Instructor’s Solutions Manual to Accompany A First Course in the Finite Element Method

564 Pages·2012·1.99 MB·English

by  Daryl L. Logan

INSTRUCTOR'S SOLUTIONS MANUAL

TO ACCOMPANY

A FIRST COURSE IN THE
FINITE
ELEMENT
METHOD

FIFTH EDITION

DARYL L. LOGAN
Contents

Chapter 1 1

Chapter 2 3

Chapter 3 23

Chapter 4 127

Chapter 5 183

Chapter 6 281

Chapter 7 319

Chapter 8 338

Chapter 9 351

Chapter 10 371

Chapter 11 390

Chapter 12 414

Chapter 13 432

Chapter 14 473

Chapter 15 492

Chapter 16 518

Appendix A 550

Appendix B 555

Appendix D 561
Chapter 1
1.1. A finite element is a small body or unit interconnected to other units to model a larger
structure or system.
1.2. Discretization means dividing the body (system) into an equivalent system of finite
elements with associated nodes and elements.
1.3. The modern development of the finite element method began in 1941 with the work of
Hrennikoff in the field of structural engineering.
1.4. The direct stiffness method was introduced in 1941 by Hrennikoff. However, it was not
commonly known as the direct stiffness method until 1956.
1.5. A matrix is a rectangular array of quantities arranged in rows and columns that is often
used to aid in expressing and solving a system of algebraic equations.
1.6. As computer developed it made possible to solve thousands of equations in a matter of
minutes.
1.7. The following are the general steps of the finite element method.
Step 1
Divide the body into an equivalent system of finite elements with associated
nodes and choose the most appropriate element type.
Step 2
Choose a displacement function within each element.
Step 3
Relate the stresses to the strains through the stress/strain law—generally
called the constitutive law.
Step 4
Derive the element stiffness matrix and equations. Use the direct equilibrium
method, a work or energy method, or a method of weighted residuals to
relate the nodal forces to nodal displacements.
Step 5
Assemble the element equations to obtain the global or total equations and
introduce boundary conditions.
Step 6
Solve for the unknown degrees of freedom (or generalized displacements).
Step 7
Solve for the element strains and stresses.
Step 8
Interpret and analyze the results for use in the design/analysis process.
1.8. The displacement method assumes displacements of the nodes as the unknowns of the
problem. The problem is formulated such that a set of simultaneous equations is solved
for nodal displacements.
1.9. Four common types of elements are: simple line elements, simple two-dimensional
elements, simple three-dimensional elements, and simple axisymmetric elements.
1.10 Three common methods used to derive the element stiffness matrix and equations are
(1) direct equilibrium method
(2) work or energy methods
(3) methods of weighted residuals
1.11. The term ‘degrees of freedom’ refers to rotations and displacements that are associated
with each node.
1
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1.12. Five typical areas where the finite element is applied are as follows.
(1) Structural/stress analysis
(2) Heat transfer analysis
(3) Fluid flow analysis
(4) Electric or magnetic potential distribution analysis
(5) Biomechanical engineering
1.13. Five advantages of the finite element method are the ability to
(1) Model irregularly shaped bodies quite easily
(2) Handle general load conditions without difficulty
(3) Model bodies composed of several different materials because element equations
are evaluated individually
(4) Handle unlimited numbers and kinds of boundary conditions
(5) Vary the size of the elements to make it possible to use small elements where
necessary

2
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Chapter 2
2.1
(a)

