Formal Methods for Deriving Element Eqations And the importance of Shape Fnctions
Formal Methods In previos lectres we obtained a bar element s stiffness eqations sing the Direct Method to obtain eact soltions for a simplified specific sitation i.e., constant cross-section; loads at end only. Normally the elemental stiffness eqations are derived throgh more general formal methods in which the eqations are obtained by sing approimate soltions for comple general sitations e.g., varying cross-section; distribted loads.
Aially oaded Members General Problem Review A p Displacement: Strain: Stress: Force: ε d σ Eε d P Aσ dp p d Eqilibrim: Bondary Conditions: d d EA d d σ and p or or σ Slide from B.S. Altan, Michigan Technological U.
Shape Fnctions All eisting formal methods reqire making an assmption abot the displacement within an element and how it shold depend on the nodal displacements and. et s assme:
Shape Fnctions 5 The shape fnctions in N are sed to interpolate the displacement N is called the shape fnction matri. Each element in the matri is the eqation of the shape fnction for the corresponding DOF. [ ] Nd N N Then:
Shape Fnctions The aial strain is therefore given by: I.e., B d N d Ths. If E is the elastic modls, then: σ ε EBd E [ ] 6 σ Eε E The fnctions in B are sed to interpolate the strain ε and stress σ.
Higher-Order Shape Fnctions To get more accracy in complicated sitations, higherorder polynomial shape fnctions shold be sed. et s assme: / 7
Higher-Order Shape Fnctions Then: 8 [ ] Nd N N N
Higher-Order Shape Fnctions The shape fnctions corresponding to each DOF can be graphed as: N N N N N N The overall shape fnction looks like this: N N N The displacement is a qadratic fnction of as specified in the N matri. 9
Higher-Order Shape Fnctions The aial strain is therefore given by: Ths: Bd d N 8 d d d d ε 8 ε The strain is a linear fnction of as specified in the B matri.
Higher-Order Shape Fnctions E E E E 8 8 Bd ε σ If the displacement shape fnction N is qadratic nd order, then the stress and strain are linear st order. In a bar element, the stress and strain are always one order less than the displacement.
The Problem with Shape Fnctions The shape fnctions restrict how the stress and strain can vary as a fnction of. This makes it impossible in general to eactly satisfy: The eqilibrim condition and The bondary conditions d d d EA d p or or σ σ and Different formal methods try to make the soltion accrate in different ways.
Formal Method : Weighted Residal Method In this method we try to create elemental k and r matrices that minimize the error R when we solve for d. For a bar element with linear shape fnction: d p F A E d p F d d A E d p F d d A E d p d d EA d d R It wold eqal zero if there was no approimation error in the shape fnction. Sbstitte from the shape fnction.
Weighted Residal Method The challenge is to compte vales for: Stiffness k to accont for E,A,and, and Eqivalent loads r to accont for p so that and are calclated in a way that always minimizes the error the residal. Keep in mind that: k and r contain constant vales not fnctions of the displacement bondary conditions mst be satisfied eactly.
Weighted Residal Method There are several poplar Weighted Residal Methods that try to minimize the residal in different ways: Collocation Method set the residal to be zero at specific locations R E A F R E A F 5
Weighted Residal Method Sb-domain Method Integrate the error over part of each element and set that eqal to zero. Galerkin Method Integrate over the whole domain, bt se weighting fnctions that are eactly the same as the shape fnctions. R d R d N R d N R d 6
n e E e Formal Method : Minimize Potential Energy In the Minimm Potential Energy Formlation, we try to create K and R matrices that minimize the potential energy with respect to the soltion D. The potential energy for a body Π is given by: Π Strain Energy n e n e e e σεad Eε m i Ad Work Energy F m i m Bd Ad i i F i i i 7 F i i Sbstitte ε from the shape fnction derivative.
Minimize Potential Energy Potential Energy is at a minimm when: Some math happens, which yields: k B / E / AE T E BAd A d Π δ 8
Uniformly Distribted oads The effect of the per-nit-length distribted load p is represented by eqivalent point loads r applied on the element s nodes. For niformly distribted aial loads: 9
Uniformly Distribted oads For a general aial load p, from the Minimization of Potential Energy method: r For a linear shape fnction: r p N p p T d d d