人N FIG. 2: The forces acting on the lower portion of the chimney, due to the upper part and the action of the constraint at the base, are shown here. The two insets explain the definition of the bending moment in terms of a couple of forces. The resulting deformation of the structure is also shown for the two possible cases which, for r=H, splits into radial and angular equations, H m Fr- mg cos 8 (7 mo-b= Fo+mg sin g Using egs. 3 and 4, the two components of the force F are easily determined 5 3 F Fe 'ng sin (8b) III INTERNAL FORCES AND BENDING MOMENT We now move to the analysis of the internal forces which develop inside the structure of the falling chimney. The resulting stresses and bending moment are the causes of the rupture of the toppling chimney. Consider, as in Fig. 2, an arbitrary lower portion of the himney of height r(as opposed to the total height H)and the forces acting on this part of the structure due to the upper portion and the base The weight of the lower portion is nowb a Nb -f f -f f Nb Nb X Y S 0 W(r) r F P FIG. 2: The forces acting on the lower portion of the chimney, due to the upper part and the action of the constraint at the base, are shown here. The two insets explain the definition of the bending moment in terms of a couple of forces. The resulting deformation of the structure is also shown for the two possible cases. which, for r = H 2 , splits into radial and angular equations, −m H 2 . θ 2 = Fr − mg cos θ (7a) m H 2 .. θ = Fθ + mg sin θ. (7b) Using Eqs. 3 and 4, the two components of the force F are easily determined: Fr = 5 2 mg  cos θ − 3 5  (8a) Fθ = − 1 4 mg sin θ. (8b) III. INTERNAL FORCES AND BENDING MOMENT We now move to the analysis of the internal forces which develop inside the structure of the falling chimney. The resulting stresses and bending moment are the causes of the rupture of the toppling chimney. Consider, as in Fig. 2, an arbitrary lower portion of the chimney of height r (as opposed to the total height H) and the forces acting on this part of the structure due to the upper portion and the base. The weight of the lower portion is now 5