A. Loading diagram and design data
External loading diagram
|
Loading diagram of joint 13-2 |
 |
Design data
External forces: FX, FY, FZ act at distances Lc (Ex.3.1), X1, X2, X3, Y1, Z1 (Ex.1.2.2); the forces and moments at the centre of joint 13-2 (13′-2′) are calculated (Calc.0):
FXs = FXd = FX/2,
FYs = FYd = FY/2,
FZs = FZd = FZ/2;
MX2s = FY (Lc + Z1)/2 + FZ Y1/2,
MY2s = FX (Lc + Z1)/2 + FZ (X1 + X2 + X3)/2,
MZ2s = FX Y1/2 + FY (X1 + X2 + X3)/2.
Note. Since the external forces may act in either direction in service, the maximum possible loads were determined (the moments were added in the relations).
Dimensional constraints: width of element 2, b2 = b13; thickness of element 13, g13; thickness of element 2, g2 (Ex.8.1).
Operating constraints: temperature T = (−30…40) °C; outdoor working environment with high humidity.
Design constraints: steels for threaded fasteners (bolts, nuts) are used.
B. Choice of materials and heat treatments
The steels of the threaded fasteners (bolts, nuts) may be chosen from property classes 6.6, 6.8, 8.6, 8.8. These steels are not heat-treated by the user. For the adopted class, the yield strength σ02 and the ultimate tensile strength σr are retained for the calculations; they follow from the class symbol (e.g. for class 6.8, σ02 ≈ 6·8·10 = 480 MPa and σr ≈ 6·100 = 600 MPa).
C. Design of the constructive shape
Constructive model
Constructive features and constraints
– flat bar of width b2 = b13 and thicknesses g13 and g2 was used for elements 13 and 2;
– the clearance holes of diameter D1 are placed at the corners of a square of side L, located at distance a from the edges (Ex.7.1, Ex.8.1);
– the clearance holes of diameter D2 are placed at the corners of a rectangle of sides L1, L2, located at distance a1 from the edges; these values are determined later (Sec.8.4, Ex.8.4).
D. Calculation model, sizing and checking
Calculation model
Calculation assumptions and loads
– the loads are transmitted from tie rod 13 to bracket 2 through the shanks of the four bolts and their threads and through friction on the contact surfaces; for sizing, the most heavily loaded bolt must be identified, using the superposition principle;
– the force FYs loads the joint axially and centrally; the moments MZ2s and MX2s load it axially and eccentrically; the forces FXs, FZs and the moment MY2s load it transversely;
– the bolts are fitted with clearance, and the main load on their shanks is tension;
– the most heavily loaded bolt is sized from the tensile strength condition of its shank.
Calculation data
– loading: FXs, FYs, FZs; MX2s, MY2s, MZ2s;
– dimensions: L; a (see stage C); r1 = r2 = r3 = r4 = r = L
/2;
– materials: property class m.n (possible classes: 6.6, 6.8, 8.6, 8.8), σ02 = m·n·10 MPa – the yield strength; c = 2…3 – the safety factor; σat = σ02/c – the allowable tensile stress;
– friction parameters: µ = 0.15…0.25 – the sliding friction coefficient between the thread turns; µ1 = 0.15…0.25 – the sliding friction coefficient between elements 13 and 2; µ2 = 0.15…0.25 – the sliding friction coefficient between nut and washer.
Calculation relations
Identifying the most heavily loaded bolt
Axial forces in the bolts for each load:
– bolt forces due to the force FYs, which loads the joint axially and centrally:
= 
= 
= 
= 
(Calc.0);
– bolt forces due to the moment MZ2s, which loads the joint axially and eccentrically:
= 
= 
, 
= 
= 
(Calc.8.2.1);
– bolt forces due to the moment MX2s, which loads the joint axially and eccentrically:
= 
= 
, 
= 
= (Calc.8.2.2);
– bolt forces due to the resultant of the forces FXs and FZs, which load the joint transversely:
= 
= 
= 
= 
(Calc.8.2.3),
where βa = 1.1…1.3 is the safety factor against slipping and µ1 = 0.15…0.25 the sliding friction coefficient between elements 13 and 2;
– bolt forces due to the moment MY2s, which loads the joint transversely:
= 
= 
= 
= 
(Calc.8.2.4),
where r is the half-diagonal of the square on which the bolt axes lie (r1 = r2 = r3 = r4 = r = L/2).
