CE 382 L7 - Deflections

8/9/2019 CE 382 L7 - Deflections

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StructuralDisplacements

Beam Displacement

1


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StructuralDisplacements

P

2

Truss Displacements


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-

ing structures under the action ofusual design loads are known to

be small in relation to both the

overall dimensions and member

. y udeflections? Basically, the

that the predicted design loads

will not result in large deflectionsthat may lead to structural failure,

impede serviceability, or result in

distorted structure.

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Several examples which demon-

analysis include (Tartaglione,1991):

1. Wind forces on tall buildings

have been known to produce

excessive lateral deflections that

have resulted in cracked windows

,

the occupants.

. arge oor e ec ons n a

building are aesthetically

,

confidence, may crack brittlefinishes or cause other damage,

4and can be unsafe.


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3. Floor systems are often

designed to support motor-

driven machines or sensitiveequipment that will run satisfac-

undergoes limited deflections.

. arge e ec ons on a ra wayor highway structural support

,

cause passenger discomfort,

and be unsafe.

5.Deflection control and camber

behavior of re-stressed con-

crete beams during variousstages of construction and load-

5

ng are v a or a success u

design.


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6.Deflection computations serve to

dynamic characteristics ofstructures that must withstand

moving loads, vibration, and

shock environment -- inclusive of

.

Elastic Deformations structuree ec ons sappear an e

structure regains its original

the deformations are removed.

structures are referred to asinelastic or plastic deformations.

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This course will focus on linear.

deformations vary linearly with

a lied loads and the rinci le of

superposition is valid for such

structures. Furthermore, since

the deflections are expected to be

small, deflections are measured

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formed or reference geometry.

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Work-Energy Methods

Work-energy methods for truss,beam and frame structures are

considered. Such methods are

based on the principle of

,states that the work done by a

structure (W) equals the strain

energy stored (U) in the structure.

This statement is based on slowly

applied loads that do not produce

kinetic energy, which can bewritten as

8W = U


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A disadvantage of work-energymethods is that only one dis-

rotation can be computed with

each a lication.

Work force (moment) times

displacement (rotation) in the

Differential work of Fig. 1 can be

expresse as

dW = P (d)

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.

Displacement Curves

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ForP = F (force), equals

= F d

ForP = M (moment), equals

=

0

Eqs. (1, 2) indicate that work issimply the area under the force

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sp acemen or momen ro a-

tion) diagrams shown in Fig. 1.


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Linear Elastic Structure

W = 1 F

W =

1

M

Com lementar Work

The area above the load- .

known as complementary work,as shown in Fig 2. For aW

linear-elastic system:112

2


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Fig. 2. Complementary Work

Load complementary

work

W

-

Displacement

calculations is only capable of

calculating displacements at thelocation of an applied point force

and rotations at the point of

obviously a very restrictive

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. ,

work principles are developed in

the subsequent sections.


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Virtual Work

Virtual (virtual imaginary, notreal or in essence but not in fact

work procedures can produce a

single displacement component at

any desired location on the

structure. To calculate the desired

,

load(normally of unit magnitude)

is a lied at the location and in

the direction of the desired

displacement component. Forces

associated with this virtual force

are subscripted with a V.

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Use of a virtual force in calcula-

principle of virtual forces (which

will be the focus of this cha ter :

Principle of Virtual ForcesIf a deformable structure is in

equilibrium under a virtual system

,

done by the virtual forces going

throu h the real dis lacements

equals the internal virtual work

done by the virtual stress

resultants going through the real

displacement differentials.

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Alternatively, if virtual displace-

work is defined as the principle of

virtual dis lacements:

Principle of VirtualDisplacementsIf a deformable structure is in

equilibrium while it is subject to a

virtual distortion the external

virtual work done by the external

forces acting on the structure is

equal to the internal virtual work

done by the stress resultants.

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The virtual work principles (forces

conserving the change in energy

due to the a lied virtual load or

displacement, which can be

expressed mathematically as

=V VW U

for the principle of virtual forces,

which is the focus of this chapter,

complementary energy. The realand virtual com lementar exter-

nal work is shown schematically inFig. 3.

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Pv = virtual force or moment

= real displacement orrotation P + Pv

WP

Fig. 3. Complementary Real

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an r ua or s


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Complementary Axial

Strain EnergyFor a sin le axial force

member subjected to a real

force F, the com lementarstrain energy (internal work)

isF

U2

=

F L =

2F L=

192 E A


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For a single axial force

member in equilibriumsubjected to a virtual force

FV, the virtual complemen-

tary strain energy (virtual

complementary internal

work) is

V VU F= F L

V VE A

=

-

tary strain energies for asingle member are shown

20schematically in Fig. 4.


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Fig. 4. Complementary Real and

Virtual Strain Ener ies for

a Single Truss Member

+

F

vUV

U

For a truss structure:

m mV Vi vi iU U F= =

21i 1 i 1= =


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ViU Complementary Virtual=

Strain Energy for Truss Member i

i Real Member i Deformation =

F Lii iE A

=

loaded truss member

linear coefficient of

thermal expansion

=

22T change in temperature =


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=

ferror of L in the truss

member

Non-mechanical i are positive

change in member length

consistent with tensionpositive forces in trussmembers.

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Example Deflection

Loaded Truss Structure

EA = constant

oc er

Calculate the horizontal displace-

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.


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Example Deflection

Loaded Truss Structure

=EA = constant

= constant

Calculate the horizontal displace-

ment at G if the top chord mem-

bers are subjected to a temper-

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.

Equation of condition at C!


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Complementary BendingStrain Energy

MdU d

2=

Md dx

E I =

1 MU M dx = 26

L


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V VdU M d=

V VMU M dxE I

=

M + Mv

MdUV

dU

Fig. 6. Complementary Real andd

27

a Differential Bending

Segment


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= rea en ng equa on ueto the external applied loading

VM = for a virtual momentVM

for a point displacement

calculation at the desired

point in the assumed direc-

tion of the displacement

= for a virtual moment

equation for a unit virtualVM

couple for a point rotation

calculation at the desired

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direction


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For a multi-segment beam:

iV Vi

MU M dx=

iL

w

segment.

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Beam Deflection Example

and rotation for the cantilever

beam.

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For a frame structure:

jV Vj

j1

U M dxE I

=

=

where m equals the number of

frame members. Note, axial

deformation has beengnore . so, a rame

member is composed of multi-

,

over the segments must also beincluded.

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Frame Deflection Example

alculate the vertical displace-

ment and rotation at C.

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CE 382 L7 - Deflections

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