Good day. In this module, we are going to cover a very important topic and that is calculating the internal forces throughout the structure. The objective of static analysis is to identify all forces within the structure in order to properly size the members to carry the applied loads. Now, the static analysis involves two steps. One step is calculating the reactions. This is something we already know how to do considering the free body diagram, and the equations of equilibrium of the structure or substructure, in this case, where we have several rigid members. The calculation of course of this reaction forces is important because these are the forces that are transmitted to the foundation of the structure, and therefore, knowledge of this forces helps us to dimensionize, to size and design the foundation so that it can safely carry these loads. However, equally important is to be able to calculate the internal forces that the structure experience throughout. Because we must make sure that at a particular section, for example, the internal forces are known and the member is sized and dimensionized in a way that it can safely carry this internal forces. So knowledge of the internal forces throughout the structure is very important and is an indispensable part of structure analysis. Up to now, the only static analysis we have learned how do completely is the static analysis of trusses. If you recall in the case of trusses, we proved that the forces within the members of a truss are only axial force, so each member was subjected only to an axial load. Therefore, the internal forces of a member where fully characterized by a single number since the axial force did not vary along the member, it was constant throughout the member. So a single positive number in case of a tensile force or negative number in case of a compressive force, fully characterized this member. Therefore, the static analysis of a truss structure, we already know how to do. However, in the case of a beam structure or in the case of a frame structure as this, the situation becomes more complicated. The reason is that besides axial forces at a particular section, beside forces which are axial that means force acting along the member which of course if excessive they might cause rupture and breaking of the member due to excessive tensile or compressive force. So besides axial internal force, we have what we call shear forces, which are acting not along the member but transverse to the member, and if excessive there will cause rupture due to excessive shear. Also, we have bending. Again, if the bending becomes excessive, if the bending at a particular section exceeds safety threshold, then again we can have failure due to excessive bending. So calculating the axial shear and bending moment is important. The other thing which complicates the situation is that this shear axial and bending moment vary as we move along the different members. So we need to calculate this internal forces throughout the entire structure. In the case of not 2D but in case of a 3D structure, the situation becomes even more complicated because besides axial and shear forces, we have also and other shear force in the orthogonal directions. That means two shear forces, one in this direction, one in this direction. Besides having one single bending moment, we have also a bending moment in this direction and also a twisting moment, that means a bending moment along the axis of the member. So in the case of 2D structure, we have three internal forces. In the case of 3D structure, we would have at this cross-section, six internal forces. Two bending moments, one twisting moment, one axial force, and two shear forces. For now, let us concentrate on 2D structures. We must realize that if we want to calculate the internal forces at a particular section as here, then we cannot do that by considering the overall equilibrium equations. Simply because internal forces once they're internal to the free body diagram we are considering, then this forces since the act on pairs and since they are action-reaction forces that means the internal forces acting on the left face of this section and the forces acting on the right face of the structure are opposite to each other. Therefore, if we consider the overall equilibrium equations, they will cancel out and they would not appear in the equations of equilibrium. So it is impossible to calculate internal forces by considering free body diagram where this section is an internal point. We need to make this a boundary point. That means we need to create a cut through this section and consider either the left or the right part of the structure. So next, let us look in the 2D case and let us show what the internal forces might look like. What we might call positive or negative axial shear and bending moment.
