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||Chair of Structural Mechanics
Workshop: nonlinear simulation using MATLAB
28.11.2016 1
Method of Finite Elements II
||Chair of Structural Mechanics 28.11.2016 2
Outline
Introduction
Numerical integration of nonlinear structures
MATLAB case study I : 1 – DOF shear building
MATLAB case study II: 100 – DOF shear building
Outlook
||Chair of Structural Mechanics 28.11.2016 3
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
||Chair of Structural Mechanics 28.11.2016 4
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
||Chair of Structural Mechanics 28.11.2016 5
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
||Chair of Structural Mechanics 28.11.2016 6
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
||Chair of Structural Mechanics 28.11.2016 7
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
||Chair of Structural Mechanics 28.11.2016 8
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.
||Chair of Structural Mechanics 28.11.2016 9
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
||Chair of Structural Mechanics 28.11.2016 10
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
||Chair of Structural Mechanics 28.11.2016 11
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
||Chair of Structural Mechanics 28.11.2016 12
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
||Chair of Structural Mechanics 28.11.2016 13
OutlookSubstructuring
+
Substructure A Substructure B
||Chair of Structural Mechanics 28.11.2016 14
OutlookModel Reduction
Original
SizeModes
Reduced
Size
||Chair of Structural Mechanics 28.11.2016 15
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)?