(cid:0) k 0 –k 0(cid:2)
1 1
(cid:3) (cid:4)
0 0 0 0
[k(1)]= (cid:3) (cid:4)
(cid:3)–k 0 k 0(cid:4)
1 1
(cid:3) (cid:4)
(cid:5) 0 0 0 0(cid:6)
(cid:0)0 0 0 0 (cid:2)
(cid:3) (cid:4)
0 0 0 0
[k(2)] = (cid:3) (cid:4)
(cid:3)0 0 k –k (cid:4)
2 2
(cid:3) (cid:4)
(cid:5)0 0 –k k (cid:6)
2 2
(cid:0)0 0 0 0 (cid:2)
(cid:3) (cid:4)
0 k 0 –k
[k (3)] = (cid:3) 3 3(cid:4)
3 (cid:3)0 0 0 0 (cid:4)
(cid:3) (cid:4)
(cid:5)0 –k 0 k (cid:6)
3 3
[K] = [k(1)] + [k(2)] + [k(3)]
(cid:0) k 0 –k 0 (cid:2)
1 1
(cid:3) (cid:4)
0 k 0 –k
[K] = (cid:3) 3 3 (cid:4)
(cid:3)–k 0 k (cid:5)k –k (cid:4)
1 1 2 2
(cid:3) (cid:4)
(cid:6) 0 –k –k k (cid:5)k (cid:7)
3 2 2 3
(b) Nodes 1 and 2 are fixed so u = 0 and u = 0 and [K] becomes
1 2
(cid:2)k (cid:0)k –k (cid:3)
[K] = (cid:4) 1 2 2 (cid:5)
(cid:6) –k k (cid:0)k (cid:7)
2 2 3
{F} = [K] {d}
(cid:0)F (cid:2) (cid:2)k (cid:0)k –k (cid:3) (cid:0)u (cid:2)
(cid:3) 3x(cid:4) = (cid:4) 1 2 2 (cid:5) (cid:3) 3(cid:4)
(cid:5)F (cid:6) (cid:6) –k k (cid:0)k (cid:7) (cid:5)u (cid:6)
4x 2 2 3 4
(cid:0)0(cid:2) (cid:2)k (cid:0)k –k (cid:3) (cid:0)u (cid:2)
⇒ (cid:3) (cid:4) = (cid:4) 1 2 2 (cid:5) (cid:3) 3(cid:4)
(cid:5)P(cid:6) (cid:6) –k k (cid:0)k (cid:7) (cid:5)u (cid:6)
2 2 3 4
{F} = [K] {d} ⇒ [K–1] {F} = [K–1] [K] {d}

⇒ [K–1] {F} = {d}

Using the adjoint method to find [K–1]

C = k + k C = (– 1)3 (– k )
11 2 3 21 2
C = (– 1)1 + 2 (– k ) = k C = k + k
12 2 2 22 1 2
3

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(cid:2)k (cid:0)k k (cid:3) (cid:2)k (cid:0)k k (cid:3)
[C] = (cid:4) 2 3 2 (cid:5) and CT = (cid:4) 2 3 2 (cid:5)
(cid:6) k k (cid:0)k (cid:7) (cid:6) k k (cid:0)k (cid:7)
2 1 2 2 1 2
det [K] = | [K] | = (k + k ) (k + k ) – ( – k ) (– k )
1 2 2 3 2 2
⇒ | [K] | = (k + k ) (k + k ) – k 2
1 2 2 3 2
[CT]
[K–1] =

