FrameElements.pdf

Nonlinear Frame Finite Elements in OpenSees
Michael H. Scott
Associate Professor
School of Civil and Construction Engineering
OpenSeesDays Users Workshop
Richmond, CA September 26, 2014
M.H. Scott (OSU) OpenSees Frame Elements OpenSeesDays 2014 1 / 40

Types of Nonlinearity
Two sources of nonlinear frame element response:
Material – yielding, strain hardening, crushing of concrete, etc.
Geometry – loss of stability due to loads acting through large
displacements
An analysis can account for each source of nonlinearity
separately, giving four possible approaches
Geometry Linear (GL) Geometry Nonlinear (GN)
Material Linear (ML) ML, GL ML, GN
Material Nonlinear (MN) MN, GL MN, GN

Steel Frame Pushover Analysis
σy =36 ksi
E=30,000 ksi
α=0.02
30 ft
Columns: W14x90
αE
ε
σy
σ
E
Vb = 2λ
150 kip 150 kip
300 kip 300 kip
Beams: W18x76
0.67λ
1.33λ
15 ft
15 ft
Simple steel frame model analyzed under four approahces
Relatively large column axial loads will intensify both material
and geometric nonlinear response for demonstration purposes

0 10 20 30
0
200
400
600
800
1000
0 5 10 15 20
0
50
100
150
200
ML, GL
ML, GN
MN, GL
MN, GL
MN, GN
MN, GN
Roof Displacement (in)
Base
Shear
(kip)
Base
Shear
(kip)
We observe the following:
Material nonlinearity kicks in well before geometric nonlinearity
Geometric nonlinearity allows for prediction of loss of stability for
increasing displacement

Section Force-Deformation Response
At each cross-section along a frame element, we must determine
the section forces for any given section deformations
Material nonlinearity eminates from the stress-strain response in
each frame element
Heuristic approach through stress-resultant section models, e.g.,
moment-curvature; or
Integrate stress-strain response via “fiber section” approach
y
z
x
Ai
(yi , zi )

Commands for Section Definition
uniaxialMaterial modelName $tag ...
Define uniaxial stress-strain models for use in Bernoulli beam
elements
Elastic, Steel01, Steel02, Concrete01, Concrete02, etc.
nDMaterial modelName $tag ...
Define multiaxial stress-strain models for use in Timoshenko
beam elements
ElasticIsotropic, J2Plasticity, ConcreteMCFT, etc.

Commands for Section Definition
General definition of Bernoulli cross-section using patches and
layers of fibers whose stress-strain response is defined by
uniaxialMaterial objects
section Fiber $tag {
patch $type $matTag ...
layer $type $matTag ...
fiber $matTag ...
...
}
Use NDFiber with nDMaterial objects instead of Fiber with
uniaxialMaterial objects for Timoshenko beams
Specific cross-sections obtained with “canned” models
section WFSection2d $tag $matTag ...
section RCSection2d $tag $matTag ...

Rectangular Steel Section
Rectangular section with EPP uniaxial stress-strain response
Compute moment-curvature response for increasing number of
fibers
Exact solution for My = fy bd2
/6 and Mp = bd2
/4
b
i
Nfiber
2
1
yi
d
d/2
.
.
.
.
.
.
σ
σy
E
ε

Rectangular Steel Section
0 0.2 0.4 0.6 0.8 1
x 10
−3
0
1
2
3
4
x 10
4
0 0.2 0.4 0.6 0.8 1
x 10
−3
0
1
2
3
4
x 10
4
0 0.2 0.4 0.6 0.8 1
x 10
−3
0
1
2
3
4
x 10
4
0 0.2 0.4 0.6 0.8 1
x 10
−3
0
1
2
3
4
x 10
4
Nfiber = 2 Nfiber = 4
Nfiber = 8 Nfiber = 16
Curvature (1/in)
Curvature (1/in)
Curvature (1/in)
Curvature (1/in)
Moment
(kip-in)
Moment
(kip-in)
Moment
(kip-in)
Moment
(kip-in)
Yield
Yield
Yield
Yield
Exact
Exact
Exact
Exact
Computed
Computed
Computed
Computed

