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14.8 f _ { 1 } = f _ { 3 } = 5 \mathrm { i n } ^ { 3 } / \mathrm { s } , \quad f _ { 2 } = 0
14.9 p _ { 2 } = p _ { 3 } = 1 2 ~ \mathrm { m } , ~ p _ { 5 } = 1 1 ~ \mathrm { m }

Chapter 15

15.1 d2x ¼ 0:021 in., d3x ¼ 0:042 in., sx ¼ 0
15.2 d _ { 2 x } = 0 , ~ \sigma _ { x } = 5 0 . 4 ~ \mathrm { M P a }
15.3 d _ { 1 x } = d _ { 1 y } = - 0 . 0 1 7 5 \mathrm { ~ i n . , } \quad \sigma ^ { ( 1 ) } = 4 3 5 0 \mathrm { ~ p s i \ : ( T ) }


\sigma^ {(2)} = - 6 1 5 0 \text {   psi   (C) }, \quad \sigma^ {(3)} = 4 3 5 0 \text {   psi   (T) }

15.4 d _ { 1 x } = - 0 . 0 2 9 1 \ \mathrm { i n . , } \quad d _ { 1 y } = - 0 . 0 0 9 5 \ \mathrm { i n . }


\sigma^ {(1)} = - 1 3 7 0 \text {   psi   (C) }, \quad \sigma^ {(2)} = 2 3 7 5 \text {   psi   (T) }, \quad \sigma^ {(3)} = - 1 3 7 0 \text {   psi   (C) }

15.5 d _ { 2 x } = 1 . 4 4 \times 1 0 ^ { - 4 } \mathrm { m } , ~ \sigma ^ { ( 1 ) } = - 2 0 . 2 \mathrm { M P a } \left( \mathrm { C } \right) , ~ \sigma ^ { ( 2 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 1 \mathrm { M P a } \left( \mathrm { C } \right)
15.6 \begin{array} { r } { d _ { 1 x } = 0 , d _ { 1 y } = 6 . 0 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 3 ) } = - 1 0 . 5 \mathrm { M P a } ( \mathrm { C } ) } \end{array}


\sigma^ {(2)} = 1 8. 2 \mathrm{MPa(T)}

15.7 d _ { 1 x } = 0 , d _ { 1 y } = - 3 . 6 \times 1 0 ^ { - 4 } \mathrm { m } , \sigma ^ { ( 1 ) } = \sigma ^ { ( 2 ) } = 0
15.8 d _ { 2 x } = 0 . 0 1 7 3 \ \mathrm { i n . , } \quad \sigma _ { s t } = 8 4 0 \ \mathrm { p s i } \ ( \mathrm { T } ) , \quad \sigma _ { b r } = 1 6 8 0 \ \mathrm { p s i } \ ( \mathrm { C } )

15.12 \mathbf { a } . \ - 0 . 0 0 1 9 0 7 \ \mathrm { i n } . \quad \mathbf { b } . \ \sigma _ { b r } = - 2 8 , 6 0 0 \ \mathrm { p s i } , \quad \sigma _ { m g } = - 1 9 , 0 6 7 \ \mathrm { p s i }
15.13 fT1x ¼ -4464 lb, fT1y ¼ -8929 lb, fT2x ¼ 4464 lb


f _ {T 2 y} = - 8 9 2 9 \mathrm{lb}, \quad f _ {T 3 x} = 0, \quad f _ {T 3 y} = 1 7, 8 5 7 \mathrm{lb}

