Stiffness Method Overview The stiffness and flexibility methods are the most common techniques for analysing statically indeterminate structures. In the flexibility method the requirement that the deflected shape of the structure being analysed be compatible with the boundary conditions of the structure is used to generate additional equations. In the flexibility approach the number of compatibility equations required (i.e. the size of the flexibility matrix) is equal to the degree of static indeterminacy of the structure. Remember, you introduce releases until the structure is statically determinate In the stiffness method the equations being solved are equilibrium equations. However, the unknown variables in the equations are not the member forces (as would be used in a traditional static analysis) but are the nodal displacements. Once the displaced shape of the structure is known the forces in the individual members can be calculated. For example, in a pin-jointed truss change in member length will dictate the member force. Similarly, in a frame structure the relative displacements, lateral and rotational, of the members ends will, in combination with the loads applied to the member, define the bending moments in the member. In the stiffness method the number of equations equals the number of displacements required to describe the displaced shape of the structure. The number of displacements required to describe the displaced shape of a structure is called the Degree of Kinematic Indeterminacy, this is also called the number of degrees of freedom. For example the statically determinate structure shown above has a degree of kinematic indeterminacy of 11. To describe the displacement of each internal requires two coordinates (i.e. a X and a Y ) plus the roller support can displace. Thus making a total of 11 degrees of freedom. Note: the structure shown below, which is statically indeterminate, has the same degree of kinematic indeterminacy, and its solution will involve solving the same number of equilibrium equations as the statically determinate structure. Stiffness and flexibility methods are related. 1 General Procedure 1. Identify the degrees of freedom required to describe the deflected shape of the structure. Assume that all the n degrees of freedom are restrained in position. 2. Apply the loads to the structure and calculate the forces FR that need to be applied at the restraints to prevent movement. In addition calculate AR m1 , the m required structural actions, for the restrained structure. 3. Generate the stiffness matrix S . The stiffness matrix is generated by, displacing each degree of freedom by one unit while preventing movement at all other degrees of freedom. This is done for each degree of freedom in turn. Thus, for each degree of freedom i we calculate sii , the force applied at i associated with a unit displacement at i (this is the force required to cause the displacement) and zero displacement at all other nodes. And we calculate a series of values s ji , the force applied at j associated with a unit displacement at i and zero displacement at all other nodes (including j ). In addition, calculate AU mn the m structural actions associated with unit displacements at each of the n degrees of freedom. 4. Find the unknown displacements D by solving the equation, S D FR 5. Calculate the member forces and moments from the nodal displacements using the equation, Am1 AR m1 AU mn Dn1 2 Example – 1 Note: i 1 m s 21 Ai Ei cos i sin i Li m Ai Ei sin i cos i Li i 1 s12 i 1 m s 22 Ai Ei cos 2 i Li i 1 Thet m s11 a For pin-jointed members Ai Ei sin 2 i Li A1 E1 L cos 1 1 A2 E 2 cos 2 Au L2 A E m m cos m Lm A1 E1 sin 1 L1 AE 2 2 sin 2 L2 Am E m sin m Lm 3