2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 1
INFLUENCE LINES
PRELIMINARIES
Moving loads -- Loads applied to a structure with points of application (including their
magnitude) can vary as a function of positions on the structure. Examples of moving loads include live load on buildings, traffic or vehicle loads on bridges, loads induced by wind and earthquake, etc. In the analysis, the moving loads can be modeled as varying distributed loads, a series of concentrated loads, or the combination of distributed loads and concentrated loads.
Figure 1
A moving unit load -- a concentrated load of unit magnitude with its point of application
varies as a function of position on the structure.
Figure 2
Responses due to moving loads -- Quantities of interest that indicate the effect of the
moving loads on a structure, e.g. internal forces, support reactions, displacements and rotations, deformations, etc.
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 2
Responses due to a moving unit load -- Quantities of interest at a particular point
within a given structure, e.g. internal forces, support reactions, deformations, displacements and rotations, due to an applied moving unit load. The quantities are given in terms of functions of a position of a moving unit load on the structure; these response functions are termed as the influence functions and their graphical representations are known as the influence lines.
Application of the influence functions (lines)
Let fA be a quantity of interest at a point A within a given structure due to applied distributed load q and a series of concentrated loads {P1, P2, …, PN} and fAI denote the influence function of the corresponding quantity at point A. By a method of superposition, we obtain the relation of fA and fAI as
N
1i
iiAIAIA Pxfdxqff )( (1)
where the integral is to be taken over the region on which load q is applied and xi indicates the location on which the load Pi is applied. For instance, assume that the influence line of the support reaction at point A (RAI) of the beam is given as shown in the Figure 3a. The support reaction at point A (RA) due to applied loads as shown in Figure 3b can then be obtained using Eqn. (1) as follow:
2P3L/4RPL/4RdxqRR AIAI
L/2
0
AIA )()(
5P/43qL/82P1/4P3/4dxx/L-1qL/2
0
Figure 3a Figure 3b
1
A Bx
L
xRAI
1-x/L
L
RAI
0
L/4L/4 L/4
Pq
L/4
RA
xRAIL
2P
3/41/2
1/4Area = 3L/8
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 3
In addition, the influence lines can also be used to predict the load pattern that maximizes responses at a particular point of the structure. For instance, let consider a two-span continuous beam subjected to both dead load (fixed load) and live load (varying load) as shown in the figure below.
To determine the maximum positive bending moment at points A, the maximum negative moment B, and the maximum positive shear at point A due to these applied dead and live loads, we construct first the influence lines MAI, MBI, and VAI as shown below.
It is evident from the influence lines that the maximum positive bending moment at point A occurs when the live load is placed only on the first span; the maximum negative moment at point B occurs when the live load is placed on both spans; and the maximum positive shear occurs when the live load is placed on the first half of the first span and on the second span. The maximum value of the responses can then be obtained using Eqn.(1) for each corresponding loading pattern. It is noted that the dead load is fixed and therefore it is applied to both spans of the beam for all cases.
1
A B
x
Dead load
Live load
A B
MAI x
MBI x
VAI x
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 4
INFLUENCE LINES FOR DETERMINATE BEAMS
Support reactions (e.g. RAI, RDI)
Bending moment at a particular section (e.g. MCI)
Shear force at a particular section (e.g. VCI)
Deflection at a particular point (e.g. BI)
Rotation at a particular point (e.g. BI)
Direct Methods for Constructing Influence Lines
Treat a structure subjected to a moving unit load (as function of positions)
Influence functions are obtained by considering all possible load locations
Support reactions
-- Equilibrium equations of the entire structure
Internal forces
-- Method of sections
-- Equilibrium equations of parts of the structure
Displacement and rotations
-- Determining support reactions and internal forces from equilibrium
-- Displacement and rotations are obtained from
Direct integration method
Moment area and conjugate structure methods
Energy methods, etc.
