MECHANICS OF MATERIALS
|
|
|
- Joy Walker
- 6 years ago
- Views:
Transcription
1 9 The cgrw-hill Compnies, Inc. All rights reserved. Fifth SI Edition CHAPTER 5 ECHANICS OF ATERIALS Ferdinnd P. Beer E. Russell Johnston, Jr. John T. DeWolf Dvid F. zurek Lecture Notes: J. Wlt Oler Texs Tech University Anlysis nd Design of Bems for Bending
2 ifthechanics OF ATERIALS Contents Beer Johnston DeWolf zurek Introduction Sher nd Bending oment Digrms Smple Problem 5. Smple Problem 5. Reltions Among Lod, Sher, nd Bending oment Smple Problem 5. Smple Problem 5.5 Design of Prismtic Bems for Bending Smple Problem The cgrw-hill Compnies, Inc. All rights reserved. 5-
3 ifthechanics OF ATERIALS Introduction Beer Johnston DeWolf zurek Objective - Anlysis nd design of bems Bems - structurl members supporting lods t vrious points long the member Trnsverse lodings of bems re clssified s concentrted lods or distributed lods Applied lods result in internl forces consisting of sher force (from the sher stress distribution) nd bending couple (from the norml stress distribution) Norml stress is often the criticl design criteri σ y σ I c I x m S Requires determintion of the loction nd mgnitude of lrgest bending moment 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
4 ifthechanics OF ATERIALS Introduction Beer Johnston DeWolf zurek Clssifiction of Bem Supports 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-4
5 ifthechanics OF ATERIALS Sher nd Bending oment Digrms Beer Johnston DeWolf zurek Determintion of mximum norml nd shering stresses requires identifiction of mximum internl sher force nd bending couple. Sher force nd bending couple t point re determined by pssing section through the bem nd pplying n equilibrium nlysis on the bem portions on either side of the section. Sign conventions for sher forces nd nd bending couples nd 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-5
6 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Treting the entire bem s rigid body, determine the rection forces For the timber bem nd loding shown, drw the sher nd bendmoment digrms nd determine the mximum norml stress due to bending. Section the bem t points ner supports nd lod ppliction points. Apply equilibrium nlyses on resulting free-bodies to determine internl sher forces nd bending couples Identify the mximum sher nd bending-moment from plots of their distributions. Apply the elstic flexure formuls to determine the corresponding mximum norml stress. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-6
7 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Treting the entire bem s rigid body, determine the rection forces from F : R 46kN R 4kN y B Section the bem nd pply equilibrium nlyses on resulting free-bodies F y kn ( kn)( m) + B D kn F y kn kn ( kn)(.5m) + 5kN m + 6kN 5kN m 4 + 6kN 4 + 8kN m 5 4kN 5 + 8kN m 6 4kN 6 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-7
8 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Identify the mximum sher nd bendingmoment from plots of their distributions. m 6kN 5kN m m B Apply the elstic flexure formuls to determine the corresponding mximum norml stress. S 6 b h 6 8. (.8 m)(.5 m) 6 m σ m S B 5 8. N m 6 m σ m 6. 6 P 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-8
9 ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Replce the 45 kn lod with n equivlent force-couple system t D. Find the rections t B by considering the bem s rigid body. The structure shown is constructed of W 5x67 rolled-steel bem. () Drw the sher nd bending-moment digrms for the bem nd the given loding. (b) determine norml stress in sections just to the right nd left of point D. Section the bem t points ner the support nd lod ppliction points. Apply equilibrium nlyses on resulting free-bodies to determine internl sher forces nd bending couples. Apply the elstic flexure formuls to determine the mximum norml stress to the left nd right of point D. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-9
10 ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Replce the 45 kn lod with equivlent forcecouple system t D. Find rections t B. Section the bem nd pply equilibrium nlyses on resulting free-bodies. From A to C : Fy 45x 45x kn ( 45x)( x) +.5 knm x From C to D : Fy 8 8 8kN ( x.) + ( )knm x From D to B : 5kN ( 5. 5x)kNm 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
11 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Apply the elstic flexure formuls to determine the mximum norml stress to the left nd right of point D. From Appendix C for W5x67 rolled steel shpe, S.8x - m bout the X-X xis. To the left of D : σ To the right of D : σ m m S S Nm m Nm m σ m σ m 9P 96 P 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
12 ifthechanics OF ATERIALS Beer Johnston DeWolf zurek Reltions Among Lod, Sher, nd Bending oment Reltionship between lod nd sher: : ( + ) w x F y w x C d dx D w C xd w dx xc Reltionship between sher nd bending moment: : ( + ) d dx D x x + w x x w C x x D C ( x) dx 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
