Principles of Finite Element for Design Engineers and Analysts Ayman Shama, Ph.D., P.E., F.ASCE
Outline Principles of Engineering Analysis The development of the finite element method Types of elements Types of analyses Applications of the method in: failure investigations Construction Design
Basic Principles of Engineering Analysis Analysis of an Engineering system Formulation of mathematical model Lumped-parameter (discrete systems) Response is described by solution of a set of algebraic equations Continuum-mechanics (continuous systems) Response is described by Solution of a set of differential equations Matrix stiffness method Finite element method
The Predecessor of the Finite Element Method The matrix stiffness method was first developed by Collar and Duncan between 1934 and 1938 Further contributions by John Argyris between 1952 and 1958 Limited to one dimensional elements only Deals with discrete mathematical models
Overview of the Matrix Stiffness Method A structure consists of one dimensional line elements each has two nodes
We define the stiffness as the force applied at a certain node to produce a unit deformation at this node or at another node. Therefore: P = the applied force P K K= the stiffness = the displacement For a two node member with only one degree of freedom P 1 K11 1 K12 2 P K K 2 21 1 22 2 P P 1 2 K K 11 21 K K 12 22 1 2
Step 1: Determine the stiffness matrices of all elements EA/L 0 0 0 0 0 0 12EI z L 3 0 0 0 6EI z L 2 K 11 12EI 0 0 y 6EI L 3 0 y L 2 0 0 0 0 GJ/L 0 0 0 0 6EI y L 2 0 4EI y /L 0 0 6EI z L 2 0 0 0 4EI z /L EA/L 0 0 0 0 0 0 12EI z L 3 0 0 0 6EI z L 2 K 12 12EI 0 0 y 6EI L 3 0 y L 2 0 0 0 0 GJ/L 0 0 0 0 6EI y L 2 0 2EI y /L 0 0 6EI z L 2 0 0 0 2EI z /L
Step 2: Build up the overall structure stiffness matrix for the whole structure Step 3: Solve for the nodal displacements K 11 K 12 K 13 0 0 0 K 21 K 22 K 23 0 0 0 K 31 K 32 K 33 K 34 K 35 0 0 0 K 43 K 44 K 45 0 0 0 K 53 K 54 K 55 K 56 0 0 0 0 K 65 K 66 Step 4: Knowing nodal displacements, determine internal forces in each member and reactions
The Boeing Project 1952-1953 Collaboration between Berkley and Boeing One-dimensional elements failed to model the experimental behavior of wing structures Jon Turner proposed 2-dimensional elements Ray Clough provided analysis and calculations The study resulted in a formulation of the constant strain triangular membrane element
The Birth of the Finite Element Method Clough coined the terminology finite element method in his paper Presented a FE approach of two-dimensional elements to solve problems in continuum mechanics
How Finite Element Method Works? Step 1: The continuum is idealized as an assemblage of a number of discrete elements connected at the nodes only
Step 2: Assume a displacement field: Polynomials used to specify the relationship between the internal displacements of each element and its nodal displacements The displacement field is used as a basis for the element stiffness matrix Example (triangular element): u x, y v x, y = a 0 + a 1 x + a 2 y = b 0 + b 1 x + b 2 y Constant stress element Provides stresses, strains, and displacements at one point inside the element
Displacements: In matrix form Apply at nodes u x, y v x, y u i n v i n u j n v j n u k n v k n u v = a 0 + a 1 x + a 2 y = b 0 + b 1 x + b 2 y = 1 x y 0 0 0 0 0 0 1 x y = φ { } = 1 x i y i 0 0 0 0 0 0 1 x i y i 1 x j y j 0 0 0 0 0 0 1 x j y j 1 x k y k 0 0 0 0 0 0 1 x k y k a 0 a 1 a 2 b 0 b 1 b 2 a 0 a 1 a 2 b 0 b 1 b 2 n = A { } = A 1 { n } or Therefore = φ A 1 { n } or = N { n } N
Strain: From theory of elasticity strain is evaluated ε = ε = 0 1 0 0 0 0 0 0 0 0 0 1 0 0 1 0 1 0 Stain displacement transformation matrix ε x ε y γ xy = u x v y u y + v x 1 x i y i 0 0 0 0 0 0 1 x j y i 1 x j y j 0 0 0 0 0 0 1 x j y j 1 x k y k 0 0 0 0 0 0 1 x k y k {ε} = B {Δ n } 1 u i n v i n u j n v j n u k n v k n
Stress: In isotropic material of Young s modulus E and Poisson s ν ratio stresses and strains are related by σ x σ y = E 1 υ 0 σ 1 v υ ε x 0 x E υ 1 0 σ υ τ xy 1 υ 2 0 0 1 1 υ vε y 0 y = τ γ xy xy (1 + v)(1 2v) 1 2υ 0 2 0 2 {σ} = D {ε} The stiffness matrix of the plate element is obtained using the principle of virtual displacements The virtual work done by the external forces is equal to the virtual work done by internal stresses P = ε x ε y γ xy x P i P i y P j x P j y P k x P k y
Evaluate element stiffness matrix: The virtual work done by nodal external forces W e = { n v } T {P} The virtual work done by internal stresses Therefore Element stiffness matrix
