Problems 4158.The stiffness matrixkof finite members in local coordinate connects unknown
internal forceSand displacement of the end of the element. These matrices
may be presented in truncated or in expanded form. In truncated form, the stiff-
ness matrices for truss member and bending fixed-pinned member presents a
scalar, while for fixed-fixed member ( 2 2 ) matrix. The stiffness matrix in local
coordinate for whole structurekQpresents matrix which contains on the princi-
pal diagonal the stiffness matrices of separate members. This matrix is square,
symmetrical, and all entries are positive.
9.JuxtaposeZ-PandJ-Ldiagrams leads to the vector of external forcesP.The
number of entries of this vector equals to the number of primary unknown of
the displacement method. For truss, the number of entries of this vector equals
to the number of the possible displacementsof the joints. If external joint load
at the direction of the possible displacement is absent then corresponding entry
of the vectorPis zero. If a structure is subjected to different groups of external
loads, thenPwill present a matrix. The number of the columns equals to the
number of the set loading.
10.The stiffness matrixKof whole structure in global coordinate is symmetrical
square (nn) matrix wherenis a number of primary unknowns of displacement
method. The members of this matrix are the unit reactionsrikof the displace-
ment method in canonical form; equationKZ = Pis exactly canonical equations
of displacement method.
11.JuxtaposeS-eandMP^0 diagrams leads to the vector of internal forcesS 1 of the
first state. The number of entries of this vector equals to the number of unknown
bending moments at the rigid joints. For trussS 1 D 0.
12.The final results may be presented using two matrices as follows:
The vector of joint displacement isZDK^1 Pand the internal forces isSD
S 1 CS 2 , where the vector of unknown internal forces of the second state is
S 2 DkAQ
T
Z. Thus, the joint displacements and the distribution of internal
forces of any structure are defined only by three matrices. They are the static
matrixAof a structure, stiffness matrixkQof a structure in local coordinates,
and the vector of external forcesP.
13.To plot the final bending moment diagram, the signs of obtained final moments
should be juxtaposed withS-ediagram. The shear force can be calculated on
the basis of the bending moment diagram; the axial forces can be calculated on
the basis of the shear diagram. Reactions of supports can be calculated on the
basis of the axial and shear forces and bending moment diagrams.
Problems.......................................................................
11.1a-c.The uniform two-span beam is subjected to fixed load as shown in
Fig.P11.1. The flexural rigidity of the beam isEI. Determine the angle of rota-
tion at support 1and the bending moments at specified points.