13 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek SOLUTION: Tking the entire bem s free body, determine the rections t A nd D. Apply the reltionship between sher nd lod to develop the sher digrm. Drw the sher nd bending moment digrms for the bem nd loding shown. Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
14 ifthechanics OF ATERIALS Smple Problem 5. SOLUTION: Beer Johnston DeWolf zurek Tking the entire bem s free body, determine the rections t A nd D. F A ( 7.m) ( 9kN)(.8m) ( 54 kn)( 4. m) ( 5.8 kn)( 8.4 m) D D 5.6 kn y A A y y 9kN 54kN kn 5.8 kn 8. kn Apply the reltionship between sher nd lod to develop the sher digrm. d dx w d w dx - zero slope between concentrted lods - liner vrition over uniform lod segment 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-4
15 ifthechanics OF ATERIALS Smple Problem 5. Beer Johnston DeWolf zurek Apply the reltionship between bending moment nd sher to develop the bending moment digrm. d dx d dx - bending moment t A nd E is zero - bending moment vrition between A, B, C nd D is liner - bending moment vrition between D nd E is qudrtic - net chnge in bending moment is equl to res under sher distribution segments - totl of ll bending moment chnges cross the bem should be zero 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-5
16 ifthechanics OF ATERIALS Smple Problem 5.5 Beer Johnston DeWolf zurek SOLUTION: Tking the entire bem s free body, determine the rections t C. Apply the reltionship between sher nd lod to develop the sher digrm. Drw the sher nd bending moment digrms for the bem nd loding shown. Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-6
17 ifthechanics OF ATERIALS Smple Problem 5.5 SOLUTION: Beer Johnston DeWolf zurek Tking the entire bem s free body, determine the rections t C. F y C w w + R C L + C R C C w w L Results from integrtion of the lod nd sher distributions should be equivlent. Apply the reltionship between sher nd lod to develop the sher digrm. B B A w w x dx w x x ( re under lod curve) - No chnge in sher between B nd C. - Comptible with free body nlysis 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-7
18 9 The cgrw-hill Compnies, Inc. All rights reserved. ifthechanics OF ATERIALS dition Beer Johnston DeWolf zurek 5-8 Smple Problem 5.5 Apply the reltionship between bending moment nd sher to develop the bending moment digrm. 6 w x x w dx x x w B A B ( ) ( ) ( ) 6 L w L w L w dx w C L C B Results t C re comptible with free-body nlysis
19 ifthechanics OF ATERIALS Design of Prismtic Bems for Bending The lrgest norml stress is found t the surfce where the mximum bending moment occurs. σ m I c mx mx S A sfe design requires tht the mximum norml stress be less thn the llowble stress for the mteril used. This criteri leds to the determintion of the minimum cceptble section modulus. σ S m σ min ll σ mx ll Among bem section choices which hve n cceptble section modulus, the one with the smllest weight per unit length or cross sectionl re will be the lest expensive nd the best choice. Beer Johnston DeWolf zurek 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-9
20 ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek SOLUTION: Considering the entire bem s freebody, determine the rections t A nd D. A simply supported steel bem is to crry the distributed nd concentrted lods shown. Knowing tht the llowble norml stress for the grde of steel to be used is 6 P, select the wide-flnge shpe tht should be used. Develop the sher digrm for the bem nd lod distribution. From the digrm, determine the mximum bending moment. Determine the minimum cceptble bem section modulus. Choose the best stndrd section which meets this criteri. 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
21 ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek Considering the entire bem s free-body, determine the rections t A nd D. D( 5m) ( 6kN)(.5m) ( 5kN)( 4m) F A A D 58. kn y y A y 5. kn kn 6kN 5kN Develop the sher digrm nd determine the mximum bending moment. A B B A y A 8kN 5. kn ( re under lod curve) 6kN ximum bending moment occurs t or x.6 m. ( re under sher curve, Ato E) mx 67.6 kn 9 The cgrw-hill Compnies, Inc. All rights reserved. 5-
22 ifthechanics OF ATERIALS Smple Problem 5.8 Beer Johnston DeWolf zurek Determine the minimum cceptble bem section modulus. S min σ mx ll kn m 6P 6 m 4.5 mm Shpe W4 8.8 W6.9 W 8.7 W W 46. S mm Choose the best stndrd section which meets this criteri. W The cgrw-hill Compnies, Inc. All rights reserved. 5-
CE 160 Lab 2 Notes: Shear and Moment Diagrams for Beams
E 160 Lb 2 Notes: Sher nd oment Digrms for ems Sher nd moment digrms re plots of how the internl bending moment nd sher vry long the length of the bem. Sign onvention for nd onsider the rbitrrily loded
Chapter 5 Bending Moments and Shear Force Diagrams for Beams
Chpter 5 ending Moments nd Sher Force Digrms for ems n ddition to illy loded brs/rods (e.g. truss) nd torsionl shfts, the structurl members my eperience some lods perpendiculr to the is of the bem nd will
STATICS. Vector Mechanics for Engineers: Statics VECTOR MECHANICS FOR ENGINEERS: Centroids and Centers of Gravity.
