Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method •...
Transcript of Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method •...
![Page 1: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/1.jpg)
Buckling of Rigid Frames – I
Prof. Tzuyang Yu Structural Engineering Research Group (SERG)
Department of Civil and Environmental Engineering University of Massachusetts Lowell
Lowell, Massachusetts
CIVE.5120 Structural Stability (3-0-3) 02/28/17
![Page 2: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/2.jpg)
2
Outline
• Effect of geometrical imperfection – P-δ and P-Δ effects
• Load-deflection behavior of frames • Analysis approaches
– D.E. method – Slope-deflection method – Matrix stiffness method
• Slope-deflection method • Elastic critical loads – D.E. method
– Non-sway case – Sway case
• Summary • References
![Page 3: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/3.jpg)
3
Buckling of Rigid Frame – I
(Source: AutoFEM Analysis)
![Page 4: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/4.jpg)
4
Buckling of Rigid Frame – I
(Source: Super Structures Associates, UK)
• Moment distribution
![Page 5: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/5.jpg)
5
Buckling of Rigid Frame – I
(Source: J.E. Steel, AZ)
![Page 6: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/6.jpg)
6
Buckling of Rigid Frame – I
(Source: AutoFEM Analysis)
![Page 7: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/7.jpg)
7
Buckling of Rigid Frame – I
(Source: T-Flex Analysis)
![Page 8: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/8.jpg)
8
Rigid Frames – I
• Effects of geometric imperfection – P-δ and P-Δ effects
![Page 9: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/9.jpg)
9
Rigid Frames – I
• Load-deflection behavior of frames – Elastic buckling load, Pcr – Plastic collapse load, Pp – Actual failure load, Pf
![Page 10: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/10.jpg)
10
Rigid Frames – I
• Analysis approaches – D.E. method
• Force based • Second-order and fourth-order
– Slope-deflection method • Displacement based • Matrix form
– Matrix stiffness method • Force based • Matrix form
![Page 11: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/11.jpg)
11
Slope Deflection Method
• Slope deflection method is a displacement-based analysis for indeterminate structures
• Unknown displacements are first written in terms of the loads by using load-displacement relationships; then these equations are solved for the displacements.
• Once the displacements are obtained, unknown loads are determined from the compatibility equations using load-displacement relationships.
– Nodes: Specified points on the structure that undergo displacements (and rotations).
– Degrees of Freedom (DOF): These displacements (and rotations)
are referred to as degrees of freedom.
![Page 12: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/12.jpg)
12
To clarify these concepts we will consider some examples, beginning with the beam in Fig. 1(a). Here any load P applied to the beam will cause node A only to rotate (neglecting axial deformation), while node B is completely restricted from moving. Hence the beam has only one degree of freedom, θA.
Slope Deflection Method
Fig. 1 (a)
![Page 13: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/13.jpg)
13
The beam in Fig. 1(b) has node at A, B, and C, and so has four degrees of freedom, designed by the rotational displacements θA, θB, θC, and the vertical displacement ΔC.
Slope Deflection Method
Fig. 1 (b)
![Page 14: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/14.jpg)
14
Consider now the frame in Fig. 1(c). Again, if we neglect axial deformation of the members, an arbitrary loading P applied to the frame can cause nodes B and C to rotate nodes can be displaced horizontally by an equal amount. The frame therefore has three degrees of freedom, θA, θB, ΔB.
Slope Deflection Method
Fig. 1 (c)
![Page 15: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/15.jpg)
15
Consider portion AB of a continuous beam, shown below, subjected to a distributed load w(x) per unit length and a support settlement of Δ at B; EI of the beam is constant.
Δ A B
θB ψ
θA
ψ = rigid body motion = Δ/L
=
FEMAB FEMBA
Mʹ′BA=(2EIθA)/L Mʹ′AB=(4EIθA)/L
θA
i) Due to externally applied loads
ii) Due to rotation θA at support A
+
Slope Deflection Method
![Page 16: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/16.jpg)
16
Mʹ′ʹ′BA=(4EIθB)/L θB
Mʹ′ʹ′AB=(2EIθB)/L
+
A B
iii) Due to rotation θB at support B
L
Mʹ′ʹ′ʹ′AB=(-6EIΔ)/L2 A B
L
Mʹ′ʹ′ʹ′BA=(-6EIΔ)/L2
Δ
+ iv) Due to differential settlement of Δ (between A and B)
Slope Deflection Method
![Page 17: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/17.jpg)
17
Slope Deflection Method
• Governing equation –
![Page 18: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/18.jpg)
18
θA
θB
Δ
=
(FEM)AB
(FEM)BA
(FEM)BA/2
(FEM)BA +
Slope Deflection Method
![Page 19: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/19.jpg)
19
+
θA
Mʹ′ʹ′AB=(3EIθA)/L +
Δ Mʹ′ʹ′ʹ′AB=(3EIΔ)/L2
MAB = [(FEM)AB - (FEM)BA/2]+(3EIθA)/L -(3EIΔ)/L2
Modified FEM at end A
Δ = PL3/(3EI), M = PL = (3EIΔ/L3)(L) = 3EIΔ/L2
Slope Deflection Method
![Page 20: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/20.jpg)
20
Rigid Frames – I
• Elastic critical load – Differential equation approach – Non-sway case 1. Governing equations
1.1 For the beam 1.2 For the column
2. Displacement solution & boundary conditions
![Page 21: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/21.jpg)
21
Rigid Frames – I
• Elastic critical load – Differential equation approach – Non-sway case 3. Compatibility condition
4. Characteristic equation à Critical load, Pcr
![Page 22: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/22.jpg)
22
Rigid Frames – I
• Elastic critical load – Differential equation approach – Sway case 1. Governing equations
1.1 For the beam 1.2 For the column
2. Displacement solution & boundary conditions
![Page 23: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/23.jpg)
23
Rigid Frames – I
• Elastic critical load – Differential equation approach – Sway case 3. Compatibility condition
4. Characteristic equation à Critical load, Pcr
![Page 24: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/24.jpg)
24
Summary
• Real structures, such as buildings, behave like frames. Boundary conditions and joint conditions become critical in determining the critical load of the structures.
• Geometric imperfection plays a key role in the critical load.
• Secondary effects (P-δ and P-Δ effects) will make the elastic load-deflection behavior become nonlinear (geometric nonlinearity)
![Page 25: Buckling of Rigid Frames – I...Rigid Frames – I • Analysis approaches – D.E. method • Force based • Second-order and fourth-order – Slope-deflection method • Displacement](https://reader035.fdocuments.net/reader035/viewer/2022071512/61332a7cdfd10f4dd73ae994/html5/thumbnails/25.jpg)
25
References
• Y-Y Hsieh, S.T. Mau (1995), Elementary Theory of Structures, Prentice Hall, Upper Saddle River, NJ. à Chapter 8
• E.C. Rossow (1998), Analysis and Behavior of Structures, Prentice Hall, Upper
Saddle River, NJ. à Chapter 11
• K.M. Leet, C-M Uang, A.M. Gilbert (2011), Fundamentals of Structural Analysis, 4th ed., McGraw-Hill, Ne York, NY. à Chapter 12
• R.C. Hibbeler (2012), Structural Analysis, 8th ed., Prentice Hall, Upper Saddle River, NJ. à Chapter 11