Computer Aided Engineering Design
FINITE ELEMENT METHOD 323 ui vi θi uj vj θj ktruss = 00– 00 000000 000000 –00 00 000000 000000 AE l AE l AE l AE l u u i i i j j ...
324 COMPUTER AIDED ENGINEERING DESIGN ui vi θi uj vj θj k = 00– 00 0 12 6 0 – 12 6 06 4 0–6 2 –00 00 0 – 12 – 6 0 12 – 6 06 2 0– ...
FINITE ELEMENT METHOD 325 ke = λλλλλTkλλλλλ (11.9j) withk and λλλλλ defined in Eqs. (11.9h) and (11.9i), respectively. Example 1 ...
326 COMPUTER AIDED ENGINEERING DESIGN Note that the interpolation above is equally biased along the x and y directions. To deter ...
FINITE ELEMENT METHOD 327 =Ai/A,Nj(x,y) = Aj/Aand Nk(x,y) = Ak/A are all greater than or equal to zero. Further, if P is at node ...
328 COMPUTER AIDED ENGINEERING DESIGN whereB is the strain-displacement matrix which is a constant and depends on the position o ...
FINITE ELEMENT METHOD 329 Example 11.4. Consider a rectangular plate cantilevered at one edge as shown in Figure 11.13. The load ...
330 COMPUTER AIDED ENGINEERING DESIGN For element 2 B 2 =^1 2 0010–10 0 –200 0 2 –20012–2 ⎛ ⎝ ⎜ ⎜ ⎜ ⎞ ⎠ ⎟ ⎟ ⎟ Hence 12 5 6 7 8 k ...
FINITE ELEMENT METHOD 331 Solving for displacements gives U =^15 16 × 10 –3× [0 0 0.18 –0.24 0.18 –0.24 0 0]T. The displaced pla ...
332 COMPUTER AIDED ENGINEERING DESIGN u = a 0 + a 1 x+a 2 y+a 3 xy (11.11a) Note again that the polynomial basis [1 xyxy] is cho ...
FINITE ELEMENT METHOD 333 Since the interpolation functions for the coordinates (x,y) and displacements (u,v) are the same, the ...
334 COMPUTER AIDED ENGINEERING DESIGN where the weights wi = 1, i = 1,... , 4 and ξηii = =^1 3 ± are the four Gauss points. Exam ...
FINITE ELEMENT METHOD 335 For Gauss point –^1 3 ,^1 3 ,^2 ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ ke is ke^26 =^16 15 10 2.7375 0.5833 0.5750 –0.2875 –1.166 ...
336 COMPUTER AIDED ENGINEERING DESIGN ke =^16 15 10 (1) (2) (3) (4) (5) (6) (7) (8) 0.9333 0.3500 –0.0933 –0.0700 –0.4667 –0.350 ...
FINITE ELEMENT METHOD 337 Determine the horizontal and vertical displacements at node 2 for the truss assemblage shown. Conside ...
338 COMPUTER AIDED ENGINEERING DESIGN Gauss point integration is used extensively in computing the integral for the stiffness m ...
Chapter 12 Optimization In design, construction and maintenance of any engineering system, engineers have to take many technolog ...
340 COMPUTER AIDED ENGINEERING DESIGN ofg′(x)≡ f(x) = 0. The roots may be none, one or many and multiple as well. The methods de ...
OPTIMIZATION 341 interval for which reason the method is named so. If sign (f(xl)) = sign (f(xr)), the root lies in the upper su ...
342 COMPUTER AIDED ENGINEERING DESIGN The bisection method can be improved by taking into account the magnitudes of f(xl) and f ...
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