Step 3: Build up the overall structure stiffness matrix for the whole assemblage Step 4: Solve for the nodal displacements Step 5: {Δ n } = K 1 P Knowing nodal displacements, determine internal displacements, strains, and stresses in each element = [N] { n } {ε} = B {Δ n } {σ} = D {ε} [N]= φ A 1
urther Developments of the Finite Element Method The Isoparametric element formulation (B. Irons 1966) X = 1 2 1 r X 1 + 1 2 1 + r X 2 X = h i X i 2 i=1 2 U = h i U i i=1 The formulation is achieved through the use of shape functions (interpolation functions) and natural coordinate system for the element
Shape functions of four to nine nodes two dimensional elements n n X = h i X i Y = h i Y i Coordinate interpolations i=1 i=1 n n U = h i U i V = h i V i i=1 i=1 Element displacements
Advantages of isoparametric elements provides the relationship between the element displacement at any point inside the element and the element nodal points displacements No need to evaluate the transformation matrix A -1 Matrix [N] is obtained directly in terms of the shape functions The elements can have curved boundaries
Further Developments in the Formulation of Finite Element Equations Melosh (1963) showed that the finite element formulation can be expressed using variational and energy methods using the principle of the minimum potential Barna and Lee (1969) showed that finite element nodal equations can be derived and solved numerically using Galerkin s method. Zienkiewiczs (1971) applied the energy and Galerkin methods to a class of physical problems including structural mechanics, heat transfer, and fluid mechanics. These methods are very powerful for major physical problems that can be idealized in the form of differential equations
Types of Elements Used in Finite Element Analysis Truss Element The truss element is usually a 2- node element Subjected to axial loads either tension or compression One degree of freedom, axial displacement Beam Element 2-node element with 6 degrees of freedom Comes with moment release and rigid-end capabilities
Two-Dimensional Solid Elements Also called plane element Two translational in plan degrees of freedom at each node, i.e., no rotations or moments Two options either plane stress elements or plane strain elements Plane stress element: Thin plates that are free to move in the direction normal to the plane of the plate Plane strain element: A slice of a very long solid structure such as dams, and retaining walls
Axisymmetric Elements Used to model cylindrical structures Both the structure and loading are axisymmetric Provides for the stiffness of one radian of the structure
Three-Dimensional Solid Elements Three translational degrees of freedom at each node. i.e. no rotations
Shell Elements Carry plate bending, shear and membrane loadings Forces per unit length or stresses can be reported on mid-surface Care has to be taken in selecting the element to avoid locking Plate Elements Special case of shell elements
Spring Elements It is used to connect two nodes or to attach a node to ground Forces are calculated in terms of the spring s stiffness and the relative deformation between the element end nodes A damper element provides the same purpose but the relationship is between relative velocities and damping
Types of Analyses Linear static analysis Solution of K =P Nonlinear static analysis The analysis is undertaken incrementally in time steps At each step equilibrium satisfies K (t+δt) (i 1) ΔU (i) = P (t+δt) F (t+δt) (i 1) U (t+δt) (i) = U (t+δt) (i 1) + ΔU (i) Dynamic analysis-time domain Linear on nonlinear Newmark method or Wilson method
Types of Analyses Dynamic analysis-frequency domain Fourier methods Spectral analysis method (probabilistic) Response spectrum methods Heat transfer analysis Temperature displacement Conductivity modulus of elasticity Heat flux stress or pressure Computational fluid dynamics analysis Principles of conservation of mass, momentum, and energy govern
Applications in Failure Investigations
Investigation of the Failure of Welded Steel Moment Frame Connections During Northridge Earthquake Inspection of several buildings after the Northridge Earthquake demonstrated damage of the connection. damage occurred in or near the welded joint of the girder bottom flange to the supporting column flange The connection has undergone considerable research under SAC program. A damaged building was subject of this investigation
Building Subject Investigation A comprehensive survey of the building showed that it was leaning 6 to the north at roof level
Building Subject Investigation Inspection of moment frame connections indicated fracture damage at 29 locations in the east and west frames.
