## 2342 Manual Based Approximate Analysis of Semirigid Structures

For simple semirigid framed structures, analysis using the linear connection model defined by Equation 23.1, can be conducted manually as for rigid frames. In this section, a procedure based on a modified moment distribution method is presented for the analysis of semirigid continuous beams and braced frames.

Moment distribution is a method of successive approximations that may be carried out to any desired degree of accuracy. The convergence of the method is guaranteed for stable linear elastic structures under small deformations and displacements. Its primary advantage over other hand-calculation structural analysis procedures, such as the method of consistent deformations and the slope deflection method, is that it eliminates the requirement to solve a set of simultaneous equations to determine the member end moments for statically indeterminate beams or nonsway plane frames in which the joints are restrained against translation.

Upon the introduction of the end-fixity factor defined in Equation 23.7 to take into account semirigid connection behavior based on Equation 23.1, the analysis of semirigid continuous beams and frames can be conducted through minor modification of the conventional moment distribution method for rigid continuous beams and frames, as described in the following:

• Sign convention. All loads acting on the members in the mathematical model of a semirigid continuous beam or a plane frame are referenced to the individual local member coordinate system. As indicated in Figure 23.6, positive load acts downward and positive end moment and shear act counter-clockwise and upward, respectively.

• Member end-reaction moments. The end moments of a semirigid member due to different types of applied loads are listed in Table 23.3.

• Member rotational stiffness factor Kij. For a semirigid member i, with different end-connection stiffnesses R1 and R2 determined by Equation 23.1, let the end-fixity factors defined in Equation 23.7 for the ends 1 and 2 be r1 and r2, respectively. The end-stiffness factors Ki1 and Ki2 for the member can be expressed as

3r1 4EI

in which 3r1/(4 — r1r2) and 3r2/(4 — r1r2) are factors accounting for the end-connection flexibility.

• Distribution factor Dj. The distribution factor is defined as the stiffness factor of the member divided by the sum of the stiffness factors of all members connecting at the joint,

This factor corresponds to the fraction of the total applied moments at a joint that will be resisted by any member i as the joint rotates through an angle f, where n is the number of members connected at the joint.

• Carry-over factor Cj. When a moment is applied to the near end of any member whose far end is restrained against rotation during the distribution process, a moment must be applied to the far end by the joint to resist rotation of the member end. This moment is called the carry-over moment and the ratio of the far-end moment to the near-end moment is known as the carry-over factor. The carry-over factors for a semirigid member i are

Implementation of the mathematical model of the moment distribution method for semirigid continuous beams, or semirigid braced frames in which the joints are restrained against translation, consists of the following steps:

1. Evaluate end-fixity factors r1 and r2 according to Equation 23.7 for each member for the given values of connection stiffness Rj (j = 1, 2). For cases of pinned and fixed-end connections, assign the end-fixity factor to zero and unity, respectively.

2. Calculate end-stiffness factors Kj (j = 1, 2) according to Equations 23.8 for each member i and determine moment distribution factors Dj from Equation 23.9.

3. Compute the end-reaction moments Mi(i) (j = 1, 2) due to the applied loads for each member through Table 23.3, assuming that all joints are locked by fictitious rotation restraints, and set

4. Sum the member end moments at each joint to determine the unbalanced moment

5. Release or ''unlock'' the joints by removing the fictitious rotation restraint at each joint in turn, assuming that the other joints are still restrained against rotation, and distribute the unbalanced moment at the joint to the near end of each connecting member as Mf+1)= Dj x M<k).

6. Carry the near-end moment of each member over to its other end by multiplying each moment with the carry-over factor, that is, M(k+1) = Ci2 x M(k+1) or M(k+1) = Cfl x M(1k+1).

7. If the unbalanced moment M(k) is small enough to be considered insignificant, then terminate the iteration and proceed to step 8; otherwise, set k = k + 1 and return to step 4.

8. Sum the end moments of each member. Calculate the axial forces (for nonsway frames) and the shear forces of the members satisfying equilibrium and determine the reaction forces and moments at the supports based on the joint equilibrium conditions.

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