15.14 f _ { T 1 x } = - 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 1 y } = 0 , f _ { T 2 x } = 4 3 . 1 2 5 ~ \mathrm { k N } , f _ { T 2 y } = - 8 6 . 2 5 0 ~ \mathrm { k N }


f _ {T 3 x} = 0, \quad f _ {T 3 y} = 8 6. 2 5 0 \mathrm{kN}

15.15 f _ { T 1 x } = - 6 0 . 0 \mathrm { ~ k i p } , f _ { T 1 y } = - 9 0 \mathrm { ~ k i p } , f _ { T 2 x } = 6 0 \mathrm { ~ k i p } , f _ { T 2 y } = 0 ,


f _ {T 3 x} = 0, \quad f _ {T 3 y} = 9 0 \mathrm{kip}

15.16 f _ { T 1 x } = 1 3 4 \ \mathrm { k N } , f _ { T 1 y } = 1 3 4 \ \mathrm { k N } , f _ { T 2 x } = - 1 3 4 \ \mathrm { k N } , f _ { T 2 y } = 0


f _ {T 3 x} = 0, \quad f _ {T 3 y} = - 1 3 4 \mathrm{kN}

15.17 \sigma _ { x } = \sigma _ { y } = - 8 9 2 9 \mathrm { p s i } ( \mathrm { C } ) , \tau _ { x y } = 0
15.18 \sigma _ { x } = 6 7 . 2 ~ \mathrm { M P a } , \sigma _ { y } = 6 7 . 2 ~ \mathrm { M P a } , \tau _ { x y } = 0
15.19 \{ f _ { T } \} = \frac { A E \alpha _ { 0 } } { 6 } \left\{ \begin{array} { r } { - 4 t _ { 1 } - 5 t _ { 2 } } \\ { 4 t _ { 1 } + 5 t _ { 2 } } \end{array} \right\}
15.20 \frac { A E \alpha } { 2 } \bigg \{ { - t _ { 1 } - t _ { 2 } } \bigg \}

15.21 \{f_{T}\}=\frac{2\pi\bar{r}AE\alpha(\Delta T)[\bar{B}]^{T}}{1-2\nu}\left\{\begin{array}{l}1\\ 1\\ 1\\ 0\end{array}\right\}

15.22 d_{2x} = 0.8 \times 10^{-3} in., d_{3x} = 0 , d_{3y} = 0.8 \times 10^{-3} in.

d_{4x} = d_{4y} = 0.8 \times 10^{-3} in.; stresses are zero

15.23 d_{2x} = 0.989 \times 10^{-3} in., d_{3x} = -0.756 \times 10^{-3} in.,


d _ {3 y} = 0. 9 8 9 \times 1 0 ^ {- 3} \text {   in. }, \quad d _ {4 x} = 0. 1 3 2 \times 1 0 ^ {- 2} \text {   in. },


d _ {4 y} = 0. 2 0 4 5 \times 1 0 ^ {- 2} \text {in.}, \quad \sigma_ {1} ^ {(1)} = 1 7 \mathrm{ksi}, \quad \sigma_ {2} ^ {(2)} = - 1 7 \mathrm{ksi}

Chapter 16

16.1 [M] = \frac{\rho AL}{6}\left[ \begin{array}{ccc}2 & 1 & 0\\ 1 & 4 & 1\\ 0 & 1 & 2 \end{array} \right]

16.2 a. [M] = \frac{\rho AL}{2}\left[ \begin{array}{cccc}1 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 1 \end{array} \right]

\mathbf{b.}[M] = \frac{\rho AL}{6}\left[ \begin{array}{cccc}2 & 1 & 0 & 0\\ 1 & 4 & 1 & 0\\ 0 & 1 & 4 & 1\\ 0 & 0 & 1 & 2 \end{array} \right]

16.3 \omega_{1} = 0.806\sqrt{u},\omega_{2} = 2.81\sqrt{\mu}

16.4 \omega_{1} = 5.368\times 10^{3}\mathrm{rad / s},\quad \omega_{2} = 17.556\times 10^{3}\mathrm{rad / s}

16.5 a. t(\mathrm{s}) d_{i}(\mathrm{ft}) \dot{d}_i(\mathrm{ft / s}) \ddot{d}_i(\mathrm{ft / s^2})

