Applied. Attalysis. FitBute Element. JLarry «/. Segerlind. JOHN Wn.EY & SONS New York Chichester Brisbane Toronto Singapore

Transcription

1 Applied FitBute Element Attalysis JLarry «/. Segerlind Associate Professor Department of Agricultural Engineering Michigan State University JOHN Wn.EY & SONS New York Chichester Brisbane Toronto Singapore

2 6 Tarsian af JVancircular Sections The preceding four chapters cover the discretizing of the body, the interpolating polynomial for a single element, the interpolation polynomials for a discretized body, and the derivation of the finite element equations. Each of these chapters contains basic information that is related to the finite element method. This chapter is a transition from the theory to the implementation. Its objec tive is to illustrate the steps involved in imple menting the finite element method. This objective is accomplished by obtaining a numerical solu tion for a problem that involves the torsion of a noncircular section. There are two reasons for selecting the torsion of noncircular sections to illustrate the im plementation of the finite element method. The element equations are relatively straightforward to derive. The [K] matrix is easy to calculate and the boundary integrals do not have to be evaluated because the boundary values are pre scribed. These properties prevent the evaluation of the element equations from detracting from our main objective. The other reason is that the torsion of a noncircular section involves concepts important to both mechanics-oriented and field- 87

3 88 Torsion of Nonclrcular Sections problem-oriented individuals. The theory will be familiar to an individual with a mechanics background while the governing differential equation is similar to those governing heat transfer and groundwater flow. 6.1 GENERAL THEORY, TORSION OF NONCIRCULAR SECTIONS One theory* for calculating the shear stresses in a noncircular shaft subjected to a twisting moment T about the z axis (Fig. 6.1) states that the shear stress components at any point can be calculated using 9<J) dy d(j> dx (6.1) where (p is a stress function. The governing differential equation is 1 G dx^ 1 9 G (6.2) with </) = (6.3) on the boundary. The physical parameters in (6.2) are the shear modulus of the Fig. 6.1 The shear stress components in a noncircular section subjected to a torque loading. *There are two theories for calculating the shear stresses that result when a noncircular shaft is subjected to torsion. One of them was developed by St. Venant; the other was proposed by Prantl. Both theories are discussed by Fung (1965). The minimization formulation of Prandtl's theory, which is used in this chapter, is also discussed by Temoshenko and Goodier (197).

4 General Theory, Torsion of Nonclrcular Sections 89 material, G, (N/cm^), and the angle of twist per unit length, 6 (rad/cm). This formulation does not have the applied torque, T (N m) in the governing equa tion. Instead, T is calculated once <p is known, using ^=2f <j>da ''Area (6.4) The stress function represents a surface covering the cross section of the shaft (Fig. 6.2). The twisting moment is proportional to the volume under this surface while the shear stresses are related to the gradients in the a and y coordinate directions. Fig. 6.2 The <f) surface and a related shear stress component. Equation 6.2 is usually written d^cl>^ = dx 9y (6.5) when the shaft is composed of a single material. The integral formulation of (6.5) is dv (6.6) which according to our discussion in Chapter 5 can be rewritten as = j\\{gv[d]{g]-(2ge)<^ dv (6.7)

5 9 Torsion of Nonclrcuiar Sections where 9</) dx d<j> dy and [/)] = K.. K The column vector (g) is related to the shear stress components in this application, while [D] becomes the identity matrix since K^^ = K^^ = 1. The minimization of x with respect to (O) results in the set of linear equations E E Sf S f {'^G9)dV (6.8) e = 1 I/(^) / l/(^) e=\ where is defined by (4.1) and the gradient matrix, is defined by (5.39). The evaluation of (6.8) commences as soon as we select a shape and divide it into elements. 6.2 ASSEMBLAGE OF THE ELEMENT MATRICES The square shaft (Fig. 6.3a) will be used to illustrate the evaluation and the assemblage of the element matrices into a set of linear equations. This shaft has four axes of symmetry, therefore, only one eighth of the total cross section needs to be analyzed. This fractional portion of the cross section is divided into four elements as shown in Fig. 6.3b. Four elements are not sufficient to obtain an accurate answer, but they are enough to illustrate the assemblage technique that is one of our objectives. Following the methods of Chapter 4, the element interpolating polynomials are <^( = + ^) C >(2) =, + 7V2^2^ yvi2^5 + ^ (6.9) cj>(3) = GO J -h A^2^^> A^i^^4 + A^5^^^5 + ^ =, A^i'^^4 4 A^i^^5 + yv6^^^6 The general equation for the element stiffness matrix is J y since it has already been noted that [/)] = [/] for the torsion of a noncircular

