Structural Theory II (Part 1/2)

STRUCTURAL
ANALYSIS
II
Shieh-Kung Huang
黃謝恭
1

Shieh-Kung
Huang
Copyright © 2018 by McGraw-Hill Education. All rights reserved.
DEFLECTIONS USING ENERGY METHODS
Chapter Objectives
11
Students will be able to
1. understand the concept of external work and internal
energy
2. understand the concept of and the need for work-energy
methods
3. apply the virtual work method to the trusses, beams, and
frames
4. determine the effect of temperature change, support
settlements, and fabrication errors.
5. use the Bernoulli’s Principle of virtual displacements
5. use the Maxwell-Betti Law of reciprocal deflections
CHAPTER 1
1.1 Work and Energy
1.2 Work-Energy Methods
1.3 Virtual Work: Trusses
1.4 Virtual Work: Beams and Frames
1.5 Bernoulli’s Principle of Virtual Displacements
1.6 Maxwell-Betti Law of Reciprocal Deflections
1.7 Castigliano’s Theorem
1.8 Crotti-Engesser Theorem
Chapter Outline

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1.1 WORK AND ENERGY
External Work
12
Chapter 1 Deflections using Energy Methods
Work is defined as the product of a external force times a displacement in the direction of the force. If
a force F and a moment M remain constant in magnitude as it moves from two points, the work W may
expressed as
If a force varies in magnitude during a displacement and if the functional relationship between the
force F and the collinear displacement d is known, the work can be evaluated by integration, so as
moment.
W F M
d 
= +
0 0
W F d M d
d 
d 
= +
 

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1.1 WORK AND ENERGY
Internal Energy
13
Similar to work, (strain) energy is defined as the product of a internal force times a deformation in the
direction of the force.
If a force F and a moment M remain constant in magnitude as it deforms a structure, the strain
energy U may be expressed as
If a force varies in magnitude during a displacement and if the functional relationship between the
force F and the collinear deformation L is known, the strain energy can be evaluated by integration, so as
moment.
U F L M 
=  + 
0 0
L
U F dL M d


= +
 

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1.2 WORK-ENERGY METHODS
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To establish an equation for computing the deflection of a point on a structure by the work-energy
method, we can write according to the principle of conservation of energy that
where W is the work done by the external force applied to the structure and U is the strain energy stored
in the stressed members of the structure.
This equation assumes that all work done by an external force is converted to strain energy. To
satisfy this requirement, a load theoretically must be applied slowly so that neither kinetic nor heat energy
is produced. In the design of buildings and bridges for normal design loads, we will always assume that
this condition is satisfied so that the equation is valid.
Finally, the above method—also called the method of real work—can be used compute the deflection
of a component. Since the method of real work has serious limitations (i.e., deflections can be computed
only at a point where a force acts and only a single concentrated load can be applied to the structure), the
major emphasis in this chapter will be placed on the method of virtual work.
Virtual work, one of the most useful, versatile methods of computing deflections, is applicable to
many types of structural members from simple beams and trusses to complex plates and shells. Although
virtual work can be applied to structures that behave either elastically or inelastically, the method does
require that changes in geometry be small (the method could not be applied to a cable that undergoes a
large change in geometry by application of a concentrated load). As an additional advantage, virtual work
permits the designer to include in deflection computations the influence of support settlements,
temperature changes, creep, and fabrication errors.
W U
=

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1.2 WORK-ENERGY METHODS
15

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1.3 VIRTUAL WORK: TRUSSES
Virtual Work Method
16
To compute a component of deflection by the method of virtual work, the designer applies a force to
the structure at the point and in the direction of the desired displacement. This force is often called a
dummy load because like a ventriloquist’s dummy (or puppet), the displacement it will undergo is
produced by other effects. These other effects include the real loads, temperature change, support
settlements, and so forth. The dummy load and the reactions and internal forces it creates are termed a
Q-system. Forces, work, displacements, or energy associated with the Q-system will be subscripted with
a Q. Although the analyst is free to assign any arbitrary value to a dummy load, typically we use a 1-kip or
a 1-kN force to compute a linear displacement and a 1 kip·ft or a 1 kN·m moment to determine a rotation
or slope.
With the dummy load in place, the actual loads—called the P-system, are applied to the structure.
Forces, deformations, work, and energy associated with the P-system will be subscripted with a P. As the
structure deforms under the actual loads, external virtual work WQ is done by the dummy load (or loads)
as it moves through the real displacement of the structure. In accordance with the principle of
conservation of energy, an equivalent quantity of virtual strain energy UQ is stored in the structure:
The virtual strain energy stored in the structure equals the product of the internal forces produced by
the dummy load and the distortions (changes in length of axially loaded bars, for example) of the elements
of the structure produced by the real loads (i.e., the P-system).
Q Q
W U
=

