Structure analysis cơ học kết cấu

CIVE4127 CHAPTER STRUCTURAL ANALYSIS Introduction Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis The Brooklyn Bridge Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al CIVE4127 Structural Analysis CIVE4127 Contents Overview The Design Process: Relationship of Analysis to Design Strength and Serviceability Historical Development of Structural Systems Basic Structural Elements Assembling Basic Elements to Form a Stable Structural System Analyzing by Computer Preparation of Computations Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al 1-3 Structural Analysis CIVE4127 Overview • As an engineer involved with the design of buildings, bridges, and the other structures, you will be required to make many technical decisions about structural systems such as: selecting an efficient, economical, and attractive structural form evaluating its safety, that is, its strength and stiffness planning it erection under temporary construction loads • To design a structure, you will learn to carry out a structural analysis that establishes the internal force and deflection at all points produced by the design loads Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al 1-4 Structural Analysis CIVE4127 Analyzing Basic Structural Elements • During previous courses in static and mechanics of solids, you developed some background in structural analysis • You will now broaden your background in structural analysis, in systematic way, a variety of techniques for determining the forces in and the deflections of a number of basic structural elements: beams, trusses, frames, arches, and cables • Moreover, as you work analysis problems and examine the distribution of forces in various types of structures, you will understand more about how structures are stressed and deformed by load And you will gradually develop a clear sense of which structural configuration is optimal for a particular design situation • Further, as you develop an almost intuitive sense of how a structure behaves, you wi1l learn to estimate with a few simple computations the approximate values of forces at the most critical sections of the structure Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Analyzing Two-Dimensional Structures • Structure is a complex three-dimensional system composed of beams, columns, slabs, walls, and diagonal bracing Although load applied at a particular point in a three-dimensional structure will stress all adjacent members, most of the load is typically transmitted through certain key members directly to other supporting members or into the foundation • Once the behavior and function of the various components of most three-dimensional structures are understood, the designer can typically simplify the analysis of the actual structure by subdividing it into smaller two-dimensional subsystems that act as beams, trusses, or frames This procedure also significantly reduces the complexity of the analysis because two-dimensional structures are much easier and faster to analyze than threedimensional structures Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 The Design Process: Relationship of Analysis to Design • The design of any structure is typically carried out in alternating steps of design and analysis Each step supplies new information that permits the designer to proceed to the next phase The process continues until the analysis indicates that no changes in member sizes are required The specific steps of the procedure are described below  Conceptual Design  Preliminary Design  Analysis of Preliminary Designs  Redesign of the Structures  Evaluation of Preliminary Designs  Final Design and Analysis Phases Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Strength and Serviceability • The designer must proportion structures so that they will neither fail nor deform excessively under any possible loading conditions • Although structures must be designed with an adequate factor of safety to reduce the probability of failure to an acceptable level, the engineer must also ensure that the