8.14

In structural engineering, the secant formula defines the force per unit
area, P / A, that causes a maximum stress sigma_m in a column of given
slenderness ratio L/K:

P/A = sigma_m / 1 + (ec/ k^2) sec[.5* sqrt( P/(EA)(L/k)]
  
 where ec/ k^2 = the eccentricity ratio an E = the modulus of elasticity.
If for a steel beam, E = 200,000 MPa, ec/k^2 = .4, and sigma_m = 250 MPa,
 compute P/ A for L / k = 50. Recall that sec x = 1/ cos(x).
Matlab Code:

E = 200000; 

eck = .4;

sigma_m = 250; 

LK = 50;

PA = 0:300;

%PA = sigma_m / 1 + (eck) * sec(.5 * sqrt( P/(EA) * (LK))); 

f = @(PA) (sigma_m / 1 + (eck) * sec(.5 * sqrt( (PA * 1/E )* (LK)))) - PA; 

hold on

plot(PA,f(PA))

PA = 250;

delta = .0001;

%Modified secant method

for i = 1:40

    PA_new = PA - (delta * PA * f(PA)) ./ (f(PA + delta*PA) - f(PA));

    PA = PA_new;

    

    plot(PA,f(PA), 'r*')

end

hold off

This produces

which makes sense given that the equation is 250 divided by a value close to 1. The actual zero lies at 250.4032 as proven by plugging the result of applying the modified secant method (the red dot) back into the equation (graphed in blue) as such:

EDU>> PA

PA =

  250.4032

EDU>> f(PA)

ans =

     0