Design of tall buildings lecture-1-part-2-dr-shafiul-bari-sir
- 1. ANALYSIS AND DESIGN OF TALL BUILDINGS (CE 6114) Professor Dr. Shafiul Bari Professor, Department of Civil Engineering Bangladesh University of Engineering and Technology, BUET. Professor Dr. Shafiul Bari Professor, Department of Civil Engineering Bangladesh University of Engineering and Technology, BUET.
- 2. BOOKS Two Books (No 1 & 2) are Mandatory 1. B. Stafford & A. Coull (1991): Tall Building Structures: Analysis and Design , John Wiley & Sons. 2. W. Schueller (1977): High Rise Building Structures , John Wiley & Sons. 3. BNBC: Bangladesh National Building Code-1993 4.The Council on Tall Buildings and Urban Habitat (1995): Structural Systems for Tall Buildings, McGraw Hill. 5. Taranath, B. (1988): Structural Analysis and Design of Tall Buildings, 2nd. Ed., McGraw Hill. 6. White & Salmon, (1987): Building Structural Design Handbook, John Wiley & Sons. 7. Schueller (1996): The Design of Building Structures, Prentice Hall, New Jersey. 8. Nilson et el: Design of concrete structures . 9. Fintel: Handbook of concrete engineering 1. B. Stafford & A. Coull (1991): Tall Building Structures: Analysis and Design , John Wiley & Sons. 2. W. Schueller (1977): High Rise Building Structures , John Wiley & Sons. 3. BNBC: Bangladesh National Building Code-1993 4.The Council on Tall Buildings and Urban Habitat (1995): Structural Systems for Tall Buildings, McGraw Hill. 5. Taranath, B. (1988): Structural Analysis and Design of Tall Buildings, 2nd. Ed., McGraw Hill. 6. White & Salmon, (1987): Building Structural Design Handbook, John Wiley & Sons. 7. Schueller (1996): The Design of Building Structures, Prentice Hall, New Jersey. 8. Nilson et el: Design of concrete structures . 9. Fintel: Handbook of concrete engineering
- 3. Class Lectures (Total 12 Nos) • Lecture – 1 : Introduction • Lecture – 2 : Tall Building Criteria and Loading , Structural Form • Lecture – 3 : Architectural Planning (FAR) & HVAC system. • Lecture – 4 : Floor System (Behavior Under Gravity Load) • Lecture – 5 : Structural System (Behavior Under Lateral Load) • Lecture – 6 : Structural Modeling and Analysis of Tall Buildings • Lecture – 7 : Rigid Frame & Drift Control • Lecture – 8 : Shear Wall & Coupled Shear Wall • Lecture – 9 : Wall-Frame & Tubular System • Lecture – 10 : Design of Structural Members & Foundation Design • Lecture – 11 : Fire Safety Engineering • Lecture – 12 : Construction and Project Management Aspects • Lecture – 1 : Introduction • Lecture – 2 : Tall Building Criteria and Loading , Structural Form • Lecture – 3 : Architectural Planning (FAR) & HVAC system. • Lecture – 4 : Floor System (Behavior Under Gravity Load) • Lecture – 5 : Structural System (Behavior Under Lateral Load) • Lecture – 6 : Structural Modeling and Analysis of Tall Buildings • Lecture – 7 : Rigid Frame & Drift Control • Lecture – 8 : Shear Wall & Coupled Shear Wall • Lecture – 9 : Wall-Frame & Tubular System • Lecture – 10 : Design of Structural Members & Foundation Design • Lecture – 11 : Fire Safety Engineering • Lecture – 12 : Construction and Project Management Aspects
- 4. Lecture - 1 • What is “Tall Building” ? • Classification can be done in many ways: – Based on total height – Based on “Relative Height” – Based on number of floors – Based on structural response • From Structural Point of View: – Tall building is the one in which selection, behavior and design of lateral load resisting system significantly effects the overall design of the building • What is “Tall Building” ? • Classification can be done in many ways: – Based on total height – Based on “Relative Height” – Based on number of floors – Based on structural response • From Structural Point of View: – Tall building is the one in which selection, behavior and design of lateral load resisting system significantly effects the overall design of the building
- 5. Lecture - 1 • For any structural/floor systems, we will discuss: – How to select the system – What is the behavior of the system – How to model the system for structural analysis • How to integrate the systems and analyze the entire building model • How to interpret the analysis results for system components • How to design the components based on analysis results • How to use ETABS for modeling, analysis and design of building systems and components • For any structural/floor systems, we will discuss: – How to select the system – What is the behavior of the system – How to model the system for structural analysis • How to integrate the systems and analyze the entire building model • How to interpret the analysis results for system components • How to design the components based on analysis results • How to use ETABS for modeling, analysis and design of building systems and components
