Design of elements 11, 12, 12′ (central bar, left/right bar) — 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, Y₁ = 550 mm, Z₁ = 180 mm; the moments at the centre of the square section are:

M_X11 = F_Y (L_c + Z₁) = 75000 Nmm,

M_Y11 = F_X (L_c + Z₁) + F_Z (X₁ + X₂) = 585000 Nmm,

M_Z11 = F_Y (X₁ + X₂) = 90000 Nmm.

Dimensional constraints: square tube 12 slides into square tube 11 with clearance fits.

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

Design constraints: weldable materials for elements 11, 12 and 12′.

Materials and heat treatments

Since elements 11, 12, 12′ (central, left and right bar) are welded parts, S235 according to SR EN 10025-2 (STAS 500/2) is adopted.

Material properties

Sizing and checking

Calculation data

– loading: F_X = 1500 N, F_Y = 300 N, F_Z = 700 N, M_X11 = 75000 Nmm, M_Y11 = 585000 Nmm, M_Z11 = 90000 Nmm;

– shape and dimensions: inner square tube with dimensions h₁₂, t₁₂; outer square tube with dimensions h₁₁, t₁₁;

– material: σ₀₂ = 235 MPa – the yield strength; c = 2…3 – the safety factor; σ_at = σ₀₂/c – the allowable tensile stress (σ_at = 100 MPa adopted); σ_aî = (1.05…1.1)σ_at – the allowable bending stress (σ_aî = 110 MPa adopted); τ_at = 60 MPa – the allowable torsional stress;

– bending moments in the critical sections (simplified model):

M_îmaxH = M_Z11 + F_X Y₁/2 = 502500 Nmm,

M_îmaxV = M_X11 + F_Z Y₁/2 = 267500 Nmm,

in section S₁, and M_îmaxH/2, M_îmaxV/2 in section S₂.

Sizing

Sizing the central bar 11

From the bending strength condition, with F_X = 1500 N, F_Z = 700 N, M_îmaxH = 502500 Nmm, M_îmaxV = 267500 Nmm, k = 0.1 and σ_aî = 110 MPa, = 4.15 mm; a standard square tube with h₁₁ = 60 mm, t₁₁ = 5 mm is adopted, so h₁₂ = 50 mm.

Sizing the left/right bar 12/12′

From the bending strength condition, with F_X = 1500 N, F_Z = 700 N, M_îmaxH = 502500 Nmm, M_îmaxV = 267500 Nmm, k = 0.1 and σ_aî = 110 MPa, = 3.3 mm; a standard square tube with h₁₂ = 50 mm (see above) and t₁₂ = 5 mm is adopted.

Checking

Torsional strength of bars 12/12′

From the torsion check of bars 12/12′, with M_Y11 = 585000 Nmm, h₁₂ = 50 mm and t₁₂ = 5 mm, the maximum torsional stress is τ_max = 26 MPa.

τ_max < τ_at; 26 MPa < 60 MPa (satisfied).

Constructive-technological shape

Constructive-technological drawing

Adopted dimensions and constructive-technological parameters

– section dimensions: h₁₁ = 60 mm, t₁₁ = 5 mm, h₁₂ = 50 mm, t₁₂ = 5 mm; L₁₁ = 550 mm, L₁₂ = Y₁/2 + L_11-12, l = L_11-12/2; the dimension L_11-12 is determined from the tube-in-tube joint calculation; the bore diameter D_b is determined from the pin joint calculation;

– the number of adjustment pin holes, and hence the adjustment travel, is set when drawing the assembly;

– technological operations: milling (the bar ends) and drilling;

– sharp edges are chamfered 1×45°;

– roughness of the surfaces machined by milling (bar ends) and drilling: R_a1 = 12.5 µm, R_a2 = 6.3 µm.