structure has sufficient stiffness to function usefully under all loading conditions Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Collapse of Cần Thơ Bridge Leaning Tower of Pisa Structural Analysis CIVE4127 Historical Development of Structural Systems • The evolution of structural forms is closely related to:  the materials available,  the state of construction technology,  the designer's knowledge of structural behavior (and much later, analysis),  the skill of the construction worker Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Historical Development of Structural Systems • The early Egyptian builders used stone quarried from sites along the Nile to construct temples and pyramids • Since the tensile strength of stone, a brittle material, is low and highly variable (because of a multitude of internal cracks and voids), beam spans in temples had to be short Luxor Temple Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Derivation of the Slope-Deflection Equation • Analyze span AB of the continuous beam to develop the slope-deflection equation, which relates the moments at the ends of members to the end displacements and the applied loads • Since differential settlements of supports in continuous members also create end moments, this effect will be included in the derivation  The beam, which is initially straight, has a constant cross section  The distributed load w(x), which can vary in any arbitrary manner along the beam's axis  Supports A and B settle, respectively, by amounts A and B to A’ and B’ Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis Derivation of the Slope-Deflection Equation • Assume that moments acting at the ends of members in the clockwise direction are positive Clockwise rotations of the ends of members will also be considered positive • The moment curves produced by both the w(x) and the end moments MAB and MBA are drawn by parts • The moment curve associated with the distributed load is called the simple beam moment curve • The moment curve for each force has been plotted on the side of the beam that is placed in compression by that particular force Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al CIVE4127 Structural Analysis CIVE4127 Derivation of the Slope-Deflection Equation • All angles and rotations are shown in the positive sense; that is, all have undergone clockwise rotations from the original horizontal position of the axis • The second moment-area theorem is used to establish the relationship between the member end moments MAB and MBA and the rotational deformations of the elastic curve 𝛾𝐴 = 𝑡𝐵𝐴 𝐿 𝛾𝐵 = 𝑡𝐴𝐵 𝐿 𝜓𝐴𝐵 = Δ𝐵 −Δ𝐴 𝐿 𝛾𝐴 = 𝜃𝐴 − 𝜓𝐴𝐵 and 𝛾𝐵 = 𝜃𝐵 − 𝜓𝐴𝐵 𝑡𝐵𝐴 𝜃𝐴 − 𝜓𝐴𝐵 = 𝐿 and 𝑡𝐴𝐵 𝜃𝐵 − 𝜓𝐴𝐵 = 𝐿 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis Derivation of the Slope-Deflection Equation  𝑡𝐴𝐵 =  𝑡𝐵𝐴 = 𝑀𝐵𝐴 𝐿 2𝐿 𝑀𝐴𝐵 𝐿 𝐿 𝐴𝑀 𝑥ҧ 𝐴 − 𝐸𝐼 𝐸𝐼 𝐸𝐼 𝑀𝐴𝐵 𝐿 2𝐿 𝑀𝐵𝐴 𝐿 𝐿 𝐴𝑀 𝑥ҧ 𝐵 + 𝐸𝐼 𝐸𝐼 𝐸𝐼 • The 𝐴𝑀 𝑥ҧ 𝐴 and 𝐴𝑀 𝑥ҧ 𝐵 are the first moment of the area under the simple beam moment curve about the ends of the beam • Assume that the contribution of each, moment curve to the tangential deviation is positive if it increases the tangential deviation and negative if it decreases the tangential deviation Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al CIVE4127 Structural Analysis CIVE4127 Derivation of the Slope-Deflection Equation • To illustrate the computation of 𝐴𝑀 𝑥ҧ 𝐴 , consider a beam carrying a uniformly distributed load w  𝐴𝑀 𝑥ҧ 𝐴 =area.𝑥ҧ = 2𝐿 𝑤𝐿2 𝐿 𝑤𝐿4 = 24 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Derivation of the Slope-Deflection Equation 𝑀𝐵𝐴 𝐿 2𝐿 𝑀𝐴𝐵 𝐿 𝐿 𝐴𝑀 𝑥ҧ 𝐴  𝜃𝐴 − 𝜓𝐴𝐵 = − − 𝐿 𝐸𝐼 𝐸𝐼 𝐸𝐼 𝑀𝐴𝐵 𝐿 2𝐿 𝑀𝐵𝐴 𝐿 𝐿 𝐴 𝑥ҧ  𝜃𝐵 − 𝜓𝐴𝐵 = − + 𝑀 𝐵 𝐿 𝐸𝐼 𝐸𝐼 𝐸𝐼 2𝐸𝐼 𝐴 𝑥ҧ 𝐴 𝑥ҧ  𝑀𝐴𝐵 = 2𝜃𝐴 + 𝜃𝐵 − 3𝜓𝐴𝐵 + 𝑀2 𝐴 − 𝑀2 𝐵 𝐿 𝐿 𝐿 2𝐸𝐼 𝐴𝑀 𝑥ҧ 𝐴 𝐴𝑀 𝑥ҧ 𝐵  𝑀𝐵𝐴 = 2𝜃𝐵 + 𝜃𝐴 − 3𝜓𝐴𝐵 + − 𝐿 𝐿2 𝐿2 • The last two terms that contain the quantities 𝐴𝑀 𝑥ҧ 𝐴 and 𝐴𝑀 𝑥ҧ 𝐵 are a function of the loads applied between ends of the member only • The member end moments MAB and MBA which are also termed fixed-end moments, may be designated FEMAB and FEMBA Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Derivation