Indeterminate pin-jointed frames (trusses)

Transcription

1 Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all nodes, ncludng the supports, d s the number of truss members, a s the number of external restrants. The truss s statcally ndetermnate when w<, the number of redundant members s equal to the negatve value of w. The soluton of ndetermnate pn-jonted frames by the method of forces wll be demonstrated on the numercal example presented bellow. II. Numercal example.. Member forces of truss subjected to external loads. The degree of freedom of the truss shown n g. s: w= k d a = =. The truss contans one redundant member; wth other words structure s redundant to the frst degree O O O 3 O 4 O 5 O 6 D U D D 4 D 3 U U 3 U 4 U 5 U t c =3 α=. -4 Z Z D 5 D 6 S S S 4 gure Numercal example 6 6 4V V V V3 V4 V5 V6 Geometrcal dmensons of member cross sectons: = - for upper and lower chords and renforcng chan /.5=8 - for the web members We may obtan the prmary statcally determnate system by cuttng the horzontal member of the hnged chan attached n order to stffen the upper truss. Snce the upper part of the system s statcally determnate truss supported by three sngle restrants, the elmnated constrant should belong to the renforcng chan. The chosen prmary system s gven n g.. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 8

2 X gure Statcally determnate prmary system The canoncal equaton wll be of the standard form, takng nto account the fact that n the actual system the relatve movement of the ponts of applcaton of X s zero: δ X +Δ f =. The dsplacements of hnged structures are computed usng only that term of Mohr s equaton whch takes nto account the normal nternal forces, snce bendng moments and shear forces reman nl n all members of the truss. The normal forces, cross secton area (A) and Young s modulus (E) reman constant wthn the lmts of each member, n whch case the requred dsplacements become: N N N, l δ = ds =, f, f, N N N N l Δ f = ds =. In these expressons N, X are the normal forces nduced n the dfferent bars by the unt load X =, and N f, are the forces due to the appled loads n the prmary statcally determnate system. These member forces wth the correspondng actons are gven n fgures 3 and X = X =.75 N gure 3 Member forces due to the unt load S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 8

3 f N gure 4 Member forces due to the appled external loads All data necessary for calculatons of the deflectons δ and Δ f are gven n table. The relatve movement of the ponts of applcaton of X due to the unt load, δ, s obtaned by summaton of the values of column 5. The mutual axal dsplacement Δ f s derved by the summaton of all the values of column 7 (see table ). The canoncal equaton wth the obtaned values of X s gven below the table. The fnal member forces, caused by the appled loads are receved by usng the superposton formula: Nf, = Nf, + N, X. These member forces are derved n column 9 of the table. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 83

4 Table Member forces nduced by the external loads Member forces due to temperature changes member l N N l/ N f N N f l/ N X N f N N f l/ t atn l N t N t N l/ renforcng chan S.5.565E-4.E E S 8 8.E-5.E E S E-4.E E Z E-5.E E Z E-5.E E U 4-4.E-5.E E E lower chord U E E E E U E E E E U E E E E U E E E E U E-5.E E E O E E E upper chord O E E E O E E E O E E E O E E E O E E E S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 84

5 D E E E D E E E D E+ 7.7.E+ 7.7.E+. D E E E+. web members D E E E D E E E V E E E V 4 8.E+.E+.E+. V E-5.E E V E+ -8.E+ -8.E+. V E-5.E E V E+ -8.E+ -8.E+. V E E E δ N, l = Δ f = N, N f, l N N l, t, δ =.49E-3 Δ f = -.58 Δ t = Determnaton of X due to appled external loads Dervaton of X due to temperature changes δ X +Δ = f 3.49 X.58 = X =.9 δ X, t+δ = t 3.49 X.78 = X, t = 77., t S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 85

6 . Member forces n redundant trusses due to temperature changes In the event of temperature changes the standard canoncal equaton for redundant structure of frst degree s: δ X, t+δ =. t Δ t s the relatve movement of the ponts of applcaton of X n the prmary system caused by the temperature changes. Snce bendng moments of all truss members are zero the temperature gradent, Δ h, does not affect on Δ. The unform temperature t c leads to the extenson or t / t flexon of the heated bars. In that respect the dsplacement Δ t may be obtaned usng the followng expresson: l Δ = α t N dx= α t N, l. t c c In our case the coeffcent of temperature expanson s α=. -4, t c =3. All the necessary calculatons concernng temperature changes are also gven n table. The relatve axal movement of the ponts of applcaton of X caused by the temperature changes Δ t s derved by the summaton of all the values of column. The canoncal equaton wth the obtaned values of X,t s gven below the table. The fnal nternal normal force of any member, nduced by temperature changes can be obtaned from: N = N X. t,,, t These member forces are obtaned n column 3 of the table. Compatblty verfcaton The prncple of geometrcal compatblty postulates that the dsplacement of any pont of a structure due to a system of appled forces must be compatble wth the deformatons of the ndvdual members. In that respect the relatve movement of ponts of applcaton of X must be zero, because a constrant between these ponts s avalable n the real redundant structure. Ths requrement for the case of appled external loads could be wrtten as: N Nf N, Nf, l Δ = ds = =. Δ The magntude of dsplacement can be obtaned by the summaton of all the values of column n table. In order to estmate the numercal error due to rounds n numbers we should sum up all the negatve values and all the postve values separately. In ths case, takng fve sgnfcant dgts, we wll obtan: Δ = The relatve movement of ponts of applcaton of X of the redundant structure due to temperature changes has the followng appearance: l N N t t c N dx ds tcn, l Δt N, Nt, l Δ = α + = α + =, S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 86

