Advanced Methods of Structural Analysis

13.6 Compressed Rods with Lateral Loading 503

13.6.4.1 Simply Supported Beam-Column

The bending moment diagrams in unit and actual states are shown in Fig.13.26a.
The bending moment diagramMPdue to transversal load (this load is not shown)
is curvilinear.

M

b

d

MP l

a

c

P

M 0

Mtr

l

jA

M 0

P A P

M(m)

M= 1

1

B

ab

Fig. 13.26 (a) Graph multiplication method for simply supported beam-column; (b) Calculation
of the slope'A

Graph multiplication method leads to the following result

D

l

6 EI

Œ.2abC2cd/ ̨ ./C.adCbc/ ˇ . / ; DnlDl

r

P

EI

;

̨./D

3





1





1

tan



;ˇ./D

6





1

sin



1





:

In case ofPD 0 this formula and trapezoid rule (6.22) coincide.
Let a simply supported beamABis loaded by coupleM 0 at supportBand com-
pressed forceP(Fig.13.26b). For computation of the slope at the supportAwe
need to construct the bending moment diagram.Mtr/for transversal loadwithout
compressive load and bending moment diagram for unit stateM. Graph multiplica-
tion method leads to the following result

'AD

MtrM

EI

D

l

6 EI

Œ.2 0  1 C 2 M 0 0/ ̨ . /C.0 0 CM 0 1/ ˇ . /D

lM 0

6 EI

ˇ./:

13.6.4.2 Fixed-Free Beam-Column

The beam of lengthlis loaded by any transversal load and compressed by forceP
(Fig.13.27a). Notation of the bending moment ordinates at the free end and clamped
support caused by transversal loads and unit load are shown in Fig.13.27a.