Shape functions in 1D
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1 MAE 44 & CIV 44 Introuction to Finite Elements Reaing assignment: ecture notes, ogan.,. Summary: Prof. Suvranu De Shape functions in D inear shape functions in D Quaratic an higher orer shape functions Approimation of strains an stresses in an element Aially loae elastic bar y F A() cross section at b() boy force istribution (force per unit length) E() Young s moulus Potential energy of the aially loae bar corresponing to the eact solution u() Π(u) u EA bu Fu( ) Finite element formulation, takes as its starting point, not the strong formulation, but the Principle of Minimum Potential Energy. Task is to fin the function w that minimizes the potential energy of the system Π(w) EA bw Fw( ) From the Principle of Minimum Potential Energy, that function w is the eact solution.
2 Rayleigh-Ritz Principle Step. Assume a solution w( ) a ϕo ( ) + aϕ ( ) + aϕ ( ) +... Where ϕ o (), ϕ (), are amissible functions an a o, a, etc are constants to be etermine. Step. Plug the approimate solution into the potential energy The approimate solution is u( ) a ϕo ( ) + aϕ ( ) + aϕ ( ) +... Where the coefficients have been obtaine from step Π(w) EA bw Fw( ) Step. Obtain the coefficients a o, a, etc by setting Π(w), a i i,,,... ee to fin a systematic way of choosing the approimation functions. One iea: Choose polynomials! w ( ) a Is this goo? (Is an amissible function?) Finite element iea: Step : Divie the truss into finite elements connecte to each other through special points ( noes ) 4 w ( ) a Is this goo? (Is an amissible function?) El # El # El # Total potential energysum of potential energies of the elements Π(w) EA bw Fw( )
3 Total potential energy Π(w) EA 4 El # El # El # Potential energy of element : Π(w) EA bw Potential energy of element : Π (w) EA bw bw Fw( ) 4 El # El # El # Potential energy of element : 4 4 Π (w) EA bw Fw( ) Total potential energysum of potential energies of the elements Π(w) Π(w) + Π (w) + Π (w) Step : Describe the behavior of each element Recall that in the irect stiffness approach for a bar element, we erive the stiffness matri of each element irectly (See lecture on Trusses) using the following steps: TASK : Approimate the isplacement within each bar as a straight line TASK : Approimate the strains an stresses an realize that a bar (with the approimation state in Task ) is eactly like a spring with kea/ TASK : Use the principle of force equilibrium to generate the stiffness matri ow, we will show you a systematic way of eriving the stiffness matri (sections. an. of ogan). TASK : APPROXIMATE THE DISPACEMET WITHI EACH EEMET TASK : APPROXIMATE THE STRAI an STRESS WITHI EACH EEMET TASK : DERIVE THE STIFFESS MATRIX OF EACH EEMET (net class) USIG THE PRICIPE OF MI. POT EERGY otice that the first two tasks are similar in the two methos. The only ifference is that now we are going to use the principle of minimum potential energy, rather than force equilibrium, to erive the stiffness matri.
4 TASK : APPROXIMATE THE DISPACEMET WITHI EACH EEMET Simplest assumption: isplacement varying linearly insie each bar w() a + a How to obtain a an a? w( ) a + a El # w( ) a + a w( ) a + a w( ) a + a Solve simultaneously a a Hence w() a a + () + () () () Shape functions () an () In matri notation, we write w() () Vector of noal shape functions - - [ () ()] Vector of noal isplacements OTES: PROPERTIES OF THE SHAPE FUCTIOS. Kronecker elta property: The shape function at any noe has a value of at that noe an a value of zero at A other noes. - () - () Check - () - ( ) - an ( ) El # 4
5 . Compatibility: The isplacement approimation is continuous across element bounaries () - - w () + El # El # At () - - w ( ) + () - - w () + () - - w ( ) + Hence the isplacement approimation is continuous across elements. Completeness () + () for all () + () for all - Use the epressions () ; - () An check - - () + () an () + () + Rigi boy moe () + () for all What o we mean by rigi boy moes? Assume that, this means that the element shoul translate in the positive irection by. Hence AY point () on the bar shoul have unit isplacement. et us see whether the isplacement approimation allows this. w() () + () () + () YES! Constant strain states ()+ () at all What o we mean by constant strain states? Assume that an. The strain at AY point () within the bar is ε () et us see whether the isplacement approimation allows this. w() () + () () + () w() Hence, ε () YES! 5
6 Completeness Rigi boy moes + Constant Strain states Compatibility + Completeness Convergence Ensure that the solution gets better as more elements are introuce an, in the limit, approaches the eact answer. 4. How to write the epressions for the shape functions easily (without having to erive them each time): Start with the Kronecker elta property (the shape function at any noe has value of at that noe an a value of zero at all other noes) - () - () () () ( - ) ( - ) ( - ) ( - ) El # oe at which is otice that the length of the element - ( - ) The enominator is ( - ) the numerator evaluate at the noe itself A slightly fancier assumption: isplacement varying quaratically insie each bar () () () () () () El # ( - )( - ) ( - )( - ) w() () + () + () ( - )( - ) ( - )( - ) This is a quaratic finite element in ( - )( - ) D an it has three noes an three ( - )( - ) associate shape functions per element. TASK : APPROXIMATE THE STRAI an STRESS WITHI EACH EEMET From equation (), the isplacement within each element w() Recall that the strain in the bar Hence w ε ε B () The matri B is known as the strain-isplacement matri B 6
7 For a linear finite element Hence - - [ () ()] - B - ε B - [ ] Hence, strain is a constant within each element (only for a linear element)! Displacement is linear w() a + a El # Strain is constant - ε El # Recall that the stress in the bar u σ Eε E Hence, insie the element, the approimate stress is σ EB () For a linear element the stress is also constant insie each element. This has the implication that the stress (an strain) is iscontinuous across element bounaries in general. Summary Insie an element, the three most important approimations in terms of the noal isplacements () are: Displacement approimation in terms of shape functions u() Strain approimation in terms of strain-isplacement matri ε() B () Stress approimation in terms of strain-isplacement matri an Young s moulus σ EB () () 7
8 Summary For a linear element Displacement approimation in terms of shape functions - - u() Strain approimation Stress approimation ε [ ] E σ [ ] 8
dw 2 3(w) x 4 x 4 =L x 3 x 1 =0 x 2 El #1 El #2 El #3 Potential energy of element 3: Total potential energy Potential energy of element 1:
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