## Ei

Spring stiffness k2

Fig. 12.13. Member with rotational springs at each end.

can be shown from basic principles that the member stiffness will equal 4EI/L. To align with the member stiffness definition, the spring rotational stiffness at each end is written in a similar format as follows:

EI EI

where k1 = M1/ftr1 and k2 = M2/ftr2, and M1 and M2 are the end moments on the beam, #r1 and #r2 are the rotations of the spring at end a and b, respectively, ja and jb are the secant rotational stiffness coefficients of the joint connection at a and b, respectively, and include for all of the shear planes in each of the connections.

From equations (12.36) and (12.37), for a connection in timber structures the secant rotational stiffness coefficient can be written as;

where the symbols are as previously defined and:

K is the relevant slip modulus of the fastener type being used in the connection for the design condition being addressed, i.e. either ttttt—r when analysing at the

ULS or (1+K;; ) at the SLS. If the connection at each end is different, the value of K at each end will also differ. The value of the deformation factor(s), (kdef,c), will be obtained from Table 2.10 and reference should be made to 2.3.2 to determine the adjustments to be applied to suit the configuration being used. nsp is the number of shear planes in the connection.

Consider the beam to be loaded in a generalised manner where the area of the free bending moment diagram is A with its centre of area at a distance a1 from end a. If the rotation at end 1 of the member is ^ and at end 2 is using unit load theory, a combination of flexibility coefficients and the stiffness method of analysis, also assuming no relative movement in the direction of the y-axis between ends 1 and 2, it will be shown that:

The functions (1+(3[7I/kiL)) and (1+(3£.1J/) modify the prismatic member relationship to take account of the effects of the end springs and are referred to by Monforton and Wu [3] as the fixity factors and by Kermani [4] as the rigidity factors. They are dimensionless parameters and are functions of the respective spring rotational stiffness and the stiffness properties of the beam.

Expressing the rigidity factors in terms of the symbol y , they can be written as follows:

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