Structure Analysis I

Chapter 9

Structure Analysis I

Chapter 9

Deflection

Energy Method

Structural Analysis IDr. Mohammed Arafa

Energy Method

This principle states that the work done by all the external forces

acting on a structure Ue is transformed into internal work or strain

energy Ui which is developed when the structure deforms.

This principle can be stated mathematically as:

e iU U

The conservation of energy principle

Energy Method

PU

dxFU

e

x

e

21

0

When a force F undergoes a displacement dx

in the same direction as the force, the work

done is

If the total displacement is x the work become

dxFdUe

The force applied gradually

External Work -Force

Energy Method

' 'eU P

If P is already applied to the bar and that

another force F` is now applied

The work done by P when the bar undergoes

the further deflection ` is then

External Work -Force

If the total angle of rotation is the work become:

0

eU M d

12eU M

If the moment is applied gradually to a structure from zero to M

dMdUe

The work of a moment is defined by the product of

the magnitude of the moment M and the angle dthen:

External Work -Moment

`eU M

If the moment is already applied to the structure and other

loadings further distort the structure by an amount `

Energy Method

Strain Energy – Axial Force

L

A

N

E

AE

NL

AE

LNU

PU

NP

i2

2

21

N = internal normal force in a truss member caused

by the real load

L = length of member

A = cross-sectional area of a member

E = modulus of elasticity of a member

Energy Method

Strain Energy – Bending

MU

dxEI

Md

21

L

i

i

EI

dxMU

EI

dxMdU

0

2

2

2

2

Principle of Work and EnergyExample

22 2 3

0 0

1,

2 2 2 6

L L

e i

PxM P LU P U dx dx

EI EI EI

Consider finding the displacement at the point where the force P is applied to

the cantilever beam in

2 3

3

1 1

2 6

3

e iU U

P LP

EI

PL

EI

Principle of Virtual Work

uPWork of

External

Loads

Work of

Internal

Loads

dLu..1

Virtual Load

Real displacement

Method of Virtual Work: Trusses

External Loading

AE

NLn . .1

dLu..1

1 = external virtual unit load acting on the truss joint in the stated direction of

n = internal virtual normal force in a truss member caused by the external

virtual unit load

= external joint displacement caused by the real load on the truss

N = internal normal force in a truss member caused by the real load

L = length of member

A = cross-sectional area of a member

E = modulus of elasticity of a member

Method of Virtual Work: Trusses

Temperature

1. T Ln

1 = external virtual unit load acting on the truss joint in the stated direction of

n = internal virtual normal force in a truss member caused by the external virtual

unit load

= external joint displacement caused by the temperature change.

= coefficient of thermal expansion of member

T = change in temperature of member

L = length of member

Method of Virtual Work: Trusses

Fabrication Errors and Camber

1. Ln

1 = external virtual unit load acting on the truss joint in the stated direction of

n = internal virtual normal force in a truss member caused by the external virtual

unit load

= external joint displacement caused by fabrication errors

L = difference in length of the member from its intended size as caused by a

fabrication error.

Structural Analysis IDr. Mohammed Arafa

Example 1

The cross sectional area of each member of the truss show, is A = 400mm2

& E = 200GPa.

