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STRUCTURAL ANALYSIS BFC23102
Displacement of Statically Determinate Plane Truss
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Introduction
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Chapter 2: Displacement of statically statically determinate determinate plane truss
Work and energy principles
Application for determining displacement of structures
Virtual Work Method/ Unit Load Method
This pedestrian through-truss has been custom designed for a skywalk
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Introduction
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Chapter 2: Displacement of statically statically determinate determinate plane truss
Work and energy principles
Application for determining displacement of structures
Virtual Work Method/ Unit Load Method
This pedestrian through-truss has been custom designed for a skywalk
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Forces and actions - External, External, internal internal - Perma Permanen nent, t, variable - Stat Static, ic, dyna dynamic mic - React eactio ions ns
Introduction
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Strength - Material Material;; elastic elastic & plastic properties
Structural behaviour
- Geomet Geometry; ry; shape shape,, size, Ixx, Zxx
- Deform Deformat ation ion;; deflection, displacement, buckling
- Shear, Shear, moment moment,, stress, strain
- Mode Mode of failu failure re;; cracking, post-failure post-failure
Principles in Structural Analysis 3
Work and Energy Principles
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Greatest use in providing rapid approximate solutions of problems.
Suitable for finding the displacement at points subjected to the load.
Energy is referred to strain energy, U. P
P
Internal work
External work
Work
4
=
Force x displacement
* The internal work is called strain energy. *In a system, energy can only be transferred without deprived
Work and Energy Principles
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Spring Analogy
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Elastic energy is the potential energy of an object that is deformed under stresses (i.e. tension or compression). It arises as a consequence of a force that tries to restore the object to its original shape. The elastic potential energy stored in a stretched spring can be calculated by finding the work necessary to stretch the spring a distance x from its un-stretched length: an ideal spring will follow Hooke’s Law:
The work done (and therefore the stored potential energy) will then be: The units are in joules (J). The equation is often used in calculations of positions of mechanical equilibrium. This equation can also be stated as: 5
Work and Energy Principles
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Strain energy = Internal force Load, P
P1 a ∆1; P1 = k.∆1 Work
=
Work
=
Work
=
Work,w
∫ 0
∆1
∆1
0
∫
P1.d∆
P1
k∆1.d∆
k∆12/2
=
½ P1∆1 d∆
If P1 is located for displacement ∆ 1 Work, w 6
=
P1∆1
∆1
Deflection, ∆
Work and Energy Principles
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From conservation of energy principle External work
=
Internal Work
External work
=
Strain Energy
W
=
U
U
=
½ P∆
As we know, ∆strain = ∆disp/L, σ = P/A and substitute in Hooke’s Law σ = E ∆ Therefore; displacement, ∆ disp
U 7
=
½ P (PL/AE)
=
(PL)/(AE) =
P2L/(2AE)
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Virtual Work
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Principles of Virtual Work When the applied force is proportional to the displacement, a material is said to obey Hooke’s Law. There is a linear relationship between the force and displacement, and thus linear elastic materials obey this law.
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For statically determinate beam, the deflection and slope at specific points can be calculated using the Double Integration Method, Macaulay Method and Moment Area Method (graphical technique).
However, these methods is restricted at the conditions of support, and only suitable for finding the displacement at points subjected to the load.
For others points/locations, modification can be done by Virtual Method or so-called Unit Load Method.
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Virtual Work
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Virtual Work Method can be used to determine the displacement of truss joint under many load cases.
Consider a truss under external loadings:
C A
∆c∆
1
External work = 1.∆ 9
B
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Virtual Work
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If force, F is apply in one member of the truss:
A,E
F F L
dL dL = FL/AE
External work
1.
=
=
10
Internal work
F . L AE F . .l A.E
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Virtual Work
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Steps to find the displacement at joint C are:
Truss: Internal Forces
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For truss statically determinate, the internal (member) forces can be calculated using Joint Method, Method of Section and Alternative Method. Lets revise the Alternative Method that commonly use in virtual work.
Truss: Internal Forces
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Truss: Internal Forces
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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Truss: Displacement
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