2 T₍₁₎=P , M_g(1)=-P*(a-x₁) , T₍₁₎=∫qdx₁=q*(a-x₁) M_g(1)=∫q*(x-x₁)dx=-1/2*q*(a-x₁)²

3 p_j=lim∆W_j/∆A=dW_j/dA {N/mm²]=[Mpa] p_j=σ*e_j σ=p_j*e_j

4 σ₁₁-σ σ₁₂ σ₁₃ σ₂₁ σ₂₂-σ σ₂₃ σ₃₁ σ₃₂ σ₃₃-σ det[σ_ij-σб_ij]=σ³-I_σσ²+II_σσ-III_σ=0 I_σ=σ₁₁+σ₂₂+σ₃₃=σ_ij II_σ=1/2*(σ_iiσ_kk-σ_ikσ_ik) III_σ=Det[σ_ik]

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 (∂σ_ij/∂x_i)+Y_j=0 i,j=1,2,3

6 u=x-x⁰ u₁=x₁-x₁⁰ u'=u=du T=[t_ik]= u₁₁ u₁₂ u₁₃ u₂₁ u₂₂ u₂₃ u₃₁ u₃₂ u₃₃ u_ik=∂u_i/∂x_k =1/2*(∂u_i/∂x_k+∂u_k/∂x_i) +1/2*(∂u_i/∂x_k-∂u_k/∂x_i) 2 -część określa sztywne obroty ciała , 1-wspó. Symetrycznego tensora małych odkształceń ε_ik=ε_ki=1/2*(∂u_i/∂x_k+∂u_k/∂x_i) ε₁₁,ε₂₂,ε₃₃ ε_12,ε₂₃,ε₃₁

7 ds₀²=dx_i⁰dx_i⁰ ds²=dx_jdx_j dx_i=∂x_i/∂x_j⁰*dx_j⁰ ds²-ds₀²=(∂x_i/∂x_k⁰)*dx_k⁰*(∂x_i/∂x_l⁰)*dx_l⁰-dx_m⁰dx_m⁰ =[(∂x_i/∂x_k⁰)*(∂x_i/∂x_l⁰)-б_kl]*dx_k⁰*dx_l⁰ =(u_ik+u_ki+u_jiu_jk)* dx_i⁰*dx_k⁰ ds²-ds₀²=2*ε_ik*dx_i⁰*dx_k⁰ ε_qr'*dx_q'*dx_r'=ε_ik*dx_i⁰*dx_k⁰ α_qi=dx_i⁰/dx_q⁰ ε_qr'=ε_ik*α_qi*α_rk

8 σ_ij=C_ijkl*ε_kl ε_kl=S_klij*σ_ij 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*σ_ik*ε_ik σ_ik=s_ik+1/3*σ_jj*б_ik ε_ik=e_ik+1/3*e_ll*б_ik U=1/2*s_ik*e_ik+1/6*σ_jj*e_ll=U_p+U₀

11 W=W(σ_ik)=W(σ₁,σ₂,σ₃) i,k =1,2,3

12 W=AI_σ+BI_σ²+CII_σ+DI_σ⁴+EI_σ²II_σ+FII_σ² W=AI_σ+BI_σ²+CII_σ≤1 A=1/R_r B=1/(R_r*R_c) C=-1/R_t² (1/R_r-1/R_c)*I_σ—[(R_rR_c-3R_t²)/(3R_rR_cR_t²)]*I_σ² +(1/3R_t²)*σ_H²≤1

13 I_x1c=I_x1-(e₂)²*A I_x2c=I_x2-(e₁)²*A I_x1cx2c=I_x1x2-e₁e₂*A I_x1'c=1/2*(I_x1c+I_x2c)+1/2*(I_x1c-I_x2c)*cos2α--I_x1cx2csin2α I_x2'c=1/2*(I_x1c+I_x2c)-1/2*(I_x1c-I_x2c)*cos2α+I_x1cx2csin2α I_x1'cx2'c=1/2*(I_x1c-I_x2c)*sin2α+I_x1cx2ccos2α

