Analysis of Statically Determinate Pin -Jointed Frames

Statically Determinate Pin-Jointed Frames

Pin-Jointed Frames

The pin-jointed frames form truss system that has a triangular planar framework with straight members. The truss systems are mainly used in the construction of bridge and roof construction. A triangular formed out of 3 members will form the basic unit of a simple truss. The addition of members, two at a time will develop additional triangular units. 

Fig.1.Different Truss System Pin-Jointed Frames in Structural Analysis
Fig.1.Different Truss System Pin-Jointed Frames in Structural Analysis

The top and the bottom members of the truss are called as chords. The vertical and the sloping members are called as the web members. The combination of simple trusses will form a compound truss. This is shown in figure-2. 

Fig.2: Compound Truss Form
Fig.2: Compound Truss Form

As shown in the figure above, the two trusses are connected to the apex node and an additional member at the base.

Assumptions in Analysis of Pin-Jointed Frames

The analysis of simple trusses can be carried out with the help of equations of static equilibrium. The following are the assumptions followed while doings the analysis of pin-jointed frames.

1. The members in the truss system are connected at the nodes with frictionless hinges
2. The centroidal axis of all the members at a node will intersect at a single point. This will avoid the     chances of eccentricities in the truss system
3. All the loads – external loads, the self-weight is applied at the nodes
4. All the members are subjected to axial forces only.
5. Axial deformation caused secondary stresses are ignored.

Statically Determinate Pin-Jointed Frames

If you are given a truss system, whose member forces and the external reactions can be determined easily by the equation of equilibrium, then the truss system is called as Statically Determinate.

Consider the arrangement of a truss system. The truss may be either supported by means of a roller or a hinge support as shown in figure-3. If the support is

  • Hinge – There will be two reactions, One Vertical & One Horizontal. That means the support will restrict the movement in the horizontal and vertical direction. This will only allow rotational displacement to occur. Figure-3(i).
  • Roller – There will be a single reaction, One Vertical. This means the roller support will not let vertical displacements to occur. But this system can allow the movement in the horizontal and rotational direction. Figure-3(ii)
Fig.3: Hinge and Roller Support; Their Support Reactions
Fig.3: Hinge and Roller Support; Their Support Reactions

The equations of equilibrium are used to determine the magnitude of external restraints. There are three equations of equations of equilibrium.

Note: A truss system whose external restraints exceeds the number of equations of equilibrium, i.e. the unknowns cannot be determined by equations of equilibrium are called as externally indeterminate. The same system if have less than 3 number of external restraints, then the system is considered as unstable.

The equations of equilibrium used are ΣH=0; ΣV=0;ΣM=0 

Note: Do the truss members carry moment value?Why do they take axial and compressive forces? The connection involved in a truss system is pinned. This means the directions that are restrained are vertical and horizontal. The joint will allow rotational movement, this will not result in any kind of movement. Hence the moment value in the case of the truss system is zero. Now, this is an area on which still studies are still going on. But for instance, we are following this theory in analysis.
A small formula to find easily a whether a truss system is determinate or indeterminate is given below.

If, n = number of members of the truss system
     r = number of external restraints or reactions
     j = number of joints or nodes,

Truss system is determinate when,
n + r = 2j         eq. (1)
Truss system is Indeterminate when,
n + r > 2j         eq. (2)
Truss system in Unstable when
n + r < 2j          eq. (3)

Methods of  Analysis of Determinate Pin-Jointed Frames

There are several methods that can be used for the analysis of pin-jointed frames which are determinate. The choice of the method will depend on the usefulness and the applicability of the method. The different methods used are: 
  1. Methods of Resolution at the nodes 
  2. Method of Sections 
  3. Method of Force Coefficients 
  4. Method of Substitution of members