||Chair of Structural Mechanics

Workshop: nonlinear simulation using MATLAB

28.11.2016 1

Method of Finite Elements II

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Outline

Introduction

Numerical integration of nonlinear structures

MATLAB case study I : 1 – DOF shear building

MATLAB case study II: 100 – DOF shear building

Outlook

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IntroductionNumerical integration

Numerically solve the dynamic Equation of motion

𝐌 ሷ𝐱 𝑡 + 𝐂 ሶ𝐱 𝑡 + 𝐊𝐱 𝑡 = 𝐟(𝑡)

Aim: calculate the vibration displacements/velocities/accelerations at

successive time intervals Δ𝑡, 2Δt, …

Two classes of methods:

Explicit: Solution is based on the equilibrium equations at time 𝑡

Implicit: Solution is based on the equilibrium equations at time 𝑡 + Δ𝑡

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IntroductionExplicit integration: Runge – Kutta family of methods

Underlying theory: state – space representation

ሶ𝐱 𝑡ሷ𝐱 𝑡

=𝐎 𝐈

𝐌−𝟏𝐊 𝐌−𝟏𝐂

𝐱 𝑡ሶ𝐱 𝑡

+𝟎

𝐌−𝟏𝐟(𝑡)

𝐱 𝑡 = 𝐈 𝐎𝐱 𝑡ሶ𝐱 𝑡

Discretization of the state equation

ሶ𝐱𝑡+1ሷ𝐱𝑡+1

=𝐎 𝐈

𝐌−𝟏𝐊 𝐌−𝟏𝐂

𝐱𝑡ሶ𝐱𝑡+

𝟎𝐌−𝟏𝐟𝑡

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IntroductionImplicit integration: Newmark’s method

Underlying theory (refer to the lecture’s presentation): Taylor series

expansion to the displacement (𝐱 𝑡 ) and velocity ( ሶ𝐱 𝑡 ) vectors

Successive steps:

STEP 0: formulate 𝐌, 𝐂 and 𝐊, determine the step size 𝛥𝑡 and the initial conditions

𝐱(0) and ሶ𝐱 0

STEP 1: select integration parameters 𝛽 and 𝛾

STEP 2: calculate required integration constants

𝑏1 =1

𝛽𝛥𝑡2, 𝑏2 =

1

𝛽𝛥𝑡, 𝑏3 =

1

2𝛽− 1,

𝑏4 = 𝛾𝛥𝑡𝑏1, 𝑏5 = 1 − 𝛾𝛥𝑡𝑏2, 𝑏6 = 𝛥𝑡 1 − 𝛾𝑏3 − 𝛾

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IntroductionImplicit integration: Newmark’s method

Successive steps:

STEP 3: calculate the effective stiffness matrix

෩𝐊 = 𝐊 + 𝑏1𝐌+ 𝑏4𝐂

STEP 4: triangularize ෩𝐊 via the LDL factorization

෩𝐊 = 𝐋𝐃𝐋𝑇

STEP 5: for every integration step Δ𝑡, 2Δt, …

𝑏1 =1

𝛽𝛥𝑡2, 𝑏2 =

1

𝛽𝛥𝑡, 𝑏3 =

1

2𝛽− 1,

𝑏4 = 𝛾𝛥𝑡𝑏1, 𝑏5 = 1 − 𝛾𝛥𝑡𝑏2, 𝑏6 = 𝛥𝑡 1 − 𝛾𝑏3 − 𝛾

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IntroductionImplicit integration: Newmark’s method

Successive steps:

