CE6206 (210)_ Lecture#1_rev#2_[2013_0916]

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Methods of Structural Analysis

CE 6202-10 (210)

Professor

Amir A. Arab, P.E., Ph.D.

Fall 2013

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Course Requirements

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Contacts

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Contacts

Primary contact

[email protected]

Secondary contact

(202) 994-4901

CEE Department

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Grading

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Grading

Final grade will be assigned based on the following percentages.

10 % Homework (8)

20 % Exam 1 25 % Exam 2

15 % Design Project (1)

30 % Final Exam (TBD)

Grading in percentages: A straight scale, thus, No Curving, will be used

with grades based on:

Score (%) 90 88 85 80 78 75 70 68 < 60

Grade A A (-) B (+) B B -() C (+) C C (-) F

1. No make-up work/exam is allowed unless with proper documented

justification

2. Any students with certified disabilities should contact me as early as

possible to accommodate special arrangements.

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Class Policies

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Class Policies

All homework must be submitted onEngineering Paper

Submittals must be neat & clear; unclear submittals will not be graded

Maximum one problem per page

Graphic representations shall be done using Straight Edgeor CAD

All drawings shall be to scale

Academic Dishonesty

At any time will academic dishonesty be tolerated in this class.

Academic honesty is central to the mission of GWU

http://www.gwu.edu/~ntegrity/code.html#repeal

Attendance

Attendance is mandatory.

More than 2 (10% of classes) unexcused absences will earn an

automatic grade ofF
http://www.gwu.edu/~ntegrity/code.htmlhttp://www.gwu.edu/~ntegrity/code.html

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Energy Based MethodsSection 1

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Castiglianos Theorem

All energy methods are based on the Conservation of Energy Principle

work done by externalforces internalwork or strain energy

The external forces perform work and the energy is stored in the structure

in the form of stress and elastic deformation.

External Work

ie UU =

=0

dxFUe

Given an axially loaded bar with linear

elastic response, deflected gradually

from 0to due to force P:

= PUe2

1

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Summing the effects on all elements dxalong the beam:

Similarly

Method of Virtual Work Beams

dxEI

mML

=0

.1

dx

EI

MmL

=

0

.1

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Virtual unit load is applied at PointAin the direction of deflection

Internal virtual moment mis determined

We know that

The internal virtual work by moment mis ...

Thus ...

Method of Virtual Work Beams

dxEI

Md =

( )dxEI

Mmmd =

dxEI

mML

=0

.1

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OrMethod of Least Work

To determine deflection or slope at a point in a structure

Applicable only to structures with

Constant temperature

Unyielding supports

Linear elastic material response

Castiglianos Second Theorem

niPfUU iei

1)( ===

i

i

iiii dP

P

UUdUU

+=+

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Given

Then

where denotes the additional strain energy caused by the

incremental load .

Therefore,

Castiglianos Second Theorem

ie UU =

iiiii dPUdUU +=+

iidPidP

i

i

iiiiiii dP

P

UUdPUdUU

+=+=+

i

iiP

U

=

Displacement in the direction of is equal to the first partial

derivative of the strain energy with respect toi iP

iP

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Strain Energy Terms

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Procedure

To determine a deflection yiin the direction of a real or fictitious force Ff

1. Obtain an expression for the total strain energy including the loads F,M,T,Vand a fictitious force if required

2. Obtain the linear deflection y ffrom the relationship y f= U / F f

3. If the force is fictitious set F f= 0 and solve the resulting equation

To determine an angular deflection fin the direction of a real or fictitious

moment M f

1. Obtain an expression for the total strain energy including the loads F,M,T,V

and a fictitious moment if required.

2. Obtain the angular deflection from the relationship f= U /M i

3. If the moment is fictitious set M f= 0 and solve the resulting equation

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Example 1.1

Consider a simply supported beam with a central load F. Determine the

deflection at the central load point.

L= 2m

b=0.1m

h= 0.05m

F=10 ,000NE=206 Gpa

G = 78,610

I = 4,17.10-6 m4

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Example 1.1 (contd)

The given beam is a rectangle with the dimensions of width b and depth h.

The strain energy for bending and for traverse shear is included in theconsideration.

Because the beam is symmetrical the deflection at the central point is

obtained by doubling the solution from 0 to l/2.

Assume: x = the distance from the left hand support.

We know that

Moment M = (F/2).x

Transverse Shear Force V = F/2

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Example 1.1 (contd)

1) The expression for the total strain energy =

2) From Castigliano's theorem the deflection of the Force F in the direction of

F is : y f= U / F f

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Virtual Work Method

( ) ( )f s g s ds M M ds

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( ). ( ). . .i ka a

f s g s ds M M ds

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Example 1.2

Re-do Example 1.1 using the Virtual Work Method

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Example 1.2 (contd)

C i i

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Example 1.2 (contd)

C i li Th

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Example 1.2 (contd)

C ti li Th

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Example 1.2 (contd)

M th d f St t l A l i

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Static IndeterminacySection 2

St ti I d t i

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Static Indeterminacy

Definition

Number of reactions > Number of equations of equilibrium

Works withForces

Additional equations are required Compatibility Equations

2D Structures

EQUATIONS OF EQUILIBRIUM

=

=

=

0

0

0

Z

Y

X

M

F

F

X

Y

Z

=

=

=

=

=

=

0

0

0

0

0

0

Z

Y

X

Z

Y

X

M

M

M

F

F

F

3D Structures

St ti I d t i

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External

The number of reactions

exceed the number of

equations of equilibrium

Internal

Reactions can be evaluated

based on the equations of

equilibrium. But the member

forces cannot.

