Strength of Materials
W ORK -B OOK
RM Nkgoeng
,Copyright c 2013 Mashilo Nkgoeng
S ELF - PUBLISHED
W W W.M A S H I L O N K G O E N G.C O.Z A
Licensed under the Creative Commons Attribution-Non-commercial 3.0 Unported License (the
“License”). You may not use this file except in compliance with the License. You may obtain a
copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Un-
less required by applicable law or agreed to in writing, software distributed under the License
is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF
ANY KIND, either express or implied. See the License for the specific language governing
permissions and limitations under the License.
First printing, December 2013
, Contents
1 Deflection of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 What is a Beam? 5
1.1.1 Beam terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Assumption of Classical Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Beam Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.5 Support Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.6 Stresses, strains and bending moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Notation 7
1.3 Second Order Method for Beam Deflections 7
1.4 Double Integration Using Bracket Functions 7
1.5 Examples 9
1.6 Exercises 16
2 Continuous Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Introduction 19
2.1.1 Point Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Exercise 22
2.3 Examples 23
2.4 Exercises 23
3 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Introduction 25
3.2 Strain Energy of Bars 26
3.3 Castigliano’s Theorem 26
3.4 Structures 27
3.5 Castigliano’s theorem applied to Curved Beams 27
3.6 Castigliano’s theorem applied to Beams 27
3.6.1 Cantilever beam with a Point Load at the free end . . . . . . . . . . . . . . . . . . . . . . . 27
3.6.2 S/S beam with a Point Load at mid-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Examples 29
3.8 Exercises 34
, 4 Unsymmetrical Bending of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Symmetric Member in Pure Bending 39
4.2 Unsymmetrical Bending 40
4.3 Alternative procedure for stress determination 42
4.4 Deflection 44
4.5 Notation 44
4.6 xxx 44
4.6.1 Point Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6.2 Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Examples 44
4.8 Exercises 47
5 Inelastic Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Plastic Bending of Rectangular Beams 49
5.2 Plastic Bending of Symmetrical (I-Section) Beam 52
5.3 Partially plastic Bending of Unsymmetrical Sections 53
5.4 Limit Analysis-Bending 56
5.4.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4.2 The principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Solid Shaft 65
5.6 Hollow shaft 67
5.7 Exercises 69
W ORK -B OOK
RM Nkgoeng
,Copyright c 2013 Mashilo Nkgoeng
S ELF - PUBLISHED
W W W.M A S H I L O N K G O E N G.C O.Z A
Licensed under the Creative Commons Attribution-Non-commercial 3.0 Unported License (the
“License”). You may not use this file except in compliance with the License. You may obtain a
copy of the License at http://creativecommons.org/licenses/by-nc/3.0. Un-
less required by applicable law or agreed to in writing, software distributed under the License
is distributed on an “AS IS” BASIS, WITHOUT WARRANTIES OR CONDITIONS OF
ANY KIND, either express or implied. See the License for the specific language governing
permissions and limitations under the License.
First printing, December 2013
, Contents
1 Deflection of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1 What is a Beam? 5
1.1.1 Beam terminology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.2 Mathematical Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.1.3 Assumption of Classical Beam Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.4 Beam Loading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.5 Support Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.1.6 Stresses, strains and bending moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 Notation 7
1.3 Second Order Method for Beam Deflections 7
1.4 Double Integration Using Bracket Functions 7
1.5 Examples 9
1.6 Exercises 16
2 Continuous Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1 Introduction 19
2.1.1 Point Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.1.2 Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Exercise 22
2.3 Examples 23
2.4 Exercises 23
3 Energy Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.1 Introduction 25
3.2 Strain Energy of Bars 26
3.3 Castigliano’s Theorem 26
3.4 Structures 27
3.5 Castigliano’s theorem applied to Curved Beams 27
3.6 Castigliano’s theorem applied to Beams 27
3.6.1 Cantilever beam with a Point Load at the free end . . . . . . . . . . . . . . . . . . . . . . . 27
3.6.2 S/S beam with a Point Load at mid-point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.7 Examples 29
3.8 Exercises 34
, 4 Unsymmetrical Bending of Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
4.1 Symmetric Member in Pure Bending 39
4.2 Unsymmetrical Bending 40
4.3 Alternative procedure for stress determination 42
4.4 Deflection 44
4.5 Notation 44
4.6 xxx 44
4.6.1 Point Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.6.2 Uniformly Distributed Load . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.7 Examples 44
4.8 Exercises 47
5 Inelastic Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.1 Plastic Bending of Rectangular Beams 49
5.2 Plastic Bending of Symmetrical (I-Section) Beam 52
5.3 Partially plastic Bending of Unsymmetrical Sections 53
5.4 Limit Analysis-Bending 56
5.4.1 Bending . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5.4.2 The principle of Virtual Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.5 Solid Shaft 65
5.6 Hollow shaft 67
5.7 Exercises 69
