Building Design and Construction Handbook - Merritt - Ventech!

STRUCTURAL THEORY 5.115
k1 0 ... 0
� �
0 k 2 ... 0
k � (5.162)
.. .. ... ..
0 0 ... kn Step 5. Develop a matrix b 0 that will transform the displacements � of the nodes
of the structure into the displacement vector � while maintaining geometric compatibility:
� � b � (5.163)
0
b 0 is a matrix of influence coefficients. The jth column of b 0 contains the element
nodal displacements when the node where � j occurs is given a unit displacement
in the direction of � j, and no other nodes are displaced.
Step 6. Compute the stiffness matrix K for the whole structure from
T K � bkb 0 0
(5.164)
T where b0 � transpose of b0 � matrix b0 with rows and columns interchanged
T
This equation may be derived as follows: From energy relationship, P � bS. 0
Substitution of k� for S [Eq. (5.161)] and then substitution of b0� for � [Eq.
T
(5.163)] yields P � bkb 0 0�. Comparison of this with Eq. (5.103a), P � k� leads
to Eq. (5.164).
Step 7. With the stiffness matrix K now known, solve the simultaneous equations
�1 � � K P (5.165)
for the nodal displacements �. With these determined, calculate the member forces
from
S � kb0� (5.166)
(N. M. Baran, ‘‘Finite Element Analysis on Microcomputers,’’ and H. Kardesluncer
and D. H. Norris, ‘‘Finite Element Handbook,’’ McGraw-Hill Publishing
Company, New York; K. Bathe, ‘‘Finite Element Procedures in Engineering Analysis,’’
T. R. Hughes, ‘‘The Finite Element Method,’’ W. Weaver, Jr., and P. R.
Johnston, ‘‘Structural Dynamics by Finite Elements,’’ and H. T. Y. Yang, ‘‘Finite
Element Structural Analysis,’’ Prentice-Hall, Englewood Cliffs, N.J.)
5.14 STRESSES IN ARCHES
An arch is a curved beam, the radius of curvature of which is very large relative
to the depth of the section. It differs from a straight beam in that: (1) loads induce
both bending and direct compressive stresses in an arch; (2) arch reactions have
horizontal components even though loads are all vertical; and (3) deflections have
horizontal as well as vertical components (see also Arts. 5.6.1 to 5.6.4). Names of
arch parts are given in Fig. 5.93.