Module 3 : DeflectionsDr Yan Zhuge

CIVE3011 Structural Analysis and Computer Applications

IntroductionTo learn how to determine the elastic deflections of a structureTwo types of methods will be discussedGeometrical methodEnergy methodEach of these methods has particular advantages or disadvantages

Why to calculate the deflections ?To prevent cracking of attached brittle materials To make sure the structure not deflect severely and to appear safe for its occupantsTo help analysing statically indeterminate structures

To find out deflectionsConjugate beam methodEnergy methodsGeometrical methodsMoment-area method Castiglianos theoremNewNewNewNot covered by this courseLearned beforeDouble-integration methodsVirtual work method

Conjugate-beam MethodThe method requires the same amount of computation as the moment-area theorems to determine a beams slope or deflection. However, the method relies only on the principles of statics, its application will be more familiar.

Using the similarity of equations forUnit = kNm2/EIUnit = kNm3/EI

Beam StaticsBeam deflection

Or integrating

Theorems

Theorem 1: The slope at a point in the real beam is numerically equal to the shear at the corresponding point in the conjugate beam.

Theorem 2: The displacement of a point in the real beam is numerically equal to the moment at the corresponding point in the conjugate beam.

Procedure for AnalysisDraw the conjugate beam for the real beam with a proper boundary conditionsLoad the conjugate beam with the real beams M/EI diagram. This loading is directed downward when M/EI is positive and upward when M/EI is negativeDetermine the statics of the conjugate beam: reactions, Shear force and momentsShear force V corresponds to the slope of the real beam, moment M corresponds to the displacement v of the real beam.

Conjugate-Beam supportsThe shear and moment developed at the supports of the conjugate beam account for the corresponding slope and displacement of the real beam at its supports.

REAL BEAMCONJUGATE BEAM

+++ +

Conjugate beamReal beam

moment

displacement

shear force

slope

Example 1 Conjugate beam Determine the maximum deflection of the steel beam shown in the figure. E = 200 GPa, I = 60(106) mm4.A9 m8 kNBx3 m2 kN6 kN

Example 1 Conjugate beam8 kNABx18kNmABx18/EIConjugate BeamReal Beam45/EI63/EIMaximum deflection occurs at the point where the slope is zeroThis corresponds to the same point in the conjugate beam where the shear is zero2 kN6 kN9 m3 m81/EI27/EI

Example 1 Conjugate beamABx18/EI45/EI63/EIx

Virtual Work MethodThe principle of work and energy is:=Work of External LoadsWork of Internal LoadsIt provides a general means of obtaining the displacement and slope at a specific point on a structure: a beam, frame or truss.

Virtual Work Method (Cont)Imaginary and does not actually exist as part of the real loading

Apply a virtual unit load P=1, acts in the same direction as . This cause internal loading u

Apply the real loading P1, P2, Pn. This causes displacement and internal deformation dL.

Virtual Work Equation

WhereP = 1 = external virtual unit load acting in the direction of u = internal virtual load acting on the element in the direction of dL= external displacement caused by the real loadsdL = internal deformation of the element caused by the real loads Virtual loadingsReal displacements

Method of Virtual Work: TrussesProcedure for analysis External Loading:Apply virtual load P=1 on the structure in the direction of required deflection Calculate the internal load, ie, axial forces n in each member due to the virtual loadCalculate the internal load, ie. Axial forces N in each member due to the read load.Apply the equation of virtual work: If the resultant sum is positive, the displacement is in the same direction as the unit load. Otherwise, is opposite to the unit load.

Displacements due to TemperatureProcedure for analysis:Apply virtual load P=1 on the structure in the direction of required deflection Calculate the internal load, ie, axial forces n in each member due to the virtual loadDeformation for each member due to change in temperature is Apply the equation of virtual work: If any of the members undergoes an increase in temperature, T will be positive. Otherwise, T is negative.

Displacements due to Fabrication ErrorsProcedure for analysis:Apply virtual load P=1 on the structure in the direction of required deflection Calculate the internal load, ie, axial forces n in each member due to the virtual loadDeformation for each member due to a fabrication error (lack of fit) is Apply the equation of virtual work: If a fabrication error increases in the length of a member, L is positive. Otherwise, L is negative.

Example 2 truss displacement The cross-sectional area of each member is A = 400 mm2 and E = 200 Gpa. (a) Determine the vertical displacement of joint C if a 4 kN force is applied to the truss at C. (b) if no loads act on the truss, what would be the vertical displacement of joint C if member AB were 5 mm too short ?

Example 3 Temperature Determine the vertical displacement of joint C of the truss. Due to radiant heating from the wall, member AD is subjected to an increase in temperature of T = +60C. Take = 1.08(10-5)/ C and E = 200 GPa. The cross-sectional area of each member is indicated in the figure.

No need to be computed

Method of Virtual Work: Beams and FramesTo compute at A, a virtual unit load acting in the direction of is placed on the beam at AThe internal virtual moment m is determined by the method of sections at an arbitrary location x from the left supportWhen point A is displaced , the element dx deforms or rotates d = (M/EI)dxHere M is the internal moment at x caused by the real loads The external virtual work done by the unit load is 1 and the internal virtual work done by the moment m is m d =m(M/EI)dx.Real loadsVirtual loadsd

Method of Virtual Work: Beams and FramesProcedure for analysis Bending effect only:Apply virtual load P=1 or a unit couple moment on the structure in the direction of required deflection or slope Calculate the internal load, ie, moment m due to the virtual load, expressed as a function of xCalculate the internal load, ie. moment M due to the read load, expressed as a function of xApply the equation of virtual work:

If the slope is to be determined, the equation of virtual work will become: where m is the internal moment due to a unit couple moment

Simplified Method Using the TablesValues of product integralsFrom Mechanics of materials, 3rd edition, Gere & TimoshenkoMm

Simplified Method Using the TablesValues of product integralsFrom Structural Analysis, SI edition, Hibbeler

Example 4 beam deflection Determine the deflection at point B of the steel beam. E =200GPa, I = 500(106) mm4 Virtual Moment mReal Moment M

Example 5 beam deflectionDetermine the slope at point B of the steel beam. E =200GPa, I = 60(106) mm4 Virtual Moment mReal Moment M

Example 6 frame deflections Determine the horizontal deflection at point C on the frame. E = 200GPa and I = 235(106) mm4. 2.4 m3 m60 kN/ m

Example 6 frame deflections Virtual Moment mReal Moment M