AITC Introduction to Nonlinear Modeling

Engr. Thaung Htut Aung

M. Eng. Asian Institute of Technology Deputy Project Director, AIT Consulting

• •

Nonlinear stress-strain curves Used in two applications – Fiber hinge – Layered shell element

•

Hysteresis – Elastic • Nonlinear but elastic • Loads and unloads along the same stress-strain curve • No energy is dissipated – Kinematic • Commonly observed in metals • Dissipate significant amount of energy • Appropriate for ductile materials – Takeda • Suitable for concrete and other brittle materials • Less energy is dissipated than Kinematic model

3

4

5

Concrete Stress-Strain Curve 10 0

-0.01

-0.005

Stress (MPa)

-0.015

-10

0

0.005

-20 -30 -40

Original Modified

-50 -60 Strain

-70

6

Concrete Stress-Strain Curve 10 0 -0.014

-0.012

-0.01

-0.008

-0.006

-0.004

-0.002

0

0.002

0.004

-10

Stress (MPa)

-20 -30 -40 -50 -60 Strain

-70

7

8

• Consider linear, nonlinear or mixed material behavior • Stress-strain behavior for a given layer is always defined in material coordinate system specified by material angle for that layer • For uni-axial material, σ22 = 0 • Mass and weight are computed for membrane and shell layers, not for plate layers

10

t2

Axis 1

Reference surface

t4

t1

t3

Axis 3

Concrete membrane layer Concrete plate layer Vertical rebar layer 1 Vertical rebar layer 2

11

Wall thickness = 18 in. Concrete cover = 3 in. Vertical rebar = 2% Horizontal rebar = 1%

12

• Consider nonlinearity in vertical stresses, suitable for tall shear walls • Out-of-plane behavior is assumed to be linear, thickness is reduced to account for cracking

13

14

Out of plane bending stiffness of concrete = 0.25 Ig Thk. of plate = 700 x (0.25)^(1/3) = 441 mm Thk. of reinf. = 700 x 0.008 / 2 = 2.8 mm

15

• Modeled as discrete point hinge • All plastic deformation occur within point hinge • Uncoupled hinges – – – –

Moment Torsion Axial force Shear

• Coupled Hinges – – – – –

P-M2-M3 P-M2 P-M3 M2-M3 Fiber Hinge (PMM) 17

• A: Origin • B: Yielding – No deformation occurs in the hinge up to point B. – Only plastic deformation beyond point B is shown by the hinge. – Deformation at point B will be subtracted from the deformations at points C, D

• C: Ultimate capacity • D: Residual strength • E: Total failure 18

19

20

Ref: CSi Software Verification (Example 1-026)

21

Nonlinear Option

Coupled Behavior (M2M3)

Axial-Moment Interaction (PM2-M3)

Uncoupled Hinge M2, M3 Interaction PMM Hinge Fiber PMM Hinge

X X

X X

Degrading Behavior

Ductility Estimation

X X X

X X

Numerical Stability

Low Computational Effort

X X

X

22

Nonlinear Option

2D Pushover Analysis

Uncoupled Hinge M2, M3

X X

Interaction PMM Hinge Fiber PMM Hinge

X

3D Pushover Analysis

2D NLTHA

3D NLTHA

X X

X

X

23

24

25

• Define coupled axial force and biaxial bending • Create manually or automatically for certain type of frame sections, including Section Designer sections • For each fiber material direct nonlinear stress-strain curve is used • Summing up behavior of all fibers at a cross-section and multiplying by hinge length gives axial force deformation and biaxial moment relationships • Shear behavior is not considered in fibers • Shear behavior is computed using as usual using linear shear modulus 26

Lp = 0.5 H Lp = 0.08 L + 0.022 fye dbl ≥ 0.044 fye dbl (MPa) Lp = plastic hinge length H = section depth L = critical distance from the critical section of plastic hinge to the point of contraflexure fye = expected yield strength of longitudinal reinforcement dbl = diameter of longitudinal reinforcement 27

28

• Modal analysis – Eigen analysis – Ritz analysis

• Static analysis – Linear – Nonlinear – Nonlinear staged construction

• Response spectrum analysis • Time history analysis – Linear – Nonlinear 30

Gravity Load Case (Construction Sequence) (Force Controlled)

Nonlinear Static/Time History

31

32

33

34

Linear Static

Construction Sequence

35

Linear Static

Construction Sequence

36

37

20% of 1st mode

38