1
AD
x
C
RAI RDI
VCI
MCI
BBI
BI
Deformed state
Undeformed state
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 5
Example1: Construct influence lines RAI, RBI, VCI, MCI, CI, CI of a simply supported beam
Solution Consider the beam subjected to a moving unit load as shown below.
Influence lines for reactions RAI, RBI
0MB + 0x)-(1)(L)(L)(RAI
L
x-LRAI
0MA + 0(1)(x))(L)(RBI
L
xRBI
A C B
L/3 2L/3
AC
B
L/3 2L/3
1x
A B
1x
RAI RBI
RAI
xL0
RBI
x
1
1
L0
EI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 6
Influence lines for shear and bending moment VCI, MCI
0FY + 0V1R CIAI
L
x1RV AICI
0MC + 0Mx)-(1)(L/3)(L/3)(R CIAI
3
2xx
3
1)L(RM AI
CI
0FY + 0VR CIAI
L
x1RV AICI
0MC + 0M)(L/3)(R CIAI
L
x1
3
L
3
LRM AI
CI
1
A B
x L/3
RAI RBI
C
A B
1
RAI RBI
C
x L/3
A
1x L/3
RAI
C
VCI
MCI
A
RAI
C
VCI
MCI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 7
Influence lines for deflection and rotation CI, CI
VCI
xL0
MCI
x
2/3
L0
L/3-1/3
2L/9
L/3
A B
x L/3
RAI RBI
C
x
1
L/3-x/3
BMDM
2x/3
x-L/3
A B
x L/3
RAI RBI
C
x
1
L/3-x/3
2x/3
L/3-x
A B
2/3 1/3C
x
1
2L/9
A B
1/L 1/LC
x
1/3
1
-2/3
Actual System I Actual System II
Virtual System I Virtual System II
BMD
M
BMDM
BMD
M
RAI
xL0
12/3
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 8
The deflection CI for x L/3 can be obtained using the unit load method along with the actual system I and the virtual system I; i.e.
dxEI
MML
0
CI
9
2L
L
x
3
2x
3
L
3
Lx
2EI
1
9
2L
3
2
3
L
3
xL
2EI
1
9
2L
3
2
3
2L
3
2x
2EI
1
22 9x5L81EI
x
The deflection CI for x L/3 can be obtained using the unit load method along with the actual system II and the virtual system I; i.e.
dxEI
MML
0
CI
9
2L
2L
x
6
7
3
Lxx
3
L
2EI
1
9
2L
3
2
3
L
3
xL
2EI
1
9
2L
3
2
3
2L
3
2x
2EI
1
22 9x18LxL162EI
L-x
CI
xL0
4L3/243EI
L/3
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 9
The rotation CI for x L/3 can be obtained using the unit load method along with the actual system I and the virtual system II; i.e.
dxEI
MML
0
CI
3
1
L
x
3
2x
3
L
3
Lx
2EI
1
3
1
3
2
3
L
3
xL
2EI
1
3
2
3
2
3
2L
3
2x
2EI
1
22 3xL18EIL
x
The rotation CI for x L/3 can be obtained using the unit load method along with the actual system II and the virtual system I; i.e.
dxEI
MML
0
CI
3
2
2L
x
6
7
3
Lxx
3
L
2EI
1
3
1
3
2
3
L
3
xL
2EI
1
3
2
3
2
3
2L
3
2x
2EI
1
22 3x6LxL18EIL
x-L
CI
xL0
-4L2/162EI
L/3
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 10
Example2: Construct influence lines RAI, MAI, VBI, MBI, BI, BI of a cantilever beam
Solution Consider the beam subjected to a moving unit load as shown below.
