Fixed and Continuous Beams:

You have already learnt from your studies of "Strength of Materials" how to analyse simply supported beams and cantilevers to find out their external support reactions, and internal stress resultants (Bending Moments, Shearing Forces) and displacements (deflections) under any loading. You will recall that while solving these structures you used only the conditions of statics, which in a co-planar structure and force system, consists of the subsequent well-known set of three equations of statical equilibrium that are namely,

∑ Fx ≡ 0 (i.e. resultant of all forces parallel to axis OX is zero).

∑ Fy ≡ 0 (i.e. resultant of the forces perpendicular to the OX axis and parallel to the OY axis is zero).

∑ M z ≡ 0 (i.e. resultant of all moments and couples about any point in the plane XOY is zero).

Such structures which can be analysed only with the help of the above three equations of statical equilibrium are known as Statically Determinate Structures. For such structures the unknown support reactions are three and these can be solved by statics only. Therefore, there is another category of equally common and important structures which are called Statically Indeterminate Structures, which cannot be solved by solving the set of above three conditions of equilibrium. Further restraints in the form of extra members or support constraints are imposed, that makes the number of unknowns to be solved as greater than three, thus they requires additional equations to analyse them, i.e. compatibility conditions.

Objectives

After studying this unit, you should be able to

• Conceptualise a new type of structures, which cannot be solved by equations of statics just, and are called as statically indeterminate structures,
• To identify this type of structures and learn what are the main features which distinguish them from statically determinate structures, and
• To analyse this type of structures, that is, to find their support reactions and draw the BM and SF diagrams.