detK
(cid:2)k (cid:0)k k (cid:3) (cid:2)k (cid:0)k k (cid:3)
2 3 2 2 3 2
(cid:4) (cid:5) (cid:4) (cid:5)
(cid:6) k k (cid:0)k (cid:7) (cid:6) k k (cid:0)k (cid:7)
[K–1] = 2 1 2 = 2 1 2
(k (cid:0)k )(k (cid:0)k )–k 2 k k (cid:0)k k (cid:0)k k
1 2 2 3 2 1 2 1 3 2 3
(cid:2)k (cid:0)k k (cid:3)(cid:4)0(cid:5)
2 3 2
(cid:8) (cid:9)(cid:6) (cid:7)
(cid:0)u (cid:2) (cid:10) k k (cid:0)k (cid:11)(cid:12)P(cid:13)
(cid:3) 3(cid:4) = 2 1 2
(cid:5)u (cid:6) k k (cid:0)k k (cid:0)k k
4 1 2 1 3 2 3
k P
⇒ u = 2
3 k k (cid:0)k k (cid:0)k k
1 2 1 3 2 3
(k (cid:0)k )P
⇒ u = 1 2
4 k k (cid:0)k k (cid:0)k k
1 2 1 3 2 3
(c) In order to find the reaction forces we go back to the global matrix F = [K] {d}
(cid:0)F (cid:2) (cid:4) k 0 (cid:0)k 0 (cid:5)(cid:2)u (cid:3)
1x 1 1 1
(cid:3) (cid:3) (cid:7) (cid:8)(cid:6) (cid:6)
(cid:3)F (cid:3) 0 k 0 (cid:0)k (cid:6)u (cid:6)
(cid:4) 2x(cid:5) = (cid:7) 3 3 (cid:8)(cid:9) 2(cid:10)
(cid:3)F3x(cid:3) (cid:7)(cid:0)k1 0 k1(cid:11)k2 (cid:0)k2 (cid:8)(cid:6)u3(cid:6)
(cid:7) (cid:8)
(cid:3)(cid:6)F (cid:3)(cid:7) (cid:12) 0 (cid:0)k (cid:0)k k (cid:11)k (cid:13)(cid:6)(cid:14)u (cid:6)(cid:15)
4x 3 2 2 3 4
k P
F = – k u = – k 2
1x 1 3 1 k k (cid:0)k k (cid:0)k k
1 2 1 3 2 3
(cid:0)k k P
⇒ F = 1 2
1x k k (cid:2)k k (cid:2)k k
1 2 1 3 2 3
(k (cid:0)k )P
F = – k u = – k 1 2
2x 3 4 3 k k (cid:0)k k (cid:0)k k
1 2 1 3 2 3
(cid:0)k (k (cid:2)k )P
⇒ F = 3 1 2
2x k k (cid:2)k k (cid:2)k k
1 2 1 3 2 3
2.2

lb
k = k = k = 1000
1 2 3
in.
(1) (2) (2) (3)
(cid:2) k (cid:0)k(cid:3)(1) (cid:2) k (cid:0)k(cid:3)(2)
[k(1)] = (cid:4) (cid:5) ; [k(2)] = (cid:4) (cid:5)
(cid:6)(cid:0)k k(cid:7)(2) (cid:6)(cid:0)k k(cid:7)(3)
By the method of superposition the global stiffness matrix is constructed.
4

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(1) (2) (3)
(cid:2) k (cid:0)k 0(cid:3)(1) (cid:2) k (cid:0)k 0(cid:3)
(cid:4) (cid:5) (cid:4) (cid:5)
[K] = (cid:0)k k(cid:6)k (cid:0)k (2)⇒ [K] = (cid:0)k 2k (cid:0)k
(cid:4) (cid:5) (cid:4) (cid:5)
(cid:4)(cid:7) 0 (cid:0)k k(cid:5)(cid:8)(3) (cid:4)(cid:6) 0 (cid:0)k k(cid:5)(cid:7)
Node 1 is fixed ⇒ u = 0 and u = δ
1 3
{F} = [K] {d}
(cid:2)F (cid:0)?(cid:3) (cid:0) k k 0(cid:2) (cid:2)u (cid:0)0(cid:3)
1x 1
(cid:4) (cid:4) (cid:3) (cid:4) (cid:4) (cid:4)
(cid:5)F (cid:0)0(cid:6) = (cid:5)k 2k (cid:5)k (cid:5)u (cid:0)?(cid:6)
2x (cid:3) (cid:4) 2
(cid:4) (cid:4) (cid:4) (cid:4)
(cid:7)F (cid:0)?(cid:8) (cid:6)(cid:3) 0 (cid:5)k k(cid:7)(cid:4) (cid:7)u (cid:0)(cid:0)(cid:8)
3x 3
(cid:0) 0 (cid:2) (cid:5)2k (cid:2)k(cid:6)(cid:3)u (cid:4) (cid:3)0 (cid:0) 2ku (cid:2)k(cid:0)(cid:4)
⇒ (cid:3)(cid:5)F (cid:4)(cid:6) = (cid:10)(cid:13)(cid:2)k k(cid:11)(cid:14)(cid:8)(cid:15)(cid:0)2(cid:9)(cid:16)(cid:7)(cid:8)(cid:15)F (cid:0) (cid:2)ku2 (cid:12)k(cid:0)(cid:9)(cid:16)
3x 3x 2
k(cid:0) (cid:0) 1in.
⇒ u = = = ⇒ u = 0.5″
2 2
2k 2 2
F = – k (0.5″) + k (1″)
3x
lb lb
F = (– 1000 ) (0.5″) + (1000 ) (1″)
3x
in. in.
F = 500 lbs
3x
Internal forces
Element (1)
(cid:0)(cid:3)f (1) (cid:2)(cid:3) (cid:5) k (cid:2)k(cid:6)(cid:3)u (cid:0)0 (cid:4)
(cid:4) 1x (cid:5) = (cid:9) (cid:10)(cid:7) 1 (cid:8)
(cid:3)(cid:6)f2x(2)(cid:3)(cid:7) (cid:12)(cid:2)k k(cid:13)(cid:14)u2 (cid:0)0.5(cid:11)(cid:15)
lb
⇒ f (1) = (– 1000 ) (0.5″) ⇒ f (1) = – 500 lb
1x in. 1x
lb
f (1) = (1000 ) (0.5″) ⇒ f (1) = 500 lb
2x in. 2x
Element (2)
(cid:0)(cid:3)f (2)(cid:2)(cid:3) (cid:6) k (cid:3)k(cid:7)(cid:4)u (cid:0)0.5(cid:2)(cid:5) f (2) (cid:0)–500lb
(cid:4) 2x (cid:5) = (cid:10) (cid:11)(cid:8) 2 (cid:9)⇒ 2x
(cid:3)(cid:6)f3x(2)(cid:3)(cid:7) (cid:12)(cid:3)k k(cid:13)(cid:14)u3 (cid:0)1(cid:2) (cid:15) f3x(2) (cid:0)500lb
2.3

