Each design code has its own standard geometric shape and size of concrete specimens, for example, ACI uses cylinder specimen wih dia. of 150mm and height of 300 mm, while BS uses 150mm size cube specimen.
Beside those standard specimen, many other nonstandard shapes and sizes may be used based on the available molds and used aggregate nominal size, the attached table shows some nonstandard specimen and their correction factors.
Using long specimen such as cylinder or prism eliminates the confining effects due to friction with testing machine heads, hence, its results always smaller that the same size cube specimen.
Cylinder specimen has one more advantage, it could be used to measure both compression and tension strength of concrete ( using compression test and splitting test ).
On other hand, cube specimen also has some advantage, it is smaller, lighter and don't need sulfur caps to be tested ( it can be tested on any side)
near to the area where you applicate the load (for the testing of compressive strength or tensile strength) the stress state is always three-dimensional because of the influence of load application.
You need an uniaxial stress state for your tests. This can be realized by using test specimen with sufficient slenderness ratio, the stress state in the middle of the test specimen (where the collapse starts) will be nearly uniaxial. (length:width should be minimal 2:1, e.g. a cylinder 15x30cm).
If you use a cube 15x15x15cm there will be a 3D-stress state in the whole cube, even in the area where the collapse starts. The influence of the load application will produce side pressure, so the collapse starts at a higher stress level. Cube tests will always produce higher values for compressive strength. These values could be recalculated to the uniaxial strength / cylindric strength, but in my opinion it makes more sense to do the tests with the cylinders.
Another way to realize uniaxial stress state is the usage of "load brushes", they are used in triaxial tests.
The uniaxial strength values are needed as the input parameters of failure criterias, e.g. Mohr-Coulomb.
The concrete specimens of different geometric shapes are used to study the shape effect on the properties of concrete. For example, in the case of compressive strength, the shape will have effects on the contact area and associated friction when the load is applied.
selection of test specimen size and geometry depends mainly on the property required to be determined, considering the maximum size of aggregate is essential in selecting the mold size for casting concrete specimen, the dimension of the mold and thickness of layer should be more than three times the maximum aggregate size, so if you want to use the real size of raw material, then you select the bigger geometry of the specimen to be cast in the laboratory.
Compression loads are applied in the direction of the casting for the cylinder and perpendicular to the direction of the casting for the cubes. Because both cylinder and cubes are cast and consolidated in multiple layers, the direction of loading is essential in the relationship between cylinder strength and cube strength. When cylinders are loaded, each casting layer occupies an entire cross-section are receives the total load from the testing machine. When cubes are loaded, each layer extends from top to bottom and receives a portion of the total load [50]. The cylinder specimen indicated a small change in compressive strength when the l/d ratio changed from 1.0 to 2.0; compared to cubes. Compressive strength for the cylinder increased as l/d increased, whereas there was an 18.2% reduction in compressive strength for the cube. The cube could carry a higher load than the cylinder (both cross-sections remained identical). At the same l/d ratio and shape (cube only), the compressive strength of foam concrete, compressive strength did not decrease as the specimen size increased. According to the influence of platen restraint on the mode of failure is more significant in a cube than in a standard cylinder, the cube strength is about 1.2 times the cylinder strength, but the actual relationship between the strengths of the two types of the specimen depends on the level of strength and the moisture condition of cement-based materials at the time of testing. It is reasonable to ask whether a cube or a cylinder is a better test specimen. Compared with the cube test, the advantages of the cylinder are less end restraint and more uniform distribution of stress over the cross-section; for these reasons, the cylinder strength is probably closer to the true uniaxial compressive strength of cement-based materials than the cube strength. According to all experimental data of the relation between cube and cylinder stated that the cylinder strength of hardened cement-based materials is not a constant value of 0.8 times the cube strength because it depends upon so many factors water-cement ratio, quality of sands (grading, surface texture, size, shape, strength, and stiffness), cement content and the period [50, 51].
Adam M. Neville is a great treasure to all these relationships and there also it is well described how these shapes affect the actual strength of concrete.
Here is the link to the book : https://drive.google.com/file/d/1F6DMaZXZCEBc3Zkr9cqPoG0eFI1uJ6JH/view