a) Determine the vertical displacement of joint C if a 4-kN force is applied

to the truss at C

A virtual force of 1 kN is applied at C in the vertical direction

AE

NLn . .1Solution

Member n (KN) N (KN) L (m) nNL

AB 0.667 2 8 10.67

AC -0.833 2.5 5 -10.41

CB -0.833 -2.5 5 10.41

Sum 10.67

mmm

AEAE

nNL

mkNm

kN

133.0 000133.0

1020010400

)67.1067.10 .1

)/(6

)(6

(

22

Structural Analysis IDr. Mohammed Arafa

Text book Example 8-14

Determine vertical displacement at C

A = 0.5 in2

E = 29 (10)3 ksi

Example 2

Example 3

Method of Virtual Work: Beam

dxEI

Md dLu..1

dxEI

Mm

L

0

.1

1 = external virtual unit load acting on the beam in the stated direction of Δ

m = internal virtual moment in a beam caused by the external virtual unit load

Δ = external joint displacement caused by the real load on the truss

M = internal moment in a beam caused by the real load

L = length of beam

I = moment of inertia of cross-sectional

E = modulus of elasticity of the beam

Method of Virtual Work: Beam

dxEI

MmL

mKN 0

).( .1

Similarly the rotation angle at any point on the beam can be

determine, a unit couple moment is applied at the point and the

corresponding internal moment have to be determinem

Example 4

Determine the displacement at point B of a steel beam

E = 200 Gpa , I = 500(106) mm4

Structural Analysis IDr. Mohammed Arafa

Solution

mEI

EI

x

EI

dxx

EI

dxxxdx

EI

Mm

L

15.0)10)(10(500)10(200

)10(15)10(15

4

66)6()1( .1

1266

33

10

0

410

0

310

0

2

0

mB

B

15.0

5.7)10)(10(500)10(200

20001266

Another Solution

Real Load

Virtual Load

Example 5

Example 6

Determine the Slope at point B of a steel beam

E = 200 Gpa , I = 60(106) mm4

Solution

Virtual Load

rad 0094.0)10(60)10(2002

)5(3)10(3

2

3 3)3()1()3()0( .1

66

22

10

5

210

5

10

5

5

00

B

L

EI

x

EI

dxx

EI

dxx

EI

dxxdx

EI

Mm

Real Load

rad

EIEIB

0094.0

1121

112

69 1060)10(200

112

Another Solution

Real Load

Virtual Load

Example 7

Example 8

Example 9

Determine the horizontal deflection at A

Example 4

Solution

Real Load

Virtual Load

m

EIEIEIA

031.0

12505.2

5000

100

69 10200)10(200

1250

Structural Analysis IDr. Mohammed Arafa

Virtual Strain Energy Caused by: Axial Load

n

nNLU

AE

n = internal virtual axial load caused by the external virtual unit load

N = internal normal force in the member caused by the real load

L = length of member

A = cross-sectional area of the member

E = modulus of elasticity of the material

Virtual Strain Energy Caused by: Shear

0

L

S

vVU K dx

GA

/

/

/

/S

dy dx

G

dy G dx

K V A

Vdy K dx

GA

dU vdy vK V A dx

0

L

S

vVU K dx

GA

= internal virtual shear in the member caused by the external virtual unit load

V = internal shear in the member caused by the real load

G= shear modulus of elasticity for the material

A = cross-sectional area of the member

K = form factor for the cross-sectional area

K=1.2 for rectangular cross sections

K=10/9 for circular cross sections

K=1.0 for wide-flange and I-beams where A is the area of the web.

Virtual Strain Energy Caused by: Shear

Virtual Strain Energy Caused by: Torsion

t

tTLU

GJ

/

/

t

cd dx

G

Tc

J

Td dx dx dx

c Gc GJ

tTdU td dx

GJ

Virtual Strain Energy Caused by: Torsion

t= internal virtual torque caused by the external virtual unit load

T= internal shear in the member caused by the real load

G= shear modulus of elasticity for the material

J = polar moment of inertia of the cross-sectional

L = member length

t

tTLU

GJ

Structural Analysis IDr. Mohammed Arafa

Virtual Strain Energy Caused by: Temperature

0

m

L

mtemp

Td dx

c

m TU dx

c

Virtual Strain Energy Caused by: Temperature

0

L

mtemp

m TU dx

c

m= internal virtual moment in the beam caused by the external virtual unit load or

unit moment

= coefficient of thermal expansion

Tm = change in temperature between the mean temperature and the temperature

at the top or the bottom of the beam

c = mid-depth of the beam

Example 10

Example 11