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neutral axis

Section d'= 60

neutral axis t cK

Section

Strain Diagram

The location of the plastic centroid is determined by taking moments of all the strc -resultants about an arbitrary axis such as AA in figure 4.21 so that

_ 0.5677'iA-c x 450/2 + 0.87/ykA„' x 60 + 0.87fykAs x 390 0.567fckAcc + 0.87/y,^ + 0.87/ykAs 0.567 x 25 x 350 x 4502/2 + 0.S7 x 500( 1610 x 60 4- 9S2 x 390) 0.567 x 25 x 350 x 450 -t- 0.87 x 5<K)( IMoT 982) = 212 mm from A A

The fundamental equations for calculating points on the interaction diagram with varying depths of neutral axis are:

(i) Compatibility of strains (used in table 4.3, columns 2 and 3): esc =0.0035 (

es 0.0035

Table 4.3 M-N interaction values for example 4.10

2.63d- 158 0.00217 >0.00217 0.87fyk 0.87fyk 899 275

xbüi-0.617d = 241 >0.00217 0.00217 0.87 fyk 0.87/yk 1229 292

h — 450 >0.00217 0.00047 0.87 fyk 93.3 2580 146

oo 0.00217 0.00217 0.87 fyk 0.87 fyk 3361 0

or when the neutral axis depth extends below the bottom of the section (.v > //): £sc = 0m2l^l and ct = 0.002-7(A'"d)

>ii) Stress-strain relations for the steel (table 4.3. columns 4 and 5):

liii) Equilibrium (table 4.3. columns 6 and 7):

N = F„ + Fsc + Fs O.H.v < // N = 0.561 fckb x 0.8a- + fKA's + /A 0.8* > li N = 0.567fabh +fKA[ +/A

Taking moments about the plastic centroid

0.8x<h M = Fcc{xf - 0.ix/2) + Fm(xp - d') - F,(d ,vp) 0.8a- > h M F^ - h/2) + Fk(xp - d') - Fn{d - xf)

Fs is negative when /„ is a tensile stress.

These equations have been applied to provide the values in table 4.3 for a range of key values of x. Then the M-N interaction diagram has been plotted in figure 4.22 from the values in tabic 4.3 as a series of straight lines. Of course. N and M could have been calculated for intermediate values of x to provide a more accurate curve.

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