find out load carried and compressive stressthe

Question

Find out load carried and compressive stress:

The concrete column 300mm x 300mm in section can be reinforced by 10 longitudinal 20mm diameter round steel bars. The column carries compressive load of 450 KN. Find out load carried and compressive stress produced in steel bars and concrete. Take
Es = 200GN/m²  and

Ec  = 15GN/m².

Sol.: The cross sectional Area of column = 300 × 300 = 90000mm²

Area of   steel bars As = 10 × ( Π /4)(20)²  = 3141.59mm²

Area of concrete = 90000 - 3141.59 = 86858.4 mm²

Each component shorten by the same amount under compressive load, and thus

Strain in concrete = strain in steel
σ c/Ec = σ s/Es

Where σc, σ s    are stress induced in the concrete and steel respectively
σs  = (E_s σ_s)/E_c

= (200 × 10⁹ /15 × 10⁹) σ_c  = 13.33  σ_c

Furthermore,

Total load on column =   Load carried by steel + load carried by concrete

P = σ_s.As  +     σ_c.Ac

450 × 103 = 13.33σ c × 3141.59 +σ   c. 86858.4

= 128735.8 c

σ_c = 3.59 N/mm²                                                                                                            .......ANS
σ_s  = 13.33 × 3.59 = 46.95 N/mm²                                                          .......ANS

Load carried by the concrete P_c =   σcAc    = 3.59 × 86858.4 = 311821.656 N = 311.82KN .......ANS

Load carried by the steel P_s  = σ_sA_s  = 46.95 × 3141.59 = 147497.65 N = 147.49 KN      .......ANS