of the Slope-Deflection Equation  𝐹𝐸𝑀𝐴𝐵 =  𝐹𝐸𝑀𝐵𝐴 = 2𝐸𝐼  𝑀𝐴𝐵 = 𝐿 2𝐸𝐼  𝑀𝐵𝐴 = 𝐿 𝐴𝑀 𝑥ҧ 𝐴 𝐴𝑀 𝑥ҧ 𝐵 𝑀𝐴𝐵 = − 𝐿2 𝐿2 𝐴𝑀 𝑥ҧ 𝐴 𝐴𝑀 𝑥ҧ 𝐵 𝑀𝐵𝐴 = − 𝐿2 𝐿2 2𝜃𝐴 + 𝜃𝐵 − 3𝜓𝐴𝐵 +𝐹𝐸𝑀𝐴𝐵 2𝜃𝐵 + 𝜃𝐴 − 3𝜓𝐴𝐵 +𝐹𝐸𝑀𝐵𝐴  Denote the end where the moment is being computed as the near end (N) and the opposite end as the far end (F)  𝑀𝑁𝐹 = 2𝐸𝐼 𝐿 2𝜃𝑁 + 𝜃𝐹 − 3𝜓𝑁𝐹 +𝐹𝐸𝑀𝑁𝐹 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Example 12.1 • Compute the fixed-end moments produced by a concentrated load P at midspan of the fixed-ended beam in Figure Knowing that EI is constant SOLUTION 𝐴𝑀 𝑥ҧ 𝐴 𝐴𝑀 𝑥ҧ 𝐵 𝐹𝐸𝑀𝐴𝐵 = 𝑀𝐴𝐵 = − 𝐿 𝐿2 𝐴𝑀 𝑥ҧ 𝐴 𝐴𝑀 𝑥ҧ 𝐵 𝐹𝐸𝑀𝐵𝐴 = 𝑀𝐵𝐴 = − 𝐿 𝐿2  the moment of the area under the simple beam moment curve produced by the applied load  𝐴𝑀 𝑥ҧ 𝐴 = 𝐴𝑀 𝑥ҧ 𝐵 = 𝑃𝐿 𝐿 𝐿 = 𝑃𝐿3 16 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis Derivation of the Slope-Deflection Equation Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al CIVE4127 Structural Analysis CIVE4127 Analysis of Structures by the Slope-Deflection Method • The slope-deflection method can be used to analyze any type of indeterminate beam or frame • Consider the case when 𝜓𝑁𝐹 = Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Analysis of Structures by the Slope-Deflection Method Identify all unknown joint displacements (rotations) to establish the number of unknowns Use the slope-deflection equation to express all member end moments in terms of joint rotations and the applied loads At each joint, except fixed supports, write the moment equilibrium equation, which states that the sum of the moments (applied by the members framing into the joint) equals zero – The number of equilibrium equations must equal the number of unknown displacements – Sign convention, clockwise moments on the ends of the members are assumed to be positive If a moment at the end of a member is unknown, it must be shown clockwise on the end of a member The moment applied by a member to a joint is always equal and opposite in direction to the moment acting on the end of the member Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Analysis of Structures by the Slope-Deflection Method Substitute the expressions for moments as a function of displacements (see step 2) into the equilibrium equations in step 3, and solve for the unknown displacements Substitute the values of displacement in step into the expressions for member end moment in step to establish the value of the member end moments – Once the member end moments are known, the balance of the analysis drawing shear and moment curves or computing reactions, for example-is completed by statics Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Example 12.3 • Using the slope-deflection method, determine the member end moments in the braced frame shown in Figure Also compute the reactions at support D, and draw the shear and moment curves for members AB and BD SOLUTION • 𝜃𝐴 = 0, 𝜃𝐵 and 𝜃𝐷 are the only unknown 2𝐸𝐼 𝐿 𝑀𝑁𝐹 = 2𝜃𝑁 + 𝜃𝐹 − 3𝜓𝑁𝐹 +𝐹𝐸𝑀𝑁𝐹 2𝐸 120 𝑀𝐴𝐵 = 18 12 12 18 12 𝜃𝐵 − 12 2𝐸 120 𝑀𝐵𝐴 = 18 12 2𝜃𝐵 + = 1.11E𝜃𝐵 − 648 12 18 12 12 = 2.22E𝜃𝐵 + 648 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127 Example 12.3 • 2𝐸(60) 𝑀𝐵𝐷 = 9(12) 2𝜃𝐵 + 𝜃𝐷 • 2𝐸(60) 𝑀𝐷𝐵 = 9(12) 2𝜃𝐷 + 𝜃𝐵 Equilibrium equations at joint D & B ෍ 𝑀𝐷 = → 𝑀𝐷𝐵 = ෍ 𝑀𝐵 = 𝑀𝐵𝐴 + 𝑀𝐵𝐷 − 24 12 = 2.22𝜃𝐷 + 1.11𝜃𝐵 = 2.22E𝜃𝐵 + 648 + 2.22𝜃𝐵 + 1.11𝜃𝐷 − 288 = 𝜃𝐷 = 46.33 𝐸 𝜃𝐵 = 92.66 − 𝐸 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis Example 12.3 Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al CIVE4127 ... reduces the complexity of the analysis because two-dimensional structures are much easier and faster to analyze than threedimensional structures Fundamentals of Structural Analysis, 4th Edition (2011)... al Structural Analysis CIVE4127 The Design Process: Relationship of Analysis to Design • The design of any structure is typically carried out in alternating steps of design and analysis Each... Redesign of the Structures  Evaluation of Preliminary Designs  Final Design and Analysis Phases Fundamentals of Structural Analysis, 4th Edition (2011) by K.M Leet et al Structural Analysis CIVE4127