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Chapter 1

Q. 1.17

A reinforced concrete column of 450 mm square has four steel bars of 30 mm diameter as shown in Fig. 1.27. It carries as load of 600 kN. Find the stresses in steel and concrete if their modulii of elasticity are 210 GPa and 22 GPa, respectively. Also calculate the adhesive force concrete and steel.

Screenshot 2022-10-07 013425

Step-by-Step

Verified Solution

Area of steel bars,

A_s = 4 \times \left(\pi / 4 \right) \times 30^2= 2827.4  mm^{2}

Area of concrete,

A_c = 450 \times 450 – 2827.4 = 199672.6  mm^{2}

Equilibrium condition: External load is shared by load in steel bars  F_s  and load in concrete  F_c as follows.

F_s+ F_c= 600 \times 10^{3}  N(i)

Compatibility condition: Steel and concrete will have equal extension under the load.

\Delta L_s= \Delta L_c (ii)

Substituting stress-strain relation (assuming to be linear) in Eq. (ii),

\frac{F_s \times L_s}{A_s \times E_s} =\frac{F_c \times L_c}{A_c \times E_c}

or,  F_s =\frac{210 \times 2827.4}{22 \times 199672.6} F_c

F_s =0.135  F_c

Substituting in Eq. (i),

F_c=528557.4   N

and F_s=71442.6  N

Stress in steel, \sigma_s=25.27 MPa

Stress in concrete, \sigma_c=2.65 MPa

External load/unit area

=\frac{600 \times 10^3}{450 \times 450}=2.96 MPa

Additional stress in steel = 25.27 – 2.96 = 22.31 MPa

Therefore, adhesive force required between steel and concrete = 22.31 \times 2827.4 = 63 \times  10^{3}  N