7 wherefrom: N N l, t, = Δt. The left hand part of the last expresson s the sum of all the values of column 4 from table. The magntude of ths sum, for the case of fve sgnfcant dgts, s: N N l, t, =.78, whch concde completely wth the value of Δ t taken wth reverse (counter) sgn. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 87

8 Influence lnes for member forces n statcally ndetermnate trusses I. Basc concepts The nternal normal force, N f,, n any bar of the presented truss (g. ) could be obtaned by the followng equaton: " N " = " N " + N " X ". f, f,, In ths equaton N f, s the normal force n the -th member of the prmary statcally determnate system caused by the appled loads. or nfluence lnes constructon the appled load s a unt force traversng the road. N, s the normal force of -th member of the smple truss caused by the acton X = replacng the elmnated constrant. The normal force N f, depends on the poston of the unt load. The correspondng nfluence lne for ths member of the prmary system could be easly obtaned. The force X ntroduced to replace the elmnated constrant depends on the poston of the unt load. The standard canoncal equaton, showng that the dsplacement along the lne of acton of the redundant constrant X (g. ) s zero, takes the form: δ X +Δ f =, so, we get: " Δ f " " X " =. δ The varaton of Δ f when the load unty travels along the upper chord of the truss wll concde wth the deflecton curve of the same chord of the prmary structure subjected to the acton of X =. II. Numercal example In the followng example we shall construct the nfluence lnes for normal forces n bars U, V, O and D of the truss gven n g... Influence lne for Δ f. In order to obtan the nfluence lne for the dsplacement Δ f, we should construct the elastc curve of the upper chord. or that purpose a horzontal conjugate beam must be formed. Ths beam should be loaded wth concentrated tous loads correspondng to the mutual dsplacements (rotatons and vertcal deflectons) arsng between two bars of the road. Concentrated tous forces must be appled to the magnary conjugate beam at ponts correspondng to jonts,,, 3, 4, 5 and 6 (g. 5), because a mutual rotaton s possble between each two bars connected at these ponts. Concentrated tous moment should be appled at ponts correspondng to jonts and 6, because a mutual vertcal dsplacement arses between connected members at these ponts and the ground (g. 5). The conjugate beam could be supported as a smple beam (g. 5). In ths case the unknown concentrated forces and 6 wll be calculated as support reactons. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 88

9 X M X Conjugate beam M 6 M Smply supported conjugate beam M " Δ " f " X " gure 5 Influence lnes for Δ f and X S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 89

10 Let us proceed wth the determnaton of elastc concentrated force. Ths tous load represents the angular rotaton of bar O wth reference to bar O. In order to fnd the magntude of ths load we should apply two vrtual moments, n the drecton of mutual rotaton, to bars O and O. These moments must be replaced by equvalent unt couples of forces = appled at λ 4 the truss jonts (g. 6a). Next, we should compute the normal forces nduced n the truss bars by these couples. It s readly seen that all the bars except bars O, O, V, D, and D wll reman dle. Member forces n the loaded bars are depcted n g. 6a. The tous concentrated force s: N, N, l =. In ths equaton N, s the normal force of the -th member of the smple truss nduced by the vrtual moments M =, X =. N, s the -th member force of the prmary truss caused by the forces All necessary nformaton concernng the calculaton of s gven n table. Table Dervaton of member l N N N N l/ O E-6 O E-6 V E+ D E-5 D E-5 N, N, l = = -.5E-5 In a smlar way we wll calculate the tous forces and 3. The vrtual loads wth the correspondng normal forces are gven n fgures 6b and 6c. The forces are obtaned usng the nformaton presented n tables 3 and 4. Snce the system s completely symmetrcal, elastc load 4 equa ls to and 5 =. nally, we should obtan the elastc moments M and M 6. The tous moment M represents the vertcal dsplacement of bar O at jont wth reference to the ground. In order to fnd the magntude of ths load we should apply two vrtual forces, n the drecton of the mutual vertcal dsplacement, to bar O at jont and to the ground (g. 6d). The dervaton of tous moment M s gven n table 5. The vrtual load for obtanng of M 6 s shown n g. 6e. The value of ths moment s obtaned n table 6. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 9