0	0	0	25
0.03	0.01125	0.71	22.09
0.06	0.04238	1.03	-0.715
0.09	0.07287	0.67	-22.87
0.12	0.08278	-0.35	-45.28
0.15	0.05194	-1.43	-26.94

16.6 a. t ðsÞ d _ { i } ðftÞ \dot { d } _ { i } \left( \mathrm { f t } / \mathrm { s } \right) \ddot { d } _ { i } \ ( \mathrm { f t } / \mathrm { s } ^ { 2 } )

0	0	0	10.00
0.02	0.0020	0.168	6.80
0.04	0.00672	0.256	1.968
0.06	0.01223	0.242	-3.338
0.08	0.01640	0.130	-7.84
0.10	0.01743	-0.053	-10.46

b. $t$ (s)	$d_{i}$ (ft)	$\dot{d}_{i}$ (ft/s)	$\ddot{d}_{i}$ (ft/s2)	$F(t)$ (lb)
0.00	0.00000	0.000	10.000	20.0
0.02	0.00179	0.169	6.923	16.0
0.04	0.00625	0.263	2.248	12.0
0.06	0.0115	0.254	-2.945	8.0
0.08	0.0157	0.150	-7.458	4.0
0.10	0.0169	-0.0147	-10.251	0.0

16.7 Node t ðsÞ di ðin:Þ \dot { d } _ { i } \ ( \mathrm { i n . / s } ) \ddot { d } _ { i } \ ( \mathrm { i n } . / \mathrm { s } ^ { 2 } )

2	0	0	0	0
	0.00025	2.6E-6	0.031	249.6
	0.00050	3.4E-5	0.284	1768.9
	0.00075	1.9E-4	1.085	4641.9
	0.0010	6.36E-4	2.605	7519.3
3	0	0	0	0
	0.00025	6.59E-5	0.791	6328.8
	0.00050	4.99E-4	2.817	9881.2
	0.00075	1.51E-3	5.265	9701.7
	0.0010	3.10E-3	7.369	7128.3

16.8 Using Newmark’s method with \begin{array} { r } { \gamma = \frac { 1 } { 2 } , \beta = \frac { 1 } { 6 } } \end{array}

Node	t (s)	$d_i$ (in.)	$\dot{d}_i$ (in./s)	$\ddot{d}_i$ (in./s2)	F(t) (lb)
2	0	0	0	0	0
	0.05	0.00172	0.103	4.131	0
	0.10	0.01544	0.513	12.27	0
3	0	0	0	40.0	2000
	0.05	0.0448	1.685	27.39	1800
	0.10	0.1536	2.479	4.37	1600

\begin{array} { l l l } { { { \bf 1 6 . 1 1 } } } & { { { \bf a . } { \ \omega } \omega _ { 1 } = \displaystyle \frac { 3 . 1 5 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { \omega _ { 2 } = \displaystyle \frac { 1 6 . 2 4 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } , } } & { { { \bf c . } { \omega } \omega _ { 1 } = \displaystyle \frac { 9 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } \\ { { } } & { { { \bf d . } { \omega } = \displaystyle \frac { 1 4 . 8 } { L ^ { 2 } } \left( \frac { E I } { \rho A } \right) ^ { 1 / 2 } } } & { { } } & { { { \bf d . } } } \end{array} EI 1=2 EI 1=2 EI 1=2

16.17	Node:		1	2	3	4	5	6
	i	t (s)			Temperature (°C)
	0	0	200	200	200	200	200	200
	1	8	0	159.0095	191.4441	198.2110	199.6110	199.8444
	2	16	0	135.5852	178.1491	193.6620	198.2112	199.1445
	3	24	0	120.2309	165.7003	187.3485	195.5379	197.5152
	4	32	0	109.1993	154.9587	180.4038	191.7446	194.8115
	5	40	0	100.7600	145.7784	173.4129	187.1268	191.1242
	6	48	0	94.00311	137.8529	166.6182	181.9599	186.6590
	7	56	0	88.39929	130.9034	160.1012	176.4598	181.6395
	8	64	0	83.61745	124.7101	153.8759	170.7856	176.2620
	9	72	0	79.43935	119.1075	147.9316	165.0508	170.6822
	10	80	0	75.71603	113.9733	142.2502	159.3352	165.0171