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Analysis of Trusses by Virtual Work
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1 1
2 2
1 1
2 2
Q
P
P Q
P P D Q
P P P D Q Q
F L
F L
L L
AE AE
W P W Q
U F L U F L
d d
 =  =
= =
=  = 
Dummy
Load
Actually
Load

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or
Q Q
Q P Q Q P
P Q P P Q P
P
P Q
W U
W Q U F L
Q F L Q F L
F L
Q F
AE
d
d d
d
=
= = 

=  = 

=
 
 
( )( ) ( )( )
=
 
虛外力實外變形虛內力實內變形
When Q = 1 is specified, we often refer it as unit load method.

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Truss Deflections Produced by Temperature and Fabrication Error
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• Temperature
As the temperature of a member varies, its length changes. An increase in temperature causes a
member to expand; a decrease in temperature produces a contraction. In either case the change in
length ΔLtemp can be expressed as
where a is coefficient of thermal expansion, in./in. per degree
ΔT is change in temperature
L is length of bar
• Fabrication Error
A change in bar length ΔLfabr due to a fabrication error is handled in exactly the same manner as a
temperature change. Example 8.4 illustrates the computation of a component of truss displacement for
both a temperature change and a fabrication error.
• Truss Deflections Produced by Temperature and Fabrication Error
If the bars of a truss change in length simultaneously due to load, temperature change, and a
fabrication error, then ΔLP is equal to the sum of the various effects; that is,
and the virtual work equation for trusses becomes
temp
L T L
a
 = 
P
P fabr
F L
L T L L
AE
a
 = +  + 
P
P Q fabr
F L
Q F T L L
AE
d a
 
= +  + 
 
 
 

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Computation of Displacements Produced by Support Settlements
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• Support Settlements
Structures founded on compressible soils (soft clays or loose sand, for example) often undergo
significant settlements. These settlements can produce rotation of members and displacement of
joints. If a structure is determinate, no internal stresses are created by a support movement because
the structure is free to adjust to the new position of the supports. On the other hand, differential
support settlements can induce large internal forces in indeterminate structures. The magnitude of
these forces is a function of the member’s stiffness.
• Truss Deflections Produced by Support Settlements
If the bars of a truss change in length simultaneously due to load and other effects, then ΔLP is
equal to the sum of the various effects; that is,
and the virtual work equation for trusses becomes
P
P fabr
F L
L T L L
AE
a
 = +  + 
x x y
P
P fabr
y Q
F L
Q F T L L
AE
r r
d a
 
= +  + 
 

+  + 


  

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1.4 VIRTUAL WORK: BEAMS AND FRAMES
37
Both shear and moment contribute to the deformations of beams. However, because the
deformations produced by shear forces in beams of normal proportions are small (typically, less than 1
percent of the flexural deformations), we will first neglect them (the standard practice of designers) and
consider only deformations produced by moment.
If a beam is deep (the ratio of span to depth is on the order of 2 or 3), or if a beam web is thin or
constructed from a material (wood, for example) with a low shear modulus, shear deformations may be
significant and should be investigated.
P Q Q Q Q
dx
d M dU M d U M d
EI
  
=  =  = 
0
x L Q P
P x
M M
Q dx
EI
d
=
=
=
 

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As an alternate procedure to evaluate the integral on the right-hand side of virtual work equation for a
variety of MQ and MP diagrams of simple geometric shapes and for members with a constant value of EI, a
graphical method entitled “Values of Product Integrals” is provided in the Appendix Table A.2 of the text.
For example, if both MQ and MP vary linearly within the span and EI is constant, then the integral can be
expressed as follows:
where C is constant listed in product integrals table
where M1 is magnitude of MQ
where M3 is magnitude of MP
where L is length of member
( )
1 2
0
1
x L Q P
Q x
M M
U dx CM M L
EI EI
=
=
= =


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If other forces are included, the deformations and strain energy have a very similar formulation, as
shown in the following.
Q P
P Q Q Q Q Q
Q P
P Q Q Q Q Q
Q P
P Q Q Q Q Q
Q P
P Q Q Q Q Q
N N
dx
dL N dU N dL U N dL U dx
EA EA
M M
dx
d M dU M d U M d U dx
EI EI
V V
dx
dy V dU V dy U V dy U dx
GA GA
T T
dx
d T dU T d U T d U dx
GJ GJ
  
  
=  =  =  =
=  =  =  =
=  =  =  =
=  =  =  =
 
 
 
 

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?

