A moving unit load Moving loads PRELIMINARIESpioneer.netserv.chula.ac.th/~cchatpan/2101310/MovingLoadI-student.pdf · Influence Line Dr. Jaroon Rungamornrat 9 The rotation T CI for

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


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


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


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


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


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


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


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


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


Example2: Construct influence lines RAI, MAI, VBI, MBI, BI, BI of a cantilever beam


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


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


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


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


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


Example3: Construct influence lines RAI, MAI, RBI, VCI, VBLI, MBLI, VBRI, MBRI, VDI, and MDI of a beam 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


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


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


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


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


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


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


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


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


1

x

Virtual System 1a

Virtual System 1b

Virtual System 1c


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


1

RELEASE moment constraint

1


1

Virtual System 2a

Virtual System 2b

Virtual System 3a

Virtual System 3b


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


1

Virtual System 1a

Actual system

x

v(x)

RAI

RBI

MAIA B

RAI x

1


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


1

VCI x1


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


1

MCI x1

1


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


1

RDI x

1h1=3/4

h2=3/2

h3=1/2


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.


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


VEI x

1h1=1/4

h2=1/2 h4=1/2

1

h3=-1/2


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.


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


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


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


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


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


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

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A moving unit load Moving loads PRELIMINARIESpioneer.netserv.chula.ac.th/~cchatpan/2101310/MovingLoadI-student.pdf · Influence Line Dr. Jaroon Rungamornrat 9 The rotation T CI for