3 6 2 8 2
so translation unknowns of original structure are 2.
4, Degree of indeterminacy
nn n The degree of indeterminacy (n) is:
n426
5, Primary structure
(a) Standing on the side of deformation, joint 1 can’t rotate (Z1) due to rigid arm; translation (Z2) at joint 2 can’t exit due to link in the primary structure. But , Z1, Z2 are existences in original structure. In order to eliminate the difference, a rotation Z1 and translation Z2 can be enforced at joint 1 and joint 2 respectively in primary structure.
Shearing force —— clockwise is positive about isolating body (free body).
Rotation angle —— clockwise is positive
The following sign is established: Translation —— clockwise rotation of the whole member is positive.
Suppose end B is hinged.
3
M 0 2i(2 AB)
M BA
BAl
F 0
AB
1( 3 AB1 F )
M B 2 A
l 2i BA
The following sign is established: Bending moment —— clockwise is positive relative to end of member; counterclockwise is negative relative to rigid joint or support.
C
C
C
B
D
3、How to decide the number of primary unknowns
(1)、The number of rotation displacements ( n ) is equal to the number of
rigid joints. (2)、The number of translation unknowns ( n ) The method of determining the translation unknowns is as following.
11
2
l
( l 1) 1
11
22 E I 2
3
3E I
1 (1 l 1) 1 1 l
12
21
EI 2
3
6EI
l 1 ( B )
1p E I
l
l 1 ( A )
2p
EI
l
1
(R • C ) (R
AC
1
R
BC
)
2
(1 0 1
l
l
)
AB
Fig.shows a beam, it is fixed at two ends. EI is constant. The beam is subjected
to a force P, rotation at end A is , and rotation at end B is
, the
A
B
translation at end B relative end A is A B , bending moments at end A, B are
desired.
Solution: let solve it using force method.
(1) The primary structure
Ml l l Ml l l F A B 2 2(2B A )
B F A 2 2(2A B )
if original structure is that one end is fixed and the other end is hinged, its
slope-deflection can be deduced .
All rigid joints (including fixed supports) are changed into hinges; calculate degree of freedom of the new structure. The degree of freedom of the new structure is equal to the number of translation unknown of the original structure.
AB
l
2
(R • C )
ABlLeabharlann We havel l
3
E
I
l 1 6EI
B AB
2 EI •l l
A
l
l 6EI
l 1 3EI
A AB
2 EI •l l
B
2EI (2
1
l
A
3 l l l B
AB ) l
2
2
(2
B
)
A
2
2EI l
(2
B
3 l l l A
In the last chapter (force method), the unknowns of primary structure are forces. After obtaining unknown force, displacement of the structure can be solved.
(a) A (b) A (c) A (d) A
q øB
B øB
l
l
øB B
q
øB
B øB
Bq
C Notice: (1) The flexural deformation of the flexural members in the frame is taken into account, but the shearing and axial deformations of which are
AB ) l
2
2
(2
A
)
B
let i E I
l
linear rigidity
We have
3
M 2i(2 AB)
M AB
A
B
l
F AB
M 2i(2
3 AB)
F
M BA
B
A
l
BA
is referred to as “slopedeflection equation”
Solution: (1) It is indeterminate to the second degree (2) Primary Structure:
(3)Canonical equations In order to solve this problem, we must find out the differences between original structure and primary structure at first.
The end moment, end shearing force and end reaction are listed in table for convenience.
Section 4 Analysis of indeterminate Structure using displacement method
Chapter 7 Slope-Deflection Method
Section 1 Introduction
Slope-deflection method (or simply the displacement method) is another method to analyze the statically indeterminate structure.
(2) Canonical equations
1
11 1
12 2
13 3
1p
1
A
2
21 1
22 2
23 3
2p
2
B
Because
M
3
0
11 1
12 2
1p
1
A
21
1
22
2
2p
2
B
Solution:
(3) Determine coefficients (Graph Multiplication)
When indeterminate structure is analyzed by using displacement method, every member is considered as a statically indeterminate beam with single span. So the primary structure is that every member is changed into an indeterminate beam with single span. A rigid arm is added at every rigid joint to prevent rotation of the joint (but can not prevent translation) at the same time, a link is added at joint where translation is possible. The link prevents translation of the joint.