5 Distributed CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Texs Tech Universit Forces: Centroids nd Centers of Grvit Contents Introduction
STATICS VECTOR MECHANICS FOR ENGINEERS: and Centers of Gravity. Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
007 The McGrw-Hill Compnies, Inc. All rights reserved. Eighth E CHAPTER 5 Distriuted VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinnd P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Wlt Oler Tes Tech
Module 1. Energy Methods in Structural Analysis
Module 1 Energy Methods in Structurl Anlysis Lesson 4 Theorem of Lest Work Instructionl Objectives After reding this lesson, the reder will be ble to: 1. Stte nd prove theorem of Lest Work.. Anlyse stticlly
Module 2. Analysis of Statically Indeterminate Structures by the Matrix Force Method. Version 2 CE IIT, Kharagpur
Module Anlysis of Stticlly Indeterminte Structures by the Mtrix Force Method Version CE IIT, Khrgpur esson 8 The Force Method of Anlysis: Bems Version CE IIT, Khrgpur Instructionl Objectives After reding
V. DEMENKO MECHANICS OF MATERIALS LECTURE 6 Plane Bending Deformation. Diagrams of Internal Forces (Continued)
V. DEMENKO MECHNCS OF MTERLS 015 1 LECTURE 6 Plne ending Deformtion. Digrms of nternl Forces (Continued) 1 Construction of ending Moment nd Shering Force Digrms for Two Supported ems n this mode of loding,
BME 207 Introduction to Biomechanics Spring 2018
April 6, 28 UNIVERSITY O RHODE ISAND Deprtment of Electricl, Computer nd Biomedicl Engineering BME 27 Introduction to Biomechnics Spring 28 Homework 8 Prolem 14.6 in the textook. In ddition to prts -e,
MECHANICS OF MATERIALS
2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 6 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J.
MECHANICS OF MATERIALS
Third E CHAPTER 6 Shearing MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Stresses in Beams and Thin- Walled Members Shearing
Shear and torsion interaction of hollow core slabs
Competitive nd Sustinble Growth Contrct Nº G6RD-CT--6 Sher nd torsion interction of hollow core slbs HOLCOTORS Technicl Report, Rev. Anlyses of hollow core floors December The content of the present publiction
MECHANICS OF MATERIALS
009 The McGraw-Hill Companies, nc. All rights reserved. Fifth S E CHAPTER 6 MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas
COLLEGE OF ENGINEERING AND TECHNOLOGY
COLLEGE OF ENGNEERNG ND TECHNOLOGY DEPRTMENT : Construction nd uilding Engineering COURSE : Structurl nlysis 2 COURSE NO : C 343 LECTURER : Dr. Mohmed SFN T. SSSTNT : Eng. Mostf Yossef, Eng. l-hussein
1 Bending of a beam with a rectangular section
1 Bending of bem with rectngulr section x3 Episseur b M x 2 x x 1 2h M Figure 1 : Geometry of the bem nd pplied lod The bem in figure 1 hs rectngur section (thickness 2h, width b. The pplied lod is pure
MECHANICS OF MATERIALS
CHAPTER 6 MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members
Columns and Stability
ARCH 331 Note Set 1. Su01n Columns nd Stilit Nottion: A = nme or re A36 = designtion o steel grde = nme or width C = smol or compression C c = column slenderness clssiiction constnt or steel column design
Distributed Forces: Centroids and Centers of Gravity
Distriuted Forces: Centroids nd Centers of Grvit Introduction Center of Grvit of D Bod Centroids nd First Moments of Ares nd Lines Centroids of Common Shpes of Ares Centroids of Common Shpes of Lines Composite
MECHANICS OF MATERIALS
2009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 3 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Torsion Lecture Notes:
Rigid Frames - Compression & Buckling
ARCH 614 Note Set 11.1 S014n Rigid Frmes - Compression & Buckling Nottion: A = nme or re d = nme or depth E = modulus o elsticity or Young s modulus = xil stress = ending stress z = stress in the x direction
Evaluation of Allowable Hold Loading of, Hold No. 1 with Cargo Hold No. 1 Flooded, for Existing Bulk Carriers
(997) (Rev. 997) (Rev.2 ept. 2000) (Rev.3 July 2004) Evlution of Allowble Hold Loding of Crgo, Hold No. with Crgo Hold No. Flooded, for Existing Bulk Crriers. - Appliction nd definitions These requirements
Ans. Ans. Ans. Ans. Ans. Ans.
