Design of Bolted Beam Splice Connections According to EC3

M_Ed = 610 kNm
V_Ed = 215 kN
N_Ed = 55 kN

At serviceability limit state;
M_Ed,ser = 445 kNm
V_Ed,ser = 157 kN
N_Ed,ser = 41 kN

Properties of UB 533 x 210 x 101
h = 536.7 mm
b = 210 mm
d = 476.5 mm
t_w = 10.8 mm
t_f = 17.4 mm
r = 12.7 mm
A = 129 cm²
I_y = 61500 cm⁴
I_z = 2690 cm⁴
W_el,y = 2290 cm³
W_pl,y = 2610 cm³;
h_w = h – 2t_f = 476.5 mm

Cover plates
Let us assume 20 mm thick cover plates for the flanges and 15 mm thick plates for the web. The thickness and dimension to be confirmed later in the post.

Bolts
M24 preloaded class 8.8 bolts
Diameter of bolt shank d = 24mm
Diameter of hole d₀ = 26mm
Shear area A_s = 353 mm²

Materials Strength
Beam and cover plates;
f_y,b = f_y,wp = 275 N/mm²
f_u,b = f_u,wp = 410 N/mm²

Bolts
Nominal yield strength f_yb = 640 N/mm²
Nominal ultimate strength f_ub = 800 N/mm²

Partial Factors for Resistance (BS EN 1993-1- NA.2.15)
Structural Steel
γ_m0 = 1.0
γ_m1 = 1.0
γ_m2 = 1.1

Parts in connections
γ_m2 = 1.25 (bolts, welds, plates in bearing)
γ_m3 = 1.0 (slip resistance at ULS)
γ_m3,ser = 1.1 (slip resistance as SLS)

Step 1: Internal Forces at Splice
For a splice in flexural member, the parts subject to shear (the web cover plates) must carry, in addition to the shear force and moment due to the eccentricity of the centroids of the bolt groups on each side, the proportion of moment carried by the web, without any shedding to the flanges.

The second moment of area of the web is;
I_w = [(h – 2t_f)³t_w/12]
I_w = [(476.5³ × 10.8)/12] × 10^-4 = 9737 cm⁴
Therefore the web will carry 9737/61500 = 15.8% of the moment in the beam (assuming an elastic stress distribution), while the flange will carry the remaining 84.2%.
The area of the web is;
A_w = 476.5 × 10.8 × 10^-2 = 51.462 cm²
Therefore, the web will carry 51.462/129 = 39.8% of the axial force in the beam, wile the flange will carry the remaining 60.1%.

Forces at ULS
The force in each flange due to bending is therefore given by;
F_f,M,Ed = 0.842 × [(M_Ed)/(h – t_f)]
F_f,M,Ed = 0.842 × [(610 × 10⁶)/(536.7 – 17.4)] × 10^-3 = 989 kN

And the force in each flange due to axial force is given by;
F_f,N,Ed = 0.601 × (55/2) = 16.53 kN

Therefore
F_tf,Ed = 989 – 16.53 = 972.47 kN
F_bf,Ed = 989 + 16.53 = 1005.53 kN

The moment in the web = 0.158 × 610 = 96.38 kNm
The shear force in the web = 215 kN
The axial force in the web = 0.398 × 55 = 21.89 kN

Forces at SLS
The force in each flange due to bending is therefore given by;
F_f,M,Ed = 0.842 × [(M_Ed,ser)/(h – t_f)]
F_f,M,Ed = 0.842 × [(445 × 10⁶)/(536.7 – 17.4)] × 10^-3 = 721.53 kN

And the force in each flange due to axial force is given by;
F_f,N,Ed = 0.601 × (41/2) = 12.3205 kN

Therefore
F_tf,Ed = 721.53 – 12.32 = 709.21 kN
F_bf,Ed = 721.53 + 12.32 = 733.85 kN

The moment in the web = 0.158 × 445 = 70.31 kNm
The shear force in the web = 157 kN
The axial force in the web = 0.398 × 41 = 16.318 kN

Choice of Bolt Number and Configuration
Resistances of Bolts
The shear resistance of bolts at ultimate limit state
For M24 bolts in single shear = 132 kN
For M24 bolts in double shear = 265 kN

Flange Splice
For the flanges, the force of 1005.53 kN at ULS can be provided by 8 M24 bolts in single shear.

The full bearing resistance of an M24 bolt in a 20 mm cover plate (without reduction for spacing and edge distance) is;

F_b,max,Rd = (2.5f_udt)/γ_m2
F_b,max,Rd = [(2.5 × 410 × 24 × 20)/1.25] × 10^-3 = 393.6 kN

This is much greater than the resistance of the bolt in single shear, and thus the spacings do not need to be such as to maximise the bearing distance. Four lines of 2 bolts at a convenient spacing may be used.

