莫耳圓公式推導

∑F_x′=0  ,  

σ_n=σ_xcos²θ+σ_ysin²θ+2τ_xysinθcosθ

            =1/2(σ_x+σ_y)+1/2(σ_x-σ_y) cos2θ+τ_xysin2θ

∑F_y′=0  ,  τ_n=-(σ_x-σ_y) sinθcosθ+τ_xy(cos²θ-sin²θ)

            =-1/2(σ_x-σ_y) sin2θ+τ_xycos2θ

σ_x+σ_y=σ_x′+σ_y′

[σ_n-1/2(σ_x+σ_y)]²+τ_n²=[1/2(σ_x-σ_y)]²+τ²_xy

σ_max=σ_avg+R , σ_min=σ_avg-R

τ_max=R , τ_min=R