ce6206 (210)_ lecture#1_rev#2_[2013_0916]
TRANSCRIPT
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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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Methods of Structural Analysis
Course Requirements
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Methods of Structural Analysis
Contacts
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Contacts
Primary contact
Secondary contact
(202) 994-4901
CEE Department
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Methods of Structural Analysis
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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Methods of Structural Analysis
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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Methods of Structural Analysis
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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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
Strain Energy Terms
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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
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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Castiglianos Theorem
Example 1.2
Re-do Example 1.1 using the Virtual Work Method
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Castiglianos Theorem
Example 1.2 (contd)
C i i
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Castiglianos Theorem
Example 1.2 (contd)
C i li Th
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Castiglianos Theorem
Example 1.2 (contd)
C ti li Th
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Castiglianos Theorem
Example 1.2 (contd)
M th d f St t l A l i
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Methods of Structural Analysis
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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Static Indeterminacy
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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Static Indeterminacy
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
Static Indeterminacy
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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 +=
Static Indeterminacy
2D Structures
Static Indeterminacy
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Number of
Joints: (j)
Number of
Members: (m)
Number of
Reactions: (r)
3jmri +=
Eiffel Tower
Static Indeterminacy
3D Structures
Degree of indeterminacy (i) of a 3D
TRUSS is calculated according to
the expression
St ti I d t i
Static Indeterminacy
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Compute the degree of indeterminacy for the truss shown below
Static Indeterminacy
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
Static Indeterminacy
Example 2.1 (Trusses)
Static Indeterminacy
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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
Static Indeterminacy
Example 2.1 (contd)
Static Indeterminacy
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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
Static Indeterminacy
Example 2.1 (contd)
Static Indeterminacy
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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)
Static Indeterminacy
Example 2.1 (contd)
Static Indeterminacy
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P
P
P P
a a a
b
Compute the degree of indeterminacy for the truss shown below
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
Static Indeterminacy
Example 2.2 (Trusses)
Static Indeterminacy
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Compute the degree of indeterminacy for the truss shown below
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
Static Indeterminacy
Example 2.3 (Trusses)
Structure is SISto the firstdegree But is it STABLE?
P/3
Static Indeterminacy
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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)
Static Indeterminacy
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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
Static Indeterminacy
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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
Static Indeterminacy
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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
Step 4: Calculate degree of indeterminacy (i):
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)
Static Indeterminacy
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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
Step 4: Calculate degree of indeterminacy (i):
SDS3-3x5-3x46
b-3j3mri
=+
=+=
j = 5
Step 3: Calculate b
b = 3
& Stable
Internal Hinges
M=0
y
Example 2.5 (2D Frames)
Methods of Structural Analysis
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Methods of Structural Analysis
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.