- 6. Lecture - 1 Special Considerations • Story Drift for Occupant Perception • Axial Shortening of Columns • Transfer Girders and Deep Beams • Shear Wall Design and Detailing • Construction Sequence Analysis • Seismic Performance and Dynamic Analysis Special Considerations • Story Drift for Occupant Perception • Axial Shortening of Columns • Transfer Girders and Deep Beams • Shear Wall Design and Detailing • Construction Sequence Analysis • Seismic Performance and Dynamic Analysis
- 7. Lecture - 1 Tall Building Systems • Building is an assemblage of various Systems – Basic Functional System – Structural System – HVAC System – Plumbing and Drainage System – Electrical, Electronic and Communication System – Security System – Other specialized systems Tall Building Systems • Building is an assemblage of various Systems – Basic Functional System – Structural System – HVAC System – Plumbing and Drainage System – Electrical, Electronic and Communication System – Security System – Other specialized systems
- 8. Lecture - 1 The Building Structural System - Conceptual • The Gravity Load Resisting System (GLRS) – The structural system (beams, slab, girders, columns, etc) that act primarily to support the gravity or vertical loads • The Lateral Load Resisting System (LLRS) – The structural system (columns, shear walls, bracing, etc) that primarily acts to resist the lateral loads • The Floor Diaphragm (FD) – The structural system that transfers lateral loads to the lateral load resisting system and provides in-plane floor stiffness The Building Structural System - Conceptual • The Gravity Load Resisting System (GLRS) – The structural system (beams, slab, girders, columns, etc) that act primarily to support the gravity or vertical loads • The Lateral Load Resisting System (LLRS) – The structural system (columns, shear walls, bracing, etc) that primarily acts to resist the lateral loads • The Floor Diaphragm (FD) – The structural system that transfers lateral loads to the lateral load resisting system and provides in-plane floor stiffness
- 9. Lecture - 1 Knowledge Model for System Selection • Architecture • Building Services • Construction Eng. • Value Eng. • Aesthetics • Ergonomics Eng. • Structural Eng. • Knowledge Eng. • Economics • Artificial Intelligence • System Eng. • Common Sense Knowledge Model for System Selection • Architecture • Building Services • Construction Eng. • Value Eng. • Aesthetics • Ergonomics Eng. • Structural Eng. • Knowledge Eng. • Economics • Artificial Intelligence • System Eng. • Common Sense
- 10. Lecture - 1 Selection of Structural System Function has considerable effect on the selection of structural system Based on Function/Occupancy of Tall Buildings: • Residential Buildings – Apartments – Hotels – Dormitories • Office and Commercial Buildings • Mixed Occupancy – Commercial + Residential • Industrial Buildings and Parking Garages Selection of Structural System Function has considerable effect on the selection of structural system Based on Function/Occupancy of Tall Buildings: • Residential Buildings – Apartments – Hotels – Dormitories • Office and Commercial Buildings • Mixed Occupancy – Commercial + Residential • Industrial Buildings and Parking Garages
- 11. Lecture - 1 Typical Characteristics of Residential Bldg • Known location of partitions and their load • Column lines generally matches architectural layout • Typical spans 15-22 ft • Tall buildings economy is achieved using the thinnest slab • One way pre-cast or flat slab – popular • Lateral load resistance provided by frame or shear walls • More or less fixed M/E system layouts Typical Characteristics of Residential Bldg • Known location of partitions and their load • Column lines generally matches architectural layout • Typical spans 15-22 ft • Tall buildings economy is achieved using the thinnest slab • One way pre-cast or flat slab – popular • Lateral load resistance provided by frame or shear walls • More or less fixed M/E system layouts
