Design of Castellated Beams for Use With Bs 5950

Introduction
Universal beam sections are normally employed in buildings to carry load. Loads on beams may include the load from slab, walls, building services, and their own self weight. It is necessary for structural beams to satisfy ultimate and serviceability limit state requirements.
This post gives a solved design example of a laterally restrained beam according to BS 5950.

Design example
A laterally restrained beam 9m long that is simply supported at both ends supports a dead uniformly distributed load of 15 KN/m and an imposed load uniformly distributed load of 5KN/m. It also carries a dead load of 20KN at distance of 2.5m from both ends. Provide a suitable UB to satisfy ultimate and serviceability limit state requirements (P_y = 275 N/mm²).

Solution
At ultimate limit state;
Concentrated dead load = 1.4Gk = 1.4 × 20 = 28 KN
Uniformly distributed load = 1.4Gk + 1.6Qk = 1.4(15) + 1.6(5) = 29 KN/m

Support Reactions
Let ∑M_B = 0; anticlockwise negative
(9 × Ay) – (29 × 9 × 4.5) – (28 × 6.5) – (28 × 2.5) = 0
Ay = 158.5 KN

Let ∑M_A = 0; clockwise negative
(9 × By) – (29 × 9 × 4.5) – (28 × 6.5) – (28 × 2.5) = 0
By = 158.5 KN

Internal Stresses
Moment
M_A = 0 (Hinged support)
M_C = M_D = (158.5 × 2.5) – (29 × 2.5 × 1.25) = 305.625 KNm
M_midspan = (158.5 × 4.5) – (29 × 4.5 × 2.25) – (28 ×2) = 363.625 KNm

Shear
Q_A = Ay = 158.5 KN
Q_C ^L = 158.5 – (29 × 2.5) = 86 KN
Q_C ^R = 158.5 – (29 × 2.5) – 28 = 58 KN
Q_D ^L = 158.5 – (29 × 6.5) – 28 = -58 KN
Q_D ^R = 158.5 – (29 × 6.5) – 28 – 28 = – 86 KN
Q_B = 158.5 – (29 × 9) – 28 – 28 = – 158.5 KN

Internal Forces Diagram

Structural Design to BS 5950
P_y = 275 N/mm²

Initial selection of section
Moment Capacity of section M_c = P_yS ——- (1)

Where S is the plastic modulus of the section
Which implies that S = M_c/P_y = (363.625 × 10⁶)/275 = 1320963.636 mm³ = 1320.963 cm³

With this we can go to the steel sections table and select a section that has a plastic modulus that is slightly higher than 1320.963 cm³

Try section UB 457 × 191 × 67 (S = 1470 cm³)

Properties of the section;
I_xx = 29400 cm⁴
Z_xx = 1300 cm³
Mass per metre = 67.1 kg/m
D = 453.4mm
B = 189.9mm
t = 8.5mm
T = 12.7mm
r = 10.2mm
d = 407.6mm

Strength classification
Since T = 12.7mm < 16mm, P_y = 275 N/mm²
Hence ε = √(275/P_y) = √(275/275) = 1.0

Section classification
Flange
b/T = 7.48 < 9ε; Flange is plastic class 1
Web
d/t = 48 < 80ε; Web is also plastic class 1

Shear Capacity
As d/t = 48 < 70ε, shear buckling need not be considered (clause 4.4.4)

Shear Capacity P_v = 0.6P_yA_v  —– (2)

P_v= 0.6P_ytD = 0.6 × 275 × 8.5 × 453.4 = 635893.5 N = 635.89 kN

But design shear force F_v = 158.5 KN. Since F_v(158.5) < P_v(635.89), section is ok for shear.

Now, 0.6P_v = 0.6 × 635.89 = 381.54 kN
Since F_v(158.5KN) < 0.6P_v(381.54 KN), we have low shear load.

