1 Introduction to the Finite Element Method. 1 References . 11 2 Rods and Trusses . 13 2.1 Introduction . 13 2.2 Derivation of the Governing Differential Equation . 14 2.

2.1 Kinematics . 14 2.2.2 Constitutive Equation . 15 2.2.3 Equilibrium .

16 2.2.4 Differential Equation. 18 2.3 Finite Element Solution . 21 2.3.1 Derivation of the Principal Finite Element Equation .

21 2.3.2 Derivation of Interpolation Functions . 45 2.3.3 Assembly of Elements and Consideration of Boundary Conditions . 49 2.3.

4 Post-Computation: Determination of Strain, Stress and further Quantities . 59 2.3.5 Analogies to other Field Problems . 62 2.3.6 Solved Rod Problems . 64 2.

4 Assembly of Elements to Plane Truss Structures . 77 2.4.1 Rotational Transformation in a Plane . 77 2.4.2 Solved Truss Problems . 82 2.

5 Supplementary Problems . 94 References . 113 3 Euler-Bernoulli Beams and Frames . 115 3.1 Introduction . 115 3.2 Derivation of the Governing Differential Equation . 118 3.

2.1 Kinematics . 118 xiii xiv Contents 3.2.2 Constitutive Equation . 123 3.2.3 Equilibrium .

128 3.2.4 Differential Equation. 129 3.3 Finite Element Solution . 133 3.3.1 Derivation of the Principal Finite Element Equation .

133 3.3.2 Derivation of Interpolation Functions . 149 3.3.3 Assembly of Elements and Consideration of Boundary Conditions . 155 3.3.

4 Post-Computation: Determination of Strain, Stress and further Quantities . 161 3.3.5 Solved Beam Problems . 165 3.4 Assembly of Elements to Plane Frame Structures . 195 3.4.

1 Rotation of a Beam Element . 195 3.4.2 Generalized Beam Element . 198 3.4.3 Solved Problems . 208 3.

5 Supplementary Problems . 217 References . 249 4 Timoshenko Beams . 251 4.1 Introduction . 251 4.2 Derivation of the Governing Differential Equation . 258 4.

2.1 Kinematics . 258 4.2.2 Equilibrium . 259 4.2.3 Constitutive Equation .

260 4.2.4 Differential Equation. 260 4.3 Finite Element Solution . 269 4.3.1 Derivation of the Principal Finite Element Equation .

269 4.3.2 Linear Interpolation Functions for the Displacement and Rotational Field . 286 4.3.3 Higher-Order Interpolation Functions for the Beam with Shear Contribution . 303 4.3.

4 Solved Problems . 309 4.4 Supplementary Problems . 319 References . 325 5 Plane Elements . 327 5.1 Introduction . 327 5.

2 Derivation of the Governing Differential Equation . 328 5.2.1 Kinematics . 328 5.2.2 Constitutive Equation . 329 5.

2.3 Equilibrium . 331 5.2.4 Differential Equation. 333 5.3 Finite Element Solution . 337 5.

3.1 Derivation of the Principal Finite Element Equation . 337 Contents xv 5.3.2 Four-Node Planar Element . 341 5.3.3 Solved Plane Elasticity Problems .

354 5.4 Supplementary Problems . 376 References . 383 6 Classical Plate Elements . 385 6.1 Introduction . 385 6.2 Derivation of the Governing Differential Equation .

387 6.2.1 Kinematics . 387 6.2.2 Constitutive Equation . 389 6.2.

3 Equilibrium . 390 6.2.4 Differential Equation. 395 6.3 Finite Element Solution . 398 6.3.

1 Derivation of the Principal Finite Element Equation . 398 6.3.2 Rectangular Four-Node Plate Element . 401 6.3.3 Distorted Four-Node Plate Element . 418 6.

3.4 Solved Classical Plate Element Problems . 422 6.4 Supplementary Problems . 428 References . 435 7 Shear Deformable Plate Elements . 437 7.1 Introduction .

437 7.2 Derivation of the Governing Differential Equation . 438 7.2.1 Kinematics . 438 7.2.2 Constitutive Equation .

440 7.2.3 Equilibrium . 442 7.2.4 Differential Equation. 444 7.3 Finite Element Solution .

450 7.3.1 Derivation of the Principal Finite Element Equation . 450 7.3.2 Rectangular Four-Node Plate Element . 457 7.3.

3 Solved Thick Plate Element Problems . 463 7.4 Supplementary Problems . 465 References . 471 8 Three-Dimensional Elements .