Stability and strength of slender concrete deep beams - Reinforced Concrete Deep Beams - page 251

Line 2, having a slope exceeding N crit , will not intersect the M Î e add curve.
For such a line, the external moment M t always exceeds the internal moment
M, and equilibrium is impossible.
Line 3, having a slope equal to N crit , will touch the MÎe add curve. At the
tangent point C , equilibrium exists between Mt and M. The equilibrium is
obviously unstable.
It can be concluded from the above that the instability load N crit of a
slender column is given by the slope of the line which touches the MÎe add
curve. In practice, N crit cannot be so readily found in this way, because the
MÎe add curve is itself dependent on the value of N. However, we can proceed
as follows.
A family of MÎe add curves are drawn for a range of values of N, as shown
in Figure 10.7 a . From the point A, straight lines are drawn tangential to
these curves. The instability load N crit is then obtained as the slope of the line
which simultaneously satisfies the two requirements:

i) the line touches the MÎe add curve for N = N i
ii)the line itself has a slope tan q= N i

That is, N crit = N i
Consider again the equation M t = N ( e+e add ); for computer application, it is
convenient to convert it into dimensionless form, by dividing throughout by
f cu bh 2 :

¦ t = a [eÓ+eÓ add ] (10.2)

where

(10.3)

Figure. 10. 7a expressed in dimensionless form, becomes Figure 10.7b. The
straight line a-c in Figure 10.7b simultaneously satisfies the two requirements:

i) the line touches the curve for a = a i

ii) the line itself has a slope tan f = a i

Therefore, the critical value of a , namely a crit , is given by a i .
Hence the instability load is
N crit = a crit f cu bh = a i f cu bh
It can be shown (Kong and Wong, 1987; Wong, 1988) that along any moment-
deflection curve , the concrete strain e c , i.e. the concrete strain ratio e c / e cu ,
increases with . Therefore, with reference to Figure 10.7b, it should be noted
that a i is the correct instability load, only if at the point c on the

(10.4)

curve the

concrete ultimate strain e cu has not been reached. In Figure 10.8 , the curve