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FINITE ELEMENT ANALYSIS ON ANGLE PLATE
ABSTRACT
The finite element method (FEM) (sometimes referred to as finite element analysis) is a numerical technique for finding approximate solutions of partial differential equations (PDE) as well as of integral equations. The topic of the seminar includes the study of Finite Element Analysis and a case study of finite element analysis.
A double angled web connection is modeled and analyzed to predict the behavior of the angle plate using finite element analysis. The angle plate is analyzed to investigate the effect of angle thickness, t, on the load-displacement relationship and the moment rotation relationship under axial tensile loads and shear loads.
Presented By:Jithu Thampi
CHAPTER- 1
INTRODUCTION TO FINITE ELEMENT ANALYSIS
1.1 INTRODUCTION
Many physical phenomena in engineering and science can be described in terms of partial differential equations. In general, solving these equations by classical analytical methods for arbitrary shapes is almost impossible .The Finite Element Method (FEM) is a numerical approach by which these partial differential equations can be solved approximately.
Finite Element Analysis (FEA) is the numerical simulation of the behavior of mechanical components acquired by discrediting engineering components into finite number of building blocks so called Elements and analyzing those components for its acceptability and reliability.
The basic idea of FEM is to divide the body into finite elements, often just called elements, connected by nodes, and obtains an approximate solution as shown Figure 01. This is called the finite element mesh and the process of making the mesh is called mesh generation.
Mathematically, the structure to be analyzed is subdivided into mesh of finite sized elements of simple shapes .Within each element, the variation of displacement is assumed to be determined by simple polynomial shape functions and nodal displacement. After the boundary conditions are applied, the nodal displacements are found by solving the matrix stiffness equation. Once nodal displacements are known, element stresses and strains can be calculated.
1.2 HISTORY
Finite Element Method (FEM) was first developed in 1943 by R.Courant, who utilized the Ritz method of numerical analysis and minimization of variational calculus to obtain approximate solutions to variation systems. Shortly thereafter, a paper published in 1956 by Turner, Clough, Martin and Topp established a boarder definition of numerical analysis. The paper centered on the stiffness and deflection of complex structures.
Finite Element technology was further enhanced during 70 s by Zeinkiewicz and Cheung, when they applied the technology to general problems described by Lapace and Poisson s equations.
E. Wilson developed one of the finite element programs that were widely developed. This program was limited to two dimensional stress analysis. Then in 1965 NASA created a finite element program. From that time many industrial projects started creating the programs capable of analyses real life problems.
The research in (FEA) coincided with the rapid growth of computer power, and the method quickly become widely used in nuclear power, defense, automotive , and aeronautics industries.
1.3 THEORIES BEHIND FEM
Basically, the FEM works on the various theoretical concepts which are discussed briefly in this section.
Variation Method
The calculus of variation deals with certain kind of extreme problems ,Where in attempts is made to maximize or minimize an expression ,expressed in the form of integral called functional. A functional is a quantity whose value depends on the entire shape of some functions rather than on a number of discrete variables.
Thus we seek function y(x) varying between the limits (a,c)and such that the integral
I=
is the maximum or minimum given f ,where f is the functional. Function f=y(x) is called stationary function.
1.3.1 Rayleigh-Ritz Method
In this method the functional is substituted by some assumed solutions involving adjustable parameters and finding the stationary condition with respect to these parameters. Such an approach is the Rayleigh-Ritz method .In this method y(x) is replaced by a linear combination of suitably chosen co-ordinate functions
Yn(x)=a1F(x)+a2F2(x)+ .+anFn(x).
Where a is unknown constants to be determined.
1.3.2 The Finite Element Method
Among the various numerical methods available Finite Element Method is most popular and widely used .With the introduction of new materials viz, composites fiber reinforced materials etc, the conventional method is not used .The finite element method can be considered as Rayleigh-Ritz method combined with a variational principle applied to continuum mechanics.
