Design a R. C beam of size 230 mm x 550 mm overall depth supported between an effective span of 6 m. It is subjected to a service load of 30 KN/m. Use $M_{20}$ concrete and $F_{e415}$ steel. - | $\frac{dc}{d}$ | 0.05 | 0.1 | 0.15 | 0.2 | |----------------|-------|-------|-------|-------| | $f_{sc}$ | 355.1 | 351.9 | 342.4 | 329.2 |

b = 230 mm

D = 550mm

d = 50mm (assumed)

d = 500 mm

l = 6m.

Service load = 30KN/m

$M_{20} \ \ \& \ \ F_{e415} \\ \frac{dc}{d}=\frac{50}{500}=0.1 \\ f_{sc} \ from \ \ table =351.9N/mm^2 \\ f_{cc}=0.446{\times}20=8.92N/mm^2$

$BM_{\max}=\frac{wl^2}{8}=\frac{30{\times}6^2}{8}=135KNm$

$M_d$ $=1.5{\times}BM \\ =1.5{\times}135 \\ =202.5KNm$

$M_{\max}$ $=0.138f_ckbd^2 \\ =0.138{\times}20{\times}230{\times}500^2 \\ =158.7KNm$

$\therefore M_d \gt M_{\max}$ singly ,

Design a doubly reinforced section.

$M_{u1}=158.7KNm$

$M_{u2}$ $=M_d-M_{u1} \\ =202.5-158.7$

$M_{u1}=T_{u1}{\times}L_{a1}...............(1) \\ =0.87{\times}f_yAst_1{\times}(d-0.42X_{u \max}) \\ 128.{\times}10^6=[(0.87{\times}415{\times}Ast_1){\times}(450-0.42{\times}0.48{\times}500)] \\ \therefore Ast_1=1101.08 mm^2$

$M_{u2}=T_{u2}{\times}L_{a2}.................(2) \\ =0.87{\times}10^6=[(0.87{\times}415{\times}Ast_2){\times}(500-50)] \\ \therefore Ast_1=269.58 mm^2$

$Ast$ $=Ast_1+Ast_2 \\ =1371.38mm^2$

Provide 5-20 mm Steel.

$Ast=1570.8 mm^2$

$M_{u2}$ $=C_{u2}{\times}L_{a2}...................(3) \\ =(f_{sc}-f_{cc}){\times}Asc{\times}(d-d_c)$

$43.8{\times}10^6=(351.9-8.92){\times}Asc{\times}(500-50) \\ Asc=283.78mm^2$

Provide 3 - 12 mm

$Asc=339.29 mm^2$