Design of the output joints E (E′) (bolted joints) — example

Design data

External forces: F_X = 1500 N, F_Y = 300 N, F_Z = 700 N act at distances L_c = 70 mm, X₁ = 80 mm, X₂ = 220 mm, X₃ = 150 mm, Y₁ = 550 mm, Z₁ = 180 mm, Z₂ = 200 mm, Y_E = 60 mm; the forces and moments at the centre of joint E are:

F_Xs = F_X/2 = 750 N,

F_Ys = F_Y/2 = 150 N,

F_Zs = F_Z/2 = 350 N,

M_XEs = F_Y/2 · max(|L_c + Z₁ − Z₂|, |Z₁ + Z₂ − L_c|) + F_Z (Y₁ + Y_E)/2 = 260000 Nmm,

M_YEs = F_X/2 · max(|L_c + Z₁ − Z₂|, |Z₁ + Z₂ − L_c|) + F_Z (X₁ + X₂ + X₃)/2 = 390000 Nmm,

M_ZEs = F_X (Y₁ + Y_E)/2 + F_Y (X₁ + X₂ + X₃)/2 = 525500 Nmm.

Dimensional constraints: b₂ = 120 mm, g₂ = 10 mm, R = 20 mm.

Operating constraints: temperature T = (−30…40) °C; outdoor working environment with high humidity.

Design constraints: steels for threaded fasteners (bolts, nuts) are used.

Materials and heat treatments

Property class 6.8 is adopted for the bolts.

Material properties

Sizing and checks

Calculation data

– loading: F_Xs = 750 N, F_Ys = 150 N, F_Zs = 350 N; M_XEs = 260000 Nmm, M_YEs = 390000 Nmm, M_ZEs = 525500 Nmm;

– dimensions: a₁ = 15 mm, L₁ = 70 mm, L₂ = 90 mm; r₁ = r₂ = r₃ = r₄ = r = 57 mm;

– materials: property class 6.8, σ₀₂ = 480 MPa – the yield strength; c = 2…3 – the safety factor; σ_at = σ₀₂/c = 480/(2…3) = (160…240) MPa – the allowable tensile stress (σ_at = 200 MPa is adopted);

– friction parameters: µ = 0.2 – the sliding friction coefficient in the thread; µ₁ = 0.2 – the sliding friction coefficient between element 2 and the side member; µ₂ = 0.2 – the sliding friction coefficient between nut and washer.

Determining the loads

Identifying the most heavily loaded bolt

Axial forces in the bolts for each load:

– bolt forces due to the force F_Zs, which loads the joint axially and centrally:

= = = = = 87.5 N;

– bolt forces due to the moment M_XEs, which loads the joint axially and eccentrically; with M_XEs = 260000 Nmm, L₁ = 70 mm and a₁ = 15 mm: = = 261.7 N, = = 1483.2 N;

– bolt forces due to the moment M_YEs, which loads the joint axially and eccentrically; with M_YEs = 195000 Nmm, L₂ = 90 mm and a₁ = 15 mm: = = 260 N, = = 1820 N;

– bolt forces due to the resultant of the forces F_Xs and F_Ys, which load the joint transversely; with F_Xs = 750 N, F_Ys = 150 N, β_a = 1.2 and µ₁ = 0.2: = = = = 1147.3 N;

– bolt forces due to the moment M_ZEs, which loads the joint transversely; with M_ZEs = 172500 Nmm, r = 57 mm, β_a = 1.3 and µ₁ = 0.2: = = = = 13829 N;

where r is the half-diagonal of the rectangle on which the bolt axes lie.

Total axial forces in the bolts

F_aș1 = + + + + = 87.5 + 261.7 + 260 + 1147.3 + 13829 = 15325.5 N,

F_aș2 = + + + + = 87.5 + 1483.2 + 260 + 1147.3 + 13829 = 16547 N,

F_aș3 = + + + + = 87.5 + 1483.2 + 1820 + 1147.3 + 13829 = 18367 N,

F_aș4 = + + + + = 87.5 + 261.7 + 1820 + 1147.3 + 13829 = 17145.5 N.

The maximum load F_aș = max(F_aș1, F_aș2, F_aș3, F_aș4) = 18367 N shows that bolt 3 is the most heavily loaded; it is sized by the strength calculation.

Axial tightening force

F_st = max( + , + , + , + ) = 14976.3 N.

Sizing

– sizing the most heavily loaded bolt: from the tensile strength condition, with F_aș = 18367 N, β = 1.2 and σ_at = 200 MPa, the required minor thread diameter is = 11.85 mm;

– an M16 bolt is adopted: pitch p = 2 mm, nominal diameter d = 16 mm, minor diameter d₁ = 13.836 mm, pitch diameter d₂ = 14.701 mm; clearance hole diameter D₁ = 18 mm.

Tightening torques of the joint

– thread torque (friction in the thread): with F_st = 14976.3 N, d₂ = 14.701 mm, p = 2 mm, µ = 0.2 and α = 60°, M_î = 30494.7 Nmm;

– friction torque between nut and washer: with F_st = 14976.3 N, D₁ = 18 mm, S = 24 mm and µ₂ = 0.2, M_p = 31664.2 Nmm;

– bolt tightening torque (applied with the wrench):

M_c = M_î + M_p = 30494.7 + 31664.2 = 62158.9 Nmm.

Checks

– check of the threaded shank under combined loading (tension and torsion) during tightening: with F_aș = 18367 N, d₁ = 13.836 mm and M_î = 30494.7 Nmm, σ_e = 169.34 MPa,

and 169.34 < 160…240 MPa (satisfied);

– self-locking check: with p = 2 mm, d₂ = 14.701 mm, μ = 0.2 and α = 60°, β₂ = 2.48°, φ′ = 13°,

and 2.48° < 13° (satisfied).

Constructive-technological shape

Constructive-technological drawing

Adopted dimensions and constructive-technological parameters

– dimensions: Z₂ = 150 mm, Y_E = 60 mm, b₂ = 120 mm, g₂ = 10 mm, a = 15 mm, L = 90 mm, L_Z = 255 mm, L_Y = 120 mm, D₁ = 14 mm, a₁ = 15 mm, L₁ = 70 mm, L₂ = 90 mm, D₂ = 18 mm;

– the number of sets of adjustment holes is set when drawing the assembly;

– technological operations: milling, bending, drilling; fillet radius in the bending zone R = 20 mm;

– sharp edges are chamfered 1×45°;

– roughness of the surfaces machined by milling or drilling: R_a1 = 12.5 µm, R_a2 = 6.3 µm.

Adopted fastener dimensions

Joint with Bolt 1

– hexagon head bolt: M12, d = 12 mm, S = 19 mm, D = 21.1 mm, k = 8 mm, l_ș = 40 mm, l_f = 25 mm;

– hexagon nut: M12, S = 19 mm, m = 10 mm;

– plain washer: d = 12 mm, d₁ = 13 mm, d₂ = 24 mm, h = 2.5 mm;

– Grower spring washer: MN 12, d = 12 mm, d₁ = 12.2 mm, d₂ = 18.2 mm, g = 3 mm.

Joint with Bolt 2

– hexagon head bolt: M16, d = 16 mm, S = 24 mm, D = 26.75 mm, k = 8 mm, l_ș = 40 mm, l_f = 25 mm;

– hexagon nut: M16, S = 24 mm, m = 13 mm;

– plain washer: d = 16 mm, d₁ = 17 mm, d₂ = 36 mm, h = 3 mm;

– Grower spring washer: MN 16, d = 16 mm, d₁ = 16.3 mm, d₂ = 23.5 mm, g = 3.5 mm.