Design of joint 5-6 (tube-in-tube joint)

A. Loading diagram and design data

External loading diagram	Loading diagram of joint 5-6
Design data External forces: FX, FY, FZ act at distances Lc (Ex.3.1), X1, Z1 (Ex.1.2.2); the moments at the centre of the joint section (tube-in-tube) are calculated (Calc.0): MX6 = FY (Lc + Z1), MY6 = FX (Lc + Z1) + FZ X1, MZ6 = FY X1. Dimensional constraints: the square tube 5, of size h5 and wall thickness t5 (Ex.3.4), is assembled with the square tube 6, of size h6 and wall thickness t6. Operating constraints: temperature T = (−30…40) °C; outdoor working environment with high humidity. Design constraints: tube 5 is inserted into tube 6, with loose clearance fits on the sides of the inner and outer squares.

B. Choice of materials and heat treatments

Since elements 5 and 6 are welded to other elements, weldable steels are adopted. For the adopted steels, the yield strength σ₀₂ and the ultimate tensile strength σ_r are retained for the calculations. For conforming steel-on-steel contacts, the allowable crushing stress is σ_as = (60…90) MPa.

C. Design of the constructive shape

Constructive model

Constructive features and constraints

For correct operation, conforming contacts (over large surfaces) are required between tubes 5 and 6; they are obtained by a clearance fit, ensured by the tube dimensions.

D. Calculation model, sizing and checking

Calculation assumptions and loads

– the forces F_X, F_Y and the moments M_X and M_Y are transmitted from tube 5 to tube 6 by direct contact between the two tubes fitted with clearance (linear distribution of the contact pressures) over the rectangular surfaces I, II, III and IV;

– the force F_Z is transmitted from tube 5 to tube 6 by a pin with a clearance fit in tube 5 and a light interference fit in tube 6; the pin holes are not considered in the tube-in-tube calculation;

– the moment M_Z generates linearly distributed pressures on the contact surfaces I, II, III and IV, with maximum p_maxMZ;

– the forces F_X and F_Y generate uniformly distributed crushing pressures p_FX and p_FY on the contact surfaces I and II, respectively;

– the moments M_X and M_Y generate linearly distributed crushing pressures on the contact surfaces I, III and II, IV, with maxima p_maxMX and p_maxMY, respectively.

Known data

– loads: F_X, F_Y, M_X = M_X6, M_Y = M_Y6, M_Z = M_Z6;

– shapes and dimensions: the size of the square bore H = h₅; the wall thickness of the inner tube t₅ and of the outer tube t₆; the length L is unknown (to be determined);

– materials: σ_as = 60…90 MPa – the allowable crushing stress.

Calculation relations

Contact pressures

– the contact pressures on surfaces I and II generated by the forces F_X and F_Y respectively (figs. c, d) are:

p_FX = =

and, respectively,

p_FY = = .

– the maximum contact pressures on surfaces I, III and II, IV generated by the moments M_Y and M_X respectively (figs. c, d) are:

p_maxMY = 2 = 2 = 2 =

and, respectively,

p_maxMX = 2 = 2 = 2 =

– the maximum contact pressure on surfaces I, II, III, IV generated by the moment M_Z (fig. b),

p_maxMZ = 2 = 2 = 2 = .

Crushing strength condition of the tube-in-tube joint

The maximum contact pressures, obtained by adding the pressures generated by the loads, are limited to the allowable crushing stress of the materials in contact,

σ_max = max(p_max4/I + p_max1/II) = max(p_FX + p_maxMY + p_maxMZ, p_FY + p_maxMX + p_maxMZ) =

= max( + + , + + ) ≤ σ_as.

Sizing

From the relation above, with H known, the following equations are obtained:

σ_as H² L² – (F_X H + 3 M_Z) L − 12 H M_Y = 0,

σ_as H² L² – (F_Y H + 3 M_Z) L − 12 H M_X = 0,

which, after substituting the known parameters and solving (Calc.4.1.1), usually give two positive solutions.

Finally, L₆ is taken as the largest of the positive values of L (a larger whole value is adopted).

Choosing the tube section dimensions

Since the section of tube 5 has been determined by calculation (Ex.3.4), the dimensions of tube 6 can be taken from Annex.ST.02 with larger values, observing the constraints t₆ ≥ t₅, h₆ = h₅ + 2 t₆.

E. Design of the constructive-technological shape

Constructive-technological drawings

Recommendations for choosing the dimensions and constructive-technological parameters:

– dimensions: h₅, t₅; h₆, t₆, L₆, D_b, L₅ = Z₁ – h₄ + L₆/2 + 2D_b, l₁ = 1.5D_b; l₂ = 2D_b; the bore diameter D_b is determined from the pin joint calculation (Sec.4.2, Ex.4.2);

– technological operations: milling the tube ends, drilling/reaming the pin holes;

– roughness of the machined surfaces: R_a1 = 6.3 µm or R_a1 = 12.5 µm; R_a2 = 6.3 µm or R_a2 = 3.2 µm;

– sharp edges are chamfered 1×45°;

– the position of the pin hole (and, if needed, of the adjustment pin hole) is set when drawing the assembly.