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Author: Admin | 2025-04-28
Curve differential equation of the AB section rock beam can be expressed as: d4y2(x2)dx24+4β4y2(x2)=kγHEI(12) Among them, β = (K/4EI)1/4, K is the foundation coefficient. The general solution of the deflection curve differential equation of the AB section rock beam: e−βx2(C2cosβx2+D2sinβx2)+kγHK(13) Among them, A2, B2, C2, and D2 are constants; when x2 = 0 or x2 = L2, the conditions of the two boundaries can be simultaneously obtained to obtain the constants A2, B2, C2, and D2. The deflection and rotation angle of the left end of the second beam section is equal to those of the right end of the first beam section (continuous + smooth). -EId2y2(x2)dx22x2=0=M0-EId3y2(x2)dx23x2=0=F0-EId2y2(x2)dx22x2=L2=Mb-EId3y2(x2)dx23x2=L2=Fby2(x2)x2=0=y0(x0)x0=L0dy2(x2)dx2x2=0=dy0(x0)dx0x0=L0 (14) The length of the rock beam in the BC section is L3, and the magnitude of the distributed load increases linearly from 0 to γH from the left to right. The shear force and bending moment at the left end are the reaction forces at the right end of the AB section of the rock beam, while those at the right end section are Fc (downward) and Mc (counterclockwise), as shown in Figure 16(D). Deflection of the rock beam in the BC section can be obtained using the following expression: d4y3(x3)dx34+4β4y3(x3)=γHx3EIL3(15) The general solution of the deflection curve differential equation of the rock beam in the BC section: y3(x3)=e−βx3(A3cosβx3+B2sinβx3)+e−βx3(C3cosβx3+D2sinβx3)+γHx3KL3(16) Among them, A3, B3, C3, and D3 are constants; when x3 = 0 or x3 = L3, the conditions of the two boundaries can be simultaneously obtained to obtain the constants A3, B3, C3, and D3. The deflection and rotation angle of the left end of the BC section of the rock beam is equal to those of the right end of the AB section beam (continuous + smooth); -EId2y3(x3)dx32x3=0=Mb-EId3y3(x3)dx33x3=0=Fb-EId2y3(x3)dx32x3=L3=Mc-EId3y3(x3)dx33x3=L3=Fcy3(x3)x3=0=y2(x2)x2=L2dy3(x3)dx3x3=0=dy2(x2)dx2x2=L2 (17) The uniformly distributed load on the rock beam on the right side of the C section is γH, and the shear force and bending moment at the left end are the reaction forces at the right end of the BC section beam, as shown in Figure 16(E). Deflection of the rock beam on the right side of the C section can be obtained through the following expression: d4y4(x4)dx44+4β4y4(x4)=γHEI (18) The general solution of the differential equation of the deflection curve of the rock beam on the right side of the C section: e−βx4(C4cosβx4+D4sinβx4)+γHK(19) Among them A4, B4, C4, and D4 are constants; when x4+∞, the deflection y4 tends to be constant, then
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