Advanced Methods of Structural Analysis

374 11 Matrix Stiffness Method

M 1  5 D

ql 12  5

8

D

3  42

8

D 6 kNm

M 7  2 D

3

16

P 2 l 2  7 D 18 kNm

M 2  3 D

P 1 l 2  3

8

D

10  8

8

D 10 kNm

M 3  2 DM 2  3 D 10 kNm

Figure11.5d shows the horizontal forces which arise in the vertical loaded members
1–5 and 2–7. Both of these forces are transmitted on the cross bar, so final horizontal
joint load equalsPj4DR 1  5 CR 2  7 D7:5C2:5D12:5kN. This force may be
applied at any joint of the frame (1, 2, or 3).

R 1  5 D

5ql 1  5

8

D

5  3  4

8

D7:5kNI

R 2  7 D

5P 2

16

D

5  16

16

D 5 kN:

Thus the entire load may be presented as equivalent momentsMj1;Mj2;Mj3at
joints 1–3 and forcePj4in horizontal direction; subscriptjmeans that entire loads
are transformed to joint load. The finalJ-Ldiagram is shown in Fig.11.5e

Mj1D 6 kNmIMj2D 10 kNmIMj3D 10 kNmIPj4D7:5C 5 D12:5kN

Note again that joint load presents theequivalentbending moments and forces,
which are merelytransportedon the joints and on the cross bar on thesame
direction.
In case of truss, all loads are applied at the joints; therefore, the joint-load dia-
gram coincides with entire design diagram.

Example 11.1.The continuous beam is subjected to change of temperature as
shown in Fig.11.6a. Construct the joint-load diagram.

Solution.The primary system of the displacement method and bending mo-
ment diagram caused by given temperature exposure is shown in Fig.11.6b(first
state).
The bending moments for pinned-fixed beam and fixed-fixed beam are

M 1  0 DM 3  4 D

3 EI ̨t

2h

D

3 EI ̨ 24

2 0:4

D90 ̨EII

M 1  2 DM 3  2 D

EI ̨t

h

D60 ̨EI: