CE 137 Structural Dynamics PDF Lecture Notes - University of the Philippines

For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. CE 137 – Structural...

For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. CE 137 – Structural Dynamics and Earthquake Engineering Chapter 1 – Introduction to Structural Dynamics First Semester A.Y. 2023-2024 1 1st Semester, AY 2022-2023 LEARNING OBJECTIVES At the end of this chapter, the student should be able to: 1. Differentiate between static loads and dynamic loads; 2. Discuss the different types of dynamic loads; 3. Define structural dynamics; 4. Relate structural dynamics to structural analysis; CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 2 1 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 LEARNING OBJECTIVES 5. Explain the dynamic response of typical structures. 6. Define vibration; 7. Identify the types of vibratory motion; 8. Explain the functional elements of SDOF systems; and 9. Derive the equations of motion. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 3 1st Semester, AY 2022-2023 Structural Dynamics Branch of structural engineering which studies the dynamic properties of structures (natural frequency and damping mechanism) and the effects of the dynamic loads (or excitations) to these structures in terms of stresses and deformations (or velocities or acceleration) Analysis can be linear or nonlinear  Linear – Material stresses are within elastic limit  Nonlinear – material exhibits permanent sets CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 8 2 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.1 Static vs. Dynamic Load Static Loads Dynamic Loads  Loads that are not  Loads that are accelerating accelerating and produces vibration 𝐹⃗ = 0  Can be moving but the 𝐹⃗ = 𝑚𝑢̈ velocity is constant  Time dependent  Do not vary with time CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 9 1st Semester, AY 2022-2023 1.2 Types of Dynamic Loads www.abroadlanguages.com blog.smartbear.com johnlewis.com Live Loads CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 10 3 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.2 Types of Dynamic Loads Blast Impact Source: Khan and Javed, 2017 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 11 1st Semester, AY 2022-2023 1.2 Types of Dynamic Loads Earthquake Wind Source: Khan and Javed, 2017 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 12 4 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.3 Overview of Structural Dynamics Source: Lynch, 2004 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 13 1st Semester, AY 2022-2023 1.3 Overview of Structural Dynamics What is Dynamic Response? 1. Deformations (displacement, velocity and acceleration) 2. Element Forces (stresses, strains) Source: Sullivan, 2018 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 14 5 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.4 Relation of Structural Dynamics to Structural Analysis Static ES 11, ES 13, Static Linear CE 131, CE132 Analysis Undergraduate Static Thesis, Master Structural Nonlinear Course Analysis Dynamic CE 137 Dynamic Linear Analysis Dynamic Undergraduate Nonlinear Thesis, Master Course Source: Aguirre, 2015 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 15 1st Semester, AY 2022-2023 1. 5 Structures and their Vibration https://www.youtube.com/watch?v=HB2jgJJG2is CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 16 6 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1. 5 Structures and their Vibration https://www.youtube.com/watch?v=FnKd-wC57Lk CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 17 1st Semester, AY 2022-2023 1.6 Definition of Vibration Vibration – a type of dynamic behavior in which a system oscillates about a certain equilibrium position “oscillate” – forward and backward translation or rotation of a structure around a certain position “dynamic” – may be defined as time-varying y  u 0 sin  n t    A T CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 18 7 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.7 Types of Vibration Periodic Motion Non-periodic Motion  motion which  does not repeat itself repeats itself at at constant interval regular intervals of time CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 19 1st Semester, AY 2022-2023 Simple Harmonic Motion  motion that can be characterized by sinusoidal motion at constant frequency  define the response as the projection of P on the y-axis CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 20 8 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Simple Harmonic Motion hyperphysics.phy-astr.gsu.edu Amplitude (A) Angular frequency (ω)  Maximum response of a  Rate at which the motion periodic motion oscillates in one complete Period (T) revolution (rad/s)  Time it takes to Frequency (f) complete one cycle of  Number of cycles per unit time motion (cps or Hz) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 21 1st Semester, AY 2022-2023 1.8 Types of Vibrating System Rigid System Compliant Systems  one in which no strains  different points within the occur system may move  all points in the system differently (and out of move in phase with each phase) from each other other  characterized by distributed mass CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 22 9 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Compliant Systems Discrete Continuous/Distributed  mass can be considered to  mass is distributed be concentrated at a finite throughout the system number of locations Source: Khan and Javed, 2017 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 23 1st Semester, AY 2022-2023 Discrete Systems Single Degree of Freedom Systems (SDOF) Multi-Degree of Freedom Systems (MDOF) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 24 10 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 1.9 Elements of Discrete Systems  Spring element (k)  The stiffness of the structure  Damping element (c)  The energy dissipator of the structure  Mass element (m)  The inertial characteristics of the structure CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 25 1st Semester, AY 2022-2023 M SDOF System Massless frame with stiffness, k Dashpot representing damping (a) Original state Displacement, Applied force, u p(t) M (unknown parameter) (b) After applying force CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 26 11 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 SDOF System A system whose position can be described completely by a single coordinate (i. e. translational displacement or rotational displacement) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 27 1st Semester, AY 2022-2023 Degrees of Freedom (DOF) Defined as the number of independent coordinates necessary to describe the position of all significant masses of a system Discrete systems Finite DOF Compliant systems Infinite DOF CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 28 12 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Inertia Force ( ) and Mass Element (m) Inertia: “object’s amount of resistance to a change in velocity (which is quantified by its mass), or sometimes its momentum” (Newton’s first law of motion) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 29 1st Semester, AY 2022-2023 Inertia Force ( ) and Mass Element (m) Inertia force D’Alembert’s principle of fI dynamic equilibrium: M fictitious inertia force which acts in the direction opposite to the acceleration. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 30 13 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Inertia Force ( ) and Mass Element (m) Mass element - represents the mass and inertial characteristics of the structure (1-1) where: m = mass 𝑢̈ = acceleration CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 31 1st Semester, AY 2022-2023 Structural resisting or Spring Force (fs) Deformation, u fs Static force, fs Structural resisting force (FREE-BODY DIAGRAM) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 32 14 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Structural resisting or Spring Force (fs) Linear relations Nonlinear relations 𝑓 =ku 𝑓 = f 𝑢, 𝑢̇ Source: Chopra, A.(1995) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 33 1st Semester, AY 2022-2023 Structural resisting or Spring Force (fs) 𝒇𝒔 = 𝒌𝒖 (1-2) u = structural deformation k = lateral stiffness of the frame (unit: force/length) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 34 15 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Spring Element (k) Represents the restoring force and the potential energy capacity of a structure CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 35 1st Semester, AY 2022-2023 Damping When allowed to freely oscillate, this pendulum will come to a stop because of damping. Image source: https://commons.wikimedia.org/wiki/File:Simple_P endulum_Oscillator.gif With damping, the Image source: Chopra, A.(1995) amplitudes decreases. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 36 16 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Damping Element (c) Represents the energy dissipation in a structure Most of the energy dissipation presumably arises from: thermal effect of repeated elastic straining of the material internal friction when a solid is deformed CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 37 1st Semester, AY 2022-2023 Damping Element (c) Represents the energy dissipation in a structure Most of the energy dissipation presumably arises from: thermal effect of repeated elastic straining of the material internal friction when a solid is deformed CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 38 17 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Damping In actual building structures, mechanisms of energy dissipation include:  friction at steel connections  opening and closing of microcracks in concrete  friction between structure and non-structural elements such as partition walls Very difficult to quantify per mechanism, for an actual structural system CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 39 1st Semester, AY 2022-2023 Damping Force ( ) Image source: Chopra, A.