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 M_s=∫ρσ_sdA dx dρ γ(ρ)=ρ*dφ/dx γ(x)=σ_s/G σ_s=G*dφ/dx*ρ M_s=G*dφ/dx*∫ρ²dA dφ/dx=M_s/GI_s φ=M_sl/GI_s σ_s=M_s/I_s*ρ

16 σ_{s max}=M_s/W_s φ(x)=M_sx/GI_s W_s=2Fб_min I_s=4F²/∫(ds/б) =4F²/∑(s_i/б_i)

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18 dx₁/ρ=[(1+ε)dx₁/(ρ+x₂] =>ε=x₂/ρ ε=σ_g/E => σ_g=E/ρ*x₂ E/ρ*∫x₂²dA=M_g ∫x₂²dA=I₃ 1/ρ=M_g/EI₃ σ_g=M_g/I₃*x₂ σ_g(x₁)=M_g(x₁)/I₃*x₂

19 ∫x₂σ_gdA=M_g(x₁) ∫σ_TdA=T(x₁) σ_Tb(x₂)dx₁=∫(σ_g+dσ_g)dA'-∫σ_gdA' σ_Tb(x₂)=(dM_g(x₁)/dx₁)*[(∫x₂dA')/I₃] dM_g(x₁)/dx₁=T(x₁) S=∫x₂dA' σ_T=(T(x₁)/I₃)*(S/b(x₂))

20 χ_g(x₁)=1/ρ(x₁)=M_g(x₁)/EI₃ ψ(x₁)=dv/dx₁ 1/ρ(x₁)=(-d²v/dx₁²)/√[1+(dv/dx₁)²]³≈-(d²v/dx₁²) (d²v/dx₁²)=-(M_g(x₁)/EI₃) EI₃*(dv/dx₁)=EI₃ψ(x₁)=-∫M_g(x₁)dx₁+C EI₃v(x₁)=-∫[∫M_g(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)] σ_g=E*[(r/r₁)-1]*[x₂/(r+x₂)] ∫σ_gdA=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) M_g==∫x₂σ_gdA=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 : M_g=E*[(r/r₁)-1]*S σ_g=(M_g/S)*[x₂/(r+x₂)]

22 d²v/dx₁²=M_g(x₁)/EI₃ => EI₃*d³v/dx₁³=dM_g/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+e^βx1*(Acosβx₁+Bsinβx₁)+e^-βx1*(Ccocβx₁+Dsinβx₁) ψ=dv/dx₁ M_g(x₁)=-EI₃*(d²v/dx₁²) T(x₁)=-EI₃*(d³v/dx₁³)

23 σ_zred=√[σ_n²+3σ_t²] σ_zred=√[σ_n²+4σ_t²] σ_n=σ+σ_g σ_t=σ_s+σ_T

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25 M_g(x₁)=Pv(x₁) d²v/dx₁²=-(M_g/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 P_kr=(π²EI)/l_s²

26 P_kr=π²EI/l_s² σ_kr=P_kr/A= π²EI/l_s²A i=√[I/A] λ=l_s/i σ_kr=π²E/λ² σ_kr≤R_H π²E/λ²≤R_H λ≥π*√[E/R_H]=λ_gr

27 λ<λ_gr σ_kr=A-Bλ² λ=0 σ_kr=R_e σ_kr=R_e*[1-(λ²/2λ₀²)] λ₀= π*√[2E/R_e] σ_kr=a-bλ dla λ=0 σ_kr=R_e dla λ=λ_gr σ_kr=R_H

28

29

30 M_g(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/P_kr] v_max=a*{1/[cos((π/2)*√[p/p_kr])]-1} M_{g max}=P*(a+v_max)=Pa/[cos((π/2)*√[P/P_kr])] σ_max=P/A+ Pa/[W_gcos((π/2)*√[P/P_kr])]

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