To perform the structural analysis with ease, we need to idealize a structure into a form without losing significant accuracy. Generally, they are idealized as 3-dimensional structures. But, in some cases, they are idealized as 2D structures that help to simplify the analysis.
For example, if we consider a beam structure which is 3D element, it is idealized as 2D element, as there is more considered dimension along the horizontal element (along its span) compared to the other two-dimension ( i.e. along the cross-sectional dimension) as shown in Figure-1.
|Figure-1: Idealizing an Actual Beam
|Figure-2: Idealized Plane and Space Frame
Laws of Superposition
Conditions of Equilibrium
- The summation of all forces along X, Y and Z-axis is zero.
- The summation of all the moments about any axis is also zero.
- For a 3D system, the equations of equilibrium are: ΣFx =0; ΣFy=0; ΣFz =0; and ΣMx=ΣMy=ΣMz = 0;
- For a 2D system, the equations of equilibrium are ΣFx=ΣFy=0; ΣMx=ΣMy=0;
Compatibility Conditions in Structural Analysis
- The members that meet at any joint will continue to join at the same junction even after the occurrence of any deformation.
- A rigid joint, with member meeting at different angles, maintains that angle, even after the occurrence of any deformation.