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1.5BERNOULLI’SPRINCIPLEOFVIRTUALDISPLACEMENTS
58
Bernoulli’s principle of virtual displacements, a basic structural theorem, is a variation of the principle
of virtual work. The principle is used in theoretical derivations and can also be used to compute the
deflection of points on a determinate structure that undergoes rigid body movement, for example, a
support settlement or a fabrication error. Bernoulli’s principle, which seems almost self-evident once it is
stated, says:
If a rigid body, loaded by a system of forces in equilibrium, is given a small virtual displacement by an
outside effect, the virtual work WQ done by the force system equals zero.
In this statement a virtual displacement is a real or hypothetical displacement produced by an action
that is separate from the force system acting on the structure. Also, a virtual displacement must be
sufficiently small that the geometry and magnitude of the original force system do not change significantly
as the structure is displaced from its initial to its final position. Since the body is rigid, .
0
Q
U =
0 0
Q Q P M P
W U Q Q
d 
= =  + =
 

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1.6 MAXWELL-BETTI LAW OF RECIPROCAL DEFLECTIONS
62
Using the method of real work, we can derive the Maxwell-Betti law of reciprocal deflections, a basic
structural theorem.
The Maxwell-Betti law, which applies to any stable elastic structure (a beam, truss, or frame, for
example) on unyielding supports and at constant temperature, states:
A linear deflection component at a point A in direction 1 produced by the application of a unit load at a
second point B in direction 2 is equal in magnitude to the linear deflection component at point B in
direction 2 produced by a unit load applied at A in direction 1.

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• Case 1. FB Applied Followed by FA
• Case 2. FA Applied Followed by FB
total
1
2
1
2
1 1
2 2
B B BB
A A AA B BA
B A B BB A AA B BA
W F
W F F
W W W F F F
= 
=  + 
= + =  +  +  total
1
2
1
2
1 1
2 2
A A AA
B B BB A AB
A B A AA B BB A AB
W F
W F F
W W W F F F
 = 
 =  + 
  
= + =  +  + 
Equating the total work of cases 1 and 2 and simplifying give
The Maxwell-Betti theorem also holds for rotations as well as rotations and linear
displacements, so we can also state the Maxwell-Betti law as follows:
The rotation at point A in direction 1 due to a unit couple at B in direction 2 is equal to the rotation at B
in direction 2 due to a unit couple at A in direction 1.
total total
when 1
B BA A AB BA AB A B
W W F F F F

=   =    =  = =

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As a third variation of the Maxwell-Betti law, we can also state:
Any linear component of deflection at a point A in direction 1 produced by a unit moment at B in
direction 2 is equal in magnitude to the rotation at B (in radians) in direction 2 due to a unit load at A in
direction 1.
In its most general form, the Maxwell-Betti law can also be applied to a structure that is supported in
two different ways. The previous applications of this law are subsets of the following theorem:
Given a stable linear elastic structure on which arbitrary points have been selected, forces or
moments may be acting at some of or all these points in either of two different loading systems. The
virtual work done by the forces of the first system acting through the displacements of the second system
is equal to the virtual work done by the forces of the second system acting through the corresponding
displacements of the first system. If a support displaces in either system, the work associated with the
reaction in the other system must be included. Moreover, internal forces at a given section may be
included in either system by imagining that the restraint corresponding to the forces is removed from the
structure but the internal forces are applied as external loads to each side of the section.
1 2 2 1
F F
d d
=
 

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1 2 2 1
(8.41)
F F
d d
=
 

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1.7 CASTIGLIANO’S THEOREM
67
In 1879 Alberto Castigliano, an Italian railroad engineer, published a book in which he outlined a
method for determining the deflection or slope at a point in a structure, be it a truss, beam, or frame. This
method, which is referred to as Castigliano’s second theorem, or the method of least work, applies only to
structures that have constant temperature, unyielding supports, and linear elastic material response.
Since the external work done by these loads is equal to the internal strain energy stored in the body,
we can write
It should be noted that this equation is a statement regarding the structure’s
compatibility. Also, the above derivation requires that only conservative forces be
considered for the analysis. These forces do work that is independent of the path and therefore create no
energy loss. Since forces causing a linear elastic response are conservative, the theorem is restricted to
linear elastic behavior of the material. This is unlike the method of virtual force discussed in the previous
section, which applied to both elastic and inelastic behavior.
1 2
and ( , , , )
and
n
i i i
i
i
i
W U W f P P P
U
U dU U dP U dU U dP
P
U
P
= =