08 Solutions 46060 5/28/10 8:34 M Pge 532 8 1. sphericl gs tnk hs n inner rdius of r = 1.5 m. If it is subjected to n internl pressure of p = 300 kp, determine its required thickness if the mximum norml
BEAM A horizontal or inclined structural member that is designed to resist forces acting to its axis is called a beam
BEM horizontal or inclined structural member that is designed to resist forces acting to its axis is called a beam INTERNL FORCES IN BEM Whether or not a beam will break, depend on the internal resistances
Statically indeterminate examples - axial loaded members, rod in torsion, members in bending
Elsticity nd Plsticity Stticlly indeterminte exmples - xil loded memers, rod in torsion, memers in ending Deprtment of Structurl Mechnics Fculty of Civil Engineering, VSB - Technicl University Ostrv 1
ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION
28 TH INTERNATIONAL CONGRESS OF THE AERONAUTICAL SCIENCES ENERGY-BASED METHOD FOR GAS TURBINE ENGINE DISK BURST SPEED CALCULATION Anton N. Servetnik Centrl Institute of Avition Motors, Moscow, Russi servetnik@cim.ru
i 3 i 2 Problem 8.31 Shear flow in circular section The centroidal axes are located at the center of the circle as shown above.
Problem 8.31 Sher flow in circulr section i 3 R θ s i 2 t Remove@"Globl` "D H remove ll symbols L The centroidl xes re locted t the center of the circle s shown bove. (1) Find bending stiffness: From symmetry,
Review of Calculus, cont d
Jim Lmbers MAT 460 Fll Semester 2009-10 Lecture 3 Notes These notes correspond to Section 1.1 in the text. Review of Clculus, cont d Riemnn Sums nd the Definite Integrl There re mny cses in which some
Solutions to Supplementary Problems
Solutions to Supplementry Problems Chpter 8 Solution 8.1 Step 1: Clculte the line of ction ( x ) of the totl weight ( W ).67 m W = 5 kn W 1 = 16 kn 3.5 m m W 3 = 144 kn Q 4m Figure 8.10 Tking moments bout
Theme 8 Stability and buckling of members
Elsticity nd plsticity Theme 8 Stility nd uckling o memers Euler s solution o stility o n xilly compressed stright elstic memer Deprtment o Structurl Mechnics culty o Civil Engineering, VSB - Technicl
DIRECT CURRENT CIRCUITS
DRECT CURRENT CUTS ELECTRC POWER Consider the circuit shown in the Figure where bttery is connected to resistor R. A positive chrge dq will gin potentil energy s it moves from point to point b through
Progressive failure analysis of compression-loaded composite flat panel with cutout
Interntionl Journl on Theoreticl nd Applied Reserch in Mechnicl Engineering (IJTARME) Progressive filure nlysis of compression-loded composite flt pnel with cutout 1 Guspir S. Mkndr, 2 N.K. Chhpkhne, 3
PDE Notes. Paul Carnig. January ODE s vs PDE s 1
PDE Notes Pul Crnig Jnury 2014 Contents 1 ODE s vs PDE s 1 2 Section 1.2 Het diffusion Eqution 1 2.1 Fourier s w of Het Conduction............................. 2 2.2 Energy Conservtion.....................................
eleven rigid frames: compression & buckling Rigid Frames Rigid Frames Rigid Frames ELEMENTS OF ARCHITECTURAL STRUCTURES: FORM, BEHAVIOR, AND DESIGN
ELEMENTS O RCHITECTURL STRUCTURES: ORM, BEHVIOR, ND DESIGN DR. NNE NICHOLS SRING 018 lecture eleven rigid rmes: compression & uckling Rigid rmes 1 Lecture 11 S009n http:// nisee.erkeley.edu/godden Rigid
Design Data 1M. Highway Live Loads on Concrete Pipe
Design Dt 1M Highwy Live Lods on Concrete Pipe Foreword Thick, high-strength pvements designed for hevy truck trffic substntilly reduce the pressure trnsmitted through wheel to the subgrde nd consequently,
Case (a): Ans Ans. Case (b): ; s 1 = 65(4) Ans. s 1 = pr t. = 1.04 ksi. Ans. s 2 = pr 2t ; s 2 = 65(4) = 520 psi
8 3. The thin-wlled cylinder cn be supported in one of two wys s shown. Determine the stte of stress in the wll of the cylinder for both cses if the piston P cuses the internl pressure to be 65 psi. The
E S dition event Vector Mechanics for Engineers: Dynamics h Due, next Wednesday, 07/19/2006! 1-30
Vector Mechnics for Engineers: Dynmics nnouncement Reminders Wednesdy s clss will strt t 1:00PM. Summry of the chpter 11 ws posted on website nd ws sent you by emil. For the students, who needs hrdcopy,
Design of T and L Beams in Flexure
Lecture 04 Design of T nd L Bems in Flexure By: Prof. Dr. Qisr Ali Civil Engineering Deprtment UET Peshwr drqisrli@uetpeshwr.edu.pk Prof. Dr. Qisr Ali CE 320 Reinforced Concrete Design Topics Addressed
Exam CT3109 STRUCTURAL MECHANICS april 2011, 09:00 12:00 hours
Subfculty of Civil Engineering rk ech ge with your: Structurl echnics STUDENT NUBER : NAE : Em CT309 STRUCTURAL ECHANICS 4 ril 0, 09:00 :00 hours This em consists of 4 roblems. Use for ech roblem serte
Strength Theory.