FE Model of the Connection Stress 90 80 70 60 50 40 A36 Beam 30 20 A572 Grade 50 Column 10 E70 Weld 0 0 0.05 0.1 0.15 0.2 Strain
Connection Damage Assessment Triaxiality: T=s H /s eff Hydrostatic stress: s H = (s 1 + s 2 + s 3 )/3 von Mises (effective) stress: (stress)/(yield stress) 2.0 1.8 1.6 1.4 1.2 1.0 0.8 0.6 0.4 0.2 0.0 s y s max triaxial stress state (T=1.5) uniaxial tension test, or s e in any general stress state 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 strain s ult s eff 2 2 s s s s s 2 1 2 1 2 2 3 3 s 1
Triaxiality- q p =0.011- Symmetry Plane W36x170 beam, W14x342 column
Triaxiality Stress- q p = 0.011- Through column flange 92132 102132 W36x170 beam, W14x342 column
Triaxiality Stress- q p = 0.011-Through Beam Bottom Flange W36x170 beam, W14x342 column
Triaxiality Stress- q p = 0.011- in Weld 12133 72133 82133 92133 Trillium-W36x170 beam, W14x342 column
Triaxiality Evaluation Capacity of an exterior connection located at 14th floor as dictated by stress triaxiality
Triaxiality demand Triaxiality limits
Investigation of Failure of Barrier Gates during Sandy Aluminum sections: 3.5 OD -.125 wall thickness Pipe 3.5 SCH 40 (OD 4 -wall thickness 0.226 ) Pipe 4 SCH 40 (OD 4.5 -wall thickness 0.237 ) ASTM B241, aluminum alloy 6061-T6
Performance during Sandy Barrier gate arms fractured during storm Sandy when wind gusts in the NJ area exceeded 70 mph.
Performance during Sandy weld 3.5 in. outer diameter pipe on each section had fractured at the same distance (approximately 0.15 0.25 in.) from the weld.
FE Investigation Three-Dimensional FE model 0.2 inch 3-node shell element Analysis for wind load of 70 mph
FE Investigation-continued the plastic strain reached the ultimate value at the row of elements right above the weld, i.e. at approximately 0.15 in. to 0.20 in. from the weld.
Solve issues During Construction
Suspension System/Superstructure Connection Suspender brackets are designed to transfer the load of the suspended deck to the suspender ropes. The connection consists of four (4) open sockets each with 3-1/16 diameter pin
Suspension System/Superstructure Connection Connection consists of one ¾ pin plate and two 1 doubling plates. Effective weld throat 8 mm. Doubling plate Pin Plate
Problem Encountered During Fabrication Ultrasonic testing illustrated that the effective weld throat was not kept at the minimum size of 8 mm (5/16 inches). The reduced weld throat size ranges from 3 mm (1/8 inches) to 8 mm (5/16 inches).
Finite element model 8-node brick elements to represent the plates, pin, suspender socket, and welds Constant effective weld throat of 3 mm (1/8 inches) was assumed as a worst case scenario contact gap elements for the interface of the pin with the sockets and the plates.
Sources of non-linearity in the model The nonlinear boundaries at the pin-plates interface (contact elements). The non-linear material properties for steel. Plastic multi-linear model with isotropic hardening
Analysis Results Effective stresses at the Pin plate PJP weld
Applications in Design
FE Verification of Radome Structure design Radome is a dome that protects radar equipment. The radome subject this study is located on top of an Air Port Control Tower The main purpose of the study is to independently check the design provided by the manufacturer
Summary of Radome Design The manufacturer used an in-house shell stress program The radome is constructed of polygonal panels The Panels program are made uses of sandwich input construction: wind speed: Skins are of fiberglass reinforced resin laminate. to determine the wind pressure on the radome Foam core (polyisocyanurate) to calculate membrane stress resultant Panels are assembled by bolting through reinforced panel edges
Independent study strategy Evaluate the distribution of wind pressures on the radome structure. Use basic principles of computational fluid dynamics (CFD) as implemented in the FE method simulate a boundary layer that includes the radome as located on top of the control tower.
Independent study strategy-continued The wind pressures as obtained from the CFD analysis are applied to the radome structure in a subsequent FE stress analysis Determine the maximum principle stresses, loads and moments of the structure due to wind loads. Check the design of the radome structure
FE Wind Tunnel Analysis The control tower and its adjacent structures were included in the CFD-FE wind analysis study.
FE Wind Tunnel Analysis-continued The wind computational flow domain employed: was 2,000 ft long; 1,000 ft wide; 358.6 ft high. Design wind velocity of 150 mph was used at the radome base E W S N
FE Wind Tunnel model parameters
FE Wind Tunnel Results Velocity vectors of the wind approaching the tower (ft/s)
FE Wind Tunnel Results-(continued) Rotational velocity and vortices growth
FE Wind Tunnel Results-(continued) Distribution of pressure on radome (psi)
FE Wind Tunnel Results-(continued) Distribution of kinetic energy (lb ft)
Analysis of the Radome structure to Wind Pressure A maximum tensile stress of 9,018 psi less than the 38,000 psi tensile strength of the fiberglass reinforced resin laminate factor of safety is 4.2 for tension stresses.
Analysis of the Radome structure to Wind Pressure Maximum values of principal membrane force and shear Used to check the capacity of connecting bolts (3/8 diameter steel bolts spaced at 5 ) and base bolts (32-1/2 diameter ASTM A325 bolts FOS for shear and bearing 7.3 and 3.7 (connecting bolts) FOS for shear and tension 2.7 and 2.8 (base bolts)
Acknowledgement Ray Clough 1920-2016 Robert Melosh 1926-1999 Bruce Irons 1924-1983 Olek Zienkiewicz 1921-2009 Edward Wilson Klaus Jurgen Bathe David Hibbitt John Swanson
Thank you