(4.08 x 10fi x 24 - 59.4 x !(/') 0.8(4.08 x 106/70000 - 75)
and allowing for the division by the negative denominator P{) > -2881 kN
Similarly from equations 11.18 lo 11.20: Pn < +1555 kN P0 > +557 kN P{) < +654 kN
The minimum value of preslress force is therefore 557 kN with an upper limit of 654 kN.
(b) Check the upper economic limit to preslress force From equation 11.23:
Since this is greater than the upper limit already established from equation 11.20 a design with an initial prestressing force between 557 kN and 654 kN will be acceptable.
The calculation in example ! 1.3 is based on the characteristic loads. Once a value of preslress force lying between the minimum and upper limit value is chosen, the compressive stress at the top of the section under the quasi-permanent loads should also be calculated and compared with the lesser allowable value of 0.45/C|<. If this proves to be critical then the section may have to be redesigned taking the quasi-permanent load condition as more critical than the characteristic load condition.
Stress under quasi-permanent loads
For the previous example, using minimum preslress force of 557 kN. check the stress condition under the quasi-permanent loading condition. Assume that the 3kN/m imposed load consists of a permanent load of 2kN/m as finishes and I.OkN/m variable load. Take 30 per cent of the variable load contributing to the quasi-permanent load. From the previous example:
Moment due to self-weight = 21.9 kN m
Moment due to finishes = 2 x 102/8
Moment due to variable load I x 10-/8
Stress al the lop of section is given by: _ KP0 KP{)e M
_ 0.8 x 557 x 103 0.8 x 557 x 103 x 75 50.65 x 10fe ~ 70000 4.08 x I06 + 4.08 x 106
10.59 N/mirr
Allowable compressive stress 0.45/k 0.45 x 40 l8N/mm2.
Hence the maximum compressive stress is less than the allowable figure.
Equations 11.17 to 11.20 can be used to determine a range of possible values of prestress force for a given or assumed eccentricity. For different assumed values of eccentricity further limits on the prestress force can be determined in an identical manner although the calculations would be tedious and repetitive. In addition, it is possible to assume values of eccentricity for which there is no solution for the prestress force as the upper and lower limits could overlap.
A much more useful approach to design can he developed if the equations are treated graphically as follows. Equations 11.9 to 11.12 can be rearranged into the following form:
{equation 11.12}
These equations now express linear relationships between I /P,t and e. Note that in equation 11.25 the sense of the inequality has been reversed to account for the fact that the denominator is negative (/¿in is negative according to the chosen sign convention). The relationships can be plotted as shown in figure 11.12(a) and (b) and the area of the graph to one side of each line, as defined by the inequality, can be eliminated, resulting in an area of graph within which any combination of force and eccentricity will simultaneously satisfy all four inequalities and hence will provide a satisfactory design. The lines marked 1 to 4 correspond to equations 11.24 to 11.27 respectively. This form of construction is known as a Magnel Diagram.
The additional line (5) shown on (he diagram corresponds to a possible physical limitation of the maximum eccentricity allowing for the overall depth of section, cover to the prestressing tendons, provision of shear links and so on. Two separate figures are shown as it is possible for line I. derived from equation 11.24, to have either a positive or a negative slope depending on whether fmM is greater or less than Mmay/zi.
The Magnel diagram is a powerful design tool as it covers all possible solutions of the inequality equations and enables a range of prestress force and eccentricity values to be investigated. Values of minimum and maximum prestress force can be readily read from the diagram as can intermediate values where the range of possible eccentricities for a chosen force can be easily determined. The diagram also shows that the minimum prestress force (largest value of 1 /Pq) corresponds to the maximum eccentricity, and as the eccentricity is reduced the prestress force must be increased to compensate.
Construction of Magnel diagram
Construct the Magnel diagram for the beam given in example 11.2 and determine the minimum and maximum possible values of prestress force. Assume a maximum possible eccentricity of 125 mm allowing for cover etc. to the tendons. From (he previous examples:
K = 0.8 Zb = z, = 4.08 x 106 mm3 A = 70000 mm2 From equation 11.24: 1 > K(\/A-e/z<)
v70 000 4.08 x106J /\ 4.08 x 106 which can be re-arranged to give: 106
and similarly from the other three inequalities, equations 11.25 to 11.27:
"o
10fi
"o
These inequalities are plotted on the Magnel diagram in figure 11.13 and the zone bounded by the four lines defines an area in which all possible design solutions lie. The line of maximum possible eccentricity is also plotted but, as it lies outside the zone bounded by the four inequalities, does not place any restriction on the possible solutions.
From figure 11.13 it can be seen that the maximum and minimum values of prestress force are given by-
Maximum I()VFo = 2415; hence minimum P» 4l4kN (e = 121 mm)
Minimum l06/Fo = 862; hence maximum F„ I l60kN (c = 17 mm)
The intersection of the two lines at position A on the diagram corresponds to a value of Pi) — 1050 kN. established in example 11.3 as the maximum economical value of prestress force for this section (see equation 11.23). Hence the intersection of these two lines should be taken as the maximum prestress force and. as can be seen, this information can be readily determined from the diagram without the need for further calculation.
The Magnel diagram can now be used to investigate other possible solutions for the design prestressing force and eccentricity. For a fixed value of prestress force (and hence fixed value of l/F()) the corresponding range of permissible eccentricity can be read directly from the diagram. Alternatively, if the eccentricity is fixed, (he diagram can be used to investigate the range of possible prestress force for the given eccentricity.
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