Advanced Methods of Structural Analysis

328 10 Influence Lines Method

Influence line for bending MomentMk

Should be constructed using formula (10.4). Bending moment at sectionkin the pri-
mary system due to primary unknownX 1 D 1 isMkD0:4(Fig.10.2a), therefore

IL.Mk/D0:4IL.X 1 /CIL



Mk^0



: (10.10)

Construction of IL.Mk/step by step is presented in Fig.10.3.

k

P= 1

1345268910711

0.01920.0336 0.0288 0.4⋅IL X

1

− − (factor l)

0.0384 0.02880.03840.03360.0192

a

Load P=1 in the left span Load P=1 in the right span

Mk=0 (see primary system

in Fig. 10.1.b)

0

0.120.240.160.08

Inf. line Mk

(factor l)

+

b

+

− Inf. line (factor l)Mk

0.10080.20640.12160.0512

0.02880.03840.03360.0192

IL(Mk)=0.4IL(X 1 )+IL(Mk)

c^0

0

Fig. 10.3 Construction of influence line for bending moment at sectionk

The first term0:4IL.X 1 /of (10.10) is presented in Fig.10.3a. The second
term IL



Mk^0


is the influence line of bending moment at sectionkin the pri-
mary system. If loadP D 1 is located in the left span, then the influence line
presents the triangle with maximum ordinate at the sectionk; this ordinate equals to
ab= lD.0:4l0:6l/ = lD0:24l.IfloadPD 1 is located in the right span, then
the bending moment at sectionkdoes not arise, and the influence line has zeros
ordinates. It happens because the primarysystem presents two separate beams. In-
fluence line forMk^0 is shown in Fig.10.3b. Summation of two graphs0:4IL.X 1 /and
IL



Mk^0


leads to the final influence line for bending moment at sectionk; this influ-
ence line is presented in Fig.10.3c. The maximum bending moment at the sectionk
occurs when the loadPis located at the same section. The positive sign means that
if a load is located within the left span, then extended fibers at the sectionkwill be
located below the neutral line. It is easy to show the following important property
of influence line forMk. The sum of the absolute values of the slopes at the left and
right of sectionkis equal to unity.