Boosting the hardness of the bolts based on the new data method
(1) Design variables: According to the structural characteristics of the thread shape, all parameters are established by the pitch P, so the pitch P is selected as the design variable: X=[P]=[x1].
(2) Objective function: The minimum tensile stress of the bolt section is the target. The tensile stress of the root of the thread is σca=1.3F/(0.25πD21) under the action of the total tensile force F. Where: D1=D-(53/8)P. The objective function is: f(x)=σca"min.
(3) Constraints: 1) The shear strength condition that the thread should satisfy is g1(x)=τ=F1/(πDb)=F1/(0.65πDP)≤[τ]. Where D is the large diameter of the thread. b is the thickness of the root of the thread. For trapezoidal threads, b=0.65P.F1 is the load of the first loop of thread sharing, according to the assumption that F1=F/3.
2) The bending strength condition that the thread should satisfy is: g2(x)=σb=6F1lπDb2=6F1(D-D2)/2πD(0.65P)2=6F1×33P/16πD(0.65P)2=93F18×(0.65 ) 2πDP ≤ [σb].
Where: l is the bending force arm, l=(D-D2)/2.D2 is the thread diameter, D2=D-(53%/8) P.F1 and b are as defined above.
3) Geometric constraints: In order to simplify and narrow the optimization step, select: g3(x)=P≤10.
2 Calculation example Take the M20 ordinary bolt as an example. It is known that the bolt diameter is D=20mm, and the total tensile force of the bolt is F=6000N. The bolt of the first ring thread is subjected to the penalty function F1≈F/3=2000N. The tensile strength limit [σ]=600MPa of the 45# steel bolt, the allowable shear stress [τ]=0.6[σ]≈80MPa, the allowable bending stress of the bolt material [σb]=(1.0～1.2)[σ] ≈132MPa. Substituting the known quantity into the formula, the mathematical model is: minf(x)=1.3×60000.25π×(20-53"x/8)2.
The constraint condition is: g1(x)=80-20000.65π×20x≥0.g2(x)=132-93"×20008×(0.65)2π×20x≥0.g3(x)=10-x≥0.
According to the established optimization objective function and constraint conditions, Matlab is an optimization design problem with one design variable and three constraints. The inner point penalty function method can obtain ideal results, so the Matlab optimization toolbox can be used. The optimal solution.
The following function files myfun.m and mycon.m are compiled: [myfun.m]functionf=myfun(x)f=5.2*6000/(pi*(20-1.732*5/8*x(1))^2) ;[mycon.m]function[c,ceq]=mycon(x)c=[2000/(0.65*pi*20*x(1))-80;3*2000*1.732*3*x(1)/ (8*pi*20*(0.65*x(1))^2)-132];ceq=0; enter the following statement in the command window: >>x0=2.5;% initial iteration value; >>A=1; % linear equality constraint coefficient Ax<B;>>B=10;>>Aeq=;% linear equality constraint;>>Beq=;>>lb=1; lower limit of %x;>>ub=10;% The upper limit of x>>[x,fval]=fmincon(@myfun,x0,A,B,Aeq,Beq,lb,ub,@mycon)x=1.1121fval=28.1104 This is the optimized result.
Conclusion The comparison between the optimized value and the national standard value can be concluded as follows: (1) The tensile stress of the dangerous section of the bolt rod is greatly reduced after optimization, so it can be appropriately reduced in consideration of the condition of increasing the tensile strength of the dangerous section. Small thread pitch parameters; (2) The common failure mode of threaded joint is shearing and breaking of thread teeth. The national standard GB/T192-1981 has a larger pitch value, which is the priority to ensure that the thread has sufficient bending strength and shear strength. , does not conflict with this article.
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