1、Chapter 7 Deflection/DisplacementInstructor: Dr. Jun Wu Date: Feb 22 2012 IntroductionContents1 Principle of virtual work 虛功原理2 General method-unit load 單位荷載法3 Deflection of statically determinate structure under load 荷載作用下的位移計算4 Graph-multiplication method 圖乘法Contents (Contd)5 Deflection under temp

2、erature change6 Deflection under supports settlement7 Reciprocal laws for linear elastic system線彈性變形體的互等定理8 Influence line of deflection 位移影響線 9What is deflection/displacement? 什么是結構的變形(位移)？1. IntroductionWhat is deflection/disp.?Linear disp./translationAngular disp./rotation1. Introduction1. Introd

3、uction (Contd)What cause a structure to yield deflection/disp.?Load Temperature changeSupport settlement1. IntroductionWhy to calculate the deflection/displacement?Check the serviceability and safety of structurestiffnessThe foundation of the analysis of statically indeterminate structureConstructio

4、n monitoringDynamic computation and stabilityTheoretical foundation to calculate disp.:Principle of virtual workMethod of calculationUnit load (force) method1. Introduction (Contd)Focus: Linearly elastic systemMaterials are linearly elasticDeformation is small enough to neglect the changes of origin

5、ally acting effect of its loadingsPrinciple of superposition can be applied1. Introduction (Contd)2. Principle of virtual work 虛功原理2.1 Work 功2.2 Principle of virtual work(pvw) 虛功原理2.3 Two kinds of application of pvw2. Principle of virtual work 虛功原理2.1 WorkReal work 實功Virtual work 虛功External force-in

6、duced 外力引起的Internal force-induced 內力引起的Generalized force （廣義力） and its corresponding displacementForce/moment-PCorresponding disp.-translational disp./angular disp. along the direction of P1) real work 實功ABPW= PP: from 0 to P：from 0 to 2）virtual work 虛功 the force has no relation with the displacemen

7、t undergone by the force during performing the workThe displacement is NOT caused by the force doing the workFp1Fp212AB1211 Virtual work by external force 外力虛功 Under load P，internal force （N,Q,M） and their corresponding deformation (d,d,d) will be generated-see fig.7-5 on page 121. Virtual work by i

8、nternal force內力虛功 2.2 Principle of virtual work 虛功原理 If a deformable structure is in equilibrium, for any small real deformation, the sum of the virtual external work and the virtual internal work is zero.Notes:The principles application is not limited to the elastic structure. 2.3 Two kinds of appl

9、ication of pvwWorkForce stateDisp. stateIndependent for virtual work2.3.1 virtual disp. Principle 虛位移原理2.3.2 virtual force principle 虛力原理2.3.1 Virtual disp. Principle 虛位移原理if the body is given an imaginary or virtual displacement, the work done by all the forces and couple moments must also be equal

10、 to zero. Virtual displacementimagined as unit value- Obtain the unknown force2.3.2 Virtual force Principle 虛力原理Virtual loadimagined as unit load- Obtain the unknown displacement3. General method-unit load 單位荷載法Purpose:Determine one of its displacement Only one unit load P=1 must be imposed on the p

11、osition corresponding to 3. General method-unit load 單位荷載法Virtual loadInternal forces/reaction and disp. caused by P=13. General method-unit load 單位荷載法Procedure to find a desired displacement Impose a unit load P=1 at the location and in the direction of the desired displacement Determine the reacti

12、on (R)and internal force (N,Q,M)of the structure caused by P=1 according to the static equilibrium conditionsCalculate the desired displacement by using （7-1）Notes:If has positive sign, it means that the direction of is the same as that of P=1 and vice versaEqn (7-1) is a general formula. It can be

13、applied to the structure regardless of the material properties, type of deformation and the factors causing the deformation. BAFp=1CyBxFp=1CM=1Fp=1AByFp=1Fp=1Fp=1CBCABABxM=1CDFp=1/LFp=1/LC C DLC4. Deflection of statically determinate structure under load 靜定結構在荷載作用下的位移計算4.1 General formula4.2 Feature

14、s of each type of structure4.3 Examples4.1 General formulaObject: Statically determinate elastic structure Displacement is caused by load 4.1 General formula (Contd)Caused by unit virtual loadReal deformation caused by real load4.1 General formula (Contd)Note：Applied for straight bar or curved with

15、small curvature Np,Qp and Mp are the internal forces due to real loads acting on the structure（7-2）4.1 General formula (Contd)ProcedureReal system：write out the internal loads Np,Qp and Mp due to the real external loads by static equilibrium eqn. and construct the diagram4.1 General formula (Contd)Procedure (Contd)Virtual system：Remove all the given loadsApply the unit virtual loadFind out virtual internal forces Nv,Qv and Mv, and construct diagrams 4.1 General formula (Contd)Procedure (Contd)The desired displacement can be calcu