Basic assumptions

The following basic assumptions are required for the analysis of a prestressed concrete beam section:

• Plane sections before bending remain plane after bending.

• The stress-strain relationships of the materials are known.

• The relationship between the strain in the steel and the strain in the surrounding concrete is known.

Each of the three basic assumptions are discussed in the following with regard to their impact on the analysis of prestressed concrete beam sections.

Plane sections before bending remain plane after bending.

The strain distribution over the depth of a beam in bending varies as a function of the distance from the neutral axis. The first assumption implies that a linear relationship exists between the strain at a fibre in the concrete and its distance from the neutral axis as shown in Figure 4-3. A large number

x

e, y

Centroidal axis

Figure 4-2: Axial system for section properties.

M M

ec

k y Neutral

axis

(a) Beam subjected to flexure (b) Strain distribution with depth Figure 4-3: Plane sections remain plane during bending.

of tests on reinforced concrete members (Ref. 4-1) indicate that this assumption is very nearly correct at all stages of loading up to failure, provided that good bond exists between the concrete and the steel.

The assumption proves to be accurate for the concrete in the compression zone even at high loads close to the ultimate load. After cracks have developed in the tension zone, the tensile strain in the uncracked concrete between cracks is known to vary from zero at the crack to some non-zero value at positions located some distance away from the crack because of the action of bond between the steel and the surrounding concrete. Consequently the assumption that plane sections remain plane cannot be true in a cracked member when considering individual sections. However, if the gauge length for measuring strain is large enough to include a number of cracks, this assumption will hold for this “average” tensile strain (Ref. 4-1).

The first assumption does not hold for deep beams and regions of high shear. According to SABS 0100 (Ref. 4-2) a simply supported beam should be considered as being deep when the ratio of the height of the section to the effective span length exceeds ½.

Note that the validity of the first assumption has often been questioned for a number of reasons (Ref. 4-3):

• Most of the conclusions were derived from the results of tests on beams with rectangular cross sections and measurements were made in a region of constant moment.

• The strains are usually measured on the outside of the beam and it could be argued that this situation is not representative of conditions inside the beam.

• For non-rectangular sections, disturbances occur at points where the width of the member abruptly changes.

In spite of these objections, application of this assumption by many researchers has shown that a good correlation can be obtained between calculated and measured results so that it can be considered to be accurate enough for design purposes. In lieu of an alternative, this assumption will be used.

The stress-strain relationships of the materials are known.

The stress-strain relationships of both the prestressed and non-prestressed steel, as presented and discussed in section 2.2, can be used. However, it is important to note that the concrete is acting in flexure and not in direct compression or tension, and the relationship used for the purposes of flexural analysis must therefore take this into account.

A great deal of research has been carried out to determine the stress-strain relationship of concrete flexural elements. The most notable research was carried out by Hognestad et al (Ref. 4-4) and Rüsch (Ref. 4-5), and the following results were obtained:

• A similarity exists between the stress-strain relationships for concentrically loaded cylinders and the stress-strain relationship for eccentrically loaded beam specimens.

• The maximum stresses reached in the beam specimens were lower than the cylinder strengths, with the difference increasing with an increase in cylinder strength.

• The stress-strain relationship for beam specimens could be determined for strains much larger than the strain at which the maximum stress occurs. The determination of the stress-strain relationship for concentrically loaded cylinders beyond the cylinder strength is complicated by the fact that special testing equipment is required.

• The maximum strain e_cu reached in the extreme compression fibre in bending is a function of the concrete strength, decreasing with an increase in cylinder strength. Rüsch (Ref. 4-5) has shown that the strain at the extreme compression fibre at maximum moment is also a function of the shape of the cross-section.

If the above points are kept in mind, the stress-strain relationships obtained for concrete in direct compression can be applied to beams in bending. The parabolic-rectangular stress-strain relationship recommended by the design codes of practice commonly used in South Africa (Refs. 4-2, 4-6, 4-7 and 4-8) is shown in Figure 4-4. The purpose of the 0.67 factor is to take into account the differences between the cube strength f_cu and the experimentally obtained results for beams in bending. A constant value of 0.0035 is recommended for ecu and the partial factor of safety gm is discussed later.

Because the stress-strain relationship is usually difficult to determine and to deal with computation-ally, much research has been carried out to represent the stress distribution in the compression zone of a beam at ultimate as an equivalent rectangular stress-block. The recommendations given by SABS 0100 (Ref 4-2) and BS 8110 (Ref. 4-7) are summarized in Figure 4-5. It should be noted that the equivalent rectangular stress-block is only valid at ultimate, and not when considering flexural response at other levels of loading.

When calculating the response of the section at ultimate, the tensile strength of the concrete is usually ignored because its influence on the moment of resistance is small. This follows because the concrete in the tension zone is usually cracked at ultimate, so that the remaining area in tension is small with a correspondingly small lever-arm. The tensile strength becomes more important when calculating deformations at loadings appropriate to the serviceability limit state, and its influence on behaviour should be accounted for at these load levels.

Parabolic curve Stress

Strain ec0

E_ci gm

0. 67fcu

ecu= 0.0035

e^c g

cu m

f

0

2 4 10 4

= . ´

-E f

ci

cu m

=55.

g GPa

fcuin MPa

Figure 4-4: Parabolic-rectangular stress-strain relationship for concrete in flexure (Refs. 4-2, 4-6, 4-7 and 4-8).

ec0

ecu= 0.0035 _g

m

0. 67fcu

gm

0. 67fcu

s= 0.9x x

Parabolic-rectangular stress block

Equivalent rectangular stress block Strain

distribution Neutral axis

Figure 4-5: Rectangular stress-strain relationship for concrete in flexure (Refs. 4-2 and 4-7).

The relationship between the strain in the steel and the strain in the surrounding concrete is known.

The strain in the concrete at the level of the steel is calculated by making use of the assumption that plane sections remain plane, and the change in strain in the steel is subsequently obtained by assuming that it is equal to the calculated strain in the concrete at the level of the steel. This approach applies to all bonded steel.

The distribution of the strain in unbonded tendons is assumed to be uniform along the length of the member, even though it actually varies to some degree because of the effects of friction. Under these conditions, the total change in length of the concrete at the level of the prestressing steel is assumed to be equal to the change in length of the prestressing steel. The implications of this assumption are discussed in Section 4.3.6.