Advanced Methods of Structural Analysis

376 11 Matrix Stiffness Method

According to TablesA.3andA.4, the bending moments for pinned-fixed and
fixed-fixed beams are

M 1  0 D

2 EI

l 1

'D

2 EI

5

0:05D0:02EI.kNm/;

M 1  2 D

3 EI

l 22

D

3 EI

42

0:032D0:006EI.kNm/:

Computation of equivalent moment at the joint 1 and correspondingJ-Ldiagram
are shown in Fig.11.7c, d.

11.2.2 Displacement-Load (Z-P)Diagram........................

This diagram shows a numeration of angular and independent linear displacements
Zat the joints and their positive direction. Also this diagram containstypeload,
which corresponds to displacementsZand their positive directions.
Sign rule.Assume that the positive angular displacementZoccurs clockwise; the
positive horizontal linear displacement occurs from left to right, and vertical linear
displacement occurs upward. Direction of the positive load coincides with positive
direction of the displacement.
Design diagram frame and correspondingprimary system of the displacement
method are shown in Fig.11.8a, b. Introduced constraints 1–3 prevent angular
displacementsZ 1 – Z 3 ; corresponding loads are momentsM 1 – M 3. Constraint 4 pre-
vents linear displacementZ 4 , so corresponding load isP4:; these loads and all
displacements are shown in positive directions. TheZ-Pdiagram contains only
sceleton of the entire scheme of the structure, the type of end displacements, its
positive direction and corresponding loads (Fig.11.8c). Geometrical parameters,
stiffnesses, and loading are are not shown on theZ-Pdiagram.

c

1 23

4

Z-P

diagram

12 34

Z1, M 1 Z2, M 2 Z3, M 3

Z4, P 4

a b

Fig. 11.8 Displacement-load (Z-P) diagram for frame

Let us show the construction ofZ-Pdiagram for truss in Fig.11.9a. SupportA
prevents two displacements; therefore, this joint has not possible displacement and
corresponding joint loads. JointCprevents horizontal displacement, so possible dis-
placement of this joint and corresponding load is directed vertically (labeled 1). The
possible displacement of jointBand corresponding load are directed horizontally