equal to α times the moment without

curvature reversal (i.e. for one parabola between supports A and B).

In order for the load balancing analyses to be within an acceptable margin of accuracy α

should not exceed 0.05 for the cable profile of Fig. 3.

It was mentioned before that one of the advantages of the load balancing method is that

the secondary moments need not be determined. This applies only to the elastic analysis

under service conditions. However, for the strength design the secondary moments, and

the related shear forces, have to be determined and added to the moments and shears due

to the factored loads.

Knowing the total moments and shears the secondary moments and shears are obtained

by deducting the primary forces from the total forces:

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

Page 7 of 11

ppp MMM ''' = −

ppp VVV ''' = −

Examples 1 and 2 demonstrate the application of the load balancing method to a two-span

continuous beam. The first of these examples is a hypothetical case with no cable

eccentricity at the middle support. This example is intended to demonstrate the basic idea

of load balancing. In the second example the cable eccentricity eB at the central support,

corresponds to a practical case where eB is chosen as large as possible.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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7.3.2 Force Method

To analyse an n-times statically indeterminate structure by the force method, the

following steps are followed:

1. The structure is made statically determinate by n releases.

2. The displacements D due to applied loads are determined at the releases.

3. Unknown forces F (moments or forces) are introduced at the coordinates where the

structure was released.

4. Displacements due to the unknown forces are determined.

5. The displacements (discontinuities) established in (2) have to be eliminated by those of

(4).

For a two span beam the compatibility equation is of the form

01111 + fFD =

When applied to find the forces due to prestressing, D1 is the displacement at coordinate 1

due to prestressing (primary moment) and F1 is the secondary moment. The displacement

D1 is determined by integration of the primary moments due to prestressing. The

flexibility coefficient f11, represents the displacement due to a unit moment applied at

coordinate 1.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

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Reactions In a statically determinate beam no reactions are produced by prestressing. In a statically

indeterminate beam reactions are caused by the secondary moments. For example in case

of a two span beam the reactions caused by prestressing are as follows:

- For edge support

lM

R pA

''=

- For interior support

lM

R pB

''2−=

Where M”p is the secondary moment due to prestressing at the interior support and l is the

span length.

Shear

The shear force at the edge support for the above mentioned beam would be:

lM

PVVV pppA

''sin''' +=+= α

Example 3 shows the calculations of moments and reactions due to the prestressing force

in a two span beam.

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

Page 10 of 11

7.3.3 Equivalent Loads

The force method as described in the previous section is useful for hand computation for

simple cases. For more complex cases it is useful to determine the forces exerted by the

prestressing tendons on the released concrete structure and use these forces in any

analysis. The equivalent loads are particularly for analysis by computer. The equivalent

loads for a basic case are shown in Fig. 4. The equivalent loads due to prestressing are:

1. The axial load due to prestressing

2. The transverse loads to prestressing, including the shears at the ends of the member and

at intermediate tendon anchors

3. Bending moments due to tendon eccentricities at the member ends and at intermediate

tendon anchors.

In Fig. 4 the prestressing force is assumed to be constant along the cable length. It is well

known, however, that in post-tensioned structures friction results in a variable

prestressing force along the tendons. The friction losses should be considered in the

calculation of the equivalent loads for major structures such as bridges with spans

exceeding 50 to 60 m.

In the analysis the equivalent loads should be checked carefully. For each span the

prestressing forces are self equilibrating so that the basic equilibrium equations have to be

satisfied:

∑ ∑ ∑ === 0,0,0 MFF yx

Example 4 shows the equivalent load of the prestressing force for a two span beam with

parabolic cable layout.

7.4 CABLE PROFILE

Helwan University Civil Engineering Department Faculty of Engineering-Mataria Theory & Design of Prestressed Concrete Prof. Dr. Alaa Sherif Dr. Hatem Seliem

Page 11 of 11

In most prestessed concrete bridges the live load moments are considerably smaller than

the dead load moments. Thus, the cable eccentricity should be as large as possible over

the intermediate supports and in the zones of maximum positive moments. The maximum

tendon eccentricity depends on:

1. Concrete cover (check fire rating for flat slabs)

2. Number of layers and diameter of non- prestressed reinforcement between tendon and

cover.

3. Diameter of transverse tendons. The eccentricity of the tendons is considerably

increased by placing some of the transverse tendons below the longitudinal tendons.

4. The duct diameters. In this context it should be mentioned that the centroid of the duct

does not coincide with the centroid of the prestressing steel. The strands or wires are

concentrated on the convex side of a curved duct and only at the point of inflection is

the prestressing steel approximately in the middle of the duct. Using flat ducts instead

of round ones will increase both the longitudinal and transverse cable eccentricity.

5. For box girder bridges the maximum cable eccentricity at or near midspan depends to

some extend on the location of the construction joint (see Fig. 3.5)

In the design of a prestressed concrete bridge the profile of the tendons has to be

determined. The cable profiles for an interior span consisting of three parabolas are given

in Table 7.1.