STEP 5: for every integration step Δ𝑡, 2Δt, …

STEP 5a: calculate the effective force vector

෨𝐅𝒕 = 𝐟𝒕 +𝐌 𝑏1𝐱𝑡−Δ𝑡 + 𝑏2 ሶ𝐱𝑡−Δ𝑡 + 𝑏3 ሷ𝐱𝑡−Δ𝑡 + 𝐂 𝑏4𝐱𝑡−Δ𝑡 − 𝑏5 ሶ𝐱𝑡−Δ𝑡 − 𝑏3 ሷ𝐱𝑡−Δ𝑡

STEP 5b: calculate the displacements at time 𝑡

𝐋𝐃𝐋𝑇𝐱𝑡 = ෨𝐅𝒕

STEP 5c: calculate the velocities and accelerations at time 𝑡

ሶ𝐱𝑡 = 𝑏4 𝐱𝑡 − 𝐱𝑡−Δ𝑡 + 𝑏5 ሶ𝐱𝑡−Δ𝑡 + 𝑏6 ሷ𝐱𝑡−Δ𝑡ሷ𝐱𝑡 = 𝑏1 𝐱𝑡 − 𝐱𝑡−Δ𝑡 − 𝑏2 ሶ𝐱𝑡−Δ𝑡 − 𝑏3 ሷ𝐱𝑡−Δ𝑡

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IntroductionImplicit integration: Newmark’s method

Remark: 𝛽 and 𝛾 are parameters that act as weights for calculating

the approximation of the acceleration.

Newmark originally proposed 𝛽 = 0.25, 𝛾 = 0.50, which results in an

unconditionally stable scheme.

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IntroductionExample: SDOF shear building

Equation

𝑚 ሷ𝑢 𝑡 + 𝑐 ሶx 𝑡 + 𝑘𝑢 𝑡 = −𝑚 ሷ𝑥𝑔 𝑡

Find 𝑐 for 5% damping. 𝑚 = 100 Mgr and 𝑘 = 150 kN/m

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Numerical integration of nonlinear structuresThe problem

Numerically solve the dynamic Equation of motion

𝐌 ሷ𝐱 𝑡 + 𝐂 ሶ𝐱 𝑡 + 𝐊𝐱 𝑡 + 𝐠(𝐱, ሶ𝐱, 𝑡) = 𝐟(𝑡)

𝐠(𝐱, ሶ𝐱, 𝑡): nonlinear term

Two classes of methods:

Explicit

Implicit

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Numerical integration of nonlinear structuresExplicit integration

State – space formulation of the dynamic Equation of motion

ሶ𝛏 𝑡 = 𝚨𝛏 𝑡 + 𝐁𝐟 𝑡 + 𝐊𝐡 𝛏, 𝑡𝐲 𝑡 = 𝐇𝛏 𝑡 + 𝐃𝐟(𝑡) + 𝐋𝐡(𝛏, 𝑡)

𝐠(𝐱, ሶ𝐱, 𝑡): nonlinear term

Two classes of methods:

Explicit

Implicit

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Numerical integration of nonlinear structuresImplicit integration

Modify Newmark’s method incremental formulation

𝐌 ሷ𝐱 𝑡𝑖 + 𝐂 ሶ𝐱 𝑡𝑖 + 𝐊𝐱 𝑡𝑖 + 𝐠(𝐱, ሶ𝐱, 𝑡𝑖) = 𝐟(𝑡𝑖)

𝐌 ሷ𝐱 𝑡𝑖+1 + 𝐂 ሶ𝐱 𝑡𝑖+1 + 𝐊𝐱 𝑡𝑖+1 + 𝐠(𝐱, ሶ𝐱, 𝑡𝑖+1) = 𝐟(𝑡𝑖+1)

Subtract:

𝐌𝛿 ሷ𝐱 𝑡𝑖 + 𝐂𝛿 ሶ𝐱 𝑡𝑖 + 𝐊𝛿𝐱 𝑡𝑖 + 𝛿𝐠(𝐱, ሶ𝐱, 𝑡𝑖) = 𝛿𝐟(𝑡𝑖)

Apply Newmark’s concept to the inremental formulation

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OutlookSubstructuring

+

Substructure A Substructure B

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OutlookModel Reduction

Original

SizeModes

Reduced

Size

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OutlookExercises

Study the Bouc – Wen model.

Search the literature for other types of structural nonlinearities.

Create corresponding MATLAB functions. Send us your code!

Embed them into the SDOF and MDOF structures analyzed today.

Search commercial FEM software for their implementations on

numerical integration.

How do they handle nonlinearities (for example, contact problems)?