Structural Mechanics

Computer Analysis of Pin-Jointed Frames

Degree of Static Indeterminacy (Ds)

In statics, a structure is said to be statically indeterminate when the available equilibrium equations are insufficient to determine unknown forces. Thus additional equations are written in terms of compatibility to evaluate unknown forces. The number of such additional equations defines the degree of Static Indeterminacy 

Degree of Kinematic Indeterminacy (Dk)

In Kinematics, a structure is said to be statically indeterminate when the available compatibility equations are insufficient to determine unknown displacements. Thus additional equations are written in terms of equilibrium to evaluate unknown displacements. The number of such additional equations defines the degree of kinematic Indeterminacy 

For analysis of a pin-jointed frames/trusses both flexibility and stiffness methods are employed to write a computer programs. In Practice stiffness method is preferable to the flexibility method due to;

1. The assumption of restrained structure (lock joints) is a simple approach compare to redundant forces (released structure) in flexibility approach, which has limitation in case of a complex and large structure.
2. A Displacement given at any one joint deforms only those members which are connected directly to the joint.
3. The development of the stiffness matrix involves lesser computations

The Direct stiffness method approach is used in analysis of trusses and frames. In this method stiffness matrix for a single member with respect to system coordinate is obtained first, from the stiffness matrix with respect to element coordinates by means of transformation matrices , the stiffness matrices for all members are then synthesized to obtain the stiffness matrix for the entire structure.

      In order to facilitate computer application and to obtain a standard form for transformation matrix two element coordinates are assigned to the member instead of one. 1* and 2* are the element coordinates  and 1 to 4 are system coordinates located at the near end A and far end B of member AB of a plane truss/frame as shown in figure 1 (a).

       

                  Figure 1(a)                                            Figure 1(b)
similarly 1* and 2* are the element coordinates and 1 to 6 are the system coordinates for a member of a space truss/frame as shown in figure 1 (b)
    In this method for writing the computer program, the coordinates are assigned to all unrestrained as well as restrained displacement components, thus ensuring that there are two coordinates in the direction of X and Y axes at both ends of member in plane truss/frame. Similarly there are coordinates in the direction X, Y and Z axes at each end of member in space truss/frame. Thus total number of system coordinates are;
Plane truss/ frame ---- 2j
Space truss/frame ---- 3j
where j is number of joints
     The numbering of coordinates are done starting from unconstrained displacement components first, followed by the constrained displacement components, to facilitate the partitioning of the matrices in force-displacement relationship. The matrix proportioning gives two matrix equations one of them gives the unconstrained displacement, while other gives the support reactions.  
for example:

               The degree of freedom = Number of unconstrained displacement component (13) 
               Number of constrained displacement  = 3 (14, 15, 16)

N-12: Determine the resultant of the following force system shown in figure

Design of shallow foundations