+ = + + = + 


  =


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• Castigliano’s Theorem for Trusses
where ΔL is external joint displacement of the truss
P is external force applied to the truss joint in the direction of ΔL
N is internal force in a member caused by both the force P and the loads on the truss
L is length of a member
A is cross-sectional area of a member
E is modulus of elasticity of a member.
• Castigliano’s Theorem for Beams and Frames
where Δθ is external displacement of the point caused by the real loads acting on the beam or frame
P is external force applied to the beam or frame in the direction of Δθ
M is internal moment in the beam or frame, expressed as a function of x and caused by both
the force P and the real loads on the beam
E is modulus of elasticity of beam material
I is moment of inertia of cross-sectional area computed about the neutral axis.
• What is Castigliano’s first theorem?
2 2
2
N N L NL N
U dx U L
EA EA EA P

=  =   =


2
or
M U M M
U dx dx
EI P EI P
 
 
 
=   =  =  
 
 
 
i i
i i
U U
P
P
 
=   =
 

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U*
1.8 CROTTI-ENGESSER THEOREM
77
The Crotti-Engesser theorem states that the first partial derivative of the complementary strain energy
(U*) expressed in terms of applied forces is equal to the corresponding displacement.
Now, we can derive similar equations with Castigliano’s second theorem
• Why Castigliano’s second theorem is restricted to linear elastic behavior of the material?
The following equation works if and only if the material is linear-elastic
Not Linear-elastic Linear-elastic
Energy
Castigliano’s 1st Theorem Castigliano’s 2nd Theorem
Virtual Work Theorem
Complementary Energy Crotti-Engesser Theorem
*
0 0 0 0
*
0 0 0 0
F M
F M L
W dF dM W F d M d
U LdF dM U F dL M d
d 

d  d 
 
 
= + = +
 

 
= + = +
 
 
   
   
* * *
1 2
*
* * * * * *
*
and ( , , , )
and
n
i i i
i
i
i
W U W f P P P
U
U dU U dP U dU U dP
P
U
P
= =

+ = + + = + 


  =

*
U U
=

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1.8 CROTTI-ENGESSER THEOREM
Example for Crotti-Engesser Theorem
78
The cross-sectional area of a member is A and the relationship between stress and
strain is . What is the horizontal displacement at joint B?
E
 
=
* * *
3
2
3
2 2 2
3
2 2
* * *
3
2
3
2 2 2
3
2 2
2 2
2 2 2
2
3 3
4
3
3 3
3
AB
AB
BC
BC
U dU AL du AL d
AL AL P
AL d
E E E A
P L
E A
U dU AL du AL d
AL AL P
AL d
E E E A
P L
E A
 

 
 

 
= = =
 
= = =  
 
=
= = =
−
 
= = =  

−

=
  

  

P
−
2P
3 3 3
* * *
total 2 2 2 2 2 2
* 2
total
2 2
4 5
3 3 3
5
AB BC
Bx
P L P L P L
U U U
E A E A E A
U P L
P E A
d
= + = + =

= =


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ANALYSIS OF STRUCTURES USING FLEXIBILITY METHOD
Chapter Objectives
79
Students will be able to
1. know that the additional equations are needed to analyze
indeterminate structures.
2. learn in the flexibility method to establish additional
equations by using the redundants.
3. identify redundants and then use any method to establish
the compatibility equations.
CHAPTER 2
2.1 Concept of a Redundant
2.2 Fundamentals of the Flexibility Method
2.3 Alternative View of the Flexibility Method
2.4 Analysis Using Internal Releases
2.5 Support Settlements, Temperature Change,
and Fabrication Errors
2.6 Analysis of Structures with Several Degrees
of Indeterminacy
2.7 Beam on Elastic Supports
2.8 Concept of Flexibility Method and Stiffness
Method
Chapter Outline

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2.1 CONCEPT OF A REDUNDANT
80
We have seen in “Structural Analysis I” that a minimum of three restraints, which are not equivalent to
either a parallel or a concurrent force system, are required to produce a stable structure. Since three
equations of equilibrium are available to determine the three reactions, the structure is statically
determinate.
If a third support is constructed at B, an additional reaction RB is available to support the beam. Since
the reaction at B is not absolutely essential for the stability of the structure, it is termed a redundant. In
many structures the designation of a particular reaction as a redundant is arbitrary.
Although the addition of the roller at B produces a structure that is indeterminate to the first degree
(four reactions exist but only three equations of statics are available), the roller also imposes the
geometric requirement that the vertical displacement at B be zero.
Chapter 2 Analysis of Structures using Flexibility Method