- 12. Lecture - 1 Typical Characteristics of Office and Commercial Bldg • Unknown location of partitions and their load • Typical spans 20-35 ft • Need for flexible M/E layouts • Post-tension or ribbed and flat slab with drop panel – popular • Ideal balance between vertical and lateral load resisting systems: sufficient shear walls to limit the resultant tension under gravity plus wind • Lateral load resistance varies significantly Typical Characteristics of Office and Commercial Bldg • Unknown location of partitions and their load • Typical spans 20-35 ft • Need for flexible M/E layouts • Post-tension or ribbed and flat slab with drop panel – popular • Ideal balance between vertical and lateral load resisting systems: sufficient shear walls to limit the resultant tension under gravity plus wind • Lateral load resistance varies significantly
- 13. Lecture - 1 Gravity Load Resisting Systems Purpose “ To Transfer Gravity Loads Applied at the Floor Levels down to the Foundation Level” • Direct Path Systems • Slab Supported on Load Bearing Walls • Slab Supported on Columns • Indirect Multi Path Systems • Slab Supported on Beams • Beams Supported on Other Beams • Beams Supported on Walls or Columns Gravity Load Resisting Systems Purpose “ To Transfer Gravity Loads Applied at the Floor Levels down to the Foundation Level” • Direct Path Systems • Slab Supported on Load Bearing Walls • Slab Supported on Columns • Indirect Multi Path Systems • Slab Supported on Beams • Beams Supported on Other Beams • Beams Supported on Walls or Columns
- 14. Lecture - 1 Vertical Load Resisting Systems 1. Slabs supported on Long Rigid Supports – Supported on stiff Beams or Walls – One-way and Two-way Slabs – Main consideration is flexural reinforcement 2. Slab-System supported on Small Rigid Supports – Supported on Columns directly – Flat Slab Floor systems – Main consideration is shear transfer, moment distribution in various parts, lateral load resistance 3. Slabs supported on soil – Slabs on Grade: Light, uniformly distributed loads – Footings, Mat etc. Heavy concentrated loads Vertical Load Resisting Systems 1. Slabs supported on Long Rigid Supports – Supported on stiff Beams or Walls – One-way and Two-way Slabs – Main consideration is flexural reinforcement 2. Slab-System supported on Small Rigid Supports – Supported on Columns directly – Flat Slab Floor systems – Main consideration is shear transfer, moment distribution in various parts, lateral load resistance 3. Slabs supported on soil – Slabs on Grade: Light, uniformly distributed loads – Footings, Mat etc. Heavy concentrated loads
- 15. Lecture - 1 Popular Gravity Load Resting Systems • Direct Load Transfer Systems (Single load transfer path) – Flat Slab and Flat Plate – Beam-Slab – Waffle Slab – Wall Joist • Indirect Load Transfer System (Multi step load transfer path) – Beam, Slab – Girder, Beam, Slab – Girder, Joist Popular Gravity Load Resting Systems • Direct Load Transfer Systems (Single load transfer path) – Flat Slab and Flat Plate – Beam-Slab – Waffle Slab – Wall Joist • Indirect Load Transfer System (Multi step load transfer path) – Beam, Slab – Girder, Beam, Slab – Girder, Joist
- 16. Lecture - 1 – Some Sample Floor Slab Types – Slab Only – Hollow Core Slab – Pre-Cast Slab Panels – Beam and Slab – Beam-Slab – Girder-Beam Slab – Joist Slab – Girder-Joist Slab – Pre-Cast Slab-Beam System – Some Sample Floor Slab Types – Slab Only – Hollow Core Slab – Pre-Cast Slab Panels – Beam and Slab – Beam-Slab – Girder-Beam Slab – Joist Slab – Girder-Joist Slab – Pre-Cast Slab-Beam System
- 17. Lecture - 1 • Some Sample Floor Slab Systems• Some Sample Floor Slab Systems Flat Slab Flat Slab + Capital Flat Slab + Drop Panel Flat Slab + Drop Panel + Capital Waffle Slab Band Slab Inverted Band Slab Metal Deck/ Wood Deck Composite Metal Deck Hollow Block Slab Composite Girder-Slab Composite Truss Slab Alpha Truss System Wooden Beam, Rafter Plank System Sandwich Panels Flat Plate Flat Plate
- 18. Lecture - 1 Selection of Layout and Type of Slab • Basic Consideration • Span Length: Small, Medium, Long • Panel aspect ratio: Square, rectangular, oblong • Loads: Light, Medium, Heavy • Ducts and Piping: Electrical, Mechanical, Water supply • Openings: Size and Location • Architectural consideration: Aesthetics, clearance, etc • Special elements: Drop Panel, Column Capital, Beams • Construction Considerations: Form work, time, case of considerations Selection of Layout and Type of Slab • Basic Consideration • Span Length: Small, Medium, Long • Panel aspect ratio: Square, rectangular, oblong • Loads: Light, Medium, Heavy • Ducts and Piping: Electrical, Mechanical, Water supply • Openings: Size and Location • Architectural consideration: Aesthetics, clearance, etc • Special elements: Drop Panel, Column Capital, Beams • Construction Considerations: Form work, time, case of considerations