1.4 EQUATIONS FOR A SYSTEM
Equations for a system are used throughout the FEM in even the most complex problems. We will describe the process of developing these equations by an examplewhich also gives the material properties, loads and support conditions.
The first step in applying FEM is to divide the structure into elements. The selection and generation of nodes depends on the loading conditions and structure. In the case of a discrete structure such as this, it is necessary only to put nodes wherever loads are applied and at points where the section properties or material properties change.
We will develop the element stiffness matrix which relates the element internal nodal forces to the element nodal displacements. The element internal forces denoted by F and element displacement matrix by d. For a two node element, these matrix are given by
F = [ ] d = [ ]
The stiffness equation of the elements is given by F=k.d or
[ ] = [ ] [ ]
The above equations can be summarized as follows :the sum of the internal element forces is equal to that of the external forces and reactions. This differs somewhat from the well-known equilibrium condition that the sum of forces on any point must vanish.The reason for the difference is that the element nodal forces,which are the forces that appear in th element stiffness matrix , act on the elements.The forces exerted by the elements on the nodes are equal and opposite.
The next procedure is to develop the assembly procedures in terms of equations.In this approach , compatibility between elements is enforced by relating the element nodal displacements to the global displacment matrix.
1.5 GOVERNING EQUATIONS AND BOUNDRY CONDITIONS
Many Engineering phenomena can be expressed by governing equations and boundary conditions.
Governing equations are that differential equation which contains the property of the materials and its behavior. Stiffness matrix is a matrix defining the geometric and material property of the system. At the step of analysis we must impose the particular boundary conditions of the problem being analyzed. Boundary conditions are the differential equations which determine the conditions at which the system works.
The properties of the system which are imposed on it.
Elastic Governing equation
Thermal LF + F=0
Fluid Boundary conditions
Electrostatic BF + g=0
Governing equation A set of simultaneous
+ algebraic equations.
Boundary condition [k] {u} = {f}.
[Equations of a system]
[k] {u} = {f} {u} = [k-1] {f}
property behaviour action behaviour property action
1.6 MESH AND NODES
The FEA uses complex system of points called nodes which make a grid called a mesh. The mesh is programmed to contain the materials and structural properties which defines how the structure will react to certain loading conditions.
Nodes are assigned certain density throughout the material depending on the anticipated stress level of a particular area. Regions which will receive large amounts of stress usually have a higher node density than those which experience little or no stress.
The mesh act like a spider web in that from each node there extends each mesh element to each of the adjacent node .This web vectors is what carries the material properties to the object creating many elements. The representation of a given region by a set of elements(ie discretization or mesh generation) is an important step in finite element analysis .The choice of element type ,number of elements, and the density of elements depends on the geometry of the domain , the problem to be analyzed, and the degree of accuracy desired. Of course there are no specific formulae to obtain this information. The general rules for mesh generation for finite element formulations include
(1) The elements that are selected should characterize the governing equations of the problem.
(2) The number, shape and type of the element should be such that the geometry of the domain is represented as accurately as desired.
(3) The density of elements should vary gradually from high-density region to low-density regions.
CHAPTER-2
FINITE ELEMENT ANALYSIS OF ANGLE PLATE
2.1 PROBLEM DEFINITION
To predict the complicated behavior of a double angle web connection, a 3D non linear finite element model for half of the entire connection is analyzed Double angle connections are modeled to investigate the effect of angle thickness t and the distance of centre line of bolt to back to back leg of the angle, on the load-displacement relationship and the moment rotation relationship under axial tensile loads and shear loads. The two angle sections L5x3x1/4 and L5x3x1/2were selected for this purpose. The same gage distance, g, bolt spacing, s, and bolt diameter are used for these three cases. A36 steel is used for the three angles.
2.2 GEOMETRIC PARAMETERS
Figure 02 shows the entire details of the model and the table shows the dimensions used for the model.