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2.2 FUNDAMENTALS OF THE FLEXIBILITY METHOD
Compatibility
81
In the flexibility method, one imagines that sufficient redundants (supports, for example) are removed
from an indeterminate structure to produce a stable, determinate released structure.
We next analyze the determinate released structure for the applied loads and redundants. In this step
the analysis is divided into separate cases for (1) the applied loads and (2) for each unknown redundant.
For each case, deflections are computed at each point where a redundant acts.
Since the structure is assumed to behave elastically, these individual analyses can be combined—
superimposed—to produce an analysis that includes the effect of all forces and redundants. This
procedure produces a set of compatibility equations equal in number to the redundants.
The deflection produced by the unit value of the redundant is called a flexibility coefficient. In other
words, the units of a flexibility coefficient are in distance per unit load.

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Flexibility Coefficients
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Flexibility Coefficients
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2.3 ALTERNATIVE VIEW OF THE FLEXIBILITY METHOD
84
In certain types of problems—particularly those in which we
make internal releases to establish the released structure—it may be
easier for the student to set up the compatibility equation
(or equations when several
redundants are involved) by
considering that the redundant
represents the force needed to
close a gap.
For the support settlement (e) or
the fabrication error (f)
0
0
B BB B
X
d
 + =
0
0
( ) 2
( ) 3
B BB B
B BB B
e X
f X
d
d
  + = −
  + = −

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2.4 ANALYSIS USING INTERNAL RELEASES
92
We will now extend the flexibility method to a group of structures in which the released structure is
established by removing an internal restraint.
For this condition, redundants are taken as pairs of internal forces, and the compatibility equation is
based on the geometric condition that no relative displacement (i.e., no gap) occurs between the ends of
the section on which the redundants act.
0
0 1 2
0
( ) 0
B BB
B
T
T
d
d d
 + =
  + + =

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2.5 SUPPORT SETTLEMENTS, TEMPERATURE CHANGE,AND
Support Settlements Correspond to a Redundant
101
The effects of support settlements, fabrication errors, and so forth can easily be included in the
flexibility method by modifying certain terms of the compatibility equations.
• Support Settlements Correspond to a Redundant
If a predetermined support movement occurs that corresponds to a redundant, the compatibility
equation (normally set equal to zero for the case of no support settlements) is simply set equal to the
value of the support movement.
FABRICATION ERRORS
0
B BB B B
X
d
 + = 

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Support Settlements Do Not Correspond to a Redundant
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• Support Settlements Do Not Correspond to a Redundant
If a support movement occurs that does not correspond to a redundant, its effect can be included
as part of the analysis of the released structure for the applied loads.
FABRICATION ERRORS
0
( )
B BS BB B B
X
d
 +  + = 
0”

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FABRICATION ERRORS

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FABRICATION ERRORS

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2.6ANALYSIS OF STRUCTURES WITH SEVERAL DEGREES OF
105
The analysis of a structure that is indeterminate to more than one degree follows the same format as
that for a structure with a single degree of indeterminacy.
A small saving in computational effort can be realized by using the Maxwell-Betti law (Section 1.6),
which requires that .
INDETERMINACY
0
0
0
0
B B BB B BC C
C C CB B CC C
X X
X X
d d
d d
 = =  + +


 = =  + +

CB BC
d d
=

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INDETERMINACY

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INDETERMINACY

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INDETERMINACY

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INDETERMINACY

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INDETERMINACY

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2.7 BEAM ON ELASTIC SUPPORTS
111
The supports of certain structures deform when they are loaded. The procedure to analyze a beam
on an elastic support is similar to that for a beam on an unyielding support, with one difference. If the force
X in the spring is taken as the redundant, the compatibility equation must state that the deflection Δ of the
beam at the location of the redundant equals
X
K
 = −

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2.8 CONCEPT OF FLEXIBILITYMETHODAND STIFFNESS
114
When analyzing any indeterminate structure, it is necessary to satisfy equilibrium, compatibility, and
force-displacement (constitutive law) requirements for the structure.
In general there are two different ways to satisfy these three requirements. For a statically
indeterminate structure, they are the force or flexibility method, and the displacement or stiffness method.
METHOD
Equilibrium
Eqs.
Compatibility
Eqs.
Constitutive Law
Force
Stress
Displacement
Strain
Rigid Body
Elastic Body

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Thanks for your attention!
See you next week!

Structural Theory II (Part 1/2)

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Structural Theory II (Part 1/2)