16.18
Time (s)	1	Node		(using consistent capacitance matrix)
Time (s)	1	2	3	(using consistent capacitance matrix)
Temperature (°C)
0	25	25	25
0.1	85	18.53611	26.36189
0.2	85	29.61303	21.63526
0.3	85	36.18435	22.42717
0.4	85	40.72491	25.30428
0.5	85	44.27834	28.85201
0.6	85	47.29072	32.49614
0.7	85	49.95809	36.01157
0.8	85	52.37152	39.31761
0.9	85	54.57756	42.39278

16.18
Time (s)	1	Node		(using consistent capacitance matrix)
Time (s)	1	2	3	(using consistent capacitance matrix)
Temperature (°C)
1	85	56.60353	45.23933
1.1	85	58.46814	47.86852
1.2	85	60.1859	50.29457
1.3	85	61.76908	52.53218
1.4	85	63.22852	54.59557
1.5	85	64.574	56.49814
1.6	85	65.81448	58.25235
1.7	85	66.95818	59.86974

16.18

Time (s)	Node
Time (s)	1	2	3
	Temperature (°C)
0	25	25	25
1.8	85	68.01265	61.36096
1.9	85	68.98485	62.73586
2	85	69.88121	64.0035
2.1	85	70.70765	65.17226
2.2	85	71.46961	66.24984
2.3	85	72.17214	67.24336
2.4	85	72.81986	68.15938
2.5	85	73.41705	69.00393
2.6	85	73.96766	69.78261
2.7	85	74.47531	70.50053
2.8	85	74.94336	71.16246
2.9	85	75.3749	71.77274
3	85	75.77277	72.33542

Appendix A

A1. a. \left[ { \begin{array} { r r } { 3 } & { 0 } \\ { - 3 } & { 1 2 } \end{array} } \right] b. Nonsense c. Nonsense
\left\{ { \begin{array} { l } { 1 1 } \\ { 9 } \\ { 1 1 } \end{array} } \right\} e. Nonsense f. \left[ { \begin{array} { r r r } { 1 0 } & { 7 } & { 6 } \\ { 3 } & { - 1 } & { 7 } \end{array} } \right]

\left[ \begin{array} { l l } { 1 } & { 0 } \\ { { \frac { 1 } { 4 } } } & { { \frac { 1 } { 4 } } } \end{array} \right]

A3. { \frac { 1 } { 1 7 } } \left[ { \begin{array} { r r r } { 1 2 } & { - 3 } & { - 8 } \\ { - 3 } & { 5 } & { 2 } \\ { - 8 } & { 2 } & { 1 1 } \end{array} } \right]

A4. Nonsense

A5. \left[ \begin{array} { l l } { { \frac { 1 } { 2 } } } & { 0 } \\ { { \frac { 1 } { 8 } } } & { { \frac { 1 } { 8 } } } \end{array} \right]

A6. Same as A3

A8. \left[ \begin{array} { c c } { \cos \theta } & { - \sin \theta } \\ { \sin \theta } & { \cos \theta } \end{array} \right]

Appendix B

B1. x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2

B2. x _ { 1 } = 3 . 1 5 , x _ { 2 } = 0 . 6 2

B3. x_{1} = 2.5, x_{2} = -1, x_{3} = 0.5

B4. x_{1} = 3, x_{2} = -1, x_{3} = -2

B5. a. \left\{ \begin{array}{l}x_{1}\\ x_{2} \end{array} \right\} = \left[ \begin{array}{ll}2 & -1\\ 1 & -1 \end{array} \right]\left\{ \begin{array}{l}y_{1}\\ y_{2} \end{array} \right\} b. \left\{ \begin{array}{l}z_1\\ z_2 \end{array} \right\} = \left[ \begin{array}{ll} - 3 & 2\\ 5 & -3 \end{array} \right]\left\{ \begin{array}{l}y_{1}\\ y_{2} \end{array} \right\}