Reinforced Concrete Section
EPP steel and Concrete01 concrete
Using “canned” RCSection2d command
Confined and unconfined concrete
2 in
2 in
24 in
24 in
Strain, εc
Stress,
f
c
f ′
cc
f ′
cu
εcc εcu

Reinforced Concrete Section
Moment-curvature response for increasing levels of axial load
With and without confining effects of transverse reinforcement
Modify the Concrete01 input parameters for confined concrete
0 0.5 1 1.5
x 10
−3
0
5000
10000
15000
Curvature (1/in)
Moment
(kip-in)
ν = 0.0
ν = 0.1
ν = 0.2
ν = 0.3
(c) Without Confining Effects
0 0.5 1 1.5
x 10
−3
0
2000
4000
6000
8000
10000
12000
Curvature (1/in)
Moment
(kip-in)
ν = 0.0
ν = 0.1
ν = 0.2
ν = 0.3
(d) With Confining Effects

Numerical Integration of Element Response
For most material nonlinear element formulations, cross-section
response is integrated numerically along the frame element
length in order to determine element force-deformation response
Sections located at discrete points along the element length,
each with a prescribed weight
Highly accurate Gauss-based quadrature commonly used
x1 x2 x3 x4 x5
L
w4
w1 w3 w5
w2

Displacement-Based Frame Element
element dispBeamColumn $tag $ndI $ndJ $transfTag Legendre $secTag 2
Strict compatibility
Linear axial and cubic Hermitian transverse displacement fields
Constant axial deformation and linear curvature along element
length
Weak equilibrium
Equilibrium satisfied only at the nodes, not at every section
along the element
Two-point Gauss-Legendre integration along element length
Improve numerical solution by using more elements per member
(mesh- or h-refinement)

Propped Cantilever
Constant axial load and increasing moment applied at propped
end
Fiber-discretized section response with strain-hardening
stress-strain
Two Gauss-points per element
Investigate refinement for increasing number of elements
ε
σ
E
αE
σy
θ
M
. . .
1 2 Nele
Nele-1
2 fibers each flange
N
100 in
W14x90
10 web fibers
U

Global Response
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
Rotation, θ (rad) Deflection, U (in)
Moment,
M
(kip-in)
Moment,
M
(kip-in)
Moment,
M
(kip-in)
Computed
Exact
Nele = 1
Nele = 1
Nele = 2
Nele = 2
Nele = 4
Nele = 4

Local Flexural Response
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
x (in)
M(x)
(kip-in)
M(x)
(kip-in)
M(x)
(kip-in)
x (in)
κ(x)
(1/in)
κ(x)
(1/in)
κ(x)
(1/in)
Computed
Exact
Nele = 1 Nele = 1
Nele = 2 Nele = 2
Nele = 4 Nele = 4

Local Axial Response
0 20 40 60 80 100
−400
−200
0
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
0 20 40 60 80 100
−400
−200
0
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
0 20 40 60 80 100
−400
−200
0
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
x (in)
N(x)
(kip)
N(x)
(kip)
N(x)
(kip)
x (in)
ε
a
(x)
(in/in)
ε
a
(x)
(in/in)
ε
a
(x)
(in/in)
Computed
Exact
Nele = 1 Nele = 1
Nele = 2 Nele = 2
Nele = 4 Nele = 4

Investigate refinement of load-displacement response for
increasing number of displacement-based elements per member
σy =36 ksi
E=30,000 ksi
α=0.02
30 ft
Columns: W14x90
αE
ε
σy
σ
E
Vb = 2λ
150 kip 150 kip
300 kip 300 kip
Beams: W18x76
0.67λ
1.33λ
15 ft
15 ft

Coarse mesh over-predicts strength – unconservative
Improved solution with refined mesh
0 10 20 30 40
0
50
100
150
200
Nele = 1
Nele = 2
Nele = 4
Base
Shear
(kip)