Or both

Different Cases of Static Indeterminacy

St ti I d t i

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Trussed Structures

Members can only resist axial deformations or loads

Members have no resistance to bending moments or shear forces Joints must be considered at any point where load is applied

Correct Wrong

Can be 2D or 3D

In order to determine the degree of indeterminacy one must break down the

structure into external reactions (r), number of members (m), and number

of joints (j)

Stability of a structure is evaluated by drawing line of actions from

reactions. If all line of actions intersects at a single point structure is

unstable


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Number of

Joints: (j)

Number of

Members: (m)

Number ofReactions: (r)

Eiffel Tower

Degree of indeterminacy (i) of a 2D

TRUSS is calculated according to

the expression

2jmri +=


2D Structures


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Number of

Joints: (j)

Number of

Members: (m)

Number of

Reactions: (r)

3jmri +=

Eiffel Tower


3D Structures

Degree of indeterminacy (i) of a 3D

TRUSS is calculated according to

the expression

St ti I d t i


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Compute the degree of indeterminacy for the truss shown below


Solution:

Step 1: Calculate external reactions on the structure (r):

r = 4

Step 2: Calculate number of members between joints (m)

4

6

7 9

8 11

141210

13

531

2

m = 14


Example 2.1 (Trusses)


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Step 3: Calculate number of joints (j):

4 6

7 9

8

531

2

j = 9

Step 4: Calculate degree of indeterminacy (i):

09x2-1442jmri =+=+=

r = 4 m = 14 j = 9

Solution:Structure is statically determinate (SDS)

But is it Externally and Internally STABLE ?

EXTERNALLY

STABLE


Example 2.1 (contd)


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-6y

+6x -6x

+6y

1. Structure is statically determinate (SDS)2. Structure is Externally STABLE

But is it Internally STABLE ?

3. Structure is Internally STABLE also


Example 2.1 (contd)


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Another method to determine if the structure is internally and externally stable:

a. Internally STABLE ?

Structure is Externally STABLE

iint= 3 + m 2 j = 3 + 14 2 x 9 = -1

b. Externally STABLE ?

iext= i iint = 0 (-1) = 1

Cannot tell

1. Structure is statically determinate (SDS)

2. Structure is Externally STABLE

3. Structure is Internally STABLE (by recognition)


Example 2.1 (contd)


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P

P

P P

a a a

b


P

P

Pa/b

Pa/ba

b = 0FY

Structure is SISto the firstdegree But is it STABLE?

?

0bb

PaPaMZ

=+=

Structure is also STABLEBut is it Load Dependent?

= 0MZ?

i = r + m 2 j = 3 + 14 2 x 8 = 1




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3P1

3P2PF

Y +=+=

Structure is NOT StableAND is Load Dependent

= 0FY

? = 0MZ?

P2/3

P

P/3

i = r + m 2 j = 3 + 14 2 x 8 = 1



Structure is SISto the firstdegree But is it STABLE?

P/3


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Another method to estimate if the structure is

internally and externally stable:

a. Internally STABLE ?

iint= 3 + m 2 j = 3 + 14 2 x 8 = 1

b. Externally STABLE ?

iext= i iint = 1 1 = 0

1. Structure is statically determinate (SDS)

2. Structure is Externally STABLE

3. But it is internally unstable / load dependent

Structure cannot carry shear forces within the central bay

Externally stable

S a a y

Example 2.3 (contd)


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Frame Definition:

Members can resist axial deformations or loads

Members __________ bending moments or shear forces

Joints do _____ need to be considered at every point where load is applied

Can be 1D --- Beams

Can Be 2D --- 2D Frames

Can be 3D --- 3D Frames

In order to determine the degree of indeterminacy one must break down the

structure into external reactions (r), number of members (m), number of joints

(j), and number of actions released between connecting members (b)

Correct Correct

CAN RESIST

y

Framed Structures

NOT


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Kowloon Station

Development by LERA

Degree of indeterminacy (i) of a 2D Frames

or Beams is calculated according to the

expression:

b-6j6mri +=

Degree of indeterminacy (i) of a 2D Frames

or Beams is calculated according to theexpression

b-3j3mri +=

y

2D Framed Structures

3D Framed Structures


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Compute the degree of indeterminacy for the continuous beam shown below

Solution:

Step 1: Calculate external reactions on the structure (r):

r = 7

Step 2: Calculate number of members (m) and joints (j)

m = 3


i r 3m 3j-b 7 3x3-3x4-1 3 SIS= + = + =

j = 4

Step 3: Calculate number of internal actions released (b)

b = 1

& Stable

Internal Hinge M=0

y

Example 2.4 (Beams)


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Compute the degree of indeterminacy for the frame shown below

Solution:

Step 1: Calculate r

r = 6

Step 2: Calculate m & j

m = 4


SDS3-3x5-3x46

b-3j3mri

=+

=+=

j = 5

Step 3: Calculate b

b = 3

& Stable

Internal Hinges

M=0

y

Example 2.5 (2D Frames)


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Kinematic Indeterminacy

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Kinematic Indeterminacy: Trusses

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Where

Number of joints

Number of restrained joint displacement

Degree of Kinematic Indeterminacy,K.I.(d)

2j rd n n=

j

r

n

n

=

=

Kinematic Indeterminacy:Beams

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Compute the degree of kinematic indeterminacy for the continuous

beam shown below

Solution:

Step 1: Draw external reactions on the structure

Step 2: Draw deformations on the structure at every member

end

Internal Hinge

Step 3:

Example 2.6 (Beams)

X

Z

ZZ Y

XZ

3j - r + b=3 5-7+1=9d=

Staged Construction

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HOOVER DAM

A structure may undergo various stages of indeterminacy during staged

construction. If so, stability check may be required at controlling stages.

CE6206 (210)_ Lecture#1_rev#2_[2013_0916]

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