Total axial forces in the bolts
Faș1 = + + + 
+ (Calc.0),
Faș2 = + + + 
+ (Calc.0),
Faș3 = 
+ + + 
+ 
(Calc.0),
Faș4 = + + + 
+ (Calc.0).
The maximum total axial force Faș = max(Faș1, Faș2, Faș3, Faș4) identifies the most heavily loaded bolt, which is sized by the strength calculation.
Axial tightening force in the bolts at assembly
Fst = max( + , + , + , + ) (Calc.0).
Note. This force provides the clamping between elements 13 and 2 needed for friction without slipping.
Sizing the threaded shank
– sizing the most heavily loaded bolt: from the tensile strength condition, the required minor thread diameter is
= 
(Calc.3.3.2),
where β = 1.1…1.3 is a correction factor accounting for the torsion that appears when the joint is tightened, and σat the allowable tensile stress;
– adopting the standard dimensions: from Annex.ST.03, the thread parameters d1 (d1 ≥ ), d2, d (Md), p; matching the thread size (Md), the clearance hole diameter D1 (medium series) is taken from Annex.ST.05.
Tightening torques of the joint
– thread torque (friction in the thread):
Mî = Fst 
tan(arctan 
+ arctan 
) (Calc.3.3.4),
where µ = 0.15…0.25 is the friction coefficient in the thread, α = 60° the profile angle of the metric thread and p the thread pitch;
– friction torque between nut and washer (bearing-face friction):
Mp = Fst 
μ2 
(Calc.3.3.5),
where µ2 = 0.15…0.25 is the friction coefficient between nut and washer, S the width across flats of the nut and D1 the clearance hole diameter;
– nut tightening torque (applied with the wrench):
Mc = Mî + Mp (Calc.0).
Checks
– check of the threaded shank under combined loading (tension and torsion) during tightening:
σe ≤ σat,
where the equivalent stress is
σe = 
(Calc.3.3.6);
– self-locking check:
β2 ≤ φ′,
where
β2 = arctan , φ′ = arctan 
(Calc.3.3.7),
φ′ being the reduced friction angle.
Notes.
– the strength calculation considers the most unfavourable case, in which all external forces act, so that the joint is not unloaded and the forces FXs, FZs and the moment MY2s are still transmitted by friction;
– since a nut with a standard thread was adopted, no strength calculation of its thread is needed; the nut was designed taking the thread strength into account;
– usually, for metric threads the self-locking check is not required, since self-locking is ensured by design.
E. Design of the constructive-technological shape
Constructive-technological drawings
Recommendations for choosing the dimensions and constructive-technological parameters:
– dimensions: Z2, YE, b2, g2, R, a, L, LZ, LY, D1; the dimensions a1, L1, L2, D2 result from the design of the bolted joints of joint E (Sec.8.4, Ex.8.4);
– 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 = (1.5…3)g2;
– sharp edges are chamfered 1×45°;
– roughness of the surfaces machined by milling or drilling: Ra1 = 6.3 µm or Ra1 = 12.5 µm; Ra2 = 6.3 µm or Ra2 = 12.5 µm.
Recommendations for choosing the dimensions of the fasteners (Bolt 1, Bolt 2):
– hexagon head bolt (Annex.ST.12): Md, d, S, D, k, l, lf, lș;
– hexagon nut (Annex.ST.04): Md, S, m;
– plain washer (Annex.ST.06): d, d1, d2, h;
– Grower spring washer (Annex.ST.07): MN, d, d1, d2, g.