- 19. Lecture - 1 • Flat plates Small Spans ( 4.5 to 6 m) Relatively light load < 500 Kg/m2 ( 5 kPa ) Common for residential buildings Used where Drop panel or Column capital undesirable Easy construction. Low overall height of building • Flat Slabs Medium Spans ( 6.0 to 9.0 m) Relatively heavy load > 500 Kg/m2 ( 5 kPa ) Common in industrial floors, parking areas etc. • Waffle Slab Large spans ( 7.5 – 12 m ) Relatively heavy loads Common for public buildings. More aesthetic appearance Difficult to construct. Requires special form work • Flat plates Small Spans ( 4.5 to 6 m) Relatively light load < 500 Kg/m2 ( 5 kPa ) Common for residential buildings Used where Drop panel or Column capital undesirable Easy construction. Low overall height of building • Flat Slabs Medium Spans ( 6.0 to 9.0 m) Relatively heavy load > 500 Kg/m2 ( 5 kPa ) Common in industrial floors, parking areas etc. • Waffle Slab Large spans ( 7.5 – 12 m ) Relatively heavy loads Common for public buildings. More aesthetic appearance Difficult to construct. Requires special form work
- 20. Lecture - 1 Selection of Slab System • Beam-Slabs / Beam and Slabs Medium to large spans ( 5 to 10 m) Relatively economical in concrete and steel cost Greater depth: Increase in building height • Band Slab Medium spans ( 5 to 9 m) More economical than Flat Slab Common in industrial floors, parking areas etc. • One-way Joist and Beams Large spans ( 7.5 – 12 m ) Easier to construct than Waffle slabs Suitable for high-rise office building Beams on shorter side, Joist on longer side Selection of Slab System • Beam-Slabs / Beam and Slabs Medium to large spans ( 5 to 10 m) Relatively economical in concrete and steel cost Greater depth: Increase in building height • Band Slab Medium spans ( 5 to 9 m) More economical than Flat Slab Common in industrial floors, parking areas etc. • One-way Joist and Beams Large spans ( 7.5 – 12 m ) Easier to construct than Waffle slabs Suitable for high-rise office building Beams on shorter side, Joist on longer side
- 21. Lecture - 1 Purpose “ To Transfer Lateral Loads Applied at any location in the structure down to the Foundation Level” • Single System • Moment Resisting Frames • Braced Frames • Shear Walls • Tubular Systems • Dual System • Shear Wall - Frames • Tube + Frame + Shear Wall Purpose “ To Transfer Lateral Loads Applied at any location in the structure down to the Foundation Level” • Single System • Moment Resisting Frames • Braced Frames • Shear Walls • Tubular Systems • Dual System • Shear Wall - Frames • Tube + Frame + Shear Wall
- 22. Lecture - 1 Lateral Loads • Primary Lateral Loads – Load generated by Wind Pressure – Load generated due to Seismic Excitation • Other Lateral Loads – Load generated due to horizontal component of Gravity Loads in Inclined Systems and in Un-symmetrical structures – Load due to lateral soil pressure, liquid and material retention Lateral Loads • Primary Lateral Loads – Load generated by Wind Pressure – Load generated due to Seismic Excitation • Other Lateral Loads – Load generated due to horizontal component of Gravity Loads in Inclined Systems and in Un-symmetrical structures – Load due to lateral soil pressure, liquid and material retention
- 23. Lecture - 1 Sample Lateral Load Resistance Systems • Bearing wall system – Light frames with shear panels – Load bearing shear walls • Fully Braced System (FBS) – Shear Walls (SW) – Diagonal Bracing (DB) • Moment Resisting Frames (MRF) – Special Moment-Resisting Frames (SMRF) – Concrete Intermediate Moment-Resisting Frame (IMRF) – Ordinary Moment-Resisting Frame (OMRF) • Dual Systems (DS) – Shear Walls + Frames (SWF) – Ordinary Braced Frame (OBF) – Special Braced Frame (SBF) Sample Lateral Load Resistance Systems • Bearing wall system – Light frames with shear panels – Load bearing shear walls • Fully Braced System (FBS) – Shear Walls (SW) – Diagonal Bracing (DB) • Moment Resisting Frames (MRF) – Special Moment-Resisting Frames (SMRF) – Concrete Intermediate Moment-Resisting Frame (IMRF) – Ordinary Moment-Resisting Frame (OMRF) • Dual Systems (DS) – Shear Walls + Frames (SWF) – Ordinary Braced Frame (OBF) – Special Braced Frame (SBF)
