Fundamentals of the stiffness method. Fundamentals of the stiffness method. Fundamentals of the stiffness method

Transcription

1 CHAPER 6 russ Analsis using Stiffness Metho Objectives เข าใจว ธ ของ stiffness metho ใช ว ธ stiffness metho ก บ russ, BM & rame จะพ ดถ งในบท หน า unamentals of the stiffness metho he stiffness metho: Is a isp metho of analsis Can be use to analse both staticall eterminate an ineterminate structures Yiels the isp & forces irectl It is generall much easier to formulate the necessar for the computer using the stiffness metho unamentals of the stiffness metho unamentals of the stiffness metho Application of the stiffness metho reuires subiviing the structure into a series of iscrete finite elements & ientifing their en points as noes or truss analsis, the finite elements are represente b each of the members that compose the truss & the noes represent the joints he force-isp properties of each element are etermine & then relate to one another using the force em en written at the noes hese relationships for the entire structure are then groupe together into the structure stiffness matri, K he unnown isp of the noes can then be etermine for an given loaing on the structure When these isp are nown, the eternal & internal forces in the structure can be calculate using the force-isp relations for each member

2 Member stiffness matri o establish the stiffness matri for a single truss member using local an coorinates as shown When a +ve isp is impose on the near en of the member while the far en is hel pinne he forces evelope at the ens of the members are: Member stiffness matri iewise, a +ve isp at the far en, eeping the near en pinne an results in member forces '' ' ' B superposition, the resultant forces cause b both isp are ' ' Member stiffness matri hese loa-isp en ma be written in matri form as: ' ' his matri, is calle the member stiffness matri isplacement & orce ransformation Since a truss is compose of man members, we will evelop a metho for transforming the member forces an isp efine in local coorinates to global coorinates Global coorinates convention: +ve to the right an +ve upwar an as shown + =

3 isplacement & orce ransformation isplacement & orce ransformation matri analsis as follows hese will be ientifie as or e.g. consier member of the truss as shown he coorinates of & are (, ) an (, ) ( ) ( ( ) ( ) ) isplacement & orce ransformation isp ransformation matri In global coorinates each en of the member can have egrees of freeom or inepenent isp namel joint has an Joint has an isplacement & orce ransformation isp ransformation matri When the far en is hel pinne & the near en is given a global isp, the corresponing isp along member is A isp n will cause the member to be isplace along the ais

4 isplacement & orce ransformation isp ransformation matri In matri form, et isplacement & orce ransformation orce ransformation matri If is applie to the bar, the global force components at are: Using isplacement & orce ransformation orce ransformation matri In matri form isplacement & orce ransformation orce ransformation matri In this case, transforms the local forces acting at the ens of the member into 4 global force components his force transformation matri is the transpose of the isp transformation matri

5 Member global stiffness matri Member global stiffness matri We can etermine the member s forces in terms of the global isp at its en points Performing the matri operation iels: ' Substitution iels the final result: ' or ' russ stiffness matri Once all the member stiffness are forme in the global coorinates, it becomes necessar to assemble them in the proper orer so that the stiffness matri K for the entire truss can be foun his is one b esignating the rows & columns of the matri b the 4 coe numbers use to ientif the global egrees of freeom that can occur at each en of the member he structure stiffness matri will then have an orer that will be eual to the highest coe number assigne to the truss since this rep the total no. of egree of freeom for the structure his metho of assembling the member to form the structure stiffness matri will now be emonstrate b numerical e.g. his process is somewhat teious when performe b han but is rather eas to program on computer Eample 1 etermine the structure stiffness matri for the member truss as shown. is constant.

6 Eample 1 cont Eample 1 cont Member 1 Member iviing each element b = 3m, we have: iviing each element b = 5 m, we have: Eample 1 cont his matri has an orer of 66 since there are 6 esignate egrees of freeom for the truss. Application of the stiffness metho for truss analsis he global force components acting on the truss can then be relate to its global isplacements using K his en is referre to as the structure stiffness en

7 Application of the stiffness metho for truss analsis Epaning iels u K K 11 1 Often = since the supports are not isplace hus becomes u u K K K11 u 1 Application of the stiffness metho for truss analsis Since the elements in the partitione matri K 11 represent the total resistance at a truss joint to a unit isp in either the or irection, then the above en smbolizes the collection of all the force em en applie to the joints where the eternal loas are zero or have a nown value Solving for u, we have: u 1 K11 Application of the stiffness metho for truss analsis With = iels u K1 u Eample etermine the force in each member of the - member truss as shown. is constant. he member forces can be etermine Since with = - for em,

8 Eample cont he origin of, an the numbering of the joints & members are shown. B inspection, it is seen that the nown eternal isp are 3 = 4 = 5 = 6 = Also, the nown eternal loas are 1 =, =-. Hence, Eample cont = K for the truss we have We can now ientif K 11 an thereb etermine u B matri multiplication, Eample cont B inspection one woul epect a rightwar an ownwar isp to occur at joint as inicate b the +ve & -ve signs of the answers. Using these results, Eample cont Epaning & solving for the reactions he force in each member can be foun. Using the ata for an in eample 14.1, we have:

9 oal Coorinates oal Coorinates A truss can be supporte b a roller place on a incline When this occurs, the constraint of zero eflection at the support (noe) cannot be irectl efine using a single horizontal & vertical global coorinate sstem Consier the truss he conition of zero isp at noe 1 is efine onl along the ais Because the roller can isplace along the ais this noe will have isp components along both global coorinates aes & o solve this problem, so that it can easil be incorporate into a computer analsis, we will use a set of noal coorinates, locate at the incline support hese aes are oriente such that the reactions & support isp are along each of the coorinate aes o etermine the global stiffness en for the truss, it becomes necessar to evelop force & isp transformation for each of the connecting members at this support so that the results can be summe within the same global, coorinate sstem oal Coorinates oal Coorinates Consier truss member 1 having a global coorinate sstem, at the near noe an a noal coorinate sstem, at the far noe When isp occur so that the have components along each of these aes as shown

10 oal Coorinates his en can be written in matri form as oal Coorinates his can be epresse as: he isp & force transformation in the above en are use to evelop the member stiffness matri for this situation We have ' oal Coorinates Performing the matri operation iels: his stiffness matri is use for each member that is connecte to an incline roller support he process of assembling the to form the structure stiffness matri follows the stanar proceure Eample 3 etermine the support reactions for the truss as shown.

11 Eample 3 cont etermine the support reactions for the truss as shown. he stiffness for members 1 an must be evelope. Member 1, Eample 3 cont Member,, 1,.77,.77 1,,.77,.77 Eample 3 cont Member 3,.8,.6 Eample 3 cont o etermine the structure stiffness matri, we have:

12 Eample 3 cont Carring out the matri multiplication of the upper partitione, the three unnown isp are etermine from solving the resulting simultaneous en. he unnown reactions are obtaine from the multiplication of the lower partitione.