Question

Determine the nominal compressive strength Pn of a short column which subjected to biaxial bending ex- 250 mm , ey- 150 mm h - 650 mm , b - 450 mm , reinforcement 8 28.ties 10 mm , cover - 40mm fc - 28 Mpa , fy 420 MPa

Answer 1

b = 450 mm

h = 650 mm

cover = 40 mm

f_c = 28 MPa

f_y = 420 MPa

e_x = 250 mm

e_y = 150 mm

1)calculate _g

_g = Ast/Ag = 4926/(450*650) = 0.0168

2) For x direction (bending about y axis)

Compute the ratio e_x/h

e_x/h = 250/650 = 0.385

Comput the ratio

= (d - d^' ) / h = [650-2(40)-2(10)-28]. / 650 = 0.803

Draw the line of e_x/h on diagram A-9b (for = 0.75) and diagram A-9c ( = 0.9)

Diagram A-9b(for = 0.75)

P_n/Ag = P_u/Ag =1.32

Diagram A-9c (for = 0.9)

P_n/Ag = P_u/Ag = 1.45

Use linear interpolation to compute the values for ( = 0.803)

P_n / Ag

= 1.32 + (1.45 - 1.32)*(0.803 - 0.75)/(0.9 - 0.75)

=1.366 ksi

P_n/Ag = P_u*0.145 / 650*450 = 1.366

P_uy = 1.366*650*450*10^-3 / 0.145 = 2755.4 KN

P_ny = P_uy/ = 2755.4/0.65 = 4239.1 KN

= 0.65 compression - controlled section (from interaction diagram)

3)For y direction (bending about x axis)

Compute the ratio e_y/h

e_y/h = 150/650 = 0.23

Compute the ratio

= (d - d')/h = 450-2(40)-2(10)-28/450 = 0.715

Draw line of e_x/h on diagram A-9a (=0.6) and diagram A-9b (for =0.75)

Diagram A-9a (=0.6)

P_n/Ag = P_u/Ag =1.49

Diagram A-9b (for = 0.75)

P_n/Ag = P_u/Ag = 1.6

Use linear interpolation to compute the value for (= 0.715)

P_n/Ag = 1.49 + (1.6 - 1.49)*(0.715 - 0.6)/(0.75 - 0.6) = 1.574 ksi

Pn/Ag = Pu * 0.145 / (650*450) = 1.574

P_ux = 1.574*650*450*10^-3/0.145 = 3175.14 KN

P_nx = P_ux/ = 3175.14/0.65 = 4884.83 KN

Determine P_no for the section dimensions h = 650 mm and b=450 mm and _g=0.0168

P_no=Ag(0.85f_c' (1-_g ) + _g f_y)

=450*650(0.85*28(1-0.0168) + 0.0168*420)10^-3

=8908.43 KN

Note that P_no can be defined directly from interaction diagram when e_{x =} e_y = 0

From diagram A-9 for analysis of

P_n/Ag=3

P_no = 3Ag/ = 3*650*450*10^-3/(0.415*0.65) = 9310.34 KN

Substituting P_nx,  P_ny,  P_no in bresler equation

1/P_n = 1/P_{nx +} 1/P_ny - 1/P_no

P_n= 3001.14 KN

P_{u =} P_n = 0.65*3001.14 = 1950.738 KN

Hence the nominal compressive strength = 1950.738 KN