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
Slide 2
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
Slide 3
2008 Architecture LJM3 Highlights Feflecion calculation of
beams and plane frames Objectives Stiffness check Solution of
Statically Inderminate Problems 7.1 Introduction
Slide 4
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
Slide 5
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
Slide 6
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
Slide 18
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
Slide 19
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
Slide 20
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
Slide 22
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
Slide 23
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
Slide 24
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
Slide 25
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
Slide 26
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
Slide 27
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
Slide 28
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
Slide 29
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
Slide 30
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
Slide 31
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!
Slide 32
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
Slide 33
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
Slide 34
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
Slide 35
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.
Slide 36
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
Slide 38
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
Slide 39
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
Slide 40
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
Slide 41
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
Slide 42
2008 Architecture LJM42 Deformations by superposition Check the
stiffness ( ) 7.6 Stiffness criteria of beams; Optimum design of
beams for stiffness