(cid:2) k (cid:0)k(cid:3)
(a) [k(1)] = [k(2)] = [k(3)] = [k(4)] = (cid:4) (cid:5)
(cid:6)(cid:0)k k(cid:7)
By the method of superposition we construct the global [K] and knowing {F} = [K] {d}
we have
(cid:2)F (cid:0)? (cid:3) (cid:2) k (cid:0)k 0 0 0(cid:3) (cid:2)u (cid:0)0(cid:3)
1x 1
(cid:4) (cid:4) (cid:4) (cid:5) (cid:4) (cid:4)
F (cid:0)0 (cid:0)k 2k (cid:0)k 0 0 u
(cid:4)(cid:4) 2x (cid:4)(cid:4) (cid:4) (cid:5) (cid:4)(cid:4) 2 (cid:4)(cid:4)
(cid:5)F (cid:0)P(cid:6) = (cid:4) 0 (cid:0)k 2k (cid:0)k 0(cid:5) (cid:5)u (cid:6)
3x 3
(cid:4) (cid:4) (cid:4) (cid:5) (cid:4) (cid:4)
(cid:4)F4x (cid:0)0(cid:4) (cid:4) 0 0 (cid:0)k 2k (cid:0)k(cid:5) (cid:4)u4 (cid:4)
(cid:4)(cid:7)F (cid:0)?(cid:4)(cid:8) (cid:4)(cid:6) 0 0 0 (cid:0)k k(cid:5)(cid:7) (cid:4)(cid:7)u (cid:0)0(cid:4)(cid:8)
5x 5
5

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(cid:0)0(cid:2) (cid:5)2k (cid:0)k 0 (cid:6)(cid:3)u (cid:4) 0 (cid:2) 2ku (cid:0)ku
(cid:3) (cid:3) (cid:8) (cid:9)(cid:7) 2(cid:7) 2 3
(b) (cid:4)P(cid:5) = (cid:0)k 2k (cid:0)k (cid:12)u (cid:13)(cid:10)P (cid:2) (cid:0)ku (cid:11)2ku (cid:0)ku
(cid:8) (cid:9) 3 2 3 4
(cid:3) (cid:3) (cid:7) (cid:7)
(cid:6)0(cid:7) (cid:8)(cid:14) 0 (cid:0)k 2k(cid:9)(cid:15)(cid:16)u (cid:17) 0 (cid:2) (cid:0)ku (cid:11)2ku
4 3 4
u u
⇒ u = 3 ; u = 3
2 4
2 2
Substituting in the equation in the middle
P = – k u + 2k u – k u
2 3 4
⇒ P = – k(cid:0)(cid:3)(cid:5)u23(cid:2)(cid:4)(cid:6) + 2k u3 – k(cid:0)(cid:3)(cid:5)u23(cid:2)(cid:4)(cid:6)
⇒ P = k u
3
P
⇒ u =
3
k
P P
u = ; u =
2 4
2k 2k
(c) In order to find the reactions at the fixed nodes 1 and 5 we go back to the global
equation {F} = [K] {d}
P (cid:0)P
F = – k u = –k ⇒ F =
1x 2 1x
2k 2
P (cid:0)P
F = – k u = –k ⇒ F =
5x 4 5x
2k 2
Check
ΣF = 0 ⇒ F + F + P = 0
x 1x 5x
(cid:0)P (cid:0) P(cid:2)
⇒ + (cid:4)(cid:3) (cid:5) + P = 0
2 (cid:6) 2(cid:7)
⇒ 0 = 0
2.4

