ce5601-ce5010-lecture 1

Department of Mechanical, Aerospace and Civil Engineering

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityCE5601 Nonlinear Structural Analysis & Finite Element MethodCE5010 Structural Design and FEA

Dr Zhaohui Huang

September 2015Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityIntroductionDepartment of Mechanical, Aerospace and Civil Engineering, Brunel University

The basic idea in the finite element method is to find the solution of a complicated problem by replacing it by a simpler one. Since the actual problem is replaced by a simpler one in finding the solution, we will be able to find only an approximate solution rather than the exact solution.Basic concept of Finite element methodIn the finite element method, the solution region is considered as built up of many small, interconnected subregions called finite elements. In each piece or element, a convenient approximate solution is assumed and the conditions of overall equilibrium of the structure are derived. The satisfaction of these conditions will yield an approximate solution for the displacements and stresses.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityEngineering applications of FE methodCivil engineering structures;Aircraft structures;Heat conduction;Geomechanics;Hydraulic and water resources engineering, hydrodynamics;Nuclear engineering;Biomedical design;Electrical machines and electromagnetics, and so on.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityGeneral description of the FE methodIn the finite element method, the actual continuum or body of matter, such as a solid, liquid, or gas, is represented as an assemblage of subdivisions called finite elements.These elements are considered to be interconnected at specified joints called nodes or nodal points. The nodes usually lie on the element boundaries where adjacent elements are considered to be connected.Since the actual variation of the field variable (e.g., displacement, stress, temperature, pressure, or velocity) inside the continuum is not known, we assume that the variation of the field variables inside a finite element can be approximated by a simple function. These approximating functions (also called interpolation models) are defined in terms of the values of the field variables at the nodes.

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityWhen field equations (like equilibrium equations) for the whole continuum are written, the new unknowns will be the nodal values of the field variable. By solving the finite element equations, which are generally in the form of matrix equations, the nodal values of the field variable will be known.Once these are know, the approximating functions define the field variable throughout the assemblage of elements.The solution of a general continuum problem by the finite element method always follows an orderly step-by-step process.General description of the FE method

Department of Mechanical, Aerospace and Civil Engineering, Brunel University

BeamsColumnsFloor slabsConnectionsComposite frame building

Department of Mechanical, Aerospace and Civil Engineering, Brunel University1. Stress and Strain

Beam elementsColumn elementsSlab elementsConnection elementsA composite steel-framed building can be modelled as an assembly of finite beam, column, connection and slab elements. It is assumed that the nodes of these different types of element are defined in a common reference plane that is assumed to coincide with the mid-surface of the concrete slab element.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe step-by-step FE procedure for static structural problemStep 1: Divide structure into discrete elements (discretization);Step 2: Select a proper interpolation or displacement model;Step 3: Derive element stiffness matrices and load vectors;Step 4: Assemble element equations to obtain the overall equilibrium equations. Step 5: Solve for the unknown nodal displacements;Step 6: Compute element strains and stresses.

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityPart I

Introduction to Theory of Finite ElementsDepartment of Mechanical, Aerospace and Civil Engineering, Brunel UniversityChapter 1 Basic Theory of Continuum MechanicsDepartment of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe stressThe Traction vector

Fig. 1.1 Deformable body under combined loading.Department of Mechanical, Aerospace and Civil Engineering, Brunel University

The stressConsider now an arbitrary plane that divides the volume V in two distinct volumes V1 and V2. Since V1 and V2 are in reality in contact, a set of equal and opposite forces, F + and F respectively, must exist that holds them together. Considering that this force is uniformly distributed along the surface of the section cut, the equivalent pressure load can be defined as:Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityAssumption: In Continuum Mechanics, the internal force of a deformable body is considered uniformly distributed along an arbitrary section cut. No assumption is made on the size of volume V, therefore the concepts discussed can be assumed to apply for an arbitrarily small volume V such that the deformable body degenerates to a single material point. However, as volume V decreases so does the surface of the assumed slice S +. Thus, the pressure measure introduced in equation (1.1) cannot be mathematically sound.The stress

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityDefinition 1.1. The traction vector is defined as the limit of the pressure load applied over a surface, as this surface contracts to a material point (therefore tending to a very small value equal to ).The traction vector

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe stress tensor

Fig. 1.2 Components of the stress tensor.The Traction vector has been defined as the limiting value of a force measure over the value of an arbitrary surface, as the latter tends to zero. Instead of defining Traction in terms of arbitrary surfaces, Cauchy introduced the concept of stress.Cauchy proved that the Traction vector can be decomposed in a set of nine perpendicular components that bare certain mathematical properties.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe stress tensor