- 24. Lecture - 1 Moment Resisting Frame • The Load is transferred by shear in columns, that produces moment in columns and in beams • The Beam-Column connection is crucial for the system to work • The moments and shear from later loads must be added to those from gravity loads Shear Wall and Frame • The lateral loads is primarily resisted by the shear in the walls, in turn producing bending moment • The openings in wall become areas of high stress concentration and need to be handled carefully • Partial loads is resisted by the frames • Traditionally 75/25 distribution haws been used Moment Resisting Frame • The Load is transferred by shear in columns, that produces moment in columns and in beams • The Beam-Column connection is crucial for the system to work • The moments and shear from later loads must be added to those from gravity loads Shear Wall and Frame • The lateral loads is primarily resisted by the shear in the walls, in turn producing bending moment • The openings in wall become areas of high stress concentration and need to be handled carefully • Partial loads is resisted by the frames • Traditionally 75/25 distribution haws been used
- 25. Lecture - 1 Sample Lateral Load Resistance Systems • Shear Wall - Frame • The Walls are part of the frame and act together with the frame members • The lateral loads is primarily resisted by the shear in the walls, in turn producing bending moment. • Partial loads is resisted by the frame members in moment and shear Braced Frame • The lateral loads is primarily resisted by the Axial Force in the braces, columns and beams in the braced zone. • The frame away from the braced zone does not have significant moments • Bracing does not have to be provided in every bay, but should be provided in every story Sample Lateral Load Resistance Systems • Shear Wall - Frame • The Walls are part of the frame and act together with the frame members • The lateral loads is primarily resisted by the shear in the walls, in turn producing bending moment. • Partial loads is resisted by the frame members in moment and shear Braced Frame • The lateral loads is primarily resisted by the Axial Force in the braces, columns and beams in the braced zone. • The frame away from the braced zone does not have significant moments • Bracing does not have to be provided in every bay, but should be provided in every story
- 26. Lecture - 1 Sample Lateral Load Resistance Systems • Tubular Structure • The system is formed by using closely spaced columns and deep spandrel beams • The lateral loads is primarily resisted by the entire building acting as a big cantilever with a tubular/ box cross-section • There is a “shear lag” problem between opposite faces of the tube due to in-efficiency of column beam connection • The height to width ratio should be more than 5 Braced Tube Systems • Diagonal Braces are added to the basic tubular structure • This modification of the Tubular System reduces shear lag between opposite faces Sample Lateral Load Resistance Systems • Tubular Structure • The system is formed by using closely spaced columns and deep spandrel beams • The lateral loads is primarily resisted by the entire building acting as a big cantilever with a tubular/ box cross-section • There is a “shear lag” problem between opposite faces of the tube due to in-efficiency of column beam connection • The height to width ratio should be more than 5 Braced Tube Systems • Diagonal Braces are added to the basic tubular structure • This modification of the Tubular System reduces shear lag between opposite faces
- 27. Lecture - 1 Transfer of Lateral Loads • The lateral loads are mostly applied or generated away from the lateral load resisting systems • They must be transferred to the LLRS through the “in-plane” stiffness of the floor diaphragm • The in-plane stiffness may be prided by: – Concrete Floor Slab – In-plane trusses Transfer of Lateral Loads • The lateral loads are mostly applied or generated away from the lateral load resisting systems • They must be transferred to the LLRS through the “in-plane” stiffness of the floor diaphragm • The in-plane stiffness may be prided by: – Concrete Floor Slab – In-plane trusses