Moment Capacity
Design Moment = 363.625 KNm

Moment capacity of section UB 457 × 191 × 67 (S = 1470 cm³ = 1470 × 10³ mm³)

M_c = P_yS = 275 × 1470 × 10³ = 404.25 × 106 N.mm = 404.25 KNm
1.2P_yZ = 1.2 × 275 × 1300 × 10³ = 429 × 106 N.mm = 429.00 KNm

M_c (404.25 KNm) < 1.2P_yZ (429.00 KNm) Hence section is ok

Evaluating extra moment due to self weight of the beam
Self-weight of the beam S_w = 67.1 kg/m = 0.658 KN/m (UDL on the beam)
Moment due to self weight (M_sw) = (ql²)/8 = (0.658 × 9²)/8 = 6.66 KNm

(363.625 + 6.66) < M_c (404.25 KNm) < 1.2P_yZ (429.00 KNm) Hence section is ok for moment resistance.

Deflection Check
We check deflection for the unfactored imposed load; E = 205 KN/mm² = 205 × 10⁶ KN/m²; I_xx = 29400 cm⁴ = 29400 × 10^-8 m⁴

The maximum deflection for this structure occurs at the midspan and it is given by;
δ = (5ql⁴)/384EI = (5 × 5 × 9⁴) / (384 × 205 × 10⁶ × 29400 × 10^-8) = 7.087 × 10^-3 m = 7.087mm

Permissible deflection; L/360 = 9000/360 = 25mm
7.087mm < 25mm. Hence deflection is satisfactory.

Web bearing
According to Clause 4.5.2 of BS 5950-1:2000, the bearing resistance P_bw is given by:

P_bw = (b₁ + nk)tP_yw —– (3)

Where;
b₁ is the stiff bearing length
n = 5 (at the point of concentrated loads) except at the end of a member and n = 2 + 0.6b_e/k ≤ 5 at the end of the member
b_e is the distance to the end of the member from the end of the stiff bearing
k = (T + r) for rolled I- or H-sections
T is the thickness of the flange
t is the web thickness
P_yw is the design strength of the web

Web bearing at the supports
Let us assume that beam sits on 200mm bearing, and b_e = 20mm

k = (T + r) = 12.7 + 10.2 = 22.9mm;
Hence n = 2 + 0.6(20/22.9) = 2.52mm < 5mm.
P_bw = (b₁ + nk)tP_yw
P_bw = [200 + 2.52(22.9)] × 8.5 × 275 = 602392.45 N = 602.392 KN
P_bw (602.392 KN) > F_v (158.5 KN) Hence it is ok

Contact stress at supports
P_cs = [b₁ × 2(r +T )]P_y = [200 × 2(22.9)] × 275 = 2519000N = 2519 KN
P_cs (2519 KN) > F_v (158.5 KN) Hence contact stress is ok

Web buckling

According to clause 4.5.3.1 of BS 5950, provided the distance α_e from the concentrated load or reaction to the nearer end of the member is at least 0.7d, and if the flange through which the load or reaction is applied is effectively restrained against both;

(a) rotation relative to the web
(b) lateral movement relative to the other flange

The buckling resistance of an unstiffened web is given by;

P_x = [25εt/√(b₁ + nk)d] P_bw —– (4)

When α_e < 0.7d, the buckling resistance of an unstiffened web is given by;

P_x = [(α_e + 0.7d)/1.4d] × [25εt/√(b₁ + nk)d)] P_bw ————- (5)

Therefore, α_e = 20mm + (200/2) = 120mm
0.7d = 0.7 × 407.6 = 285.32mm
α_e(120mm) < 0.7d(285.32mm). Hence equation (5) applies

P_x = [(120 + 285.32)/(1.4 × 407.6)] × [(25 × 1 × 8.5 )/√((200 + 2.52 × 22.9) × 407.6)] × 602.392 = 280.538KN

P_x(280.538 KN) > F_v(158.5 KN) Hence it is ok.

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Design of Castellated Beams for Use With Bs 5950

Source: https://structville.com/2017/08/solved-example-on-design-of-steel-beams-according-to-bs-5950-1-2000.html