lecture #11 matrix methods. methods to solve indeterminate problem 2 displacement methods force...
TRANSCRIPT
![Page 1: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/1.jpg)
Lecture #11Matrix methods
![Page 2: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/2.jpg)
METHODS TO SOLVE INDETERMINATE PROBLEM
2
Displacement methods
Force method
Small degreeof statical
indeterminacy
Large degreeof statical
indeterminacy
Displacement methodin matrix formulation
Numerical methods
![Page 3: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/3.jpg)
Disadvantages:• bulky calculations (not for hand calculations);• structural members should have some certain number of unknown nodal forces and nodal displacements; for complex members such as curved beams and arbitrary solids this requires some discretization, so no analytical solution is possible.
ADVANTAGES AND DISADVANTAGES OF MATRIX METHODS
3
Advantages:• very formalized and computer-friendly;• versatile, suitable for large problems;• applicable for both statically determinate and indeterminate problems.
![Page 4: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/4.jpg)
FLOWCHART OF MATRIX METHOD
4
Classificationof members
Stiffness matrices for members
Transformed stiffness matrices
Stiffness matrices are composed according to
member models
Stiffness matrices are transformed from local to global
coordinates
Final equationF = K · Z
Stress-strain state of structure
Unknown displacements and reaction forces are calculated
Stiffness matrices of separate members are assembled into a
single stiffness matrix K
![Page 5: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/5.jpg)
STIFFNESS MATRIX OF STRUCTURAL MEMBER
5
Stiffness matrix (K) gives the relation between vectors
of nodal forces (F) and nodal displacements (Z):
![Page 6: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/6.jpg)
EXAMPLE OF MEMBER STIFFNESS MATRIX
6
Stiffness relation for a rod:
Stiffness matrix:
i j i
EAF x x
L
![Page 7: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/7.jpg)
ASSEMBLY OF STIFFNESS MATRICES
7
To assemble stiffness matrices of separate members into a single matrix for the whole structure, we should simply add terms for corresponding displacements.Physically, this procedure represent the usage of compatibility and equilibrium equations.
![Page 8: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/8.jpg)
Let’s consider a system of two rods:
ASSEMBLY OF STIFFNESS MATRICES - EXAMPLE
8
![Page 9: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/9.jpg)
SOLUTION USING MATRIX METHOD - EXAMPLE
9
![Page 10: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/10.jpg)
SOLUTION USING MATRIX METHOD - EXAMPLE
10
i j10k
![Page 11: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/11.jpg)
SOLUTION USING MATRIX METHOD - EXAMPLE
11
i j10
k
![Page 12: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/12.jpg)
TRANSFORMATION MATRIX
12
Transformation matrix is used to transform nodal displacements and forces from local to global coordinate system (CS) and vice versa:
Transformation matrix is always orthogonal, thus, the inverse matrix is equal to transposed matrix:
1 MT T
F T F Z T Z
The transformation from local CS to global CS: T TF T F Z T Z
![Page 13: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/13.jpg)
For simplest member (rod) we get:
TRANSFORMATION MATRIX EXAMPLE
13
i
i
j
j
x
yZ
x
y
i
i
j
j
x
yZ
x
y
Z T Z
![Page 14: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/14.jpg)
TRANSFORMATION MATRIX
14
To transform the stiffness matrix from local CS to global CS, the following formula is used:
![Page 15: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/15.jpg)
EXAMPLE FOR A TRUSS
15
The truss has three members, thus 6 degrees of freedom. The stiffness matrix will be 6x6.
![Page 16: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/16.jpg)
EXAMPLE FOR A TRUSS
16
![Page 17: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/17.jpg)
EXAMPLE FOR A TRUSS
17
![Page 18: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/18.jpg)
EXAMPLE FOR A TRUSS
18
![Page 19: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/19.jpg)
EXAMPLE FOR A TRUSS
19
![Page 20: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/20.jpg)
EXAMPLE FOR A TRUSS
20
![Page 21: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/21.jpg)
EXAMPLE FOR A TRUSS
21
![Page 22: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/22.jpg)
EXAMPLE FOR A TRUSS
22
![Page 23: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/23.jpg)
EXAMPLE FOR A TRUSS
23
![Page 24: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/24.jpg)
THREE BASIC EQUATIONS
Equilibriumequations
Constitutiveequations
Compatibilityequations
Taken into account when global stiffness matrix is assembled from
member matrices
Through member stiffness matrices
Taken into account when global stiffness matrix is assembled from
member matrices
How are they implemented in matrix method
24
![Page 25: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/25.jpg)
WHERE TO FIND MORE INFORMATION?
25
Megson. Structural and Stress Analysis. 2005Chapter 17
Megson. An Introduction to Aircraft Structural Analysis. 2010Chapter 6.
… Internet is boundless …
![Page 26: Lecture #11 Matrix methods. METHODS TO SOLVE INDETERMINATE PROBLEM 2 Displacement methods Force method Small degree of statical indeterminacy Large degree](https://reader035.vdocuments.site/reader035/viewer/2022062517/56649f2c5503460f94c47201/html5/thumbnails/26.jpg)
TOPIC OF THE NEXT LECTURE
26
Stress state of sweptback wing
All materials of our course are availableat department website k102.khai.edu
1. Go to the page “Библиотека”2. Press “Structural Mechanics (lecturer Vakulenko S.V.)”