2 T(1)=P , Mg(1)=-P*(a-x1) , T(1)=∫qdx1=q*(a-x1) Mg(1)=∫q*(x-x1)dx=-1/2*q*(a-x1)2
3 pj=lim∆Wj/∆A=dWj/dA {N/mm2]=[Mpa] pj=σ*ej σ=pj*ej
4 σ11-σ σ12 σ13 σ21 σ22-σ σ23 σ31 σ32 σ33-σ det[σij-σбij]=σ3-Iσσ2+IIσσ-IIIσ=0 Iσ=σ11+σ22+σ33=σij IIσ=1/2*(σiiσkk-σikσik) IIIσ=Det[σik]
5 -σ32dx1dx2*(dx3/2)-[σ32+(∂σ32/∂x2)*dx3]*dx1dx2*(dx3/2) +σ23dx1dx3*(dx2/2)+ [σ23+(∂σ23/∂x2)*dx2]*dx1dx3*(dx2/2)=0 czyli σ23=σ32 [σ22+(∂σ22/∂x2)*dx2]*dx1dx3-σ22dx2dx1dx3+[σ32+(∂σ32/∂x3)*dx3]*dx1dx2- σ32dx3dx1dx2+[σ12+(∂σ12/∂x1)*dx1]*dx2dx3- σ12dx1dx2dx3+ Y2dx1dx2dx3=0 (∂σ12/∂x1)+(∂σ22/∂x2)+(∂σ32/∂x3)+Y2=0 (∂σij/∂xi)+Yj=0 i,j=1,2,3
6 u=x-x0 u1=x1-x10 u'=u=du T=[tik]= u11 u12 u13 u21 u22 u23 u31 u32 u33 uik=∂ui/∂xk =1/2*(∂ui/∂xk+∂uk/∂xi) +1/2*(∂ui/∂xk-∂uk/∂xi) 2 -część określa sztywne obroty ciała , 1-wspó. Symetrycznego tensora małych odkształceń εik=εki=1/2*(∂ui/∂xk+∂uk/∂xi) ε11,ε22,ε33 ε12,ε23,ε31
7 ds02=dxi0dxi0 ds2=dxjdxj dxi=∂xi/∂xj0*dxj0 ds2-ds02=(∂xi/∂xk0)*dxk0*(∂xi/∂xl0)*dxl0-dxm0dxm0 =[(∂xi/∂xk0)*(∂xi/∂xl0)-бkl]*dxk0*dxl0 =(uik+uki+ujiujk)* dxi0*dxk0 ds2-ds02=2*εik*dxi0*dxk0 εqr'*dxq'*dxr'=εik*dxi0*dxk0 αqi=dxi0/dxq0 εqr'=εik*αqi*αrk
8 σij=Cijkl*εkl εkl=Sklij*σij S1111=1/E S1122=-v/E S1212=2*(S1111-S1122)=[2*(1+v)]/E=1/G ε11=1/E*σ11 ε22=-v/E*σ11 ε33=-v/E*σ11 ε11=1/E*[σ11-v*(σ22+σ33)] ε22=1/E*[σ22-v*(σ33+σ11)] ε33=1/E*[σ33-v*(σ11+σ22)] ε12=1/2G*σ12 ε23=1/2G*σ23 ε31=1/2G*σ31
10 U=1/2*σ11*ε11 U=1/2*σ12*(2ε12)=1/2*(σ12*ε12+ε21*σ21) U=1/2*σik*εik σik=sik+1/3*σjj*бik εik=eik+1/3*ell*бik U=1/2*sik*eik+1/6*σjj*ell=Up+U0
11 W=W(σik)=W(σ1,σ2,σ3) i,k =1,2,3
12 W=AIσ+BIσ2+CIIσ+DIσ4+EIσ2IIσ+FIIσ2 W=AIσ+BIσ2+CIIσ≤1 A=1/Rr B=1/(Rr*Rc) C=-1/Rt2 (1/Rr-1/Rc)*Iσ—[(RrRc-3Rt2)/(3RrRcRt2)]*Iσ2 +(1/3Rt2)*σH2≤1
13 Ix1c=Ix1-(e2)2*A Ix2c=Ix2-(e1)2*A Ix1cx2c=Ix1x2-e1e2*A Ix1'c=1/2*(Ix1c+Ix2c)+1/2*(Ix1c-Ix2c)*cos2α--Ix1cx2csin2α Ix2'c=1/2*(Ix1c+Ix2c)-1/2*(Ix1c-Ix2c)*cos2α+Ix1cx2csin2α Ix1'cx2'c=1/2*(Ix1c-Ix2c)*sin2α+Ix1cx2ccos2α
14 σ(x)=N(x)/A u(x)=∫ε(x)dx ε(x)=σ(x)/E=N(x)/EA u(x)=∫(N(x)/EA)*dx
15 Ms=∫ρσsdA dx dρ γ(ρ)=ρ*dφ/dx γ(x)=σs/G σs=G*dφ/dx*ρ Ms=G*dφ/dx*∫ρ2dA dφ/dx=Ms/GIs φ=Msl/GIs σs=Ms/Is*ρ
16 σs max=Ms/Ws φ(x)=Msx/GIs Ws=2Fбmin Is=4F2/∫(ds/б) =4F2/∑(si/бi)
