Obras Civiles

Three methods are available for the analysis of prestressed concrete sections in flexure.

These are:

1. The section may be analysed in accordance with assumptions (a) to (f) by using the stress-strain curves given in Figures 2.1, 2.2 and 2.3 for concrete, reinforcement and tendons respectively.

2. The section may be analysed in accordance with assumptions (a) to (f) by using the rectangular stress block described in assumption (b) for concrete and the stress-strain curves given in Figures 2.2 and2.3 for reinforcement and tendons respectively.

3. The section may be designed in accordance with 4.3.7.3 and 4.3.7.4 by using design formulae and Table 4.4.

Methods 1 and 2, for which a t\’pical section, stress distribution and strain profile are shown in Figure H4.4, are applicable to sections of any shape with any distribution of tendons and reinforcement. The analysis normally involves a trial and error approach.

using strain compatibility, in order to determine the neutral axis depth for which the forces acting on the section are in equilibrium, as follows:

1. Assume a value for the neutral axis depth x and draw a strain profile by taking a value of 0.0035 at the extreme compression fibre.

2. Calculate the corresponding strain at the level of the tendons and add to this the 89

Handbook to BS8JIO:1985

Figure H4. 4: Design of prestressed concrete sections in flexure.

prestrain in the tendons after all losses have occurred. Where appropriate. also calculate the strain at the level of the reinforcement.

3. Determine the stresses in the reinforcement and the tendons from Figures 2.2 and 2.3 respectively and calculate the corresponding forces.

4. Calculate the force in the concrete in the compression zone and compare this with the total force in the reinforcement and tendons.

5. Modify the neutral axis depth and repeat steps 1 to 4 until equilibrium of forces is obtained.

6. Take moments about a convenient position. such as the neutral axis. for all the forces acting on the section.

Methods I and 2 will give very similar answers with method 2 being easier to apply.

particularly for sections with non-rectangular compresion zones. Method 3, which is directly applicable to sections with rectangular compression zones only, will give approximately the same answer as methods 1 and 2 in this case.

4.3.7.3 Design formulae

An approximate method for obtaining the resistance moment of sections. in which the tendons are effectively concentrated at one position in the tension zone, has been derived in accordance with assumptions (a) to (f). The design stress-strain curve for a high strength tendon (f~~=1860N/mm. E¶=I9SkN/m&) has been used and the non-dimensional stress terms are slightly conservative for lower strength tendons. The method uses the rectangular stress block for the concrete and is directly

that are rectangular for a depth (from the compression face) of ^- applicable to sections

x isthe neutral axis depth. not less than 0.9x. where

For a beam containing bonded tendons. equation 51 may be rewritten in a form that is suitable for plotting graphically, as follows:

[I

— —

.

-

~ ~

.

-K K ^.

^—

-—-~-‘:-‘.‘.:-.-:

-Part 1: Section⁴

0.25

0.20

0.15

4.,bd2 cia

0.05

0

0.Sx

fp

.

fp.

Figure H4. 6: Design of T beams for flexure.

Using the values of fpb/O.87fp~ andx/dgiven in Table 4.4, a graph ofM~/f~bd2against f~A~If~bd may be produced as shown in Figure H4.5.

For a flanged section in which the flange thicknessh~~0.9x. the equation or the graph may be used by taking the width of the section as the flange widthb,as shown in Figure H4.6(a). Where h~<0.9x, as shown in Figure 4.6(b), the forces acting on the section may be equated to provide:

Ap~fpb=0.45 f~[(b—bW)h~±0.9bWxI

which may be rewritten in the following form:

fpbIO.87fpU=0.52 ~ [(bIb~—1)hfld±0.9x/d]

This linear relationship between fpb/O.87f~., andx/dmay be plotted graphically, together with the appropriate relationship given in Table 4.4, as shown in Figure H4.7. The intersection of the two relationships gives the values offpb/0.87f~~andx/dto be used in equation 51 with an appropriate value of4.Alternatively, M~may be calculated from:

M~=0.45f~ [(b—bW)hf(d0.5hr)+0.9bWx(d0.45x)1

For a beam containing unbonded tendons, equation 52 has been developed from the results of tests in which the stress in the tendons and the length of the zone of inelasticity in the concrete were both determined146-^4.7)

The beam is considered to develop both elastic and inelastic zones and the length of the inelastic zone is taken to be lOx. The extension of the concrete at the level of the tendons is assumed to be negligible in the elastic zones and the extension in the inelastic zone is assumed to be taken up uniformly over the length I of the tendon. Thus, the total strain 8pb in the tendons is given by:

Cpb = e~+0.0035 [(d—x)/x](lOx/I)

I

f1,Ap

.

4~bd

Figure H4.S: Design chart for prestressed rectangular beams (bonded tendon).

b

0.9xp

(a)- Ib)

91

Handbook to 858110:1985

1.0

0-s

0.87 0.8

0.7

0.6

Figure H4. 7: Treatment of flanged section it-here H~=O. 9x.

The corresponding stress in the tendons is then given by

fpb = f~+(0.035E~) (d!I)(1—x/d) ~ 0.7f~

The neutral axis depth may be determined by equating the forces acting on the section which, for a section with a rectangular compression zone. provides:

Ap.fpb = 0.45f~b(0.9x)

This may be rewritten in the form of equation53 and substituted in the expression for

fpb to give:

fpb = f~~+(0.035EO(d/I)[1 ~2.47(~fPUAP5)/cfCIIbd))cfPb/fPU)] ~ O.7fpu

Equation 52 is then obtained by putting E, = 200kN/mm and, as an approximation.

= 0.7 in the last term.

4.3.7.4 .4llowance for additional reinforcement in the tension :one

This approximate method for taking account of additional reinforcement in the tension zone will generally under-estimate the contribution of the reinforcement. A more rigorous strain compatibility analysis ~vouldallow the full design strength of a grade 460 steel to be utilised for all values of x ~ 0.636d. for example.

4.3.8 Design shear resistance of beams