Elastic Curves of Beams

Basic Equations for Beam Deflection

xx EI

My

yx

EI

M

1

2/32

2

2

1

1

dxdv

dxvd

2

2

dx

vd

EI

Mdx

dv

Elastic Curve by Direct Integration

EI

xM

dx

vd )(2

2

)(xdx

dv

EI

xw

dx

vd )(4

4

EI

xV

dx

vd )(3

3

wdx

vdEI

4

4

13

3

CwdxVdx

vdEI

212

2

CxCwdxdxMdx

vdEI

322

12

1CxCxCwdxdxdxEI

dx

dvEI

432

23

1 2

1

6

1CxCxCxCwdxdxdxdxEIv

Alternative Coordinates

Boundary Conditions of Beams

Example I-Cantilever Beam under an End Load

x

PxMdx

vdEI

2

2

12

2

1CPx

dx

dvEI 2

1 2

10)( PLCLx

223

2

1

6

1CxPLPxEIv 3

2 3

10)( PLCLxv

323 236

LxLxEI

Pv

Example II-Simple Beam under Uniform Load

x

4334

12

1

24

1CxCwLxwxEIv

212

212

2

2

1CxCwxCxCwdxdxM

dx

vdEI

0C0)0x(M 2 wL2

1C0)Lx(M 1

00)0( 4 Cxv 33 24

10)( wLCLxv

xLLxxEI

wv 334 224

Method of Superposition

Only suitable for linear problems!!!

Easy Mistake in Superposition

Cannot remove the supporting boundary

conditions!!! X

Statically Indeterminate Beams

212

2

2

2

1CxCwx

dx

vdEI

221 2

10)( wLCLCLxM

32213

26

1CxCx

Cwx

dx

dvEI

00)0( 3 Cx

422314

2624

1Cx

Cx

CwxEIv

00)0( 4 Cxv

221 4

130)( wLCLCLxv

Alternative Solution by Superposition (I)

232 3

6Lxx

EI

Rv B

x

22341 64

24xLLxx

EI

wv

wLRvv BLxRBLxwB B 8

3

2234 35248

xLLxxEI

wv

Alternative Solution by Superposition (II)

xLLxxEI

wv 3341 2

24 xLLxx

EIL

Mv A 2232 236

2234 35248

xLLxxEI

wv

2

00 8

1wLM AxMAxwA A

Table of Beam Elastic Curves

Table of Beam Elastic Curves