Strength Theory mi@seu.edu.cn Contents Strength Condition for Simple Stress Sttes( 简单应力状态的强度理论回顾 ) Frcture Criteri for Brittle Mterils( 脆性材料强度理论 ) Yield Criteri for Ductile Mterils( 塑性材料强度理论 ) Summry of
Review of Gaussian Quadrature method
Review of Gussin Qudrture method Nsser M. Asi Spring 006 compiled on Sundy Decemer 1, 017 t 09:1 PM 1 The prolem To find numericl vlue for the integrl of rel vlued function of rel vrile over specific rnge
99/105 Comparison of OrcaFlex with standard theoretical results
99/105 Comprison of OrcFlex ith stndrd theoreticl results 1. Introduction A number of stndrd theoreticl results from literture cn be modelled in OrcFlex. Such cses re, by virtue of being theoreticlly solvble,
Job No. Sheet 1 of 8 Rev B. Made by IR Date Aug Checked by FH/NB Date Oct Revised by MEB Date April 2006
Job o. Sheet 1 of 8 Rev B 10, Route de Limours -78471 St Rémy Lès Chevreuse Cedex rnce Tel : 33 (0)1 30 85 5 00 x : 33 (0)1 30 5 75 38 CLCULTO SHEET Stinless Steel Vloristion Project Design Exmple 5 Welded
Explain shortly the meaning of the following eight words in relation to shells structures.
Delft University of Technology Fculty of Civil Engineering nd Geosciences Structurl Mechnics Section Write your nme nd study number t the top right-hnd of your work. Exm CIE4143 Shell Anlysis Tuesdy 15
Machine Design II Prof. K.Gopinath & Prof. M.M.Mayuram. Drum Brakes. Among the various types of devices to be studied, based on their practical use,
chine Design II Prof. K.Gointh & Prof...yurm Drum Brkes Among the vrious tyes of devices to be studied, bsed on their rcticl use, the discussion will be limited to Drum brkes of the following tyes which
ME311 Machine Design
ME11 Mchine Design Lecture 10: Springs (Chpter 17) W Dornfeld 9Nov018 Firfield University School of Engineering A Free Body Digrm of coil spring (cutting through nywhere on the coil) shows tht there must
Plate Theory. Section 11: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEMENTS Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s the thickness of the plte. A
Plate Theory. Section 13: PLATE BENDING ELEMENTS
Section : PLATE BENDING ELEENTS Wshkeic College of Engineering Plte Theor A plte is structurl element hose mid surfce lies in flt plne. The dimension in the direction norml to the plne is referred to s
MECHANICS OF MATERIALS
CHTER MECHNICS OF MTERILS 10 Ferdinand. Beer E. Russell Johnston, Jr. Columns John T. DeWolf cture Notes: J. Walt Oler Texas Tech University 006 The McGraw-Hill Companies, Inc. ll rights reserved. Columns
Plates on elastic foundation
Pltes on elstic foundtion Circulr elstic plte, xil-symmetric lod, Winkler soil (fter Timoshenko & Woinowsky-Krieger (1959) - Chpter 8) Prepred by Enzo Mrtinelli Drft version ( April 016) Introduction Winkler
DYNAMICS VECTOR MECHANICS FOR ENGINEERS: Plane Motion of Rigid Bodies: Forces and Accelerations. Seventh Edition CHAPTER
CHAPTER 16 VECTOR MECHANICS FOR ENGINEERS: DYNAMICS Ferdinnd P. Beer E. Ruell Johnton, Jr. Lecture Note: J. Wlt Oler Tex Tech Univerity Plne Motion of Rigid Bodie: Force nd Accelertion Content Introduction