(1995) (1-3) where: c = viscous damping coefficient (force x time/length) 𝑢̇ = velocity CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 40 18 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 2.0 Equations of Motion Source: Chopra, A.(1995) (FREE-BODY DIAGRAM OF A MOVING MASS) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 41 1st Semester, AY 2022-2023 2.0 Equations of Motion - General 𝑚𝑎 = 𝐹 𝑚𝑢̈ = 𝑝 𝑡 − 𝑓 − 𝑚𝑢̈ = 𝑝 𝑡 − 𝑘𝑢 − 𝑐𝑢̇ Equation of motion for an SDOF system subjected to an external force that varies with time, t: 𝒎𝒖̈ + 𝒄𝒖̇ + 𝒌𝒖 = 𝒑 𝒕 (1-4) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 42 19 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 2.0 Equations of Motion Can be formulated using:  Dynamic Equilibrium (D’ Alembert’s principle)  Principle of Virtual Work  Hamilton’s Principle (Lagrange’s equation) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 43 1st Semester, AY 2022-2023 Principle of Virtual Work For a system that is in equilibrium, the work done by all the forces during an assumed displacement (virtual displacement) which is compatible with the system constraints, is equal to zero.  W  0 (1-5) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 44 20 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Principle of Virtual Work The equations of motion are obtained by introducing virtual displacements corresponding to each degree of freedom and equating the resulting work done to zero. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 45 1st Semester, AY 2022-2023 Principle of Virtual Work (1-6a) (1-6b) (1-6c) (1-6d) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 46 21 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Principle of Virtual Work Since 𝜹𝒖 is arbitrarily selected as not equal to zero, the other factors in Eq. (1-6d) must be equal to zero: Hence, (1-7e) (1-7f) CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 47 1st Semester, AY 2022-2023 Ground Motion Excitation 𝑢 - No applied force - Ground itself is actually moving 𝑢 - Total structural displacement, 𝑢 : 𝑢 = relative displacement between the structure and the ground 𝑢 = ground displacement 𝑢 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 48 22 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 Ground Motion Excitation (cont.) 𝑢 = 𝑢 + 𝑢 (1-8) By differentiating twice with respect to time, 𝒖̈𝒕 = 𝒖̈ 𝒈 +𝒖̈ (1-9) 𝑚𝑎 = 𝐹 (Newton’s 2nd law of motion) 𝑚[𝑢̈ + 𝑢̈ ] = −𝑘𝑢 − 𝑐𝑢̇ (equation of motion under 𝒎𝒖̈ + 𝒌𝒖 + 𝒄𝒖̇ = −𝒎𝒖̈ 𝒈 (1-10) ground motion excitation) 𝒎𝒖̈ + 𝒌𝒖 + 𝒄𝒖̇ = 𝒑 𝒕 CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 49 1st Semester, AY 2022-2023 SUMMARY 1. Static loads vs. dynamic loads; 2. Types of dynamic loads (live, wind, earthquake, blast and impact); 3. Structural dynamics – properties of structure, response of structures under excitation; 4. Structural dynamics - focus on dynamic linear; CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 50 23 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 SUMMARY 5. Dynamic response of typical structures (deformation and stress) 6. Vibration – dynamic behavior, oscillation with respect to equilibrium position 7. Types of vibratory motion (periodic vs. nonperiodic, simple harmonic motion) 8. 3 Elements of discrete systems (mass, spring, damping) in an SDOF system 9. Equations of motion – general and with ground motion excitation CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 51 1st Semester, AY 2022-2023 References  Chopra, A. K. (1995) Dynamics of Structures: Theory and Applications to Earthquake Engineering. Prentice-Hall.  Khan, A. N. and Javed, M. Module 3: Fundamentals of Dynamic Analysis for SDOF Systems. Lecture on CE 412-Introduction to Structural Dynamics and Earthquake Engineering. Viewed 28 August 2023, https://www.slideshare.net/javeduet/module-3- spring-2020pdf  Lynch, J. (2004) Lecture on Structural Dynamics. Viewed November 10, 2004, http://www.engin.umich.edu/class/cee511/  Sullivan, T. (2018) ENCI335: Structural Dynamics Module Notes. University of Canterbury, New Zealand. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 52 24 For academic use only. Do not reproduce, 2nd Semester A. Y. 2024-2025 publish, distribute, or modify for commercial purposes. 1st Semester, AY 2022-2023 References  Tipler, P. A., & Mosca, G. (2008) Physics for scientists and engineers, 6th Ed. New York: W. H. Freeman and Company.  Zafra, R.G. (2019) CE 137 Structural Dynamics and Earthquake Engineering Lecture Notes. University of the Philippines – Los Baños. CE 137 – Structural Dynamics and Earthquake Engineering Ch. 1 – Introduction to Structural Dynamics 53 25