B6. x_{1} = 0, x_{2} = 1, x_{3} = 2, x_{4} = 2, x_{5} = 0

B7. x_{1} = 3.15, x_{2} = 0.62

B8. a. Unique b. Nonexistent c. Unique d. Nonunique

Appendix D

D1. a. f_{1y} = f_{2y} = -5 \mathrm{kip} , m_{1} = -m_{2} = -100 \mathrm{k-ft}

b. f_{1v} = f_{2v} = -5 \mathrm{kip}, \quad m_1 = -m_2 = -18.75 \mathrm{k-ft}

c. f_{1y} = f_{2y} = -15 \mathrm{kip}, \quad m_1 = -m_2 = -75 \mathrm{k-ft}

d. f_{1v} = -18.75 \text{ kip}, \quad f_{2v} = -6.25 \text{ kip}, \quad m_1 = -58.3 \text{ k-ft}, \quad m_2 = 33.3 \text{ k-ft}

e. f_{1v} = -6 kip, f_{2v} = -14 kip, m_{1} = -26.67 k-ft, m_{2} = 40 k-ft

f. f_{1y} = -0.99 \, kN , f_{2y} = -4.0 \, kN , m_{1} = -2.04 \, kN \cdot m , m_{2} = 5.10 \, kN \cdot m

g. f_{1y} = f_{2y} = -6 \, kN , m_{1} = -m_{2} = -7.5 \, kN \cdot m

h. f_{1y} = f_{2y} = -10 \, kN , m_{1} = -m_{2} = -6.67 \, kN \cdot m

Index

A

Adaptive refinement, 355

Adjoint method, 718

Admissible variation, 55

Aluminum shapes, properties of, 759–772

Amplitude, defined, 649

Approximation functions, 72–74

compatible, 73

complete, 73–74

conforming, 73

displacement, 72–74

interpolation, 74

Aspect ratio (AR), 351, 352–353

Axial symmetry, 100

Axis of revolution, 412

Axis of symmetry, 412

Axisymmetric element, 9, 412–442, 684–685

applications of, 428–433

body forces, 419–420

consistent-mass matrix, 684–685

defined, 9, 412

discretization, 423

displacement functions, 415–417

element type, selection of, 415

equations, 419–421

introduction to, 412

pressure vessel, solution of, 422–428

sti¤ness matrix, 412–422, 423–428

strain/displacement relationships, 417–419

stress/strain relationships, 417–419

surface forces, 420–421

B

Banded-symmetric method, 735–741

Bar elements, 67–72, 92–100, 109–120, 120–124, 124–127, 127–131, 444–449, 665–669, 669–674. See also Truss equations

