8/13/2019 Prestressed Cracking Moment Example

1/11

Cracking Moment Part B

Dr Antonis Michael

Department of Civil Engineering

Frederick University

What Section Shall be Chosen to ComputeI, A, yp, yb

Bonded Beams

Transformed section

Unbonded Beams

Net concrete section

Method used for effective pre-stress

(Prestressing force accounting for thelosses)

Practical Problems Use gross section

8/13/2019 Prestressed Cracking Moment Example

2/11

8/13/2019 Prestressed Cracking Moment Example

3/11

8/13/2019 Prestressed Cracking Moment Example

4/11

Ultimate Moment Capacity

Loading StagesTypical loading history and stress distribution across the depth

(a) Beam Section, (b) Initial Pre-Stressing, (c) Self Weight and Effective

Prestress, (d) Full Permanent Load and Effective Prestress, (e) Full ServiceLoad and Effective Prestress, (f) Limit State of Stress at Ultimate Load

(a) (b) (c) (d) (e) (f)

Comp. Comp. Comp.Tension

Comp. Comp. Comp. Tension

8/13/2019 Prestressed Cracking Moment Example

5/11

Initial Prestressing

Full Perm. Load

Balanced

Decompression

First Cracking Load

Service Load LimitIncluding Overload

Steel YieldingUltimate

Load

DeformationCamper Deflection

Ultimate Moment

Bonded Tendons

8/13/2019 Prestressed Cracking Moment Example

6/11

Conditions

Flexural Failure

No shear, bond or anchorage failure

Beams are bonded (Unbonded beams havedifferent ultimate strength)

Beams are statically determined (Forcontinuous beams use plastic hinge theory)

Quasi-static ultimate load (impact, fatigue or

long term loading not considered)

Method

Principle of a resisting couple in aprestressed beam (C-Line Method)

T = C

8/13/2019 Prestressed Cracking Moment Example

7/11

Modes of Failure of Prestress Concrete

Steel Steel

Failure start Concrete Failure end Concrete

General Case: Under reinforced section

Failure starts with excessive elongation of steel& ends with crashing of concrete

Uncommon Case: Over reinforced sectionConcrete is crushed before the steel is stressed

into the plastic range limited deformationbefore rupture (brittle mode of failure)

Another Uncommon Case: Too Lightlyreinforced section

Failure occurs by breaking of the steel followingthe cracking of concrete. This happens when

the tensile force in the concrete is suddenlytransferred to the steel with an area too smallto take the tension

8/13/2019 Prestressed Cracking Moment Example

8/11

There is NO SHARP LINE between thepercentage of reinforcement for an overreinforced and an under reinforced beam but agradual transition.

Why?

Prestressing steel does NOT exhibit a sharpyield point Therefore a sharp definition of

balanced condition CAN NOT be made.

Design Assumptions

- plane sections remain plane.

- the strain in bonded reinforcement or bondedprestressing tendons, whether in tension orin compression, is the same as that in thesurrounding concrete.

- the tensile strength of the concrete is ignored.

- the stresses in the concrete in compressionare derived from the design stress/strainrelationship given in 3.1.7.

8/13/2019 Prestressed Cracking Moment Example

9/11

Design Assumptions

- the stresses in the reinforcing or prestressingsteel are derived from the design curves in3.2 (Figure 3.8) and 3.3 (Figure 3.10).

- the initial strain in prestressing tendons istaken into account when assessing thestresses in the tendons.

EN 1992 1 Section 3.3.6 (Steel MaterialProperties)(1)P Structural analysis is performed on the basis of the nominal

cross-section area of the prestressing steel and the characteristicvalues fp0,1k, fpk and uk.

(2) The design value for the modulus of elasticity, Ep may beassumed equal to 205 GPa for wires and bars. The actual valuecan range from 195 to 210 GPa, depending on the manufacturingprocess. Certificates accompanying the consignment shouldgive the appropriate value.

(3) The design value for the modulus of elasticity, Ep may beassumed equal to 195 GPa for strand. The actual value canrange from 185 GPa to 205 GPa, depending on themanufacturing process. Certificates accompanying theconsignment should give the appropriate value.

8/13/2019 Prestressed Cracking Moment Example

10/11

EN 1992 1 Section 3.3.6 (Steel Material

Properties)(4) The mean density of prestressing tendons for the purposes of design maynormally be taken as 7850 kg/m3

(5) The values given above may be assumed to be valid within a temperaturerange between -40C and +100C for the prestressing steel in thefinished structure.

(6) The design value for the steel stress, fpd, is taken as fp0,1k/S (see Figure3.10).

(7) For cross-section design, either of the following assumptions may be made(see Figure 3.10):

- an inclined branch, with a strain limit ud. The design may also be basedon the actual stress/strain relationship, if this is known, with stress abovethe elastic limit reduced analogously with Figure 3.10, or

- a horizontal top branch without strain limit.

Note: The value of ud for use in a Country may be found in its NationalAnnex. The recommended value is 0,9 uk. If more accurate values are notknown the recommended values are ud = 0,02 and fp0,1k /fpk = 0,9.

EN1992 1 Section 3.3.6

A IdealisedB Design

8/13/2019 Prestressed Cracking Moment Example

11/11

EN 1992 1 Section 3.3.6 (Concrete

Material Properties)(3) A rectangular stress distribution (as given in Figure 3.5) may be assumed.

The factor , defining the effective height of the compression zone and thefactor , defining the effective strength, follow from:

= 0,8 for fck50 MPa

= 0,8 - (fck -50)/400 for 50 < fck90 MPaand

= 1,0 for fck50 MPa= 1,0 - (fck -50)/200 for 50 < fck90 MPa

EN 1992 1 Section 3.1.7 (ConcreteMaterial Properties)

for fck < 50 MPa cu3(0/00) = 3,5

for fck 50 Mpa cu3(0/00)=2,6+35[(90-fck)/100]4