Influence lines for reactions RAI, MAI
0MA + 0(1)(x)MAI
xMAI
0FY + 01RAI
1RAI
1x
A B
1x
RAI
MAI
MAI
xL0
RAI
x
-L
1
L0
A B
L/2
EI
L/2
A B
L/2
EI
L/2
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 11
Influence lines for shear and bending moment VCI, MCI
0FY + 0VBI
0VBI
0MB + 0MBI
0MBI
0FY + 0VR BIAI
1RV AIBI
0MB + 0MM)(L/2)(R BIAIAI
2
LxM
2
LRM AI
AIBI
1
A
x L/2
RAI
MAIB
BVBIMBI
A
RAI
B
VBIMBI
1
A
x L/2
RAI
BMAI MAI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 12
Influence lines for deflection and rotation BI, BI
VBI xL0
MBI x
1
L 0
L/2
-L/2
L/2
A
x L/2
RAI
B
x
1
BMDM
-x
Actual System I Actual System II
Virtual System I Virtual System II
1
MAIA
x L/2
RAI
B
1
BMDM
MAI
A
1
B
x
1
BMD
M
-L/2
-L/2A
0B
x
1
BMD
M
1
1
1
-L/2+x
L/2
-L/2
x
-xL/2-x
L/2
MAI xL0
RAI x
11
L0
-L/2
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 13
The deflection BI for x L/2 can be obtained using the unit load method along with the actual system I and the virtual system I; i.e.
dxEI
MML
0
BI
2
L
2
1
2
Lx-
EI
1
2
L
3
1
2
L
2
L
2EI
1
2
L
3L
2x
3
1x
2
Lx
2
L
2EI
1
2x3L12EI
x2
The deflection BI for x L/2 can be obtained using the unit load method along with the actual system II and the virtual system I; i.e.
dxEI
MML
0
CI
2
L
2
1
2
Lx-
EI
1
2
L
3
1
2
L
2
L
2EI
1
L6x48EI
L2
BI
xL 0
L3/24EI
L/2
5L3/48EI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 14
The rotation BI for x L/2 can be obtained using the unit load method along with the actual system I and the virtual system II; i.e.
dxEI
MML
0
CI
12
Lx-
EI
11
2
L
2
L
2EI
1
1x2
Lx
2
L
2EI
1
2EI
x2
The rotation BI for x L/2 can be obtained using the unit load method along with the actual system II and the virtual system I; i.e.
dxEI
MML
0
CI
12
Lx-
EI
11
2
L
2
L
2EI
1
4xL8EI
L
BI
xL0
-L2/8EI
L/2
-3L2/8EI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 15
Example3: Construct influence lines RAI, MAI, RBI, VCI, VBLI, MBLI, VBRI, MBRI, VDI, and MDI of a beam shown below
Solution Consider the beam subjected to a moving unit load as shown below.
Influence lines for reactions RAI, MAI, RBI and shear force VCI
From FBD II, we obtain
0MC + 0)(L)(RBI
0RBI
0FY + 0RV BICI
0RV BICI
A B
L/2 L L
1x
A B
1x L
RAI
MAI
A B
L L
C
C
RBI
C
RBI
B
C
VCI
FBD I FBD II
D
L/2
L/2 L/2
D
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 16
From FBD I, we obtain
0FY + 01RR BIAI
1R1R BIAI
0MA + 0(1)(x))(2L)(RM- BIAI
-xxL2RM BIAI
From FBD IV, we obtain
0MC + 0L)(1)(x)(L)(RBI
1L
xRBI
0FY + 01RV BICI
L
x2R1V BICI
From FBD III, we obtain
0FY + 01RR BIAI
L
x2R1R BIAI
A B
1x L
RAI
MAI
RBI
C
RBI
B
C
VCI
FBD III FBD IV
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 17
0MA + 0(1)(x))(2L)(RM- BIAI
2Lxx-L2RM BIAI
MAI x0
RAI x
-L
-1
1
L 2L
3L
L
0 L
2L 3L
1
RBI x
2
0 L 2L 3L
1
VCI x
-1
0 L
2L 3L
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 18
Influence lines for shear and bending moment VBLI, MBLI
0FY + 0RV BIBLI
BIBLI -RV
0MB + 0MBLI
0MBLI
0FY + 01RV BIBLI
BIBLI R1V
0MB + 02L)(1)(xMBLI
x2LMBLI
A B
1x L
RAI
MAI
RBI
C
RBI
BVBLIMBLI
A B
1x 2L
RAI
MAI
RBI
C
RBI
BVBLIMBLI
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 19
Influence lines for shear and bending moment VBRI, MBRI
0FY + 0VBRI
0VBRI
0MB + 0MBRI
0MBRI
RBI x
2
0 L 2L 3L
1
VBLI x
-1
0 L 2L 3L
-1
MBLI x
-L
0 L 2L 3L
A B
1x L
RAI
MAI
RBI
CVBRIMBRI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 20
