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2008 Architecture LJM1 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM2 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration methof of determining the beam deflections 7.4 Superposition methof of determining the beam deflections 7.5 Statically inderminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM3 Highlights Feflecion calculation of beams and plane frames Objectives Stiffness check Solution of Statically Inderminate Problems 7.1 Introduction

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2008 Architecture LJM4 Deflection, w vertical displacement of centriod of the cross-section. Rotational angle, , Angle rotated about neutral axis 2. Basic amounts measuring the beam deflection 3. Relation between w and 1. Deflection curve axis of the beam after deformation, smooth ever-curve F x C w C1C1 w w =w (x) 7.1 Introduction

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2008 Architecture LJM5 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM6 1. Differential equation of deflection curve w x M>0 w x M

2008 Architecture LJM17 (5) Discussion When a>b, B > A Maximum deflection: When a>b, A 0. Thus, the point of 1 (w 1 ' )=0 occurs in the longer segment of the beam. w max Maximum rotational angle: x0x0 7.3 Integration methof of determining the beam deflections F B C A

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2008 Architecture LJM18 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM19 1. Superposition by loads 2. Superposiyion using analysis of portion-by-portion Where F I is generalized force, including force and couple 7.4 Superposition method of determining the beam deflections

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2008 Architecture LJM20 Ex.7.4 Find deflection at C and rotational angle at A by superposition. Solution Exerting load alone Deformation caused by one load q F AB C aa Superposition F = A B q + AB 7.4 Superposition method of finding deflections

Slide 21

2008 Architecture LJM21 F l B A q MeMe B A MeMe q B A F B A + + w = w Me +w q +w F Ex.7.4 Find deflection and rotational angle at B by superposition. ( ) 7.4 Superposition method of determining the beam deflections Solution

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2008 Architecture LJM22 l B A Principle of portion-by-portion analysis = + Fl a A B C C B Fa w2w2 F M=Fa w1w1 Basic consideration Deflection equalization Basic theory Force transition Basic results Application directly 7.4 Superposition method of finding deflections

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2008 Architecture LJM23 A C q B F a a/2 A q F Ay F By (a) B F a/2 (b) A C B wBwB Ex.7.5 Combined beam AC EI F=qa Find w B and B. 7.4 Superposition method of finding deflections

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2008 Architecture LJM24 Ex.7.6 Determine w C. For infinitesimal portion dx From Table of deflection of beams. Superposition q0q0 0.5L x dxdx b x f C 7.4 Superposition method of finding deflections Solution

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2008 Architecture LJM25 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM26 1. Take equalization system 2. List compatibility equation 3. Introduce physical law to get supplementary 4. Solve the equation to get redundant reaction A F B A F B q q Redundant reactions( ) F By Redundant restraints( ) 7.5 Statically indeterminate beams Solution

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2008 Architecture LJM27 A F l/2 A F By F B MAMA A F Solution E.E Take equalization equation Compability condition Fond redundant reaction Change S.ID.P to S.D.P Ex.7.7 Find reactions of the beam Find other reactions A F By F MAMA F Ay ( ) 7.5 Statically indeterminate beams

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2008 Architecture LJM28 B C A F D E Example7.8 Two cantilever beams of AD and BE are joined by a steel rod CD. Determine the deflection of the cantilever beam AD, at D due to a force F applied at E. EI EA l l l l 7.5 Statically indeterminate beams

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2008 Architecture LJM29 Solution: (1) Set up a equivalent system (2) Compatible condition w D =w C - rod (a) w C = (w C ) p - (w C ) FD Where wcwc rod B C F E Fig.b FDFD A D FDFD wDwD Fig.a 7.5 Statically indeterminate beams

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2008 Architecture LJM30 B C A F D E FDFD wcwc rod wDwD Substitute w D, w C and rod into Eq.(a): (3) Find the deflection of D, w D 7.5 Statically indeterminate beams

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2008 Architecture LJM31 = q0q0 l A B l MAMA B A Ex.7.9 Shown is the beam AB of length l, EI, subjected to uniform load q. Draw M- diagram. M A, F B -- q0q0 EI q0q0 L FBFB A B 7.5 Statically indeterminate beams Solution (1) Set up a equivalent system Different Equivalent systems!

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2008 Architecture LJM32 Compatible equation + q0q0 L FBFB A B = FBFB A B q0q0 A B physical relations Supplementary equation M-diagram 7.5 Statically indeterminate beams M

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2008 Architecture LJM33 Solution Set up E.S MAMA Ex.7.9 Plot Bending-moment diagram of simple-supported beam AB shown. B A Geometry equation Reactions Bending-moment Diagram M l A B EI q0q0 FBFB FAFA MAMA B A 7.5 Statically indeterminate beams Have

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2008 Architecture LJM34 Solution E.S. Equivalent system Compatible equation Ex.7.10 Determine maximum deflection of the cantilever beam AB shown. Suppose that EI for the two beams are equal. Maximum deflection A FRFR C F B FRFR AC C B A l/2 F 7.5 Statically indeterminate beams

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2008 Architecture LJM35 = Ex.7.11 Determine axial load in rod BC for the structure shown. q0q0 FBFB FBFB A + q0q0 A L BC q0q0 L A B C EI 7.5 Statically inderminate beams Equivalent system Compatible equation Solution E.S.

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2008 Architecture LJM36 = L BC x f q0q0 L FBFB A B C FBFB A B + q0q0 A B Physical relations Supplementary Others Reactions, stresses, deflections, and so on. 7.5 Statically inderminate beams

Slide 37

2008 Architecture LJM37 7.1 Introduction 7.2 Approximately Differential equation of deflection curve 7.3 Integration method of determining the beam deflections 7.4 Superposition method of determining the beam deflections 7.5 Statically indeterminate beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Chapter 7 Bending Deformation

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2008 Architecture LJM38 [w]: Allowable deflection [ ]: Allowable rotational angle Check the stiffness; Determine allowable loads. 1. Stiffness conditions of beams Three types of stiffness calculation Design sections; 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness

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2008 Architecture LJM39 Strength Stiffness Methods 3S are related to internal forces and properties of the cross-section Reducing Bending moment M Enhancing Inertia moment I or Modulus of section W Select materials rationally Stability 2. Methods of enhancing deformations of beams 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness

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2008 Architecture LJM40 Ex.7.12 The hollow circular rod AC of in-diameter d=40mm and out-diameter D = 80mm. E = 210GPa []= 10 -5 m at C. [ ] = 0.001. at C, F 1 = 1kN, F 2 = 2kN. Check the stiffness of the overhanging beam. F2F2 B l=400 P2P2 A C a=100 200 D F1F1 B + F2F2 M = F1F1 + F2F2 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Analysis of deformations

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2008 Architecture LJM41 From Table of deformations F2F2 B C + + = ( ) C F1F1 A BD The overhang beam C F2F2 B D A M 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness Solution

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2008 Architecture LJM42 Deformations by superposition Check the stiffness ( ) 7.6 Stiffness criteria of beams; Optimum design of beams for stiffness