Deflection of Indeterminate Structure Session 03-04 Matakuliah: S0725 – Analisa Struktur Tahun: 2009

Deflection of Indeterminate Structure Session 03-04

Matakuliah : S0725 – Analisa StrukturTahun : 2009

Bina Nusantara University 3

Contents

•Elastic Beam Theory•Moment area•Conjugate Beam Method


Introduction

What is deflection?Deflection can occur from various

causes, such as loads, settlement,

temperature or fabrication error of material.

Deflection must be limited in order to prevent cracking and damaging of structure.


Introduction

What is caused of structure

deflection?

In general It caused by its internal loading such as

Normal force, shear force or bending moment


Introduction

What is caused of Beam &

Frames deflection?

It caused by internal bendingWhat is caused of truss

deflection?

It caused by internal axial forces


Introduction

Deflection Diagram represent

the Elastic Curve for the points at the centroids of cross-sectional area along the members


Introduction

Deflection on supports :

(1) Roller = 0

A


Introduction


(2) Pin = 0

A


Introduction


(3) Fixed Support = 0 ; = 0


Introduction

Deflection on supports :(4) Fixed Connected Joint causes the joint to rotate the members by the same amount of


Introduction

Deflection on supports :(5) Pin Connected Joint the members will have a different slope or rotation at pin, since the pin can’t support moment

1

2


Introduction

Sign Convention Bending Moment

M+Longitudinal axis

M+


Introduction

Sign Convention Bending Moment

M-Longitudinal axis

M-


Introduction

Deflection curve :BA

P2

P1

Moment

-

+

BA Deflection curve


xP P

y

Elastic curve

The deflection is measured from the original neutral axis to the neutral axis of the deformed beam.

The displacement y is defined as the deflection of the beam.

It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam

Importance of Beam Deflections

A designer should be able to determine deflections, i.e.

In building codes ymax <=Lbeam/300

Analyzing statically indeterminate beams involve the use of various deformation relationships.


Elastic Beam Theory

xP P

y

Elastic curve

The deflection is measured from the original neutral axis to the neutral axis of the deformed beam.

The displacement y is defined as the deflection of the beam.

It may be necessary to determine the deflection y for every value of x along the beam. This relation may be written in the form of an equation which is frequently called the equation of the deflection curve (or elastic curve) of the beam


Elastic Beam TheoryImportance of Beam

Deflections

A designer should be able to determine deflections, i.e.

ymax <=Lbeam/300

Analyzing statically indeterminate beams involve the use of various deformation relationships.


Double-Integration Method

The deflection curve of the bent beam is

Mdx

ydEI

2

2

In order to obtain y, above equation needs to be integrated twice.

y

Radius of curvature

y

x

)(Curvature1

EI

MEIM



An expression for the curvature at any point along the curve representing the deformed beam is readily available from differential calculus. The exact formula for the curvature is

2

32

2

2

1

dxdy

dxyd

small is dx

dy2

2

dx

yd M

dx

ydEI

2

2


The Integration Procedure

Integrating once yields to slope dy/dx at any point in the beam.

Integrating twice yields to deflection y for any value of x.

The bending moment M must be expressed as a function of the coordinate x before the integration

Differential equation is 2nd order, the solution must contain two constants of integration. They must be evaluated at known deflection and slope points (i.e. at a simple support deflection is zero, at a built in support both slope and deflection are zero)



Sign Convention

Positive Bending Negative Bending

Assumptions and Limitations

Deflections caused by shearing action negligibly small compared to bending

Deflections are small compared to the cross-sectional dimensions of the beam

All portions of the beam are acting in the elastic range

Beam is straight prior to the application of loads



First Moment –Area Theorem

The first moment are theorem states that: The angle between the tangents at A and B is equal to the area of the bending moment diagram between these two points, divided by the product EI.