0FY + 01VBRI
1VBRI
0MB + 02L)(1)(xMBRI
x2LMBRI
A B
1x 2L
RAI
MAI
RBI
CVBRIMBRI
1
VBRI x
1
0 L 2L 3L
MBRI x
-L
0 L 2L 3L
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 21
Influence lines for shear and bending moment VDI, MDI
0FY + 0RV BIDI
BIDI -RV
0MB + 0)(3L/2)(RM BIDI
/23LRM BIDI
0FY + 01RV BIDI
BIBLI R1V
A B
1x L/2
RAI
MAI
RBI
C
VDIMDI
D
B
RBI
C
D
A B
1x L/2
RAI
MAI
RBI
C
VDIMDI
D
B
RBI
C
D
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 22
0MB + 0)(3L/2)(RL/2)(1)(xM BIDI
xL/2/23LRM BIDI
Remarks
1. The influences lines of support reactions and internal forces (shear force and bending moment) for statically determinate beams are piecewise linear; i.e. they consists of only straight line segments.
2. The influence functions of the internal forces can be obtained in terms of the influence functions of the support reactions; therefore, the influence lines of internal forces can be readily obtained from those for support reactions.
3. The influence lines of the deflection and rotation at any points of the statically determinate beam generally consist of curve segments.
RBI x
21
VDI x
-1
0 L 2L
3L
1
MDI x
L/2
1
L/2
-L/2
0 L 2L3L
L/2
0 L 2L 3L
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 23
Muller-Breslau Principle
Actual Structure. Consider a statically determinate beam subjected to a moving unit load as shown in the figure below.
Virtual Displacement -- The fictitious and arbitrary displacement that is introduced to the structure. For use further below, the following three types of virtual displacement for the beam structure are considered:
Virtual displacement due to release of a support constraint.
1. Release a support constraint in the direction of interest 2. The beam becomes statically unstable (partially or completely) 3. Introduce unit virtual displacement (or unit virtual rotation if the
rotational constraint is released) in the direction that the support constraint is released.
4. The virtual displacement at all other points results from the development of the mechanism (or rigid body motion) of the entire beam
1
RELEASE displacement constraint
1
1
RELEASE rotational constraint
RELEASE displacement constraint
1
x
Virtual System 1a
Virtual System 1b
Virtual System 1c
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 24
Virtual displacement due to release of shear constraint.
1. Remove the shear constraint by introducing a shear release at point of interest
2. The beam becomes statically unstable (partially or completely) 3. Introduce unit relative virtual displacement between the two ends of
the shear release with their slope remaining the same (provided that the moment constraint exists at that point)
4. The virtual displacement at all other points results from the development of the mechanism (or rigid body motion) of the entire beam.
Virtual displacement due to release of bending moment constraint.
1. Remove the moment constraint by introducing a hinge at point of interest 2. The beam becomes statically unstable (partially or completely) 3. Introduce unit relative virtual rotation at the hinge without separation
(provided that the shear constraint exists at that point). 4. The virtual displacement at all other points results from the development
of the mechanism (or rigid body motion) of the entire beam.