Force-Based Frame Element
element forceBeamColumn $tag $ndI $ndJ $transfTag Lobatto $secTag $Np
Average compatibility
Nodal displacements are balanced by weighted integral of
section deformations
Complex state determination
Use Gauss-Lobatto integration so that extreme flexural response
captured at element ends
Strong equilibrium
Equilibrum of nodal and section forces satisfied at all points
along element
Constant axial force and linear bending moment in absence of
member loads
Straightforward to include member loads
Improve numerical solution by using more integration points per
element while maintaining mesh of one element per member

Propped Cantilever
Constant axial load and increasing moment applied at propped
end
Fiber-discretized section response with strain-hardening
stress-strain
Investigate refinement for increasing number of Gauss-Lobatto
point using one element
ε
σ
E
αE
σy
θ
M
2 fibers each flange
N
100 in
W14x90
10 web fibers
U
1 2 Np-1 Np
. . .

Global Response
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
0 0.05 0.1 0.15 0.2
0
5000
10000
15000
−0.4 −0.3 −0.2 −0.1 0
0
5000
10000
15000
Rotation, θ (rad) Deflection, U (in)
Moment,
M
(kip-in)
Moment,
M
(kip-in)
Moment,
M
(kip-in)
Computed
Exact
Np = 3
Np = 3
Np = 4
Np = 4
Np = 5
Np = 5

Local Flexural Response
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
0 20 40 60 80 100
−1
0
1
2
x 10
4
0 20 40 60 80 100
−5
0
5
10
x 10
−3
x (in)
M(x)
(kip-in)
M(x)
(kip-in)
M(x)
(kip-in)
x (in)
κ(x)
(1/in)
κ(x)
(1/in)
κ(x)
(1/in)
Computed
Exact
Np = 3 Np = 3
Np = 4 Np = 4
Np = 5 Np = 5

Local Axial Response
0 20 40 60 80 100
−190.8
−190.8
−190.8
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
0 20 40 60 80 100
−190.8
−190.8
−190.8
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
0 20 40 60 80 100
−190.8
−190.8
−190.8
0 20 40 60 80 100
−0.015
−0.01
−0.005
0
x (in)
N(x)
(kip)
N(x)
(kip)
N(x)
(kip)
x (in)
ε
a
(x)
(in/in)
ε
a
(x)
(in/in)
ε
a
(x)
(in/in)
Computed
Exact
Np = 3 Np = 3
Np = 4 Np = 4
Np = 5 Np = 5

Investigate refinement of load-displacement response for
increasing number of Gauss-Lobatto integration points per
element
Maintain one element per member
σy =36 ksi
E=30,000 ksi
α=0.02
30 ft
Columns: W14x90
αE
ε
σy
σ
E
Vb = 2λ
150 kip 150 kip
300 kip 300 kip
Beams: W18x76
0.67λ
1.33λ
15 ft
15 ft

Same yield point predicted in all cases
Post-yield stiffness more flexible with fewer integration points
0 10 20 30 40
0
50
100
150
200
Np = 3
Np = 4
Np = 5
Base
Shear
(kip)

Force-Based Plastic Hinge Frame Element
element forceBeamColumn $tag $ndI $ndJ $transfTag HingeRadau $secTagI $lpI
$secTagJ $lpJ $secTagE
or
element beamWithHinges $tag $ndI $ndJ $secTagI $lpI $secTagJ $lpJ $E $A $I
$transfTag
Control integration weights at element ends
Important for strain-softening section response
J
I
x1 x6
L
x2 x3 x4 x5
lpI lpJ
L − 4(lpI + lpJ)
3lpI 3lpJ

Reinforced Concrete Bridge Pier
550mm x 550mm square
12 bars, db = 20 mm
40 mm clear cover
V = Base Shear
V , U
P = 0.3f ′
c Ag
L
=
1.65
m
Specimen 7
Tanaka and Park (1990)

Displacement-Based Elements
Post-peak response is mesh-dependent
Function of element length
0 20 40 60 80
0
200
400
600
800
1
3
6
Displacement (mm)
Base
Shear
(kN)