6 Assemblage of the Element Matrices 91 Axes of symmetry Shaft is 1 cm2 G =.8 X 1^ N/cm2 = 1 deg in ICQ cm (4).25 cm f * i) (3) +.25 cm X (1). (2) A V -.25 cm cm - Fig. 6.3 Element subdivi sion for the torsion of a square shaft. section. Evaluation of involves differentiating with respect to x and y. Confining our attention to element one dx dx am" 9x 1 2^( [M" o] (6.1a, b) d(j> (1) dy dy dy dy 2A (1)[cS" 4'^ o]

7 92 Torsion of Noncircuiar Sections The gradient matrix is [5<"] = (1).(1) ^2 cv^ (6.11) The area for this element is 32 and 2/4"> = 16 The b and c coefficients are b\'^=y2-y,= -.25 c\'^=x^-x2 = b['^=y,-y,=.25 c(" = A',-X4 = M'>= 7,-72= 4" = ^2-'^i = Substituting these values into (6.11) yields [5<"] = (6.12) and the product is IS [fi<'>]^[5">]= or (6.13) The element stiffness matrix is the integral of (6.13). Since the matrix product consists of constant values, it may be removed from under the

8 Assemblage off the Element Matrices 93 integral yielding (1) assuming a unit thickness. Recalling that (6.13) yields 1/32 and combining this with (6.14) The evaluation of the volume integral (/'')= [ 2G<"l9 ^ ^ JyO) yvi'* dv is straightforward if we employ the area coordinate system discussed in Chapter 3. Defining the area coordinates as L, = Af)'\ T2=A')" and = (6.15) the volume integral becomes ^ L, ^ ' JyO) dv (6.16) Assuming a unit thickness and using the area integral for area coordinates, (3.43), yields

9 94 Torsion of Nonclrcular Sections (/")- (6.17) Substitution of the values of, and yields* in (6.18) The system of equations for element one is = (/'>} or " $ <I' (6.19a ) <^5 ^>6 A similar set of equations can be calculated for each of the other elements using an identical procedure. The resulting matrices are given below. = (/«'} ' (6.19b ) -1 1 % 29.7 ^6 *The units are N/cm^ for G, cm^ for twists 1 deg in 1 cm. and rad/cm for. ^ = 7r/18x 1/1, since the shaft

10 Assemblage of the Element Matrices 95 [ A: ]{f } = {/"*} " ' 1-1 <^ % (6.19c ) a. ' 1 $ $ (6.19d ) -1 :! -1 % ^ The final system of equations is the algebraic sum of the element equations, which is 1-1 P $, b j C.O (6.2 ) The nodal valu^ $3, $5, and are zero,.sin h these nodes are on the external boundary. Modifibaftbtrxif (672) and solution yields <I), = <f>3 = -\ o - >1 "' i- ^>2=16 <I>5 = (6.21) $4= $6 = The modification of (6.2) is discussed in the next chapter where we consider the computer techniques that related to the implementation of the finite element method. The (j) surface for this set of nodal values is presented in Fig. 6.4.

