Flanged sections

Flanged sections such as those shown in Figure 4.13aare commonly used in prestressed concrete construction, where the bending efficiency of I-, T-, and box-shaped sections can be effectively utilized. Frequently, in the construction of prestressed floor systems, beams or wide bands are poured monolithically with the slabs. In such cases, a portion of slab acts as either a top or a bottom flange of the beam, as shown in Figure 4.13b. Codes of practice generally specify the width of the slab which may be assumed to be part of the beam cross-section (i.e. the effective width of the flange, b_ef).

Both AS 3600–1988 and BS 8110: Part 1 (1985) contain the following

Figure 4.13 Typical flanged sections.

simple recommendations:

(4.33)

except that the overhanging part of the effective flange should not exceed half the clear distance to the next parallel beam. The term bwis the width of the web of the section. Lois the distance along the beam between the points of zero bending moment and may be taken as the actual span for simply supported members and 0.7 times the actual span for a continuous member.

ACI 318–83 suggests that the effective width of the flange of a T-beam should not exceed one quarter of the span length of the beam, and the effective overhanging flange width on each side of the web should not exceed either eight times the slab thickness or one half the clear distance to the next web. For L-beams with a slab on one side only, the effective

overhanging flange width should not exceed either one twelfth of the span of the beam, or six times the slab thickness, or half the clear distance to the next web.

The flexural strength theory discussed in Section 4.3can also be used to calculate the flexural strength of non-rectangular sections. The equations developed earlier for rectangular sections are directly applicable provided the depth of the idealized, rectangular stress block is less than the thickness

of the flange, i.e. provided the portion of the section subjected to the uniform compressive stress is rectangular (befwide and γdndeep). The ultimate strength Muis unaffected by the shape of the section below the compressive stress block. If the compressive stress block acts on a non-rectangular portion of the cross-section, some modifications to the formulae are necessary to calculate the resulting concrete compressive force and its line of action.

Consider the T-sections shown inFigure 4.14and the idealized rectangular stress block defined in Figure 4.2d. If , the area of the concrete in compression A' is rectangular, as shown inFigure 4.14a, and the strength of the section is identical with that of a rectangular section of width befcontaining the same tensile steel at the same effective depth. Equation 4.21 may therefore be used to calculate the strength of such a section. The depth of the neutral axis d_nmay be calculated using Equation 4.20, except that b_efreplaces b in the denominator.

If γd_n>t, the area of concrete in compression A' is T-shaped, as shown inFigure 4.14b.

Although not strictly applicable, the idealized stress block may still be used on this non-rectangular compressive zone. A uniform stress of may therefore be considered to act over the area A'.

It is convenient to separate the resultant compressive force in the concrete

Figure 4.14 Flanged section subjected to the ultimate moment.

into a force in the flange Ccfand a force in the web Ccw, as shown:

(4.34)

By equating the tensile and compressive forces on the section, the depth to the neutral axis d_n can be determined by trial and error and the ultimate moment Mucan be obtained by taking moments of the internal forces about any convenient point on the cross-section.

Example 4.9

The ultimate flexural strength of the standardized double tee section shown inFigure 4.15is to be calculated. The section contains a total of 22 12.7 mm diameter strands (11 in each cable) placed at an eccentricity of 408 mm. The effective prestressing force Pe is 2640 kN.

The stress–strain relationship for the prestressing steel is shown inFigure 4.16and the initial elastic modulus is E_p=195000 MPa. The properties of the section and other relevant material data are as follows:

Using the same procedure as was illustrated inExample 4.1, the strain components in the prestressing steel are obtained from Equations 4.8–4.10:

Figure 4.15 Standard 2400×800 double tee (CPCI 1982).

and therefore from Equation 4.11,

At this point, an assumption must be made regarding the depth of the equivalent stress block.

If γdnis less than the flange thickness, the calculation would proceed as in previous examples.

However, a simple check of horizontal equilibrium indicates that γdnis significantly greater than 50 mm. From Equation 4.34:

In this example, the web is tapering and b_w varies with the depth. The width of the web at a depth of γd_nis given by

The compressive force in the web is therefore

The resultant compression force is

and the resultant tension is

Equating C and T gives

Trial values of dnmay now be used to determine εpuand σpufrom the above expressions and the resulting points plotted on the stress–strain diagram ofFigure 4.16:

Trial dn εpu σpu Point plotted onFigure 4.16

130 0.0199 1919 (1)

110 0.0228 1832 (2)

115 0.0220 1854 (3)

Figure 4.16 Stress-strain for strands inExample 4.9.

FromFigure 4.16, the neutral axis is close enough to d_n=115 mm. The depth of the stress block is γdn=92.1 mm, which is greater than the flange thickness (as was earlier assumed).

The resultant forces on the cross-section are

For this section, dn=0.17d<0.4dpand therefore the failure is ductile. The compressive force in the flange C_cf=3570 kN acts 25 mm below the top surface and the compressive force in the web C_cw=509 kN acts at the centroid of the trapezoidal areas of the webs above γd_n, i.e. 71.0 mm below the top surface.

By taking moments of these internal compressive forces about the level of the tendons,

4.7 References

ACI 318–83 1983. Building code requirements for reinforced concrete. Detroit: American Concrete Institute.

AS 3600–1988. Australian standard for concrete structures, Sydney: Standards Association of Australia.

BS 8110: Part 1 1985. Structural use of concrete, part 1, code of practice for design and construction.

London: British Standards Institution.

Canadian Prestressed Concrete Institute (CPCI) 1982. Metric design manual—precast and prestressed concrete. Ottawa: Canadian Prestressed Concrete Institute.

Loov, R.E. 1988. A general equation for the steel stress for bonded prestressed concrete members.

Journal of the Prestressed Concrete Institute, 33, 108–37.

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