φz,0: the elastic angular deformation at time t=0
φz,∞.: the elastic angular deformation at time t=∞
The rotation angle of the supporting beam b1 in coordinate system xyz (φz,0
will be equal to the rotation angle of the supported beam
b2 in coordinate system x’y’z’ (φ
The rotation angle φz,t of the supporting beam b1 will be increasing over time t, due to creep. As φy’,t=φz,t, the equivalent angle φy’,t of the supported beam b2 will also be increasing. Consequently, the bending moment M and the equivalent torsional moment MT will be increasing, since MT=M. At
a time t=∞, the system will balance at an angle φy’, ∞=φz, ∞, which cannot exceed the angle φy’,pinnd. The φy’,pinned corresponds to the tangent of the elastic line at the left end of the supported beam b2 when the support there becomes pinned (MT=M=0).
This is why the European Standards [EC2, §188.8.131.52(2)] allow us to consider pinned supports for both beams and slabs. Otherwise, either the creep should be taken into account or the effective stiffness should be limited to a small percentage (e.g. 10%) of the full elastic stiffness.
The assumption of zero torsional stiffness provides a solution in the case of a simply supported beam. However, in the case of a cantilever beam, this turns out to be invalid, because the isostatic structure becomes a mechanism by diminishing its rotational restraint and thus transforming the support from fixed to pinned.
The assumption of effective torsional stiffness limited to 1% of the elastic stiffness gives the correct results for both frames and thus for all types of frames.