structure will bend accordingly, with the concavity on the side of the trailing edge, and will eventually break in the way shown by the many existing photos. We can see that, for any angle, the bending moment is obviously zero at the bottom and at the top of the chimney, while it assumes its maximum absolute value at exactly one third of the height H. The bending moment alone would therefore induce a rupture at one third of the structure, but the total longitudinal stress at the leading edge is also due to the force P, as we will show next Another interesting relation between the bending moment Nb and the shear force Se is at they are in general related by a simple derivative, i.e., Se=-2 as it is easy to check from Eq. 12b and Eq. 13.37 This is a well known relationship of the Statics of beams and other structural members( for a complete proof see for example Hibbeler ) It is a direct consequence of the equilibrium equations applied to an infinitesimal longitudinal portion of the beam: the change in bending moment along the beam is always equal to the shear force applied to that portion of the beam Finally, we can combine Nb and P to compute the total longitudinal stress on the cross sectional area between the lower and upper parts. We follow the theory of elasticity and deformations in beams which can be found in classical treatises such as Sommerfeld and Landau-Lifshitz, 32 or again in Bundy's paper 20 The longitudinal stress is maximum at the leading and trailing edges, located at the maximum distance from the longitudinal (centroidal) axis of the chimney which lies within the "neutral surface"of the structure, the surface which is neither stretched nor compressed For simplicity, we will only consider from here on, structures with uniform square cross section of side a, as this is the case of the toy models described in Sect. IV. In this case the stresses at the leading and trailing edge, oL and or respectively, can be evaluated from (the upper sign is for oL, the lower for ar) where A =a2 is the area of the square cross section of side a, with the factor 2 representing the distance between the longitudinal axis considered as the "neutral axis, " and the two edges. =2 is the cross sectional area computed about the neutral axis(see Sommerfeld for details). Usin also the expressions for Pr and Nb, we obtain oLIT a ng (1-)(5+3)cs-3(1+ 3H 2 a where we normalized oL/T, dividing by a, in order to obtain a dimensionless quantity which is plotted in Fig. 6, as a function of the height fraction and for several angles This quantity depends also on the ratio H, which for a real chimney is of the order H 2 10. For the toy models described in Sect. IV, the value of this ratio is even bigger c 24-61, enhancing the contribution of the second term of Eg. 16, which comes from the bending moment Nb. In Fig. 6 we show the plot for 4=24, but similar figures can be obtained for different values of the ratio In Eqs. 15-16, and in Fig. 6 the total stresses are considered positive if they represent tensions, negative if they are compressions. It is easily seen from the figure that the stress at the leading edge oL, is initially a compression, but eventually becomes a tension, constantly increasing for larger angles; ar on the contrary is usually a compression. It is therefore thestructure will bend accordingly, with the concavity on the side of the trailing edge, and will eventually break in the way shown by the many existing photos. We can see that, for any angle, the bending moment is obviously zero at the bottom and at the top of the chimney, while it assumes its maximum absolute value at exactly one third of the height H. The bending moment alone would therefore induce a rupture at one third of the structure, but the total longitudinal stress at the leading edge is also due to the force Pr, as we will show next. Another interesting relation between the bending moment Nb and the shear force Sθ is that they are in general related by a simple derivative, i.e., Sθ = − ∂Nb ∂r , as it is easy to check from Eq. 12b and Eq. 13.37 This is a well known relationship of the Statics of beams and other structural members (for a complete proof see for example Hibbeler30). It is a direct consequence of the equilibrium equations applied to an infinitesimal longitudinal portion of the beam: the change in bending moment along the beam is always equal to the shear force applied to that portion of the beam. Finally, we can combine Nb and Pr to compute the total longitudinal stress on the cross sectional area between the lower and upper parts. We follow the theory of elasticity and deformations in beams, which can be found in classical treatises such as Sommerfeld31 and Landau-Lifshitz,32 or again in Bundy’s paper.20 The longitudinal stress is maximum at the leading and trailing edges, located at the maximum distance from the longitudinal (centroidal) axis of the chimney which lies within the “neutral surface” of the structure, the surface which is neither stretched nor compressed. For simplicity, we will only consider from here on, structures with uniform square cross section of side a, as this is the case of the toy models described in Sect. IV. In this case the stresses at the leading and trailing edge, σL and σT respectively, can be evaluated from σL/T = Pr A ∓ aNb 2J (15) (the upper sign is for σL, the lower for σT ) where A = a 2 is the area of the square cross section of side a, with the factor a 2 representing the distance between the longitudinal axis, considered as the “neutral axis,” and the two edges. J = a 4 12 is the moment of inertia of the cross sectional area computed about the neutral axis (see Sommerfeld31 for details). Using also the expressions for Pr and Nb, we obtain σL/T a 2 mg = − 1 2  1 − r H  h5 + 3 r H  cos θ − 3  1 + r H i ± 3 2 H a sin θ r H  1 − r H 2 , (16) where we normalized σL/T , dividing by mg a 2 , in order to obtain a dimensionless quantity which is plotted in Fig. 6, as a function of the height fraction and for several angles. This quantity depends also on the ratio H a , which for a real chimney is of the order H a & 10. For the toy models described in Sect. IV, the value of this ratio is even bigger: H a ≃ 24 − 61, enhancing the contribution of the second term of Eq. 16, which comes from the bending moment Nb. In Fig. 6 we show the plot for H a = 24, but similar figures can be obtained for different values of the ratio. In Eqs. 15-16, and in Fig. 6 the total stresses are considered positive if they represent tensions, negative if they are compressions. It is easily seen from the figure that the stress at the leading edge σL, is initially a compression, but eventually becomes a tension, constantly increasing for larger angles; σT on the contrary is usually a compression. It is therefore the 11