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18 dx1/ρ=[(1+ε)dx1/(ρ+x2] =>ε=x2/ρ ε=σg/E => σg=E/ρ*x2 E/ρ*∫x22dA=Mg ∫x22dA=I3 1/ρ=Mg/EI3 σg=Mg/I3*x2 σg(x1)=Mg(x1)/I3*x2
19 ∫x2σgdA=Mg(x1) ∫σTdA=T(x1) σTb(x2)dx1=∫(σg+dσg)dA'-∫σgdA' σTb(x2)=(dMg(x1)/dx1)*[(∫x2dA')/I3] dMg(x1)/dx1=T(x1) S=∫x2dA' σT=(T(x1)/I3)*(S/b(x2))
20 χg(x1)=1/ρ(x1)=Mg(x1)/EI3 ψ(x1)=dv/dx1 1/ρ(x1)=(-d2v/dx12)/√[1+(dv/dx1)2]3≈-(d2v/dx12) (d2v/dx12)=-(Mg(x1)/EI3) EI3*(dv/dx1)=EI3ψ(x1)=-∫Mg(x1)dx1+C EI3v(x1)=-∫[∫Mg(x1)dx1]dx1+Cx1+D dla x1=0 v(x1)=0 dla x1=l v(x1)=0 dla x1=0 v(x1)=0 ψ(x1)=0
21 ε=[(r1+x2)dψ1-(r+x2)dψ]/[(r+x2)dψ)] σg=E*[(r/r1)-1]*[x2/(r+x2)] ∫σgdA=0 E*[(r/r1)-1]*∫[x2/(r+x2)]*dA=0 ponieważ [(r/r1)-1]≠0, bo r≠r1 to ∫[x2/(r+x2)]*dA=0 po wprowadzeniu ρ=r+x2 ∫[(ρ-r)/r]dA=A-r*∫dA/ρ=0 r=A/(∫dA/ρ) Mg==∫x2σgdA=E*[(r/r1)-1]*∫[x22/(r+x2)]dA ∫[x22/(r+x2)]dA =∫[(x22+rx2-rx2)/(r+x2)]dA= ∫x2dA-r*∫[x2/(r+x2)]dA 1-całka w wcześniejszym równaniu stanowi moment statyczny S , 2-jest równa 0 zatem : Mg=E*[(r/r1)-1]*S σg=(Mg/S)*[x2/(r+x2)]
22 d2v/dx12=Mg(x1)/EI3 => EI3*d3v/dx13=dMg/dx1=T(x1) EI3*d4v/dx14=dT/dx1=q1 q1==q-p=q-kv EI3*d4v/dx14=q-kv czyli d4v/dx14+4β4v=q/EI3 , gdzie β=4√[k/4EI3] v=q/k+eβx1*(Acosβx1+Bsinβx1)+e-βx1*(Ccocβx1+Dsinβx1) ψ=dv/dx1 Mg(x1)=-EI3*(d2v/dx12) T(x1)=-EI3*(d3v/dx13)
23 σzred=√[σn2+3σt2] σzred=√[σn2+4σt2] σn=σ+σg σt=σs+σT
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25 Mg(x1)=Pv(x1) d2v/dx12=-(Mg/EI) d2v/dx12=-k2v k2=P/EI v(x1)=Asin(kx1)+Bcos(kx1) dla x1=0 v=0 , dla x1=l v=0 B=0 , Asin(kl)=0 kl=nπ => P=(n2π2EI)/l2 Pkr=(π2EI)/ls2
26 Pkr=π2EI/ls2 σkr=Pkr/A= π2EI/ls2A i=√[I/A] λ=ls/i σkr=π2E/λ2 σkr≤RH π2E/λ2≤RH λ≥π*√[E/RH]=λgr
27 λ<λgr σkr=A-Bλ2 λ=0 σkr=Re σkr=Re*[1-(λ2/2λ02)] λ0= π*√[2E/Re] σkr=a-bλ dla λ=0 σkr=Re dla λ=λgr σkr=RH
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30 Mg(x1)=Pa+Pv (d2v/dx12)+k2v+k2a=0 k2=P/EI v(x1)=Asin(kx1)+Bcos(kx1)-a x1=0 v=0 B-a=0 B=a x1=l v=0 Asin(kl)+acos(kl)-a=0 A=a*[(1-cos(kl))/sin(kl)] v=a*{[cosk*((l/2)-x1)]/cos(kl/2)—1} k=√[P/EI]=(π/l)*√[(l2/π2)*(P/EI)]=(π/l)*√[P/Pkr] vmax=a*{1/[cos((π/2)*√[p/pkr])]-1} Mg max=P*(a+vmax)=Pa/[cos((π/2)*√[P/Pkr])] σmax=P/A+ Pa/[Wgcos((π/2)*√[P/Pkr])]