- 28. Lecture - 1 Building Response • Objective: To determine the load path gravity and lateral loads • For Gravity Loads - How Gravity Loads are Distributed – Analysis of Gravity Load Resisting System for: • Dead Load, Live Live Load, Pattern Loads, temperature, shrinkage – Important Elements: Floor slabs, beams, openings, Joists, etc. • For Lateral Loads – How Lateral Loads are Distributed – Analysis of Lateral Load Resisting System for: • Wind Loads, Seismic Loads, Structural Un-symmetry – Important elements: Columns, shear walls, bracing , beams Building Response • Objective: To determine the load path gravity and lateral loads • For Gravity Loads - How Gravity Loads are Distributed – Analysis of Gravity Load Resisting System for: • Dead Load, Live Live Load, Pattern Loads, temperature, shrinkage – Important Elements: Floor slabs, beams, openings, Joists, etc. • For Lateral Loads – How Lateral Loads are Distributed – Analysis of Lateral Load Resisting System for: • Wind Loads, Seismic Loads, Structural Un-symmetry – Important elements: Columns, shear walls, bracing , beams
- 29. Lecture - 1 Excitation Structure Response Basic Analysis Type Static Elastic Linear Linear-Elastic-Static Analysis Static Elastic Nonlinear Nonlinear-Elastic-Static Analysis Static Inelastic Linear Linear-Inelastic-Static Analysis Static Inelastic Nonlinear Nonlinear-Inelastic-Static Analysis Dynamic Elastic Linear Linear-Elastic-Dynamic Analysis Dynamic Elastic Nonlinear Nonlinear-Elastic-Dynamic Analysis Dynamic Inelastic Linear Linear-Inelastic-Dynamic Analysis Dynamic Inelastic Nonlinear Nonlinear-Inelastic-Dynamic Analysis
- 30. Lecture - 1 Some More Solution Types • Non-linear Analysis – P-Delta Analysis – Buckling Analysis – Static Pushover Analysis – Fast Non-Linear Analysis (FNA) – Large Displacement Analysis • Dynamic Analysis – Free Vibration and Modal Analysis – Response Spectrum Analysis – Steady State Dynamic Analysis Some More Solution Types • Non-linear Analysis – P-Delta Analysis – Buckling Analysis – Static Pushover Analysis – Fast Non-Linear Analysis (FNA) – Large Displacement Analysis • Dynamic Analysis – Free Vibration and Modal Analysis – Response Spectrum Analysis – Steady State Dynamic Analysis
- 31. Lecture - 1 Static Vs Dynamic • Static Excitation – When the Excitation (Load) does not vary rapidly with Time – When the Load can be assumed to be applied “Slowly” • Dynamic Excitation – When the Excitation varies rapidly with Time – When the “Inertial Force” becomes significant • Most Real Excitation are Dynamic but are considered “Quasi Static” • Most Dynamic Excitation can be converted to “Equivalent Static Loads” Static Vs Dynamic • Static Excitation – When the Excitation (Load) does not vary rapidly with Time – When the Load can be assumed to be applied “Slowly” • Dynamic Excitation – When the Excitation varies rapidly with Time – When the “Inertial Force” becomes significant • Most Real Excitation are Dynamic but are considered “Quasi Static” • Most Dynamic Excitation can be converted to “Equivalent Static Loads”
- 32. Lecture - 1 Elastic Vs Inelastic • Elastic Material – Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation • Inelastic Material – Does not follow the same path during loading and unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation • Most materials exhibit both, elastic and inelastic behavior depending upon level of loading. Elastic Vs Inelastic • Elastic Material – Follows the same path during loading and unloading and returns to initial state of deformation, stress, strain etc. after removal of load/ excitation • Inelastic Material – Does not follow the same path during loading and unloading and may not returns to initial state of deformation, stress, strain etc. after removal of load/ excitation • Most materials exhibit both, elastic and inelastic behavior depending upon level of loading.
- 33. Lecture - 1 Linear Vs Nonlinear • Linearity – The response is directly proportional to excitation – (Deflection doubles if load is doubled) • Non-Linearity – The response is not directly proportional to excitation – (deflection may become 4 times if load is doubled) • Non-linear response may be produced by: – Geometric Effects (Geometric non-linearity) – Material Effects (Material non-linearity) – Both Linear Vs Nonlinear • Linearity – The response is directly proportional to excitation – (Deflection doubles if load is doubled) • Non-Linearity – The response is not directly proportional to excitation – (deflection may become 4 times if load is doubled) • Non-linear response may be produced by: – Geometric Effects (Geometric non-linearity) – Material Effects (Material non-linearity) – Both