(cid:2) k (cid:0)k(cid:3)
(a) [k(1)] = [k(2)] = [k(3)] = [k(4)] = (cid:4) (cid:5)
(cid:6)(cid:0)k k(cid:7)
By the method of superposition the global [K] is constructed.
Also {F} = [K] {d} and u = 0 and u = δ
1 5
(cid:0)F (cid:0)?(cid:2) (cid:2) k (cid:0)k 0 0 0(cid:3) (cid:2)u (cid:0)0(cid:3)
1x 1
(cid:3) (cid:3) (cid:4) (cid:5) (cid:4) (cid:4)
F (cid:4)0 (cid:0)k 2k (cid:0)k 0 0 u (cid:0)?
(cid:3)(cid:3) 2x (cid:3)(cid:3) (cid:4) (cid:5) (cid:4)(cid:4) 2 (cid:4)(cid:4)
(cid:5)F (cid:4)0(cid:6) = (cid:4) 0 (cid:0)k 2k (cid:0)k 0(cid:5) (cid:5)u (cid:0)?(cid:6)
3x 3
(cid:3) (cid:3) (cid:4) (cid:5) (cid:4) (cid:4)
(cid:3)F4x (cid:4)0(cid:3) (cid:4) 0 0 (cid:0)k 2k (cid:0)k(cid:5) (cid:4)u4 (cid:0)?(cid:4)
(cid:3)(cid:7)F (cid:4)?(cid:3)(cid:8) (cid:4)(cid:6) 0 0 0 (cid:0)k k(cid:5)(cid:7) (cid:4)(cid:7)u (cid:0)(cid:0)(cid:4)(cid:8)
5x 5
6

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(b) 0 = 2k u – k u (1)
2 3
0 = – ku + 2k u – k u (2)
2 3 4
0 = – k u + 2k u – k δ (3)
3 4
From (2)
u = 2 u
3 2
From (3)
(cid:0)(cid:0)2u
u = 2
4
2
Substituting in Equation (2)
⇒ – k (u ) + 2k (2 u ) – k (cid:0)(cid:0)(cid:0)2(cid:0)2x(cid:2)
2 2 (cid:3) 2 (cid:4)
(cid:0) (cid:0)
⇒ – u + 4 u – u – = 0 ⇒ u =
2 2 2 2
2 4
(cid:0) (cid:0)
⇒ u =2 ⇒ u =
3 3
4 2
(cid:2) (cid:3)
(cid:0)(cid:0)2 (cid:0) 3(cid:0)
⇒ u = 4 ⇒ u =
4 4
2 4
(c) Going back to the global equation
{F} = [K] {d}
(cid:0) (cid:0) (cid:0)k(cid:0)
F = – k u = k ⇒ F =
1x 2 1x
4 4
(cid:0)3(cid:0)(cid:2)
F5x = – k u4 + k δ = – k(cid:3)(cid:5) 4 (cid:4)(cid:6) + k δ
k(cid:0)
⇒ F =
5x
4
2.5
kip kip
2 3
in. in.
2
1 2 3 4 5
1 3 x
1 kinip. 4 5 kinip.
kip
4
in.
d d d d
1 2 2 4
(cid:2) 1 (cid:0)1(cid:3) (cid:2) 2 (cid:0)2(cid:3)
[k (1)] = (cid:4) (cid:5); [k (2)] = (cid:4) (cid:5)
(cid:6)(cid:0)1 1(cid:7) (cid:6)(cid:0)2 2(cid:7)
d d d d
2 4 2 4
(cid:2) 3 (cid:0)3(cid:3) (cid:2) 4 (cid:0)4(cid:3)
[k (3)] = (cid:4) (cid:5); [k (4)] = (cid:4) (cid:5)
(cid:6)(cid:0)3 3(cid:7) (cid:6)(cid:0)4 4(cid:7)
d d
4 3
(cid:2) 5 (cid:0)5(cid:3)
[k (5)] = (cid:4) (cid:5)
(cid:6)(cid:0)5 5(cid:7)
7

