# Predicting the Compressive Strength of Concrete Using an RBF-ANN Model
## 1 Introduction  
Concrete has characteristics of durability, impermeability, fire resistance, abrasion resistance, and high compressive strength. It can be cast into any shape and size. So, it is the most used construction material in the modern world. Its basic ingredients are water, cement, gravel which indicates the coarse aggregate, and sand which indicates the fine aggregate. For some environmental and sustainable considerations, various types of by-product materials are widely used as admixtures in concrete. The most commonly seen are the fly ash and the blast furnace slag. Some may also add other materials such as waste plastic, waste glass, rice husk, etc. [1; 2; 3; 4; 5; 6]. Chemical and mineral admixtures might be added to enhance its workability or to change the time interval of the hardened state. The compressive strength is related to the proportioning of its ingredients. For economic reasons such as reducing the construction cost or saving the trial time for the strength requirement, the prediction of the compressive strength according to the proportioning design is a very important research topic. Studies have been implemented with mathematical models [7; 8; 9; 10; 11; 12]. In recent decades, a lot of attention has been paid to the application of artificial neural networks (ANNs) in determining the compressive strength of concrete [13; 14; 15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25; 26; 27]. An ANN is like a black box. No theoretical relations between the compressive strength of concrete and the proportioning of its ingredients are needed for establishing an ANN model. The only requirement is the sufficiency of data for training and testing processes. With a well-trained ANN, one can input several numerical values which represent the proportioning of the concrete ingredients such as water, cement, sand, and other admixtures, and the ANN rapidly prompts out the predicted compressive strength of the concrete. Typically, the database for the application of an ANN has to be divided into two sets, one for training and the other for testing. Usually, the testing set should contain more than10 % of the database. Otherwise, the ANN has a tendency to over-fit the training set and will perform poorly in practical uses. In [27], an ANN model for predicting the 28-day compressive strength of concrete was established. They used the database of [26] in which an ANN model for the same application was also established. It was presumed that in [26] too little data were used for testing so that the ANN model was barely acceptable in the final validation. In [27], the data were rearranged. Fewer data were for training while more were used for testing. Therefore, the ANN model in [27] outperforms the one in [26], and it only has half the number of neurons in the hidden layer. Both the ANN models in [26; 27] are the back-propagation (BP) networks. Besides over-fitting, how to avoid over-training is another important issue when employing the BP-ANN. The ANN usually fits the training set better and better as more and more iterations of training are processed. However, it does not guarantee the over-trained ANN will also perform as well in practical uses. Because the training procedure of BP-ANN is extremely time-consuming, it is difficult to determine how many iterations of training are too excessive. In addition, it sometimes converges to a state of local minimum error. Apart from the BP-ANN, there are many other types of ANNs. One of them is the Radial Basis Function (RBF) network. Unlike the BP-ANN, the solution of RBF-ANN is obtained in a rather straightforward way. The RBF centers and their shape parameters are usually determined in advance, then the weights are determined by the least-squares approach for a linear algebraic system in which the solution is unique. The only concern is how to choose the RBF centers and their shape parameters properly. Some may use the k-means approach to set the cluster centers as the RBF centers.