Fig. 1.2 Components of the stress tensor.These nine components are the components of the so called stress tensor . Without loss of generality, these can be defined on the three positive surfaces (i.e. the surfaces whose outward normal vector points to the positive axis) as presented in Fig. 1.2.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe stress tensor

Fig. 1.2 Components of the stress tensor.The stress tensor is written in matrix form as:


Fig. 1.2 Components of the stress tensor.Cauchy also proved by means of the Principle of conservation of Linear Momentum that for the cube in Fig. 1.2 to be in equilibrium, the following symmetry conditions must hold:


Fig. 1.2 Components of the stress tensor.

Therefore, the more convenient vectorial notation of the stress tensor is implemented and will be adopted in these notesDepartment of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe strain tensorSimilarly to the definition of the stress tensor, the strain tensor is defined on the grounds of the following matrix form:

where again the following symmetry conditions hold:

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe engineering notation of strainsIn Finite Elements theory, rather than using the matrix form introduced above, the so called engineering notation of strains is implemented.In this, the shear strains are substituted with their engineering counterparts .

Between the two, the following relations hold:

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityAccordingly, the following vector notation of the strain tensor is used throughout these notesThe engineering notation of strains

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityCompatibility EquationsThe compatibility equations of the theory of Continuum Mechanics are geometric relations that define the evolution of strains as a function of the displacements imposed into a deformable body. In the general case of Large Displacements the compatibility equations assume the following form:

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityRelation (1.10) is written in the following vector form:Compatibility Equations

where {e} will be herein defined as the linear part of the strain vector whereas {} will be the nonlinear component of the deformation vector.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversitySmall DisplacementsAccording to the Small Displacement assumption, the contribution of {} in equations (1.10), i.e. the nonlinear part of the deformation, is neglected leading to the following compatibility relations:

Depending on the assumption introduced (Large or Small Displacements) for the Finite Element formulation, equations (1.10) or (1.12) will be used to derive the corresponding Finite Element strain-displacement matrix.Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkConsider the case of a deformable body of volume V and surface S. The body is deforming under the impact of body loads (i.e. gravitational loads) B and surface tractions TS. The work produced by these, external, forces over a virtual displacement component u is evaluated through the following expression:

Equation (1.13) is conveniently written in the following compact form:

where the Einstein summation convention is used (e.g. a summation is assumed over all indices that appear twice in a product).Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkUsing Cauchys principle, the work done by the traction forces is written as:

Applying the Gaussian Integral theorem on the surface integral of the r.h.s of equation (1.15), the following relation is established:

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkCombining equations (1.15) and (1.16) and substituting into equation (1.14), the following relation is derived:

or, after some algebraic manipulation

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkAccording to the theory of Continuum Mechanics, a variation on the displacement field ui of a deformable body gives rise to a strain field while at the same time to body rotates as a rigid formation. That is, the theory of Continuum Mechanics assumes that the effect of variation of the displacement field is expressed through the following set of compatibility equations

where: is the i j component of the strain tensor

is the i j component of the rotation tensor . . Department of Mechanical, Aerospace and Civil Engineering, Brunel University

Since rotations are rigid body motions (e.g. they do not result into stresses within the deformable body) the work produced over a rotation component is equal to zero and therefore

Substituting relation (1.20) into relation (1.18) the following equation is derived

The Principle of Virtual WorkDepartment of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkSince the deformable body is under equilibrium the following relation holds

and therefore equation (1.21) reduces to

Department of Mechanical, Aerospace and Civil Engineering, Brunel UniversityCombining equations (1.14) and (1.23) the following relation is derived

This is the Principle of Virtual Work for a deformable body.The Principle of Virtual WorkThe work produced by the internal stress field over the strain field (e.g. the internal work of the deformable body)The work produced from the external forces over an external and virtual displacement fieldDepartment of Mechanical, Aerospace and Civil Engineering, Brunel UniversityThe Principle of Virtual WorkThe principle of Virtual work, as expressed from equation (1.24) holds provided the following assumptions are valid:1. The field of virtual displacements u does not violate any kinematical assumptions. That is, it does not violate the boundary conditions of the problem. Then and only then, the compatibility equations defined in relation (1.19) are valid.2.The externally applied forces B and TS provide a statically accepted system of forces. That is, the deformable body under the effect of such forces is under equilibrium. Then, and only then, relation (1.22) holds.