Determination of deflection of statically determinate beams and frames as a function of distance due to various types of static loads

I-Abstract: This is a method for finding deflection of statically determinate beams and frames and gets the result as a function of distance due to various types of static loads and cross sections in a simple and fast way. This method based on "Double integration method". The method contains the following branches: 1. General calculations of deflection as a function of distance: This part is the base core of the theory, and it is the main idea of it, and it talks about the way which will lead to calculate the deflection as a function of distance with statically determinate beams and frames only. 2. Calculation of deflection as a function of distance due to various types of static loads: This part contains the way to calculate deflection as a function of distance for statically determinate beams and frames due to various types of static loads. Like: 1- Loads in the shape of mathematical formulas. 2- Case of changed of normal force. 3- Case of torsion. 4- Unknown curved shape loads. 5- Effect of temperature. 6- Effect of settlement and rotation at supports. 3. Calculation of deflection as a function of distance with various types of cross sections: This part contains the calculations of deflection as a function of distance with different types of cross sections along the structure. "The changing of moment of inertia “I” gradually or suddenly, or in equation shape". II- Scientific additions and advantages: This method has the following advantages: 1- The result is in the form of a mathematical equation which gives the method the whole advantages of differentiation science, and make it easy to be developed in future. For example, we can calculate max and min deflection of any structure. 2- The result can be used to get the area of the deflection shape simply and exactly of any structure. And we can use it simply in finding influence lines of statically indeterminate structures and more. 3- The final result related to the effect of internal forces of the structure. It gets the direct relation between the internal force and the final deflection - the direct effect and amount of any lonely internal force on the deflection - . So, we can use it to understand the mechanism of deflection process. 4- The equation may be useful in the future extension of structure science. And the scientific additions of the method are: 1- The method can deal with any type of loads like curved shaped loads in a mathematical formula of any degree, and even if the load shape has unknown mathematic formula, which are considered very hard cases to study. 2- The method can deal with various types of cross sections. Like when the cross section changes gradually along the structure, and the changing is in a mathematic formula.