414 11 Matrix Stiffness Method
11.9 Summary................................................................
1.The MSM is modern effective method for analysis of any deformable structures
subjected to arbitrary actions. Among them are external loads, change of tem-
perature, settlements of supports. For trusses and frames with straight members,
the MSM leads to the exact results. Curvilinear member should be replaced by
the set of inscribed straight members.In this case, MSM leads to the approxi-
mate results.
2.Design diagram of the MSM presents the set of uniform straight members con-
nected by hinged or fixed joints. Any action should be replaced by the equivalent
loads (moments and forces), which should be applied at the joints. The primary
unknowns of the MSM and their number are same as at displacement method in
canonical form, i.e., the angular displacements of the fixed joints and indepen-
dent linear displacement of joints. Theunknown forces (in the simplest version
of the MSM) are axial forces for truss members and bending moment at the
fixed ends for the bending members.
3.Arbitrary external exposure (loads, change of temperature, and settlement of
supports) should be transformed into equivalent joint loads and presented in the
form of theJ-Ldiagram.
4.TheZ-Pdiagram contains information about possible displacements of the
joints and type of corresponding external load. In case of truss theZ-Pdia-
gram shows linear displacement of joints and concentrated forces along these
displacements. For rigid joint of a frame, theZ-Pdiagram shows angular dis-
placement of the joint and couple; in case of linear displacement of the joints,
theZ-Pdiagram shows independent linear displacement and force.
5.TheS-ediagram contains information about location and signs of the required
internal forces S. For truss the unknownforces are axial forces at the each mem-
ber; for bending member the unknown forces are bending moment at the fixed
joint of the primary system.
6.The static matrixAconnects the possible external loadsPand required internal
forcesS. For computation of members of this matrix, it is necessary to express
each possible external loadPin terms of unknown internal forcesS. The mem-
beraikis coefficient at unknown forceSkin an equation forPi. The number
mof rows of matrixAequals to the number of possible external forcesP;the
numbernof columns equals to the number of the unknown internal forces. If
m>nthen structure is geometrically changeable, ifm=nthen structure is
statically determinate; ifm<nthen structure is statically indeterminate. The
entriesaikmay be positive, negative, or zero.
7.Deformation matrixBconnects the end deformation of each finite element in
the primary system of the displacement method and unit displacement of in-
troduced constraints. The memberbikis displacement in direction of unknown
forceSidue to unit displacement of introduced constraintk. The number of
rows of matrixBequals to the number of required internal forcesS; the number
of columns equals to the number of the introduced constraints of displacement
method. The deformation and static matrices obey to equationBDAT.