The Influence of Shear - Flexibility on the Elastic

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The Influence of Shear-Flexibility on the Elastic
Critical Load for Frames of Pultruded Fibre
Reinforced Plastic Section
A. E. Abbaker and J. T. Mottram
School of Engineering
Advanced Polymer Composites for Structural
Applications in Construction
ACIC 2004
PFRP Frames
Objectives
Analysis of PFRP plane
frames with moment
resistant joints
Analysis Approach
Non-linear numerical
frame analysis
Non-linear Numerical Frame Analysis
Matrix stiffness method
Points to be considered
•
Shear flexible element
•
Semi-rigid connections
•
Second-order effects
•
Shear flexible stability
functions
•
Stability analysis
Shear Flexibility
Deformation components
• Bending controlled by EI
• Shear controlled by GA
Moduli ratio (E/G)
• Steel is about 2.5
M A = k22θ A
• FRP from 6 to 80
Modified slope–deflection
equation
Shear flexible stiffness
k
k22 =
1
3β ⎞
⎛ l
+
⎜
⎟
⎝ 4 EI 2 AG ⎠
Connection Behaviour and Modelling
Moment-rotation curves
Modelling connections
•
Rigid
•
Semi-rigid
•
Pinned
Semi-Rigid Connections
• PFRP sections have low
Young’s modulus at 20-25
GPa and are relatively
expensive
• Design loading limited by
deflection; not resistance
• Practical moment-resistant
connections will half
deflection (or double
loading)
Second-order Effects
Assumptions
•
Linear elastic material
•
Small deformation
Axial load effect
•
Beam differential equation
•
Modified slope- deflection
equation
•
Stiffness and carry-over
factors (s and c)
2
d y
1
( M ( x ) + PA y )
=−
2
EI
dx
QA ⎞
⎛
MA = k22⎜ sθA + scθB − s(1+ c) ⎟
L⎠
⎝
Shear Effect on φ Functions
φ1 = α cot α
α = 0.5π ρ
ρ
ρ =
(1 − μρ
)
Increasing shear flexibility
μ=
β Pe
GA
φ functions decreases with increase of shear-flexible parameter μ
Stability Analysis
•
Due to non-linearity the stiffness matrix becomes
function of load and can be used as stability
criterion
•
Joint equilibrium equations
•
Positive determinant implies stable state
•
Negative determinant implies unstable state
•
Zero determinant
equilibrium
•
Non-trivial solution of state of neutral equilibrium
equation
implies
state
of
neutral
Validation of Overall Stability Analysis
PB and FB are
Pinned Base
and
Fixed Base
boundary
conditions
Comparison of elastic critical load of isotropic portal frame
Previous
Frame problem
Author
ρcr
Abbaker
Difference
ρ cr
%
Sway with PB
Timoshenko
0.184
0.183
-0.10
Sway with FB
Chajes
0.74 4
0.745
0.10
No sway with PB
Horne
1.285
1.283
-0.15
No sway with FB
Coates
2.550
2.548
-0.10
Influence of Shear Flexible Parameter on
Elastic Critical Load
ρ
ρsh
0.6
0.5
0.4
0.3
0.2
0.1
0
Increasing shear flexibility
μ=
0
β Pe
GA
20
=
βπ
2
E
2
⎛l⎞ G
⎜ ⎟
⎝r⎠
40
E/G
Rigid joints
60
80
100
• As (E/G) increases from 2.5 to 80
elastic critical load decreases from
0.504 to 0.325
• Reduction in critical load is up to 35%
Influence of Joint Action on Elastic Critical Load
0.8
0.7
0.6
0.5
ρρ cr 0.4
0.3
0.2
0.1
0
pinned joint
Rigid
joint
Semi-rigid joint
0
5
10
15
S jL /EI
20
25
30
Sj is connection rotational stiffness
Joint classification
• As the relative stiffness factor (SjL/EI) is increased from 0.29 to 25
the elastic critical load for overall isotropic frame instability
increases from 0.302 to 0.671
• Increase in elastic critical load by up to 122%
Influence of Shear Flexibility and Semi-rigid Action
0.7 ρ
crsh
0.6
0.5
ρ crsh
0.4
0.3
0.2
E/G = 6
E/G = 20
E/G = 80
Increasing shear flexibility
0
10
20
S jL /EI
30
Sj is connection rotational stiffness
• High fixity achieved at low relative joint stiffness for very
shear flexible members
• Use Sj/ k rather than SjL/EI for the relative joint stiffness,
where k is the shear flexible member stiffness
Conclusions
• Elastic critical load reduces with increase in member
shear-flexibility.
• It is necessarily to include the effect of shearflexibility when analysing the overall stability of
frames of PFRP section.
• There is no practical benefit to be gained in term of
the load carrying capacity of frames when the
connection relative stiffness is increase beyond 25,
as specified in Annex J of Eurocode 3.
Thank you for your attention
Any questions?
Email: [email protected]
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