清华大学土木工程系结构力学(英)Chapter8_582602260

4 EI 2 EI 6 EI M AB A B 2 D AB M AB F l l l 2 EI 4 EI 6 EI M BA A B 2 D AB M BA F l l l
Slope-Deflection Equations: General (2) 线刚度 i EI


A rigid joint provides unknown reactive couple, which may be incurred by given angular displacement of that joint. (angular displacement → reactive couple) A hinged joint undergoes unknown angular displacement, which may be incurred by applied couple at that joint. (applied couple→ angular displacement)
One end of a member is supported by a pin or roller at its far end The angular displacement at the pin-supported end is not primary unknown, and does not have to be determined ΔA A’ l EI is constant A
M AB Pab
2 2
M BA
P A
l Pa 2b
VAB
l2
VBA
Pb 2 3 (l 2a) l Pa 2 3 (l 2b) l
A V
（A: Near end B: Far end） （B: Near end A: Far end）
B
B
P A MAB＜0
M
P B MBA＞0 A VAB＞0
An Example

Assume an angular displacement q Fixed end moments given q
D EI A EI，l l E
EI M EA 4 q l EI M EB M ED 4 q l EI Pl M EC 4 q l 8

P

பைடு நூலகம்
Slope-deflection equation is the most general relation, which is independent of end supports A member with independent disp. at both ends is the most general (fundamental) working element in the disp. method For convenience of manual computation, three other specific elements are developed as our working elements
Neglecting axial deformation P X1 X2
Primary Structure
l/2
l/2
Fixed End Moment of Bending Members (2)
Pab2 l
2
Pa 2b l2
Pb2 l
3
( l 2a )
A a l b
B
A
Pa 2 l3
B
( l 2b )
P 1
2
3
The added link implies the existing of linear displacement
Identification of Unknown Displacements (3)
Slope-Deflection Equations: General (1)

The displacement method is also called slope-deflection method since it relates the unknown slopes and deflections to the applied load on a structure
Example 1:
Unknowns: θ B Disassemble into members Fixed-end moments
F M BA 4iq B M BA 4iq B 15
20kN A B
2kN/m C
3m 3m
6m
M BC ?
Pin-Supported End Span
M
V
Sign Convention:

FEM: Positive when it acts clockwise on the span FES: Positive when it causes the span rotate clockwise
Fixed End Moment of Bending Members (3)
Chapter 8
the Displacement Method
1
Fixed End Moment of Bending Members (1)

Under concentrated forces
a A
P
b B l
a
P
b
EI
SI to the 2nd degree
SI to the 3rd degree
A EI, l
A
B φB
DAB
M
B
M AB M BA
4 EI 2 EI 6 EI A B 2 D AB l l l 2 EI 4 EI 6 EI A B 2 D AB l l l
Difference between a rigid joint and a hinge connection
EI，l
C
EI l 4iq 4iq E 4iq Pl/8 4iq B
Joint Equilibrium
q=
Pl2
/ (128EI)
A
MEA
E
8
An Example-continued

Fixed end moments
D
EI Pl M EA M EB M ED 4 q l 32 EI Pl 3Pl M EC 4 q l 8 32 EI Pl M AE M BE M DE 2 q l 64 EI Pl 9 Pl M CE 2 q l 8 64
ql 2 12
M AB
Pab2 l2
ql 2 12
A
ql 2 8
B
M
Refer to 《结构力学》page 281~283 for FEMs for other loadings
Fixed End Moment of Bending Members (5)

Under support movements
A A
B
D
D1C R1i C1i B A , D 2C
C
lA lB R2i C2i D AB 2 2
B
Fixed End Moment of Bending Members (6)
φA

To produce a unit slope at joint B, how large a couple should be applied?

Identification of Unknown Displacements (1)

Kinematically indeterminate degree The number of unknown displacements occurring at the nodes (joints) of the structure is referred to as the degree in which the structure is kinmatically indeterminate
q = Pl2 / (128EI)
A E P C
B
9
The Displacement Method

When analyzing any indeterminate structures, it is necessary to satisfy equilibrium, compatibility, and force-displacement requirements.
P 1
2
3
θ1
1
θ2
2
θ3
3
Δ3
Neglecting axial deformation
Nodal Disp.：θ1 ,θ2 ,θ3 ,Δ3
Identification of Unknown Displacements (2)

The unknown nodal displacement is classified into angular displacement and linear displacement Each rigid joint has a angular displacement (2 dimensions) Neglecting axial deformation, the linear displacement can be recognized by replacing all rigid joint by hinge connections
l
ΔA A’ A φA φB l