Solution Manual. for. Fracture Mechanics. C.T. Sun and Z.-H. Jin
Solution Mnul for Frcture Mechnics by C.T. Sun nd Z.-H. Jin Chpter rob.: ) 4 No lod is crried by rt nd rt 4. There is no strin energy stored in them. Constnt Force Boundry Condition The totl strin energy
MECHANICS OF MATERIALS
00 The cgraw-hill Copanies, Inc. All rights reserved. Third E CHAPTER 8 Principle ECHANICS OF ATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University
MECHANICS OF MATERIALS
Fifth SI Edition CHTER 1 MECHNICS OF MTERILS Ferdinand. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Introduction Concept of Stress Lecture Notes: J. Walt Oler Teas Tech University Contents
ME 141. Lecture 10: Kinetics of particles: Newton s 2 nd Law
ME 141 Engineering Mechnics Lecture 10: Kinetics of prticles: Newton s nd Lw Ahmd Shhedi Shkil Lecturer, Dept. of Mechnicl Engg, BUET E-mil: sshkil@me.buet.c.bd, shkil6791@gmil.com Website: techer.buet.c.bd/sshkil
Calculus of Variations
Clculus of Vritions Com S 477/577 Notes) Yn-Bin Ji Dec 4, 2017 1 Introduction A functionl ssigns rel number to ech function or curve) in some clss. One might sy tht functionl is function of nother function
PROBLEM 5.1. wl x. M ( Lx x )
w PROE 5.1 For the beam and loading shown, (a) draw the shear and bending-moment diagrams, (b) determine the equations of the shear and bending-moment curves. SOUTION Reactions: 0: 0 0: 0 Free bod diagram
Design of T and L Beams in Flexure
Lecture 04 Design of T nd L Bems in Flexure By: Prof. Dr. Qisr Ali Civil Engineering Deprtment UET Peshwr drqisrli@uetpeshwr.edu.pk Prof. Dr. Qisr Ali CE 320 Reinforced Concrete Design Topics Addressed
CMDA 4604: Intermediate Topics in Mathematical Modeling Lecture 19: Interpolation and Quadrature
CMDA 4604: Intermedite Topics in Mthemticl Modeling Lecture 19: Interpoltion nd Qudrture In this lecture we mke brief diversion into the res of interpoltion nd qudrture. Given function f C[, b], we sy
Chapter 5 Weight function method
Chpter 5 Weight function method The weight functions re powerful method in liner elstic frcture mechnics (Anderson, 1995; Td, Pris & rwin, 2). nitilly they were used for clculting the. The underlying hypothesis
Partial Derivatives. Limits. For a single variable function f (x), the limit lim
Limits Prtil Derivtives For single vrible function f (x), the limit lim x f (x) exists only if the right-hnd side limit equls to the left-hnd side limit, i.e., lim f (x) = lim f (x). x x + For two vribles
MECHANICS OF MATERIALS. Analysis of Beams for Bending
MECHANICS OF MATERIALS Analysis of Beams for Bending By NUR FARHAYU ARIFFIN Faculty of Civil Engineering & Earth Resources Chapter Description Expected Outcomes Define the elastic deformation of an axially
MTE 119 STATICS FINAL HELP SESSION REVIEW PROBLEMS PAGE 1 9 NAME & ID DATE. Example Problem P.1
MTE STATICS Example Problem P. Beer & Johnston, 004 by Mc Graw-Hill Companies, Inc. The structure shown consists of a beam of rectangular cross section (4in width, 8in height. (a Draw the shear and bending
Shear Force V: Positive shear tends to rotate the segment clockwise.