11 Table 3 Dervaton of member l N N N N l/ V E+ V E+ D E-5 D E+ U E-5 U E-5 Table 4 Dervaton of 3 N, N, l = = E-5 member l N N 3 N N 3 l/ O E-5 O E-5 V E+ D E+ D E+ Table 5 Dervaton of M N, N3, l 3 = = -3.E-5 member l N N N N l/ V E-5 Table 6 Dervaton of M 6 N, N, l M = = 3.75E-5 member l N N 6 N N 6 l/ V E-5 N, N6, l M6 = = -3.75E-5 S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 9

12 The smply supported conjugate beam, loaded wth the obtaned tous concentrated loads, s gven n g. 5. The bendng moment dagram of the conjugate beam s the requred nfluence lne for the dsplacement along the lne of acton of the redundant constrant X - Δ f. The nfluence lne for X s " X" = " Δf "/ δ. Both the nfluence lnes for Δ f and X are depcted n g a) N b) N.5.5 c) N. d). N.. e). N 6 gure 6 Vrtual loads for dervaton of concentrated tous forces. S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 9

13 Verfcaton Snce the deflecton Δ f and the force n the elmnated constrant X were obtaned usng the method of forces for the gven external loads, we can verfy them now usng ther nfluence lnes. Ths can be regarded as a verfcaton of the constructed nfluence lnes. Recall the dervaton of any functon by usng nfluence lnes: the value of any functon arsng n a gven secton as a result of applcaton of some concentrated forces, a number of dfferent unform loads and some moments, could be obtaned by: S = η + q ω + M tgα, where: η s the ordnate to the nfluence lne for ths functon below the force ; ω represents the area bounded by the nfluence lne, the ordnates correspondng to the lmts of loadng q, and the x axs; α s the rotaton angle between x axs and nfluence lne. or the dsplacement Δ f one can wrte (g. for external loads and g. 5 for nfluence lne): Δ = f =-.58 ( Δ f =-.58 from table ), X = =.9 (X =.9 from table ).. Influence lnes for the requred member forces n statcally determnate smple system. Dervaton of nfluence lnes for member forces n statcally determnate trusses was examned n detals n the second lecture. Here, the nfluence lnes for requred member forces are gven n g. 7 wthout detaled explanaton. Verfcaton Let us obtan the normal forces n members nto consderaton by usng ther nfluence lnes and compare the results wth the normal forces obtaned n g. 4. In such a way we shall verfy the nfluences lnes constructed for smple system (g. for appled loads and g. 7 for nfluence lnes). O = =-4 ( O =-4 from g. 4); D = = ( D = from g. 4); V = ; U =. 5 = S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 93

14 O road lane V D U "O " "D ". - "V " "U " gure 7 Influence lnes for members n the prmary statcally determnate system 3. Influence lnes for the requred member forces n the gven statcally ndetermnate truss. nally, the nfluence lnes for member forces n statcally ndetermnate truss can be derved by the followng general equaton: " Nf, " = " Nf, " + N, " X ". or the truss members nto consderaton ths expresson takes the form gven bellow (the values of N, are taken from g. 3): f,, f,, " O " = " O " + O " X " = " O " +.75 " X "; " D " = " D " + D " X " = " D " +.6 " X " ; " V " = " Vf," + V, " X " = " V " + " X " = " V " ; " U " = " U f," + U, " X " = " U " " X " = " X ". The correspondng nfluence lnes are shown n g. 8. Verfcaton In order to verfy the obtaned nfluence lnes for ndetermnate truss we shall derve the normal forces n members nto consderaton by usng ther nfluence lnes and compare the results wth those n table (g. for appled loads and g. 8 for nfluence lne ordnates). S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 94

15 O = = ( O = from Table ) Δ=.%; D = = ( D = from Table ) Δ=.%; V = ( V =from Table ); U = =-.9 ( U =-.9 from Table ) O V D U "O " "D " "V " "U " gure 8 Influence lnes for members n statcally ndetermnate truss S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 95

16 References DARKOV, A. AND V. KUZNETSOV. Structural mechancs. MIR publshers, Moscow, 969 WILLIAMS, А. Structural analyss n theory and practce. Butterworth-Henemann s an mprnt of Elsever, 9 HIBBELER, R. C. Structural analyss. Prentce-Hall, Inc., Sngapore, 6 S. Parvanova, Unversty of Archtecture, Cvl Engneerng and Geodesy - Sofa 96