11 96 Torsion of Nonclrcular Sections 4), = = Fig. 6.4 The nodal values for the four-element torsion problem. The determination of the nodal values is a major step in the solution of the problem. In most problems, however, a set of element resultants must be calculated. The shear stress values in each element and the twisting torque T which produces the angle of twist 6 are of interest in the present example. The calculation of the element resultants is discussed in the next section. 6.3 CONVENTIONAL ELEMENT RESULTANTS The gradients of the nodal parameter, (f>, are the important quantities in the torsion problem because the shear stresses are related to these gradients by d<f) dy and dxj) a;c The shear stress values are readily calculated because the gradient matrix for

12 Conventional Element Resultants 97 each element has already been evaluated. The gradient matrix for element one is (6.11). d(j> dx d<j) dy 2A^'^ 6^" c{" c^" 4" Combining (6.12) and (6.21) { = therefore, 4^)= = N/cm^ dy ' 4=-.^ dx N/cm^ The stress components for the other elements are Element 2 = ON/cm^. T = N/cm^ Element 3 = N/cm^, t = 494. N/cm^ Element 4 = ON/cm^. T- =494. N/cm^ These values are shown schematically in Fig. 6.5.

13 98 Torsion of Nonclrcular Sections Positive shear stress components Fig. 6.5 The element shear stress components for the to u r- element torsion problem. All values are in Newtons per square centimeter. The shear stress values calculated for each element are constant over the region of the element because the element interpolation polynomial was linear in j\: andy. The failure to obtain gradients that vary with x andy within an element is a definite disadvantage of the simplex elements. There are three ways to improve the stress values obtained for this example. First, a larger number of elements can be used. As the size of the element decreases, the existence of a constant value within the element becomes more realistic. An alternative approach is to use a triangular element with more nodes and an interpolating polynomial with quadratic and cubic terms. Differentiation will then yield gradients that are a function of the coordinate directions. A third approach is to utilize conjugate approximation theory. This theory makes possible the determination of the stress values at the nodal points and as a function of x and y within the element. The implementation of this theory is discussed in the next section. Another resultant of interest is the magnitude of the twisting torque T, which is given in (6.4) as This integral is equivalent to ^=2f (i>da ''Area =2/ Ȧ rea da (6.22)

14 Conventional Element Resultants 99 where the equations for are given in (6.9). Starting with element one if <l>^'^da=2f fyvl" /VjO A'i" Ol J^O) $2 <1.3 da (6.23) or if <p('>da=l[^vf [N^'^VdA Ja(^) (6.24) which is identical to the integral in (6.16). We can immediately conclude that if <#,('>^^ = ^[<1)] ^(1) 2A (1) -(, + (^2 + ^4) (6.25) Substituting the nodal values yields 2 I = ^4 ( )= ^4 3 3 (51.79) Likewise, for the other elements, 2 2A^^^ = ^($2 + <E'3 + ^5)= ^(16) if < > ^(^>2 + <f5+ <i>4) = ^(283.63) Ja O) J r (A\ 2A^^^ 2A^^^ 2jy^dA = ^(4.4 +$5 + $,)= ^(123.63) Summing these relationships and noting that the area of each element is the

15 1 Torsion of Nonclrcuiar Sections same produces E T=y f = ^( ) r= ^(169.5)=^^ We must also recall that the region we have subdivided into elements is only one eighth of the total cross section; therefore, the total twisting moment is (169) Moment = 87=8 = N-cm 48 It takes a torque of 178 N-cm to produce a twist of 1 deg in a 1-cm square steel shaft that is 1 cm long. The accuracy of this result is highly questionable, however, because of the coarseness of the grid. In fact, the theoretical value* is N-cm. Our answer is 9.5 percent below this value. 6.4 CONSISTENT ELEMENT RESULTANTS The failure to obtain gradients that are a function of ;c and y is a definite disadvantage of the linear interpolation functions. The gradient and any quant ity related to the gradient has a constant value within an element. Various averaging techniques have been used to obtain a value of the desired quantity at a node. One procedure uses the average of the gradient values in each element that surrounds the node. The correct procedure for obtaining nodal values of the element resultants is to employ conjugate approximation theory (Oden and Brauchli, 1971). This theory yields element resultant values consistent with the approximating polynomial(s) for a scalar or vector quantity.. The derivation of the conjugate approximation theory is beyond the scope of this book. The implementation of this theory, however, is straightforward and will be illustrated relative to the four-element torsion problem that we have been analyzing. The nodal values of the element resultants are obtained by solving the system of equations [C]{a} = {7?} (6.26) *The relationship between the applied torque and the angle of twist for a square of dimension 2a is given by 7=.146 (la)"^(timoshenko and Goodier, 197, equation 17, page 313). For our example, 2a = 1 and 7=.146 Gff= N-cm.

