# Analysis of Optimal Loading Angle in Dynamic Flattened Brazilian Disc Splitting Test for Concrete
## 1 Introduction  
Currently, explosions are common across the globe, posing a serious threat to human life and causing dramatic damage to homes and infrastructure. The latter has received attention from scholars in the field of structural engineering who study the effects of blast and impact on different RC members (beams, columns, slabs, silos, etc.) [1; 2; 3; 4]. When a pressure pulse is reflected as a tensile pulse in a column or on the free surface of a slab, it will likely cause a fairly high tensile stress somewhere adjacent to the free surface, and once a certain dynamic fracture criterion is met, it will cause the tensile rupture of the material at that location. When the crack is large enough, the whole lobe flies away with the momentum caught in it. This dynamic tensile fracture on the back side caused by the reflection of pressure pulses on the free surface is called spalling. The lobes that fly out are called scab. In the above case, once a lamellar crack appears, a new free surface is also formed at the same time. Continued incident pressure pulses are then reflected on this new free surface, which may cause a second spalling of lamination cracks. By analogy, a series of multi-layer lamellar fractures can be formed under certain conditions to produce a series of multi-spalling lamellar fractures. Therefore, it is of great importance to obtain the accurate tensile strength of concrete materials for the safe design of civil engineering structures [5; 6; 7]. Testing methods to determine the tensile strength of brittle materials can be divided into direct tensile methods [8; 9; 10] and indirect tensile methods [11; 12; 13; 14; 15]. The direct tensilemethod obtains the tensile strength of concrete in a straightforward manner, but the difficulty in achieving a uniaxial tensile stress state and the irregularity of the experimental operation may lead to local stress concentration and cause a non-standard damage pattern. On the other hand, indirect tensile testing methods, such as the Brazilian disc (BD) test [16], Ring method [17], and the Luong method [18], have received extensive attention due to their feasibility and simplicity of operation. Among them, the BD test has been adopted as the suggested testing method by the ISRM [19] and ISO standards [20] for testing the tensile strength of rock materials, and it has been designated as the standard method in ASTM C496 and DIN EN 12390-6 for testing the tensile strength of concrete materials [21; 22]. However, previous research [23; 24; 25; 26] has pointed out that nonnegligible stress concentration can easily be generated at the interface between BD specimen and loading platen, which results in cracks forming at the loading edge rather than the center of the specimen. The crack initiation at the loading edge does not satisfy the Griffith strength criterion, bringing out unreliable tensile strength. Therefore, efforts have been made to improve the loading configuration of BD splitting experiments to ensure the validity of the results. The improvement to reduce the stress concentration basically falls into two categories: the arc-loaded BD test and the flattened BD test. In terms of the arc-loaded BD test, [PERSON] et al. [27] gave theoretical solutions of the whole stress and displacement fields for the specimen under radial loading on a finite arc, which turned out to be superior to the original concentrated load. However, in-depth error analysis is absent for the approximate expressions describing the relevant stress components, and it is difficult to achieve the required uniform radial loads distributed over a pair of circular arcs in practical experiments [28; 29]. Regarding the flattened BD test, the loading angle greatly affects the validity of the results. With the help of the finite element method, it was found that to ensure the central cracking of the sample, the loading angle should be no less than 19.5\({}^{\circ}\)[30; 31]. Meanwhile, [PERSON] et al.