Finite Element Truss Problem

Transcription

1 6. rue Uing FEA Finite Element ru Problem We tarted thi erie of lecture looking at tru problem. We limited the dicuion to tatically determinate tructure and olved for the force in element and reaction at upport uing baic concept from tatic. Next, we developed ome baic one dimenional finite element concept by looking at pring. We developed a baic ytem of node numbering that allow u to olve problem involving everal pring by imply adding the tiffne matrice for each pring. In the next two lecture, we will extend the baic one dimenional finite element development to allow u to olve generalized tru problem. he techniue we will develop i a little more complex than that originally ued to olve tru problem, but it allow u to olve problem involving tatically indeterminate tructure. 6. ocal and Global Coordinate We can extend the one dimenional finite element analyi by looking at a one dimenional problem in a two dimenional pace. Below we have a finite element could be a pring that attache to node and. y x y ocal coordinate ytem Global coordinate Sytem x Figure - ocal and global coordinate ytem he x, y coordinate are the local coordinate for the element and x, y are the global coordinate. he local coordinate ytem look much like the one dimenional coordinate ytem we developed in the lat lecture. We can convert the diplacement hown in the local coordinate ytem by looking at the following diagram. We will let and repreent diplacement in the local coordinate ytem and,, 3, and 4 repreent diplacement in the x-y global

2 coordinate ytem. Note that the odd ubcripted diplacement are in the x direction and the even one are in the y direction a hown in the following diagram. Un-deformed element in θ 3 4 coθ Deformed element? Figure - he deformation of an element in both local and global coordinate ytem. In a previou lecture we looked at the deformation of pring by looking at the diplacement at the end of the pring. Here we are going a tep farther into finite element development by looking at the train energy of the element. he element could be a pring but in thi cae we will generalize and look at it a any olid material element. he only retriction we will place on the element i that the deformation i mall compared to it total length. We know from Hook law that the force i directly proportional to the deformation. F k x 6. We can compute the energy by integrating over the deformation u kq Q k xdx 6. where k the element tiffne, A the cro ectional area of the element, E Young modulu for the material, and the length of the element. Q i the total change in length of the element. Note that we are auming the deformation i linear over the element. All eual length egment of the element will deform the ame amount. We call thi a contant train deformation of the element. We can rewrite thi change in length a

3 ' ' Q 6.3 Subtituting thi into euation 6. give u k u 6.4 or k u 6.5 Rewriting thi in vector form we let 6.6 and k 6.7 With thi we can rewrite euation 6.5 a: k u 6.8 We can do the indicated operation in 6.8 to ee how the vector notation work. We do thi by firt expanding the term then doing the multiplication. { } u 6.9 { } u 6. u 6. u 6.

4 u 6.3 Which i the ame a euation 6.5. Euation 6.7 i the tiffne matrix for a one dimenional problem. It bear very cloe reemblance to euation 5.7 ued in our one dimenional pring development. k k K 5.7 k k [ ] k wo Dimenional Stiffne Matrix We know for local coordinate that 6.6 and for global coordinate See Figure We can tranform the global coordinate to local coordinate with the euation and coθ in θ coθ 4 in θ 6.6 hi can be rewritten in vector notation a: M 6.7 where

5 M c c, 6.8 Uing c coθ, and in θ. u k 6.8 we can ubtitute in euation 6.7 u [ M k M ] 6.9 Now we will let k M k M 6. and doing the multiplication, k our tiffne matrix for global two dimenional coordinate become where: c c c c c c k 6. c c c c c c E Young modulu for the element material A the cro ectional area of the element the length of the element c coθ in θ 6.3 Stre Computation he tre can be written a σ Eε 6. where ε i the train, the change in length per unit of length. We can rewrite thi a:

6 total deformation σ E 6.3 length of element In vector form we can write the euation a E σ { } 6.4 From our previou dicuion, we know that in local coordinate 6.5 and in global coordinate From euation 6.7 we know that M 6.7 where M c c 6.8 Subtituting thi in to the euation 6.4 yield E σ { } M 6.7 Now we multiply M by the vector E σ { c c } 6.8