16 Consistent Element Resultants 11 where [C] and (/?) are the sum of element matrices defined by [cw]= r J y (6.27) and [/^n= f Jy (6.28) where a is the conventional element resultant. The evaluation of presents no problems because it is similar to (6.8), since a is constant within an element. The evaluation of is performed most easily using area coordinates, Confining our attention to element one, yields [,yv<'>] = [L, L, L, ] The product becomes L.Lj [;V<')]^[7V<'>] = L,L, Ll ^2^3 L1L3 L2L3 Ll Integration over the area using (3.43) yields ['"'1 = 1? (6.29) assuming a unit thickness. The column vector for element one is {,(.)j (1)1 = r r '77.67" = ^( (6.3)

17 12 Torsion of Nonclrcular Sections The shear stress component was used in the previous calculations because it has the largest numerical value within each element. The shear stress com ponent could also be considered. The only quantity that changes is the column vector (/?), which would have to be reevaluated using the numerical values for The element matrices for each element are summarized in (6.29) to (6.33). [c ] = 4(2) w ' ,. (6.31) [C(3)].(3)1 = ^ 12 " ' = ^(3) (6.32) 4(4) IT ' ' (6.33) The final system of equations is obtained by summing the element equations which yields J_ ' < < < (6.34) The element areas cancel since all elements have the same area.

18 Consistent Element Resultants 13 The nodal resultant values are (a) ^ = [ 7.9,436.5,724.1,353.6,671.4,475.5 ] It is also possible to obtain equations for the element resultants that give the variation of these values over the element area. There is a very limited use for this information and the details will not be presented here. The use of conjugate approximation theory requires the solution of another system of algebraic equations that is identical in size to those used to obtain the nodal values. This is a definite disadvantage when we are solving problems involving a large number of elements. An alternative to solving the complete system of equations is discussed in Chapter 7 when we reconsider the torsion of a square shaft with an increased number of elements. The final steps of the finite element method have been illustrated in this chapter while we solved a specific problem. We have shown how the element matrices are evaluated and how the element resultants are determined once the nodal values are known. We shall discuss some of the computer techniques used when we implement the finite element method in the next chapter. We shall consider specific application areas in Chapters 8 to 12. PROBLEMS 41. Verify the element stiffness and force matrix in equation (a) 6.19b, (b) 6.19c, and (c) 6.19d. 42. Verify the shear stress values for element (a) two, (b) three, and (c) four of the example discussed in Section Determine the numerical values in {R} for the consistent element re sultants when the values of are desired at the nodes. References Fung, Y. C., Foundations of Solid Mechanics, Prentice-Hall, Englewood Cliffs, N.J., 1965, pp Oden, J. T., and H. J. Brauchli, "On the Calculation of Consistent Stress Distributions in Finite Element Approximations," International Journal for Numerical Methods in Engineering, 1971, Vol. 3, pp Timoshenko, S. P., and J. N. Goodier, Theory of Elasticity, McGraw-Hill, New York, 197, pp

19 14 Torsion of Nonclrcuiar Sections Uncited Reference Owen, D. R. J., and O. C. Zienkiewicz, "Torsion of Axi-Symmetric Solids of Variable Diameter Including Acceleration Effects," International Journal for Numerical Methods in Engineering, 1974, Vol. 8, pp Yamada, Y., T. Kawai, and N. Yoshemura, "Analysis of the Elastic-Plastic- Problems by the Matrix Displacement Method," Proceedings of the Second Conference on Matrix Methods in Structural Mechanics (AFFDL-TR-68-15), Wright-Patterson Air Force Base, Dayton, Ohio, 1968.