Force-Based Element
Post-peak response depends on number of integration points
Function of integration weight at base of column
0 20 40 60 80
0
200
400
600
800
4
5
6
Displacement (mm)
Base
Shear
(kN)

Force-Based Plastic Hinge Element
Post-peak response controlled by plastic hinge length
lp = 0.22L from empirical equation
0 20 40 60 80
0
200
400
600
800
Displacement (mm)
Base
Shear
(kN)

Drawback to Force-Based Plastic Hinge Element
For strain-hardening section behavior, post-peak response is too
flexible
0 5 10 15
0
5
10
15
Computed
Exact
Displacement
Load

Modeling Recommendations
There’s no silver bullet
Strain-Hardening Section Response
Use mesh of displacement-based elements
Use one force-based elements with 4 to 6 Gauss-Lobatto points
Plastic hinge element not recommended because post-peak
response will be too flexible
Strain-Softening Section Response
Use force-based plastic hinge element
Response with displacement-based elements is mesh dependent
Response with Gauss-Lobatto force-based element depends on
number of integration points

Geometric Transformation of Element Response
Element formulation of material nonlinearity inside the basic
system (free or rigid body displacement modes)
Element formulation of geometric nonlinearity outside the basic
system
β
β
I J
I
J
Basic System
Local Coordinate System
∆uly
ul2
ul1
L
ul4
ul5
L + ∆ulx
ul3
ul6
Ln
ub3
ub2

Geometric Transformation
geomTransf Linear $tag
Small displacement assumptions in local to basic transformation
Linear transformation of forces and displacements
geomTransf PDelta $tag
Small displacement assumption transformation of displacements
Account for transverse displacement of axial load in equilibrium
relationship
geomTransf Corotational $tag
Fully nonlinear transformation of displacements and forces
Exact in 2D but some approximations in 3D

Examine pushover response for different levels of gravity load
σy =36 ksi
E=30,000 ksi
α=0.02
12 ft
12 ft
λ50 kip
λ50 kip
λ100 kip λ100 kip
V
V
Beams: W18x76
Columns: W14x90

P − ∆ and Corotational – similar results for lateral displacement
0 20 40 60 80 100
0
100
200
300
400
0 20 40 60 80 100
0
100
200
300
400
0 20 40 60 80 100
0
100
200
300
400
0 20 40 60 80 100
0
100
200
300
400
λ = 1 λ = 2
λ = 3 λ = 4
Roof Disp (in)
Roof Disp (in)
Roof Disp (in)
Roof Disp (in)
Base
Shear
(kip)
Base
Shear
(kip)
Base
Shear
(kip)
Base
Shear
(kip)

“Exact” Corotational predicts change in vertical displacement
Important for collapse prediction and post-buckling capacity
0 5 10 15 20
0
100
200
300
400
0 5 10 15 20
0
100
200
300
400
0 5 10 15 20
0
100
200
300
400
0 5 10 15 20
0
100
200
300
400
λ = 1 λ = 2
λ = 3 λ = 4
Roof Disp (in)
Roof Disp (in)
Roof Disp (in)
Roof Disp (in)
Base
Shear
(kip)
Base
Shear
(kip)
Base
Shear
(kip)
Base
Shear
(kip)
P-∆
P-∆
Corotational

Elastic Buckling
Use mesh of corotational frame elements to simulate buckling
Simply-supported W14x90, L=100 in, Pcr =29579 kip
L/r=16.26: short column, but demonstrates point
Imperfection applied to nodes, u(t) = 0.1 sin(πx/L) in
0 2 4 6 8 10
0
1
2
3
4
x 10
4
Nele=2
Nele=4
Nele=8
29579 kip
Axial Deflection (in)
Axial
Load
(kip)
Concept works well for inelastic buckling too

Summary
Material and geometric nonlinearity treated separately for frame
finite elements in OpenSees
Only scratching the surface – other element formulations and
models of nonlinear stress-strain response
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