Model t (in) d (in) L (in) g (in)
L5x3x1/4 0.25 17.7 240 7.5
L5x3x1/2 0.50 17.7 240 7.5
Fig 02: Geometric parameters of the double angle connections.
2.3 FINITE ELEMENT MODEL
One half of the entire double angle web connection is modeled and the model is meshed
For the entire 3D finite element model with an L5x3x1/4 angle, 1,330 elements, which
have 6,136 nodes, were used. For the entire 3D finite element model with an L5x3x1/2 angle, 1,305 elements, which have 5,956 nodes, are used.A 20node quadratic brick element types are used for the angle specimen.
Bolts are used for joining and washers are not used. The contact and bearing problems
between bolt shanks and bolt holes are neglected due to computer time and cost required
for such an analysis. The back to back angle leg of the specimen is constrained.
Prestressing forces are applied to each bolt as initial stresses to simulate the fully tightened bolts with minimum bolt tension. The prestress force applied is 28 kips. Elastic-perfectly plastic material behavior is considered for each element and the von Mises yield criterion is used to represent the yielding of steel in this analysis.
2.4 LOADING CONDITIONS
Two types of loading conditions are used in the analysis ;shear load and tensile load. These conditions and the reference displacement and angle change under each loading conditions are shown in the figure 03.
Tensile Loading Shear Loading
Fig 03: Loading conditions and measurements of displacement and angle change .
2.5 Von Mises Stress
The Von Mises stress is often used as a failure criterion in design of non linear static problems. The failure criterion states that the Von Mises stress svm should be less than the yield stress sy of the material.
The Von Mises stress svm may be expressed as
svm = v(s1-s2)+(s2-s3)+(s3-s1)
where s1, s2, s3 are the principle stresses
The Von Mises stress is given by
svm = vI12-3I2
Where I1 and I2are first and second invariants of the stress tensor.
2.6 L5x3x1/4 Angle Plate Model.
2.6.1 Angle under tensile loading.
To establish the load-displacement relationship of an L5x3x1/4 double angle connection, increasing axial tensile loads are applied to the end of beam elements in the positive Z-direction. Figure 04 depicts a deformed shape of the angle connection at the final loading stage (total of 12.0 kips to the end of the beam). Figure 2.6 presents the load-displacement relationship of an L5x3x1/4 double angle connection that has been subjected to an axial tensile load that is increased until the displacement reaches 0.5 in. The load-displacement curve shows a linear relationship initially, followed by a gradual decrease in stiffness. The initial stiffness of this angle model is 116.7 kips/in., while the final stiffness is approximately 8.6 kips/in. Table 2.1 summarizes the above load displacement relationship at each loading stage.
Fig 04: Deformed Shape of L5x3x1/4 angle plate due to tensile loading.
Fig: 05 Load displacement relationships for L5x3x1/4 angle plate due to tension
Figure 2.7 shows the von Mises stress diagram of an L5x3x1/4 angle specimen at the
final loading stage. Yielding zones are observed in the outstanding leg of the angle near
the bolt heads and close to the corner of the angle. These yielding zones are propagated
toward the centers of the bolt holes as the applied loads increase.
Figure 06: Von Mises stress diagram of an L5x3x1/4 angle due to tension loading
2.6.2 Angle Plate Under Shear Loading
An increasing, uniformly distributed load is applied to the beam in the negative Ydirection
(downward) as shown in Figure 2.4 to investigate the moment-rotation relationship. Figure 2.11 presents the moment-rotation relationship of the L5x3x1/4 double angle connection. The moment-rotation curve shows almost a linear relationship after the second loading stage (at the applied load of 2 kips) and flattens out gradually as the moment increases. The initial rotational stiffness of the angle connection is approximately 3,559 in.-kips/rad.
Figure 07: Deformed shape of an l5x3x1/4 double angle connection
due to shear loading
Figure 08: Moment-rotation relationship of an l5x3x1/4 double angle connection
due to shear loading
Figure 09 shows the von Mises stress diagram of an L5x3x1/4 angle at the applied load of 80.1 kips. The yielding zones are formed along the corner of the angle in addition to the top areas of the angle. The bottom areas of Figure 2.12 are the top areas of the angle.
