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Steel Designers' Manual - 6th Edition (2003)
Fig. 9.20 Stiffness coefficients
Analysis of skeletal structures 307
or {D} = [F]{P}
where {D} = displacement matrix
[F] = flexibility matrix relating displacements to forces
{P} = force matrix
Hence {P} = [F] -1 {D}
Stiffness is the inverse of flexibility and gives a measure of the forces corresponding
to a given set of displacements. Considering the spring illustrated in
Fig. 9.18(a), it is noted that the deflection response is directly proportional to
the applied load, P. The force corresponding to unit displacement is obviously P/D.
Likewise in Fig. 9.18(b) the load to be applied on the beam to cause a unit displacement
at a point below the load is P/D. In its simplest form, stiffness coefficient
refers to the load corresponding to a unit displacement at a given point and can be
seen to be the reciprocal of flexibility. The concept is explained further using
Fig. 9.20.
First the locations 2 and 3 are restrained from movement and a unit displacement
is given at 1. This implies a downward force k11 at 1, an upward force k21 at 2 and a
downward force k31 at 3. The forces at points 2 and 3 are necessary as otherwise
there will be displacements at the locations 2 and 3.
The forces k11, k21 and k31 are designated as stiffness coefficients. In a similar
manner, the stiffness coefficients corresponding to unit displacements at points 2
and 3 are obtained.
k