Table of contents
Lecture notes: Structural Analysis II
Influence lines for displacements
Definition: The influence line for a displacement is a graph representing the variation of the
displacement in a fixed section at a given direction, due to a unit load moving along the road
lane.
II. Influence line construction according to its definition.
To determine the influence line for the horizontal displacement at section i (Fig. 1), a unit load is
placed in a number of successive points of the road. At each point the horizontal displacement at
section i must be computed.
Fk=1
Fk=1
i
δik
δik
i
Figure 1 Influence line construction for horizontal displacement at section i.
Fk=1
To determine the horizontal displacement at section i for the
given frame structure, we should construct the bending
1
2 3
4
moment diagram Mk produced by the unit load at point k.
ii
Point k is placed consecutively at sections 1 to 4 of the road
lane. The bending moment diagram Mk is caused by the
external load for the considered case.
Next, a virtual horizontal load should be applied in order to obtain the virtual moment diagram
Mi. The required displacement is:
MM
δ i ,k = Σ ∫ i k ds .
EI
1. k=1, Fk=1 is placed at section 1
1
Fk=1
ii
M1
Mi ≡M
i i
δ i ,1 = Σ ∫
M i M1
ds .
EI
2. k=2, Fk=1 is put at section 2
2
M2
Fk=1
ii
δ i ,2 = Σ ∫
MiM 2
ds .
EI
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
43
Lecture notes: Structural Analysis II
3. k=3, Fk=1 is placed at section 3
Fk=1
3
δ i,3 = Σ ∫
MiM3
ds = 0 .
EI
δ i ,4 = Σ ∫
MiM 4
ds .
EI
ii
M3
4. k=4, Fk=1 is set at section 4
Fk=1
4
ii
M4
Finally, the obtained displacements are put as ordinates on the horizontal base line in sections
relevant to the position of the unit load. The required influence line is obtained as follows:
3
1
2
δi,1
δi,2
4
δi,4
“δi,h”
II. Maxwell theorem
An alternative way of influence line construction is based on the theorem of Maxwell.
Let us take two different states of one and the same system. The first state corresponding to the
application of a vertical unit load, Fk=1, at point k, where k is any point from the road lane.
Fk=1
k
δk,i
Fi=1
i
1
2
δi,1
δi,2
3
δi,k
4
δi,4
“δi,h”
The second state corresponds to a unit load, Fi=1, in the direction of the required displacement, in
our case horizontal force at point i. In that respect δ i,k is the displacement along the direction of
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
44
Lecture notes: Structural Analysis II
load Fi due to the unit load Fk, δ k ,i indicates the displacement along the line of action of the force
Fk due to the application of load unity Fi=1.
According to Betty’s theorem of reciprocal works:
1 ⋅ δ i ,k = 1⋅ δ k ,i .
This equation postulates the Maxwell’s theorem: In any elastic system the displacements caused
by a load unity along the line of action of another load unity are always equal to the
displacements due to this second load unity along the line of action of the first one.
Therefore, based on the theorem of reciprocal displacement the following conclusion may be
written:
The influence line for a displacement in a given section, at certain direction, coincides with the
vertical displacements diagram of the points of road lane caused by a unit load Fi=1 along the
direction of the required displacement. The graphics of vertical deflections of points belonging to
the road is the elastic curve of the road lane.
III. Influence line construction by conjugate beam method (Mohr’s analogy)
If we consider a beam subjected to a distributed load of intensity q, the bending moment is:
d 2M
= −q .
dx 2
In addition the curvature of the elastic curve at any section is given by:
d 2w
M
.
EI
dx
It follows from the mathematical equivalency of both the expressions that as we construct the
bending moment diagrams without integration of the first differential equation, we may obtain
the elastic curve with no integration of the second expression. For that purpose the right hand side
of the second equation, M / EI , should be regarded as the fictitious distributed load,
q fict = M / EI , wich must be applied in an analogous (fictitious) beam, known as a conjugate
beam. The conjugate beam is the horizontal projection of the plates belonging to the road lane.
The bending moment of any section of the conjugate beam, Mfict, is the vertical deflection of the
same section in the real beam:
w( x) = M fict ( x) .
The shear force at any section of the conjugate beam, Qfict, is the slope at the corresponding
section in the real beam:
⎛ dM fict ( x)
⎞
dw( x)
fict
ϕ ( x) = Q ( x) ⎜
= Q fict ,
= ϕ ( x) ⎟ .
⎜
⎟
dx
dx
⎝
⎠
Thus, the slope and deflection at any section in the real beam are given by the shear force and
bending moment at the corresponding section in the conjugate beam, and the elastic curve of any
real beam is given by the bending moment diagram of the conjugate beam.
In order to construct the bending moment diagram, the conjugate beam, must be loaded by
fictitious loads.