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Assembling global [K] using direct stiffness method
(cid:2)1 (cid:0)1 0 0 (cid:3)
(cid:4) (cid:5)
(cid:0)1 1(cid:6)2(cid:6)3(cid:6)4 0 (cid:0)2(cid:0)3(cid:0)4
(cid:4) (cid:5)
[K] =
(cid:4) 0 0 5 (cid:0)5 (cid:5)
(cid:4) (cid:5)
(cid:7) 0 (cid:0)2(cid:0)3(cid:0)4 (cid:0)5 2(cid:6)3(cid:6)4(cid:6)5(cid:8)
Simplifying
(cid:2)1 (cid:0)1 0 0 (cid:3)
(cid:4) (cid:5)
(cid:4)(cid:0)1 10 0 (cid:0)9(cid:5)kip
[K] =
(cid:4) 0 0 5 (cid:0)5(cid:5) in.
(cid:4) (cid:5)
(cid:6) 0 (cid:0)9 (cid:0)5 14(cid:7)
2.6 Now apply + 2 kip at node 2 in spring assemblage of P 2.5.
∴ F = 2 kip
2x
[K]{d} = {F}
[K] from P 2.5
(cid:2)1 (cid:0)1 0 0(cid:3) (cid:2)u (cid:0)0(cid:3) (cid:0)F (cid:2)
1 1
(cid:4) (cid:5) (cid:4) (cid:4) (cid:3) (cid:3)
(cid:0)1 10 0 (cid:0)9 (cid:4)u (cid:4) (cid:3)2 (cid:3)
(cid:4) (cid:5) (cid:5) 2 (cid:6) = (cid:4) (cid:5) (A)
(cid:4) 0 0 5 (cid:0)5(cid:5) (cid:4)u3(cid:0)0(cid:4) (cid:3)F3(cid:3)
(cid:4) (cid:5)
(cid:6) 0 (cid:0)9 (cid:0)5 14(cid:7) (cid:4)(cid:7)u (cid:4)(cid:8) (cid:3)(cid:6)0 (cid:3)(cid:7)
4
where u = 0, u = 0 as nodes 1 and 3 are fixed.
1 3
Using Equations (1) and (3) of (A)
(cid:4)10 (cid:0)9(cid:5)(cid:2)u (cid:3) (cid:0)2(cid:2)
(cid:8) (cid:9)(cid:6) 2(cid:7) = (cid:3) (cid:4)
(cid:10)(cid:0)9 14(cid:11)(cid:12)u (cid:13) (cid:5)0(cid:6)
4
Solving
u = 0.475 in., u = 0.305 in.
2 4
2.7
conv.
f = C, f = – C
1x 2x
f = – kδ = – k(u – u )
2 1
∴ f = – k(u – u )
1x 2 1
f = – (– k) (u – u )
2x 2 1
(cid:0)f (cid:2) (cid:0) k –k(cid:2) (cid:0)u (cid:2)
(cid:3) 1x(cid:4) = (cid:3) (cid:4) (cid:3) 1(cid:4)
(cid:5)f (cid:6) (cid:5)–k k(cid:6) (cid:5)u (cid:6)
2x 2
(cid:0) k –k(cid:2) sameasfor
∴ [K] = (cid:3) (cid:4)
(cid:5)–k k(cid:6) tensileelement
8

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