analysis of, 665–669, 669–674

collocation method, 129

consistent-mass matrix, 651–653

displacement function, 68, 446, 650

dynamic analysis of, 649–653, 665–669, 669–674

equations, 124–127, 447–449, 649–653

exact solution, 120–124

finite element solution, 120–124

Galerkin’s residual method, 124–127, 131

isoparametric formulation, 444–449

least squares method, 130

local coordinates for, 66–72

lumped-mass matrix, 651

mass matrix, 650–653

natural frequencies, 665–669

one-dimensional problems, 127–131, 665–669, 669–674

potential energy approach, 109–120

residual methods, 124–127, 127–131

selection of, 67, 444–446, 650

sti¤ness matrix, 66–72, 92–100, 444–449, 650–653

strain/displacement relationships, 69, 446–447, 650

stress, computation of, 82–83

stress/strain relationships, 69, 446–447, 650

subdomain method, 129–130

three-dimensional space, 92–100

time-dependent (dynamic) stress analysis, 649–653

time-dependent problem, 669–674

transformation matrix, 92–100

Beam element, 152–161, 161–163, 194–199, 214–218, 218–236, 255–269, 674–681

arbitrarily oriented, 214–218, 255–269

bending, 153–158, 255–260

boundary conditions, 161–163

defined, 152

deformations, 153–158

displacement function, 155–156

equations, 157–158, 161–163

mass matrices, 674–681

natural frequencies, 674–681

nodal hinge, 194–199

rigid plane frames, 218–236

selection of, 154

shape functions, 155–156

sign conventions, 152, 256–257

space, arbitrarily oriented in, 255–269

sti¤ness, 152–161

sti¤ness matrix, 153–158, 158–161

strain/displacement relationships, 156–157

stress/strain relationships, 156–157

transformation matrix, 216, 259–260

Beam element (Continued )

transverse shear deformations, 158–161

two-dimensional, arbitrarily oriented, 214–218

Beam equations, 151–213

bending deformations, 153–158

boundary conditions, 161–163

direct sti¤ness method, 163–175

displacement functions, 155–156

distributed loading, 175–188

Euler-Bernouli theory, 153–158

exact solution, 188–194

finite element solution, 188–194

fixed-end reactions, 175

Galerkin’s method, 201–203

introduction to, 151–152

load replacement, 177–178

nodal hinge, element with a, 194–199

potential energy approach, 199–201

sign conventions, 152

sti¤ness matrix, 153–158, 158–161, 161–163

sti¤ness of element, 152–161

strain/displacement relationships, 156–157

stress/strain relationships, 156–157

Timoshenko theory, 158–161

transverse shear deformations, 158–161

work-equivalence method, 176–177

Bending, 153–158, 255–260, 514–518

beam elements in arbitrary space, 255–260

deformations in beam elements, 153–158

plate element, 514–518

rigidity of a plate, 517

Body forces, 324–326, 419–420, 448, 460, 497–498

axisymmetric elements, 419–420

bar element, 448

centrifugal, 325

natural coordinate system, 448

plane element, 460

tetrahedral element, 497–498

treatment of, 324–326

Boundary conditions, 13–14, 34, 39–52, 103–109, 161–163, 320–322, 601

beam elements, 161–163

constant-strain triangular (CST) element, 320–322

fluid flow, 601

homogeneous, 39–40

inclined supports, 103–109

introduction to, 13–14, 34

nonhomogeneous, 39, 40–41

penalty method, 50–52

skewed supports, 103–109

sti¤ness method, 39–52

Castigliano’s theorem, 12

Central di¤erence method, 653, 654–659

Centrifugal body force, 325

Circular frequency, natural, 649

Coarse-mesh generation, 310

Coe‰cient matrix, inversion of, 726

Coe‰cient of thermal expansion, 618

Cofactor method, 716–717

Collocation method, 129

Column matrices, 4, 708

Compatibility, 35, 363–367, 746–748

condition of, 748

equations, 746–748

finite element results, 363–367

requirement, 35

Compatible displacements, 755

Compatible functions, 73

Complete, approximation functions, 73–74

Computer programs, 6–7, 23–24, 374–380, 524–528, 693–701

finite element method, 23–24

plate bending element, solution for, 524–528

role of, 6–7

step-by-step solutions, 374–380

structural dynamics, 693–701

Concentrated loads, 360–361

Condensation, see Static condensation

Conduction,535–538,542–546,557–558

element conduction matrix, 542–546, 557–558

heat, one-dimensional, 535–537

heat, two-dimensional, 537–538

Conforming functions, 73

Connecting (mixing) di¤erent kinds of elements, 361–362

Consistent-mass matrix, 651–653, 682–685

Constant-strain triangular (CST) element, 304–305, 310–324, 324–329, 342, 406–408