Figure 09:Von Mises stress diagram of an l5x3x1/4 angle due to shear loading.
2.7 L5X3X1/2 ANGLE MODEL
2.7.1 Angle under Axial Tensile Loading
Angle Under Axial Tensile Loading The load-displacement relationship of this angle model was obtained by increasing the load to 2.26 kips/in. (total 40kips) at one end of the beam in the positive Z-direction. Figure 2.30 presents the load-displacement relationship of an 5x3x1/2 double angle connection. The load-displacement curve shows a linear relationship initially, followed by a rapid decrease in stiffness like the previous load-displacement curves. The initial stiffness of this angle model is 1,013 kips/in., while the final stiffness is approximately 6.8 kips/in.
Figure 13: Deformed Shape of an L5x3x1/2 Double Angle Connection due to Tension Loading.
Figure 14: Load-displacement relationship of an l5x3x1/2 double angle connection due to tension loading.
Figure 15 shows the von Mises stress diagram of the L5x3x1/2 angle specimen at the final loading stage. Yielding zones are formed in the outstanding leg of the angle near the bolt heads and close to the corner of the angle. Yielding zones are also formed near the center of each bolt hole area. The stress fields are more widely spread than in the previous case .
Figure 15: Von Mises Stress Diagram of an L5x3x1/2 Angle due to Tension Loading.
2.7.2 Angle Under Shear Loading
An increasing, uniformly distributed load up to 0.3334kips/in. (total 80 kips) is applied to a beam element in the negative Y-direction (downward) as shown in Figure 2.4 to investigate the moment-rotation relationship. Figure 2.35 presents the moment-rotation relationship of the L5x3x1/2 double angle connection. The moment-rotation curve of figure 17 shows almost a linear relationship after the second loading stage (at the applied load of 0.8 kips) and flattens out as the moment increases. The initial rotational stiffness of the angle connection is approximately 14,606 in.-kips/rad.
Figure 16: Deformed shape of an l5x3x1/2 double angle connection due to shear loading.
Figure 17: Moment-Rotation Relationship of an L5x3x1/2 Double Angle Connection due to Shear Loading.
Figure 18 shows the von Mises stress diagram of the L5x3x1/2 angle at the applied load of 80.9 kips. Yielding zones are formed along the corner of the angle in addition to the top areas of the angle. Stress fields are also formed around each bolt hole.
Figure 18: Von mises stress diagram of an l5x3x1/2 angle due to shear loading.
CHAPTER 3
3.1 SUMMARY
A 3D nonlinear finite element model has been executed and analyzed to investigate the behavior of double angle connections under axial tensile loads and shear loads respectively.
To confirm the acceptance of this 3D nonlinear finite element model again, experimental tests were performed for a double angle connection using L5x3x1/4 and L5x3x1/2 angle sections.
From the established load-displacement relationship and the established moment rotation relationship of the 3D nonlinear finite element model, it can be shown that the angle thickness, t, and the distance, b, play important roles in the initial stiffness, K, of a double angle connection. The parameter, b, defines the distance from the center of a bolt hole to the center line of the back-to-back angle leg of the angle as shown in Figure 02. The initial stiffness of a double angle connection is mainly dependent on the value of (t/ b) 3.
4. REFERENCES.
1. Jacob Fish and Ted Belytschko (2002), A First Course in Finite Elements,
John Willey and Sons LTD .
2. J N Reddy (2005), An Introduction to Finite Element Analysis,
TATA McGRAW Hill.
3. O. C. Zienkiewicz (2000), Finite Element Method,
TATA McGRAW Hill.
4. Tirupathi R Chandrupatia and Ashok D Belegunda, (2004), Introduction to Finite
Element Engineering, Prentic Hall of India.