CIVL3310 STRUCTURAL ANALYSIS

Professor CC Chang

Chapter 8: Chapter 8: DeflectionsDeflections

Deflection- Displacement & Rotation

Deflections maybe due to loads, temperature, fabrication errors or settlement

Linear elastic behavior: linear stress-strain relationship and small deflection

Need to: Plot qualitative deflection shapes before/after the

analysis Calculate deflections at any location of a

structure

Deflection Diagrams/Shapes

EI

M curvature

+M-Mtensile side

tensile side

Example 8.1Draw the deflected shape of each of the beams.

Need to show 1st order (slope) and 2nd order (curvature) information

tensile side

tensile side

Inflection point

Straight line(why?) Straight line

(why?)

Example 8.2

All members are axially inextensible!

M=0No curvature

M≠0

Calculation of Deflections

Direct integration Moment-area method Conjugate-beam method Energy methods (in Chapter 9)

Calculation of Deflections

Note: Superposition can be used for linear structures

)x(y

)x(y1

)x(y2

)x(y)x(y 21 =

Direct Integration

Bernoulli-Euler beam

EI

M

dx

yd

2

2

EI

M

dx

dθ

dxdxEI

My dx

EI

Mθ

apply boundary conditions

+ disp

Example 8.3The cantilevered beam is subjected to a couple moment Mo at its end. Determine the eqn of the elastic curve. EI is constant.

x

21

2

1

2

2

2 CxC

xMvEI

CxMdx

dvEI

Mdx

vdEI

o

o

o

EI

xMv

EI

xM oo

2 ;

2

EI

LMv

EI

LM oA

oA 2

;2

at x = 0dv/dx = 0 & v = 0C1 = C2 =0.

Max

Direct Integration

20m 20m 50m 20m 20m

5 kN/m3 kN/m

ACB

FD E

0

50 kN 30149241

50

-150-59

91 90

-30V

M1 M2

M3

M

dxdxEI

My

+ _ +

y

Moment-Area Theorems

dxEI

Md

EI

M

dx

yd

2

2

EI

M

dx

dθ

dxEI

MBAAB

AB / 1st moment-area theorem+

BA

Moment-Area Theorems

2nd moment-area theorem

dxEI

Mxdx

EI

Mxt

B

A

B

ABA /

dxEI

Mxdxdt

Horizontal dis btw A & centroid of M/EIA-B

M/EI area btw A & B

d

EI

M

dx

dθ

Moment-Area Theorems

dxEI

Mxdx

EI

Mxt B

ABABA /

dxEI

Mxdx

EI

Mxt A

BABAB ' /

'x

Moment-Area Theorems

+

dxEI

MBAAB /

1st moment-area theorem 2nd moment-area theorem

dxEI

Mxt BABA /

Example 8.5

Determine the slope at points B & C of the beam. Take E = 200GPa, I = 360(106)mm4

ACCABB // ;

radEI

kNm

mEI

kNm

EI

kNmm

EI

kNmABB

00521.0375

)5(50100

2

1)5(

50

2

/

dxEI

MBAAB /

radACC 00694.0/

Conjugate-Beam Method

Mathematical analogy

-slope-deflection Load-shear-momentEI

M

EI

M

dx

dθ

θdx

dy

EI

M

dx

yd

2

2

wdx

dV

Vdx

dM

wdx

Md

2

2

Conjugate-Beam Method

Mathematical equivalence

-slope-deflection Load-shear-momentEI

M

EI

M

θ

y

w

V

M

Actual beam Conjugate beam

Conjugate-Beam Method

P

EI

MEI

PL

Actual beam Conjugate beam

0

0

y

θ

0

0

y

θ

0

0

M

V

0

0

M

V

EI

PL

w

V

L

x

EI

PL1

L

xx

EI

PL

2

2

L

xx

EI

PL

62

32

P

My

θ

?

or

Conjugate-Beam Method

Conjugate-Beam Method

Conjugate beams are mathematical/imaginative beams

No need to worry about their stability

Just use EQ concept to obtain V and M from w

Example 8.5

Determine the max deflection of the steel beam. The reactions have been computed. Take E = 200GPa, I = 60(106)mm4

Max disp

Max M

Solution

OKmxmx

xEI

x

EI

)90( 71.6

02

2

145

Conjugate beam: Max Moment 0Vdx

dMNote:

when maxMM

xEI

2 0')71.6(3

171.6

)71.6(2

2

1)71.6(

45

MEIEI

6.71

mmm

mmmmmmkN

kNmEI

kNmM

8.160168.0

)])10/(1()10(60][/)10(200[

2.201

2.201'

44344626

3

3

max

Conjugate-Beam

20m 20m 50m 20m 20m

5 kN/m3 kN/m

ACB

FD E

MMEI

Conjugatebeam

CBF

D E

MEI

V=

M= y

Summary

Can you plot a qualitative deflection shape for a structure under loads (need to show 1st & 2nd order information)?

Can you calculate structural deformation (displacement & rotation) using Direct integration? Moment-area principle? Conjugate-beam method?