Web Splice
For the web splice, consider one or two lines of 4 bolts on either side of the centreline. The full bearing resistance on the 10.8 mm web is;
F_b,max,Rd = (2.5f_udt)/γ_m2
F_b,max,Rd = [(2.5 × 410 × 24 × 10.8)/1.25] × 10^-3 = 212.54 kN

This is less than the resistance in double shear, and will therefore determine the resistance at ULS. To achieve this value, the spacings will need to be;
e₁ ≥ 3d₀ = 3 × 26 = 78mm
p₁ ≥ 15d₀/4 = (15 × 26)/4 = 97.5 mm
e₂ ≥ 1.5d₀ = 1.5 × 26 = 39 mm
p₂ ≥ 3d₀ = 3 × 26 = 78mm

Initially, try 4 bolts at a vertical spacing of 120mm at a distance of 80mm from the centreline of the splice.

The additional moment due to the eccentricity of the bolt group is;
M_add = 215 × 0.08 = 17.2 kNm

Bolt forces at ULS
Force/bolt due to vertical shear = 215/4 = 53.75 kN
Force/bolt due to axial compression = 21.89/4 = 5.47 kN
Force/bolt due to moment (considering top and bottom bolts only) = (96.38 + 17.2)/0.36 = 315.5 kN
Thus, a single row is adequate.

Considering the second line of bolts at a horizontal pitch of 100mm
Force/bolt due to vertical shear = 215/4 = 53.75 kN
Force/bolt due to axial compression = 21.89/4 = 5.47 kN
Force/bolt due to moment (considering top and bottom bolts only) = (96.38 + 17.2)/0.36 = 315.5 kN

The additional moment due to eccentricity of this bolt group is;
M_add = 215 × (0.08 + 0.1/2) = 27.95 kN.m

The polar moment of inertia of the blot group is given by;
I_bolts = 6 × 120² + 8 × (100/2)² = 106400 mm²

The horizontal component of the force on each top and bottom bolt is;
F_M,horiz = {[(96.38 + 27.95) × 120]/106400} × 10³ = 140.22 kN

The vertical component of the force on each top and bottom bolt is;
F_M,vert = {[(96.38 + 27.95) × 50]/106400} × 10³ = 58.425 kN

Therefore, the resultant force on the most highly loaded bolt is;
F_v,Ed = sqrt[(53.75 + 58.425)² + (140.22 + 5.47)²] = 183.87 kN

This is less than the full bearing resistance, and is therefore satisfactory for such a bolt spacing.

Chosen Joint Configuration

Summary of cover plate dimensions and bolt spacing

Flange cover plates; 
Thickness t_fp = 20 mm
Length h_fp = 660 mm
Width b_fp = 200 mm
End distance e_1,fp = 70 mm
Edge distance e_2,fp = 30 mm

Spacing
In the direction of the force   p_1,f  = 100 mm
Transverse to direction of force   p_2,f  = 120 mm
Across the joint in direction of force   p_1,f,j  = 120 mm

Web cover plates; 
Thickness t_wp = 12 mm
Length h_wp = 460 mm
Width b_wp = 460 mm
End distance e_1,wp = 50 mm
Edge distance e_2,wp = 50 mm

Spacing
Vertically   p_1,w  = 120 mm
Horizontally  p_2,w = 100 mm
Horizontally across the joint   p_1,w,j  = 160 mm

Resistance of flange splices
Resistance of net section
The resistance of a flange cover plate in tension (N_t,fp,Rd) is the lesser of N_pl,Rd and N_u,Rd.

Here;
N_u,Rd = (0.9A_net,tpf_u,fp)/γ_M2

Where;
A_net,fp = (b_fp – 2d₀)t_fp = [200 – (2 × 26)] × 20 = 2960 mm
Therefore;
N_u,Rd = [(0.9 × 2960 × 410 )/1.1] × 10^-3 = 992.945 kN

N_pl,Rd = (A_fpf_y,fp)/γ_M0 = [(200 × 20 × 275)/1.0] × 10^-3 = 1100 kN

Since 1100 kN > 992.945 KN
N_t,fp,Rd = 992.945 kN

For the flange in tension;
F_tf,Ed = 989 – 16.53 = 972.47 kN

Therefore, the tension resistance of a flange cover plate is adequate.

