Issue 50

A. Kakaliagos et alii, Frattura ed Integrità Strutturale, 50 (2019) 481-496; DOI: 10.3221/IGF-ESIS.50.40 487 ௥௘ௗ ൌ ሺ ௪ ൅ ሻሺ ௪ െ ሻ ௦ (6) 0.5 ଶ ൌ ௥௘ௗ where: ൌ 0.5 ௪ െ 0.5ඥ ௪ ሺ ௪ െ 4 ∗ and ∗ ൌ ஻௩ మ ଶ௚௏ ೘ (7) (a) (b) (c) Figure 2: (a) Punching shear cylinder at cannonball impact area; (b) formation of punching shear cylinder; (c) wall breach. The above equations can be used to simulate continuous bombardment on the wall. To do this, in the follow up shots, wall thickness is set equal to the reduced wall thickness and V m in Eq.(7) is replaced with V red from the corresponding previous shot. In case wall thickness is reduced at or below 4s* in Eq.(7), equilibrium is not produced. This serves as an indication that the projectile has opened a breach in the wall. To simulate the first shot of Orban’s gun, input values with t w =5.0 m, d=0.752 m, f s =0.12 MPa, projectile weight B=6.0 KN, g=9.81 m/sec 2 and v=191 m/sec were considered. The result of the first shot, calculated with Eq.(7), yields: s=1.45 m, V red =7,694 kN and a reduced wall thickness at 3.55 m. It was concluded that after the first shot the outer wall masonry skin was destroyed and the projectile penetrated into the wall solid. Considering s=1.45 m as wall load eccentricity in general, this value is close to 1/3 of the wall thickness of 1.67m, where wall stability limit is reached. This result confirms eyewitness report that the wall started tilting after the first shot [5]. To simulate the second shot, wall thickness was set at t w =3.55 m and wall punching shear capacity at V m =7,694 kN. Equilibrium was not delivered with Eq.(7) and it was concluded that the projectile had opened a breach in the wall (Fig. 2c). Herein, the total breach height at 8.88m confirms eye witness report, where the height of the breach was at 9.15 m (Iskanter [5]). During the procedure of continuous bombardment wall overturning did not occur. The Inner Wall was considered as cantilever structure, whereby the connection to adjacent Towers and associated arching effects were not mobilized. A typical portion of the Inner Walls between adjacent Towers with a mean total length at 50 m was considered. Cannonball impact on the wall with a force at 10,837 KN, thus at wall punching shear capacity on the impact area, yields a maximum compressive stress at wall base was at 0.97 MPa. The associated vertical load eccentricity due to imposed overturning moment was at 1.35 m, fair below 1.67 m, which corresponds to the general acceptable wall stability eccentricity limit at 30% of wall thickness. Orban’s gun projectile penetration into soil The cannonball would impact on ground and penetrate into soil in case aiming was not appropriate. To simulate this effect, cannonball was treated as a single mass m, punching the ground with an impact velocity v, with soil stiffness simulated by a single translational spring. Considering energy equilibrium during cannonball impact on ground together with cannonball weight B, diameter d and soil subgrade modulus K S , projectile penetration into soil δ F results with Eq.(8): 2 2 2 S F Bv πd K δ 2g 4  and 0 5 F S v B δ 0 255 d K . .        (8) During gun fire testing as reported by Doukas [4], the cannonball touched down after 1 mile (1479m) and penetrated one fathom into the ground. Deploying impact velocity on ground at v=153 m/sec (Table 4), B=6.0 KN, g=9.81 m/sec 2 , d=0.752 m and K S =10,000 KN/m 3 for loose sand, δ F results at 1.27 m. This result corresponds approximately to one fathom (1.83 m). Soil subgrade modulus was selected accordingly to reflect loose soil conditions close to earth surface, such as crop fields with top organic soil layer.