INTERNL FORCES IN EM efore a structural element can be designed, it is necessary to determine the internal forces that act within the element. The internal forces for a beam section will consist of a shear
Lecture notes. Fundamental inequalities: techniques and applications
Lecture notes Fundmentl inequlities: techniques nd pplictions Mnh Hong Duong Mthemtics Institute, University of Wrwick Emil: m.h.duong@wrwick.c.uk Februry 8, 207 2 Abstrct Inequlities re ubiquitous in
Math 1B, lecture 4: Error bounds for numerical methods
Mth B, lecture 4: Error bounds for numericl methods Nthn Pflueger 4 September 0 Introduction The five numericl methods descried in the previous lecture ll operte by the sme principle: they pproximte the
Design Against Fatigue Failure 2/3/2015 1
Design Aginst Ftigue Filure /3/015 1 Ftigue is the filure of mechnicl element by the growth of crck within mteril under vrible, repeted, lternting, or fluctuting stresses. Generlly, ftigue crck growth
31.2. Numerical Integration. Introduction. Prerequisites. Learning Outcomes
Numericl Integrtion 3. Introduction In this Section we will present some methods tht cn be used to pproximte integrls. Attention will be pid to how we ensure tht such pproximtions cn be gurnteed to be
MECHANICS OF MATERIALS
CHAPTER MECHANCS OF MATERALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Shearing Stresses in Beams and Thin- Walled Members 006 The McGraw-Hill
MECHANICS OF MATERIALS
Third E CHAPTER 1 Introduction MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University Concept of Stress Contents Concept of Stress
10 Deflections due to Bending
1 Deflections due to Bending 1.1 The Moment/Curvture Reltion Just s we took the pure bending construction to be ccurte enough to produce useful estimtes of the norml stress due to bending for lodings tht
13.4 Work done by Constant Forces
13.4 Work done by Constnt Forces We will begin our discussion of the concept of work by nlyzing the motion of n object in one dimension cted on by constnt forces. Let s consider the following exmple: push
Problem Set 9. Figure 1: Diagram. This picture is a rough sketch of the 4 parabolas that give us the area that we need to find. The equations are:
(x + y ) = y + (x + y ) = x + Problem Set 9 Discussion: Nov., Nov. 8, Nov. (on probbility nd binomil coefficients) The nme fter the problem is the designted writer of the solution of tht problem. (No one
MA Handout 2: Notation and Background Concepts from Analysis
MA350059 Hndout 2: Nottion nd Bckground Concepts from Anlysis This hndout summrises some nottion we will use nd lso gives recp of some concepts from other units (MA20023: PDEs nd CM, MA20218: Anlysis 2A,
KINEMATICS OF RIGID BODIES
KINEMTICS OF RIGID ODIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. Description
200 points 5 Problems on 4 Pages and 20 Multiple Choice/Short Answer Questions on 5 pages 1 hour, 48 minutes
PHYSICS 132 Smple Finl 200 points 5 Problems on 4 Pges nd 20 Multiple Choice/Short Answer Questions on 5 pges 1 hour, 48 minutes Student Nme: Recittion Instructor (circle one): nme1 nme2 nme3 nme4 Write
6.5 Plate Problems in Rectangular Coordinates
6.5 lte rolems in Rectngulr Coordintes In this section numer of importnt plte prolems ill e emined ug Crte coordintes. 6.5. Uniform ressure producing Bending in One irection Consider first the cse of plte
( ) where f ( x ) is a. AB/BC Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB/ Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 Find the intersection of f ( x) nd g( x). A3 Show tht f ( x) is even. A4 Show tht
Exam 1 Solutions (1) C, D, A, B (2) C, A, D, B (3) C, B, D, A (4) A, C, D, B (5) D, C, A, B
PHY 249, Fll 216 Exm 1 Solutions nswer 1 is correct for ll problems. 1. Two uniformly chrged spheres, nd B, re plced t lrge distnce from ech other, with their centers on the x xis. The chrge on sphere
Dynamics: Newton s Laws of Motion
Lecture 7 Chpter 4 Physics I 09.25.2013 Dynmics: Newton s Lws of Motion Solving Problems using Newton s lws Course website: http://fculty.uml.edu/andriy_dnylov/teching/physicsi Lecture Cpture: http://echo360.uml.edu/dnylov2013/physics1fll.html
Equations of Motion. Figure 1.1.1: a differential element under the action of surface and body forces
Equtions of Motion In Prt I, lnce of forces nd moments cting on n component ws enforced in order to ensure tht the component ws in equilirium. Here, llownce is mde for stresses which vr continuousl throughout
MECHANICS OF MATERIALS
009 The McGraw-Hill Companies, Inc. All rights reserved. Fifth SI Edition CHAPTER 7 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Transformations of
INVESTIGATION OF ASSESSMENT METHODS FOR RAILWAY MASONRY ARCH BRIDGES
th Interntionl onference on Arch Bridges October -7, 06, Wrocłw, Polnd INVESTIGATION OF ASSESSMENT METODS FOR RAIWAY MASONRY AR BRIDGES J. Wng, J. ynes,. Melbourne University of Slford, Directorte of ivil
20 MATHEMATICS POLYNOMIALS
0 MATHEMATICS POLYNOMIALS.1 Introduction In Clss IX, you hve studied polynomils in one vrible nd their degrees. Recll tht if p(x) is polynomil in x, the highest power of x in p(x) is clled the degree of
STATICS. Bodies VECTOR MECHANICS FOR ENGINEERS: Ninth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
N E 4 Equilibrium CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Lecture Notes: J. Walt Oler Texas Tech University of Rigid Bodies 2010 The McGraw-Hill Companies,
( ) as a fraction. Determine location of the highest
AB Clculus Exm Review Sheet - Solutions A. Preclculus Type prolems A1 A2 A3 A4 A5 A6 A7 This is wht you think of doing Find the zeros of f ( x). Set function equl to 0. Fctor or use qudrtic eqution if
Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.
![Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.](/thumbs/79/78941097.jpg "Homework 4. (1) If f R[a, b], show that f 3 R[a, b]. If f + (x) = max{f(x), 0}, is f + R[a, b]? Justify your answer.")
Homework 4 (1) If f R[, b], show tht f 3 R[, b]. If f + (x) = mx{f(x), 0}, is f + R[, b]? Justify your nswer. (2) Let f be continuous function on [, b] tht is strictly positive except finitely mny points
AB Calculus Review Sheet
AB Clculus Review Sheet Legend: A Preclculus, B Limits, C Differentil Clculus, D Applictions of Differentil Clculus, E Integrl Clculus, F Applictions of Integrl Clculus, G Prticle Motion nd Rtes This is
Finite Element Determination of Critical Zones in Composite Structures
Finite Element Determintion of Criticl Zones in Composite Structures Alexey I. Borovkov Dmitriy V. Klimshin Denis V. Shevchenko Computtionl Mechnics Lb., St. Petersburg Stte Polytechnicl University, Russi
Stress Analysis Lecture 4 ME 276 Spring Dr./ Ahmed Mohamed Nagib Elmekawy
Stress Analysis Lecture 4 ME 76 Spring 017-018 Dr./ Ahmed Mohamed Nagib Elmekawy Shear and Moment Diagrams Beam Sign Convention The positive directions are as follows: The internal shear force causes a
( ) where f ( x ) is a. AB Calculus Exam Review Sheet. A. Precalculus Type problems. Find the zeros of f ( x).
AB Clculus Exm Review Sheet A. Preclculus Type prolems A1 Find the zeros of f ( x). This is wht you think of doing A2 A3 Find the intersection of f ( x) nd g( x). Show tht f ( x) is even. A4 Show tht f
6.2 CONCEPTS FOR ADVANCED MATHEMATICS, C2 (4752) AS
6. CONCEPTS FOR ADVANCED MATHEMATICS, C (475) AS Objectives To introduce students to number of topics which re fundmentl to the dvnced study of mthemtics. Assessment Emintion (7 mrks) 1 hour 30 minutes.
Kirchhoff and Mindlin Plates
Kirchhoff nd Mindlin Pltes A plte significntly longer in two directions compred with the third, nd it crries lod perpendiculr to tht plne. The theory for pltes cn be regrded s n extension of bem theory,
MECHANICS OF MATERIALS
00 The McGraw-Hill Copanies, Inc. All rights reserved. T Edition CHAPTER MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf Lecture Notes: J. Walt Oler Texas Tech University
Week 10: Line Integrals
Week 10: Line Integrls Introduction In this finl week we return to prmetrised curves nd consider integrtion long such curves. We lredy sw this in Week 2 when we integrted long curve to find its length.
Polynomials and Division Theory
Higher Checklist (Unit ) Higher Checklist (Unit ) Polynomils nd Division Theory Skill Achieved? Know tht polynomil (expression) is of the form: n x + n x n + n x n + + n x + x + 0 where the i R re the
KINEMATICS OF RIGID BODIES
KINEMTICS OF RIGI OIES Introduction In rigid body kinemtics, e use the reltionships governing the displcement, velocity nd ccelertion, but must lso ccount for the rottionl motion of the body. escription
MECHANICS OF MATERIALS
GE SI CHAPTER 3 MECHANICS OF MATERIALS Ferdinand P. Beer E. Russell Johnston, Jr. John T. DeWolf David F. Mazurek Torsion Lecture Notes: J. Walt Oler Texas Tech University Torsional Loads on Circular Shafts
Lecture 4: Piecewise Cubic Interpolation
Lecture notes on Vritionl nd Approximte Methods in Applied Mthemtics - A Peirce UBC Lecture 4: Piecewise Cubic Interpoltion Compiled 5 September In this lecture we consider piecewise cubic interpoltion
Introduction to Finite Element Method
Introduction to Finite Element Method Introductory Course on Multiphysics Modelling TOMASZ G. ZIELIŃSKI bluebox.ippt.pn.pl/ tzielins/ Tble of Contents 1 Introduction 1 1.1 Motivtion nd generl concepts.............
k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.
Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.
Numerical Analysis: Trapezoidal and Simpson s Rule
nd Simpson s Mthemticl question we re interested in numericlly nswering How to we evlute I = f (x) dx? Clculus tells us tht if F(x) is the ntiderivtive of function f (x) on the intervl [, b], then I =