Block Tearing Resistance;
n_1,fp = 3 and n_2,fp = 2;

The resistance to block tearing (N_t,Rd,fp) is given by:
N_t,Rd,fp = {(f_u,fpA_nt,fp)/γ_m2} + {[A_nv,fp(f_y,fp/√3)]/γ_m0}

Where:
A_nt,fp = t_fp(2e_2,fp – d₀)
A_nt,fp = 20 × [(2 × 40) – 26] = 1080 mm²

A_nv,fp = 2t_fp[(n_1,fp – 1)p_1,fp + e_1,fp – (n_1,fp – 0.5)d₀]
A_nv,fp = 2 × 20[(3 – 1) × 100 + 70 – (3 – 0.5) × 26] =  8200 mm²

Therefore;
N_t,fp,Rd = {[(410 × 1080)/1.1] + [(8200 × 265/√3)/1.0]} × 10^-3 = 1657.127 kN

Resistance of cover plate to compression flange
Local buckling resistance between the bolts need not be considered if:
p₁/t ≤ 9ε
ε = sqrt(235/f_y,fp) = sqrt(235/265) = 0.94
9ε = 9 × 0.94 = 8.46

Here, the maximum spacing of bolt across the centreline of the splice p_1,f,j  = 120 mm
p_1,f,j/t_fp = 120/20 = 6

Since 8.46 > 6, no local buckling verification required.

RESISTANCE OF WEB SPLICE
Resistance of web cover plate in shear;
The gross shear area is given by:
V_wp,g,Rd = (h_wpt_wpf_y,wp/1.27γ_m0√3)

For two web cover plates;
V_wp,g,Rd = 2{[(460 × 12)/1.27] × [275/√3]} × 10^-3 = 1380.185 kN
V_Ed = 215 kN. Therefore, the shear resistance is adequate.

The net shear area is given by:

V_wp,net,Rd = A_v,wp,net(f_u,wp/√3)/γ_m2
A_v,net = (h_wp – 3d₀)t_wp
= (460 – 3 × 26) × 12 = 4584 mm²

For two web cover plates;
V_n,Rd = {2 × [4584 × (410/√3)]/1.1} × 10^-3 = 1972.9 kN
Therefore, shear resistance is adequate.

Resistance to block tearing

V_b,Rd = (f_u,wpA_nt)/γ_m2 + (f_y,wpA_nv.A_nv)/γ_m0√3

For a single vertical line of bolts;
A_nt = t_wp(e₂ – d₀/2)
A_nt = 12 × (50 – 26/2) = 444 mm²
A_nv = t_wp – e₁ – (n₁ – 0,5)d₀)
A_nv = 12 × [460 – 50 – (3 – 0.5)26] = 4140 mm²

For two web plates
V_b,Rd = 2 × {[(410 × 460)/1.1] + [(275 × 4140)/(√3 × 1.0)]} × 10^-3 = 1657.53 kN

V_Rd = min{1657.53,  1972.9, 1380.185 }
V_Rd = 1380.185 kN

V_Ed/V_Rd = 215/1380.185 = 0.1557 < 1.0
This is ok.

Resistance of beam web
Resistance of net shear area
V_n,w,Rd = [A_v,net(f_u/√3)]/γ_m2

Where;
A_v,net = A_v – 3d₀t_w
A_v = A – 2bt_f + (t_w + 2r)t_f but not less than ηh_wt_w

A_v = 12900 – (2 × 210 × 17.4) + (10.8 + 2 × 12.7) × 17.4 = 6221.88 mm²
A_v = 6221.88 – (3 × 26 × 10.8) = 5379.48 mm²

Thus;
V_n,w,Rd = {[5379.48 × (410/√3)]/1.1} × 10^-3 = 1157.632 kN
This is ok.

Resistance of web cover plate to combined bending, shear, and axial force

V_wp,Rd = 1380.185 kN
Vsub>Ed = 215 kN < 1380.185 kN

Therefore, the effects of shear can be neglected
A_wp = 12 × 460 = 5520 mm²

Modulus of the cover plate
= (12 × 460²)/6 = 423200 mm³

N_wp,Rd = 12 × 460 × 275 × 10^-3 = 1518 kN

Therefore, for two web cover plates
M_c,wp,Rd = [(2 × 423200 × 275)/1.0] × 10^-6 = 232.76 kNm

For two web cover plates;
N_pl,Rd = 2 × 1518 = 3036 kN

M_wp,Ed = 96.38 + 27.95 = 124.33 kNm
N_wp,Ed = 21.89 kN

M_wp,Ed/M_c,wp,Rd + N_wp,Ed/N_wp,Rd
124.33/232.76 + (21.89/3036) = 0.541 < 1.0

Therefore, the bending resistance of the web cover plates is adequate.

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