(a) Design a floor slab to cover a room with internal dimensions of 4·5 m × 6·0 m. The slab is simply supported on all the sides…

(a) Two-way slab, 4.5 m × 6.0 m, corners held down (IS 456)

Trial depth D = 180 mm, cover 20 mm, 10 mm bars: d = 155 mm for the short span and 145 mm for the long span. Effective spans: l_x = 4.5 + 0.155 = 4.655 m (less than the c/c span 4.73 m) and l_y = 6.155 m, so l_y/l_x = 1.322.

Load: self-weight 0.18 × 25 = 4.5 kN/m², finishes 1.0, live 4.0, so w = 9.5 kN/m² and w_u = 1.5 × 9.5 = 14.25 kN/m².

Moments: α_x = 0.079 + 0.22 × (0.085 − 0.079) = 0.0803 and α_y = 0.056. M_x = 0.0803 × 14.25 × 4.655² = 24.8 kN·m/m and M_y = 0.056 × 14.25 × 4.655² = 17.3 kN·m/m. M_u,lim = 0.138 f_ck b d² = 66.3 kN·m/m, above M_x, so the section is under-reinforced.

Steel, from M_u = 0.87 f_y A_st d (1 − A_st f_y/(b d f_ck)): short span A_st = 473 mm²/m, so provide 10 mm bars at 150 mm c/c (524 mm²). Long span (d = 145) A_st = 348 mm²/m, so provide 10 mm at 200 mm c/c (393 mm²). Both exceed 0.12% of bD = 216 mm²/m, and the spacings are within 3d and 300 mm.

Deflection (short span): f_s = 0.58 × 415 × 473/524 = 217 N/mm² and p_t = 524/(1000 × 155) = 0.34%. The chart gives a modification factor of about 1.6. Allowable l/d = 20 × 1.6 = 32, against an actual 4655/155 = 30.0, so the check is satisfied.

Shear: V_u = 14.25 × 4.655/2 = 33.2 kN/m, τ_v = 33 200/(1000 × 155) = 0.21 N/mm². For p_t = 0.34%, τ_c ≈ 0.40 and k = 1.24, so kτ_c = 0.50 N/mm², which exceeds τ_v; safe.

Corner torsion steel: over l_x/5 = 0.93 m at each corner, provide top and bottom mats of 0.75 × 473 = 355 mm²/m each way, that is 8 mm bars at 140 mm c/c (359 mm²).

Final slab: 180 mm thick; short span 10 mm at 150 mm c/c; long span 10 mm at 200 mm c/c; 8 mm at 140 mm c/c corner mats, top and bottom.

(b) Battened column of two ISMC 300 channels

Economical spacing: the z–z axis passes through both webs, so I_zz = 2 × 6420 × 10⁴ = 12 840 × 10⁴ mm⁴. With the centroid of each channel 23.5 mm from its web back, I_yy = 2[313 × 10⁴ + 4630 (S/2 + 23.5)²]. Equating I_yy = I_zz: (S/2 + 23.5)² = (6420 − 313) × 10⁴/4630 = 13 190, so S/2 + 23.5 = 114.85 and S = 182.7 mm. Adopt S = 190 mm, which gives I_yy = 13 629 × 10⁴ mm⁴, slightly above I_zz, so z–z governs.

Strength: r_zz = 118 mm, λ = 10 000/118 = 84.7, and for a battened column the effective slenderness is 1.1λ = 93.2. With buckling class c (built-up channels) and f_y = 250 MPa, f_cd = 116.4 N/mm². P_d = 116.4 × 9260 = 1078 kN. This is about 2% below 1100 kN, so the pair is only marginally adequate, and since z–z governs a wider S cannot help; a slightly heavier channel would be needed for strict compliance. The battens are designed below for the specified load.

Batten spacing: C/r_yy ≤ 50 gives C ≤ 50 × 26 = 1300 mm, and C/r_yy ≤ 0.7λ_e = 65 is also met. Adopt C = 1250 mm c/c (8 equal panels in 10 m); C/r_yy = 48.1.

Forces: V_t = 2.5% × 1100 = 27.5 kN. The bolt line is 90 − 32 = 58 mm from the web back, so S′ = 190 + 2 × 58 = 306 mm. With N = 2 batten planes, longitudinal shear V_b = V_t C/(N S′) = 27.5 × 1250/(2 × 306) = 56.2 kN and moment M_b = V_t C/(2N) = 27.5 × 1250/4 = 8.59 kN·m.

Batten size: the distance between channel centroids is a = 190 + 2 × 23.5 = 237 mm. End battens need an effective depth of at least a = 237 mm, so with 2 × 32 mm edge distance the overall depth is 310 mm. Intermediate battens need at least 0.75a = 178 mm and 2 × 90 = 180 mm, so the overall depth is 250 mm. Thickness t ≥ S′/50 = 6.1 mm, so provide 8 mm. Length = S′ + 2 × 32 = 370 mm. Check of the 250 × 8 batten: shear capacity 250 × 8 × 250/(√3 × 1.1) = 262 kN exceeds 56.2 kN, and moment capacity (8 × 250²/6) × 250/1.1 = 18.9 kN·m exceeds 8.59 kN·m.

Adopt end battens 310 × 370 × 8 mm and intermediate battens 250 × 370 × 8 mm at 1250 mm c/c on both faces.

(c) Portal frame by slope deflection

Fixed-end moments on BC: FEM_BC = −wL²/12 = −16 × 4²/12 = −21.33 kN·m and FEM_CB = +21.33 kN·m; the columns carry no load. The frame and load are symmetric, so there is no sway and θ_C = −θ_B, with θ_A = θ_D = 0.

With EI constant and clockwise moments positive: M_AB = (2EI/3)θ_B; M_BA = (4EI/3)θ_B; M_BC = (2EI/4)(2θ_B + θ_C) − 21.33 = (EI/2)θ_B − 21.33; M_CB = −(EI/2)θ_B + 21.33; M_CD = −(4EI/3)θ_B; M_DC = −(2EI/3)θ_B.

Joint B: M_BA + M_BC = 0 gives (4/3 + 1/2)EIθ_B = 21.33, so EIθ_B = 11.64.

Final end moments (kN·m): M_AB = +7.76, M_BA = +15.52, M_BC = −15.52, M_CB = +15.52, M_CD = −15.52, M_DC = −7.76. Joint B and joint C are in equilibrium (15.52 − 15.52 = 0). The beam hogs 15.52 kN·m at each end and sags wL²/8 − 15.52 = 16.48 kN·m at mid-span.