body forces, 324–326

boundary conditions, 320–322

coarse-mesh generation, 310 defects, 342

displacement function, 311–315

equations, 310–324

forces (stresses), 322–324

global equations, 320–322

introduction to, 304–305

LST elements, comparison of, 406–408

matrix, 310–324, 329–331

nodal displacements, 322

penalty formulation, 331

selection of, 310–311

strain/displacement relationships, 315–320

stress/strain relationships, 315–320

surface forces, 326–329

Constitutive law, 11

Constitutive matrix, 309, 522

Continuity, 35, 73

requirement, 35 symbol, 73

Convection, heat transfer with, 538–539, 540

Convergence of finite element solution, 367–368

Coordinates, 66–72, 444–446

bar elements, 67–72, 444–446

intrinsic system, 444 natural system, 444

Coulomb-Mohr theory, 342

Cramer’s rule, 724–725

CST, see Constant-strain triangular (CST) element

Cubic elements, 9

Curvature matrix, 521–522

D

D’Alembert’s principle, 755–756

Defects, CST elements, 342

Deformation, 33, 153–158, 158–161, 514–518

bending in beams, 153–158

bending rigidity of a plate, 517 defined, 33

Kirchho¤ assumptions, 515–516

plate bending, 514–518

potential energy, 518

stress/strain relationships, 517–518

transverse shear in beams, 158–161

Degrees of freedom, 14, 15, 29 defined, 15

spring element, 29

unknown, 14

Determinant, defined, 716

Di¤erential equations, 535–538, 594–596, 744–746

elasticity theory, 744–746

equilibrium, 744–746

fluid flow, 594–598

heat transfer, 535–538

Direct equilibrium method, 11

Direct integration, 653

Direct sti¤ness method, 2–4, 13–14, 28, 37–39, 163–175.

E

E¤ective stress, 341

Elasticity theory, 744–751

compatibility equations, 746–748

condition of compatibility, 748

di¤erential equations of equilibrium, 744–746

equilibrium, di¤erential equations of, 744–746

introduction to, 744

modulus of elasticity, 748

strain/displacement, 746–748

stress/strain relationships, 748–751

Elements, 8–10, 11, 13–14, 30–34, 65–150, 151–213, 304–305, 310–324, 342, 351–362, 398–403, 444–449, 449–452, 480–482, 493–500, 501–508, 514–533

aspect ratio (AR), 351

axisymmetric, 9

bar, 65–150, 444–449

beam, 151–213

coarse-mesh generation, 310

connecting (mixing), modeling, 361–362

constant-strain triangular (CST), 304–305, 310–324, 342

cubic, 9

defects, CST, 324

equations, 11, 13–14, 34, 69–70, 402–403, 451–452, 522–523

finite, 8

forces, 34, 70

heterosis, 523

isoparametric, 446

LaGrange, 482

linear hexahedral, 501–504

linear-strain triangle (LST), 398–403

plane stress, 449–452

plate bending, 514–533

Q8, 480

Q9, 482

quadratic, 9

quadratic hexahedral, 504–508

refinement, methods of, 355–356, 358–359

selection of, 8–10, 30–31, 310–311, 399, 444–446, 449, 519

serendipity, 481

shapes, modeling, 351

sizing, 355–356, 358–359

spring, 30–34

sti¤ness matrix, 11, 33–34, 66–72, 402–403, 447–449, 451–452, 522–523

tetrahedral, 493–500

transition triangles, 359–360

Energy method, 12

Equations, 11, 13–14, 34, 52–60, 65–149, 151–213, 214–237, 238–255, 310–324, 398–411, 419–422, 447–449, 451–452, 459–460, 497–498, 522–523, 535–538, 542–546, 557–558, 594–596, 599–601, 608, 659–661, 664–665, 722–743, 744–751.

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