RELEASE shear constraint
1
RELEASE shear constraint
1
RELEASE moment constraint
1
RELEASE moment constraint
1
Virtual System 2a
Virtual System 2b
Virtual System 3a
Virtual System 3b
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 25
Principle of Virtual Work: Consider a system or structure subjected to external applied loads. The support reactions and internal forces at any locations within the structure are in equilibrium with the applied loads if and only if the external virtual work (work done by the external applied loads) is the same as the internal virtual work (work done by the internal forces) for all admissible virtual displacements, i.e.
IE WW (2)
It is important to note that the portion of the structure that undergoes virtual rigid body motion (virtual displacement that produces no deformation) produces zero internal virtual work.
Influence Line for Support Reactions. To clearly illustrate the strategy, let assume that the influence line of the support reaction RAI is to be determined. By applying the principle of virtual work to the actual system with a special choice of the virtual displacement as indicated in the virtual system 1a (the virtual displacement associated with the rigid body motion of the beam resulting from the release of the displacement constraint at A) , we obtain
)()( xvRxv11RW AIAIE ; 0WI
IE WW
)(xvRAI (3)
1
RELEASE displacement constraint
1
Virtual System 1a
Actual system
x
v(x)
RAI
RBI
MAIA B
RAI x
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 26
Muller-Breslau Principle: “The influence line of a particular support reaction has an identical shape to the virtual displacement obtained from releasing the support constraint in the direction of the support reaction (under consideration) and introducing a rigid body motion with unit displacement/unit rotation in the direction of the released constraint.”
Influence Line for Shear Force. Let assume that the influence line of the shear force at point C, VCI, is to be determined. By applying the principle of virtual work to the actual system with a special choice of the virtual displacement as indicated in the virtual system 2a (the virtual displacement associated with the rigid body motion of the beam resulting from the release of the shear constraint at C) , we obtain
)()( xvxv1WE ; CICII V1VW
IE WW
)(xvVCI (4)
Muller-Breslau Principle: “The influence line of the shear force at a particular point has an identical shape to the virtual displacement obtained from releasing the shear constraint at that point and introducing a rigid body motion with unit relative virtual displacement between the two ends of the shear release with their slope remaining the same.”
1
Virtual System 2a
Actual system
x
v(x)
RAI
RBI
MAIA B
C
VCIMCI
RELEASE shear constraint
1
VCI x1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 27
Influence Line for Bending Moment. Let assume that the influence line of the bending at point C, MCI, is to be determined. By applying the principle of virtual work to the actual system with a special choice of the virtual displacement as indicated in the virtual system 3a (the virtual displacement associated with the rigid body motion of the beam resulting from the release of the bending moment constraint at C) , we obtain
)()( xvxv1WE ; CICII M1MW
IE WW
)(xvMCI (5)
Muller-Breslau Principle: “The influence line of the shear force at a particular point has an identical shape to the virtual displacement obtained from releasing the shear constraint at that point and introducing a rigid body motion with unit relative virtual displacement between the two ends of the shear release with their slope remaining the same.”
1
Virtual System 3a
Actual system
x
v(x)
RAI
RBI
MAIA B
C
VCIMCI
RELEASE moment constraint
1
MCI x1
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 28
Example4: Use Muller-Breslau principle to construct influence lines RAI, RDI, RFI, VBI, VCLI,VCRI, VDLI, VDRI,VEI, MBI, MDI, and MEI of a statically determinate beam shown below
Solution The influence line of the support reaction RDI is obtained as follow: 1) release the displacement constraint at point D, 2) introduce a rigid body motion, 3) impose unit displacement at point D, and 4) the resulting virtual displacement is the influence line of RDI.