Moment Area


B

A

dxEI

MTangent at A

A B

Tangent at B

d

d

xdx

ds

M

Moment Area

dxEI

MxB

A

d

dsdds

EIM

dx with ds replace sdeflection lateral small isit dsEI

Md

dxEI

Mddx

EI

Md

B

A give willgintegratin

B

A

dxEI

Mxdx

EI

Mxxd


The second moment area theorem states that: The vertical distance of point B on a deflection curve from the tangent drawn to the curve at A is equal to the moment with respect to the vertical through B of the area of the bending diagram between A and B, divided by the product EI.

dxEI

MxB

A

d

dsdds

EIM dx with ds replace sdeflection lateral small isit ds

EI

Md

dxEI

Mddx

EI

Md

B

A give willgintegratin

B

A

dxEI

Mxdx

EI

Mxxd

Moment AreaSecond Moment –Area Theorem


Procedure1. The reactions of the beam are determined

2. An approximate deflection curve is drawn. This curve must be consistent with the known conditions at the supports, such as zero slope or zero deflection

3. The bending moment diagram is drawn for the beam. Construct M/EI diagram

4. Convenient points A and B are selected and a tangent is drawn to the assumed deflection curve at one of these points, say A

5. The deflection of point B from the tangent at A is then calculated by the second moment area theorem

Moment Area


P

PL L

P

A

B

Tangent at A

Tangent at B

Moment Area

Problem


P

PL L

P

A

B

Tangent at A

Tangent at B

PL

M

33

2

2

3PLLPL

LEI

EI

PL

3

3

PLL

EI 2 EI

PL

2

2

Moment Area


WL2

2WL

Tangent A

L

A W N per unit length

B

= ?

2

2WL

xL

WLA

23

1 2

Lx4

3

84

3

23

42 WL

LLWL

EI

EI

WL

8

4

Moment Area


Example

L

aP

aP

P P

aaL

2

Pa

Tangent A

A = ?

Moment Area


aPaa

aaL

aL

PaEI3

2

2242

3

22

32448a

PaLaLaLPa

3

3332 43

2468 L

a

L

aPLPaPaL

3

33 43

24 L

a

L

a

EI

PL

Moment Area


Conjugate Beam

The method requires the same amount of computation as the moment-area theorems to determine a beam’s slope or deflection. However, the method relies only on the principles of

statics, its application will be more familiar

Dr Yan Zhuge lecturer notesDr Yan Zhuge lecturer notes


Using the similarity of equations for

Beam Statics Beam deflection

Or integrating

wdxV

dxEI

M)(

dxdx

EI

Mv )(

dxwdxM

Unit = kN·m2/EI Unit = kN·m3/EIDr Yan Zhuge lecturer notesDr Yan Zhuge lecturer notes


Theorem 1: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam.

Theorem 2: The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam.

Conjugate Beam



Conjugate Beam•Draw the conjugate beam for the real beam with a proper boundary conditions

•Load the conjugate beam with the real beam’s M/EI diagram. This loading is directed downward when

M/EI is positive and upward when M/EI is negative•Determine the statics of the conjugate beam: reactions, Shear force and moments•Shear force V corresponds to the slope of the real beam, moment M corresponds to the displacement v of the real beam.



Conjugate BeamREAL BEAM CONJUGATE BEAM










++++ ++++

++++

++++



Determine the maximum deflection of the steel beam shown in the figure. E = 200 GPa, I = 60(106) mm4.

A

9 m

8 kN

B

x

3 m

2 kN 6 kN

Conjugate Beam



8 kN

A B

x

18kNm

A’ B’

x

18/EI

Conjugate Beam

Real Beam

45/EI 63/EI

Maximum deflection occurs at the point

where the slope is zero

This corresponds to the same point in the

conjugate beam where the shear is zero

2 kN

6 kN

9 m 3 m

81/EI 27/EI

Conjugate Beam


Documents

Deflection of Indeterminate Structure Session 03-04 Matakuliah: S0725 – Analisa Struktur Tahun: 2009