2
=−
IV. Types of fictitious loads.
1. Concentrated loads
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
45
Lecture notes: Structural Analysis II
Concentrated loads in a conjugate beam arise when a mutual displacement or rotation between
two adjacent sections belonging to two connected plates (or two sections which are not
neibouring but placed one bellow another) is posible.
1.1. Concentrated fictitious force.
Concentrated fictitious force arises when a mutual rotation is allowed between two adjacent
sections (Fig. 2).
vertical
deflections
real structure
fict
F
φleft
φright
Ffict
φleft=Ql,fict
Δφ
conjugate beam
φright= Qr,fict
Figure 2 Concentrated fictitious force
When a mutual rotation of both the plates is possible, the vertical deflection of these plates is as
shown in Fig. 2. The bending moment diagram in the conjugate beam must take the same shape
as the elastic curve of the real beam. In that respect in order to obtain a kink in the fictitious
moment diagram we must put a concentrated fictitious force in the respective section of the
conjugate beam (the section corresponding to the hinge in the real beam).
The phisical meaning of the concentrated force is a mutual rotation of both the plates of the real
structure caused by the external loads:
F fict = Q l , fict − Q r , fict = ϕ left − ϕ right = Δϕ
The fictiotious force, which is the mutual rotation, can be obtained as any other elastic
displacement of the real structure.
M=1
real structure
F fict = Σ ∫
MfM
EI
ds
⎛
Nf N ⎞
ds ⎟
⎜⎜ +Σ ∫
⎟
EA
⎝
⎠
when the influence of axial forces
is taken into account
1.2 Concentrated fictitious moment
Concentrated fictitious moment arises when a mutual vertical displacement is posible between
two plates of the real structure.
When a mutual vertical displacement of both the plates is possible, the vertical deflections of
these plates are as shown in Fig. 3. The bending moment diagram in the conjugate beam must
take the similar appearance as the elastic curve of the real beam. In that respect in order to obtain
a jump in the fictitious moment diagram we must introduce a concentrated fictitious bending
moment in the respective section of the conjugate beam.
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
46
Lecture notes: Structural Analysis II
real structure
deflected
shape
M
fict
conjugate beam
Mfict
wleft=Ml,fict
wright=Mr,fict
Figure 3 Concentrated fictitious moment.
The phisical meaning of the concentrated moment is a mutual vertical displacement of both the
plates of the real structure caused by the external loads:
M fict = M l , fict − M r , fict = wleft − wright = Δw
The magnitude of the concentrated moment is obtained as a relative vertical displacement of the
real structure. The sign of the fictitious moment (clockwise or counter-clockwise) depends on the
direction of fictitious unit forces introduced for the determination of the mutual displacement in
the real structure. The concentrated moment in the conjugate beam must correspond to the couple
of fictitious unit forces in the real structure.
F=1
real structure
M
fict
= Σ∫
MfM
EI
ds
⎛
Nf N ⎞
ds ⎟
⎜⎜ +Σ ∫
EA ⎟⎠
⎝
when the influence of axial forces
is taken into account
2. Distributed loads
2.1. Distributed transverse fictitious load
Distributed transverse fictitious load arises when a mutual rotation is posible between two
sections of a single plate, of a unit length distance one another.
dφ
ds
MF
MF
ρ
real structure
MF
dx
qfict
ds
conjugate beam
Figure 4 Distributed fictitious load
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
47
Lecture notes: Structural Analysis II
Let us consider a differential element of length ds belonging to the road lane of the real structure
(Fig. 4). The mutual rotation of both the sections at the limits of the differential element is:
ds M F
dϕ =
=
ds = dF fict .
ρ
EI
The intensity of the fictitious distributed load in the conjugate beam can be derived as follow:
MF
dF M F
1 MF
1
.
q fict =
=
ds
=
ds
=
dx
EI
dx EI
ds ⋅ cos α EI ⋅ cos α
2.2 Distributed fictitious moment.
Distributed fictitious moment arises when a mutual vertical displacement is posible between two
sections of a single plate, of a unit length distance one another (Fig. 5).
ds
Δds
dMf
NF
real structure
dx
mfict
conjugate beam
Figure 5 Distributed fictitious moment
The relative extension of a differential element of length ds is Δds (Fig. 5) and can be expressed
as:
N
Δds = F ds .
EA
The mutual vertical displacement of both the sections at the limits of the differential element,
which is the required fictitious moment dMf, reads:
N
dM fict = Δds ⋅ sin α = F ds ⋅ sin α .
EA
The intensity of the fictitious distributed moment in the conjugate beam can be derived as
follows:
dm fict =
N
dM fict N F
1 NF
1
=
ds ⋅ sin α
=
ds ⋅ sin α
= F tgα .
dx
EA
dx EA
ds ⋅ cos α EA
The fictitious conjugate beam is in equilibrium if all the fictitious loads are properly determined
and applied. The conjugate beam could be restrained as staticaly determined, the supports must
replace the concentrated fictitious loads. In this case, the support reactions, derived by
equilibrium conditions, are equal to the concentrated fictitious loads, obtained as mutual
displacements.
