Main concepts

Questions you should be able to answer:

  • What is stress?
  • What are the types of stress?
  • What is the factor of safety?
  • What is strain?
  • Differentiate between Deformation, Elongation, and Displacement
  • What is the strain-displacement relation?
  • What are yield and ultimate stresses?
  • What is the difference between brittle and ductile materials?

Deformation and Deflection

Strictly speaking, ALL solids deform when subjected to loading. A tree deforms with air blowing; your seat deforms as you sit on it; the road deforms as a car passes on it; and the great pyramids of Giza deform if you stood on the top of them! The difference between all those solid deformations lies in their stiffness (their ability to resist deformation). That is why we may call some solids rigid (do not deform) when the loads applied on them cause extremely small deformations that will make it very hard for us to measure.

On the other hand, when we mention deflection, we are concerned, usually, with how a single point on, or inside, the solid moves. Hence, we may say that the tip of the tree branch moved 20 cm to the north, the tip of the pyramid moved down 3 nm when you stood on it, or the end of the shaft rotated 5 degrees when you twisted it. Thus, sometimes, it is convenient to present the displacements in a vector form as well as scalar forms. In contrast with rigid bodies, we call solids that deform under loading elastic bodies. Let me remind you at this point that there is no real boundary between elastic and rigid bodies, rather, “does the load create a noticeable deformation to the solid?” should be the question we ask.

Compatibility of Structures

The compatibility means that the structure deforms in a way that keaps it in one piece (no fracture), it does not detach from the support (zero or fixed value deflections at the supports), and its adjacent points deform in a manner that can be described by a continuous function.

Input-Output Relations for Structures

If a piece of wood is just sitting with no external effects on it, nothing happens, that is not an interesting problem for us! However, if you stand on it or put some loads on it, it may deform or even break down, that is interesting.

It is not common to talk about structures in terms of inputs and outputs, though, I prefer to use this analogy to describe the load-deformation relations for the structures. The piece of wood we mentioned earlier, does not deform or break down unless a load is applied. Hence, we may consider that the load is an input to the structure. For equilibrium, the structure generates internal stresses which, in turn, causes strain. The strain is not measured directly, rather, it is manifested in deformation of the structure which may, then, be measured. From that, we may refer to the “deformations” as the output of the structure.

This concept, though considered alien to structural analysis, is quite important from my point of view. Since every problem involving structures, starts with an applied load and ends with a measured deformation, we can always see the direction in which we are progressing.

Input-Output relation for structures

What is Stress?

Stress is a concept relates to structures. Stresses occur inside the structure. Basically, the structure needs to stay under equilibrium, so, when subjected to external loads, internal forces between material particles are generated to reserve the equilibrium of the structure. These forces are extremely hard to measure, and may vary from one particle to the other, but, when all those forces add-up, the structure maintains equilibrium. Thus, we assume an average force per unit area inside the structure which we call stress. Therefore, we may claim that stress is not a natural phenomenon, rather, a presentation of the forces that are created between the particles to keep the structure from falling apart.

Two Types of Stresses

Since stress is not a real quantity, rather, it is an average force per unit area, the stress is always associated with the area it is acting upon. For any given planar area, you may define a normal direction and two tangential one (in 3-dimensions). The naming convention of the stresses follow the following rule:

  • Axial stresses take the name  with a subscript indicating the direction normal to the plane
  • Shear stresses take the name  with two subscripts, the first indicating the direction normal to the plane, and the second, the direction of the shear stress.

Axial and shear stresses

Axial and shear stresses in 2-D

What is Strain?

Similar to the stress, strain is a measure of the deformation of the structure that may be described at a local level. When subjected to loading, a structure creates internal forces to keep its particles from getting separated (falling apart) but, those particles move relative to one another in different directions due to that load. That motion is extremely hard to measure, but we may be able to measure the displacements of different points of the structure which, in turn, can be translated into a description of the structure deformation. When two nearby points move relative to each other we say that that part of the structure is under strain. Again, we may claim that the strain is not a physical phenomenon by itself, rather, it is a measure of the local relative motion of different points of the structure relative to one another.