The value of the influence line at other points can be readily determined from the geometry, for instance,
3/2L3L/21h2 )/())((
1/2LL/21h3 )/())((
3/4L/2L/43/2h1 )/())((
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
A
D
L/4 L/2 L/2
CB
L/4
E F
L/2
RELEASE displacement constraint
1
RDI x
1h1=3/4
h2=3/2
h3=1/2
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 29
The influence line of the shear force VEI is obtained as follow: 1) release the shear constraint at point E, 2) introduce a rigid body motion, 3) impose unit relative displacement at point Eand 4) the resulting virtual displacement is the influence line of VEI.
The value of the influence line at other points can be readily determined from the geometry, for instance,
4343 hh2Lh2Lh )//()//(
1/2h12hhh1hh 444434 )(
1/2hh 43
1/2L/2L/2hh 32 )/())((
1/4L/2L/4hh 21 )/())((
A D
L/4 L/2 L/2
CB
L/4
EF
L/2
RELEASE shear constraint
VEI x
1h1=1/4
h2=1/2 h4=1/2
1
h3=-1/2
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 30
The influence line of the bending moment MEI is obtained as follow: 1) release the bending moment constraint at point E, 2) introduce a rigid body motion, 3) impose unit relative rotation at point E without separation and 4) the resulting virtual displacement is the influence line of MEI.
The value of the influence line at other points can be readily determined from the geometry, for instance,
L/4h12Lh2Lh 333 )//()//(
L/4L/2L/2hh 32 )/())((
L/8L/2L/4hh 21 )/())((
The rest of the influence lines can be determined in the same manner and results are given below.
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
RELEASE moment
constraint
MEI x
h1=-L/8
h2=-L/4
h3=L/4
1
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 31
A D
L/4 L/2 L/2
C
B
L/4
E F
L/2
RAI x
1
RFI x
1/2
1
1/2
1/21/4
VBI x
1/2
-1/2
VCLI x
-1-1/2
VCRI x
-1-1/2
VDLI x
-1
-1/2
-1
VDRI x
1/21/4
1
MBI x
L/8
MDIx
L/4
L/8
1x
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 32
Example5: Use Muller-Breslau principle to construct influence lines RAI, MAI, RDI, VBI, VCI,VDI,VELI,VERI, MBI, MDI, and MEI of a statically determinate beam shown below.
Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release the constraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) impose unit virtual displacement/rotation in the direction of released constraint, and 4) the resulting virtual displacement is the influence line to be determined. It is noted that values at points on the influence line can be readily determined from the geometry.
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
1x
RAI x
1 1
1/2
-1/2
MAI x
L/2
L/4
-L/4
L/4
1
xREI
1
1/2
3/2
1
1/2
-1/2
1
VBI x
VCI x
1
1/2
-1/2
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 33
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
1x
-1/2
VDI x
1/2
-1/2
x
-1/2-1
-1/2
VELI
x
1
VERI
1
x
-L/4
-L/8
L/8
MBI
xMDI
L/4
-L/4
x
-L/2
MEI
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 34
Example6: Use Muller-Breslau principle to construct influence lines RAI, MAI, RDI, RFI, VBI,VCI, VDLI, VDRI,VEI, MBI, and MDI of a statically determinate beam shown below.
Solution By Muller-Breslau principle, we obtain the influence lines as follow: 1) release the constraint associated with the quantity of interest, 2) introduce a rigid body motion, 3) impose unit virtual displacement/rotation in the direction of released constraint, and 4) the resulting virtual displacement is the influence line to be determined. It is noted that values at points on the influence line can be readily determined from the geometry.
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
1x
RAI x
1 1
-1
MAI x
L/2
-L/2
L/4
1
xREI
1
2
xREI
1
2101-301 Structural Analysis I Influence Line Dr. Jaroon Rungamornrat 35
A D
L/4 L/2 L/2
CB
L/4
E F
L/2
1x
VBI x
1
-1
VCI x
1
-1
1
VDLI x
-1 -1
VDRI x
1 1
VEI x
1
MBI x
-L/4
L/4
MBI
-L/2