Structure Analysis I Chapter 9, Slides of Acting

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Structure Analysis IStructure Analysis I

Chapter 9Chapter 9

DeflectionDeflection

Energy MethodEnergy

Method

Energy MethodEnergy

Method

External Work

If P is already applied to the bar and that

h

f

F` i

li d

External

Work

another force F` is now appliedThe work done by P when the bar undergoesthe further deflection

` is then

the further deflection

is then

'

U

P

=

∆

e

U

P

∆

The work of a moment is defined by theproduct of the magnitude of the moment

M

product of the magnitude of the moment

M

and the angle

then

θ

d

If

h

l

l

f

i

i

h

k

θ

d

M

dU

e

θ

If

the total angle of rotation is

the work

become:

θ

θ

θ

d

M

U

e

0

∫

θ

M

U

e

1 2

The moment applied gradually

Energy MethodEnergy

Method

Strain Energy

Bending

Strain Energy – Bending

M

θ

θ

M

U

dx

M EI

d

1 2

=

i

EI

dx

M

dU

2

2

∫

=

L

i

EI

dx

M

U

0

2

2

0

Principle of Virtual WorkPrinciple

of Virtual Work

∑

∑

∆

δ

P

∑

∑

=

∆

δ

u

P

Work ofE t rn l

Work ofInt rn l

ExternalLoads

InternalLoads

Virtual Load

∑

=

∆

dL

u

1

∑

=

∆

dL

u

.

.

1

Real displacementReal

displacement

Method of Virtual Work: Trusses

Temperature

T L

n

α

∑∑

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.

α

=

coefficient of thermal expansion of member

∆

T

=

change in temperature of member

∆

T

change

in temperature of member

L =

length of member

Method of Virtual Work: Trusses

Fabrication Errors and Camber

L

n

∑∑

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.fabrication error.

∑

NL

n

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

∑

AE

n

pp

Member

n (KN)

N (KN)

L (m)

nNL

Member

n

(KN)

N

(KN)

L

(m)

nNL

AB

0.

2

8

10.

AC

‐

0.

2.

5

‐

10.

CB

‐

0.

‐

2.

5

10.

Sum

10.

AE

AE

nNL

kN

10

200

10

400

)

67

.

10

67

.

10

.

1

6

6

(

=

=

=

∆

−

∑

mm

m

AE

AE

m

kN

m

133 .

0

000133

.

0

10

200

10

400

) / ( 6 ) ( 6

2

2

=

=

∆

×

×

×

∑

Method of Virtual Work: BeamMethod

of Virtual Work: Beam

M

∑

∆

dL

u

1

dx

M EI

d

θ

∑

=

∆

dL

u

.

.

1

dx

M

m

L ∫

=

∆ .

1

dx

EI

m

∫

∆

0

.

1

1

=

t

l

i t

l

it l

d

ti

th

b

i

th

t t d di

ti

f

∆

1

=

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

∆

m =

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

∆

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

l

b

d b

h

l l

d

M =

i

nternal 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