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
48
Lecture notes: Structural Analysis II
Algorithm of displacement influence line construction by using the conjugate
beam method
1. Introduce a unit load in the direction of required displacement in the real frame structure.
2. Construct the bending moment diagram in the real frame, this diagram is called main –
3.
4.
5.
M
Mf.
Form a horizontal conjugate beam, which is the horizontal projection of the plates
belonging to the road lane. This conjugate beam must be loaded with distributed
transverse loads q fict = M F / ( EI ⋅ cos α ) . This distributed load arise from the plates
which are part of the road only.
In case of hinges, Q or N releases, disconnections of the road lane, jumps in the road, or
on the boundary between end of the plates as a part of the structure and the ground,
concentrated fictitious loads appear.These loads could be obtained according to their
phisical meaning – mutual displacements between two neighbouring sections. A part of
concentrated loads or all fictitious concentrated loads can be replaced by supports, in such
a way that the conjugate beam becomes staticaly determinate. The beam is in equilibrium
due to the other distributed and concentrated fictitious loads.
Sections from the road lane for which the vertical displacements are zero could be
replaced by hinges in the conjugate beam (w=0 respectively Mfict=0).
Construct the bending moment diagram in staticaly determinate conjugate beam. This
diagram is the required displacement influence line, because:
fict
≡ w ≡ " Δi , F " .
Alternative approach
The conjugate beam could be loaded by distributed loads only. The bending moments at the
beginning and the end of the distributed loads (the ordinates passing through the load limits) can
be derived as vertical displacements in the corresponding sections of the real structure. As far as
the elastic curve is a graph of the vertical displacements of points belonging to the road lane,
every ordinate of this graph can be obtained as a vertical displacement of the corresponding
section of the real frame.
In such a way we obtain a sequence of simply supported beams with known bending moments at
the limits of each simple beam. In this case M fict = M reference + M base .
Numerical example
3
2
I1
I1
I3
I1
3
1
I2
EI1=200000
EI2=100000
EI3=50000
I1
4
6
2
2
I3
I3
4
4
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
49
Lecture notes: Structural Analysis II
In the following example we shall construct the influence line for vertical displacement at section
4 in the given frame structure.
Main bending moment diagram caused from the unit load in the direction of the required
displacement
3.14 2.00
2.10
1.14
Fi=1
1.60
Mi
2.10
φ1
∆φ2
Conjugate beam
∆φ3
w1
φ4
w4
Staticaly determinate conjugate beam
∆φ2
∆φ3
w4
Alternative approach – conjugate beam loaded with distributed loads only
w2
w3
w4
w1
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
50
Lecture notes: Structural Analysis II
w1, w2, w3 and w4 are the required vertical displacements
Virtual unit bending moment diagrams for the derivation of vertical displacements
Fk= F1=1
4.00
M1
2.86
Fk= F2=1
1.71
2.86
1.71
2.86
M2
1.43
2.00
Fk= F3=1
0.95
0.95
0.57
M3
0.95
So, we have:
MM
w1 = Σ ∫ 1 i ds = −55.87 ⋅10−6 ,
EI
M M
w3 = Σ ∫ 3 i ds = 138.3 ⋅10−6 ,
EI
M 2M i
ds = −186.3 ⋅10−6 ,
EI
M 3M i
M i2
w3 = Σ ∫
ds = Σ ∫
ds = 419.9 ⋅10−6 .
EI
EI
w2 = Σ ∫
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
51
Lecture notes: Structural Analysis II
Influence line construction based on the alternative approach
3.14
200000
2.10
200000
55.87 ⋅10
−6
2.00
200000
138.3 ⋅10−6
1.60
100000 ⋅ 0.8
419.9 ⋅10−6
186.3 ⋅10−6
4
6
2
35.325 ⋅10−6
4
2.5 ⋅10−6
Mbase
10.5 ⋅10−6
20 ⋅10−6
186.3 ⋅10−6
55.87 ⋅10−6
Mreference
186.3 ⋅10
110.585 ⋅10−6
138.3 ⋅10−6
−6
419.9 ⋅10−6
128.48 ⋅10−6
55.87 ⋅10−6
66.65 ⋅10−6
M≡ " Δi ,F "
138.3 ⋅10−6
299.1⋅10−6
419.9 ⋅10−6
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
52
Lecture notes: Structural Analysis II
References
DARKOV, A. AND V. KUZNETSOV. Structural mechanics. MIR publishers, Moscow, 1969
WILLIAMS, А. Structural analysis in theory and practice. Butterworth-Heinemann is an imprint
of Elsevier , 2009
HIBBELER, R. C. Structural analysis. Prentice-Hall, Inc., Singapore, 2006
KARNOVSKY, I. A., OLGA LEBED. Advanced Methods of Structural Analysis. Springer
Science+Business Media, LLC 2010
2011 S. Parvanova, University of Architecture, Civil Engineering and Geodesy - Sofia
53