Etextbook 978 0134319650 mechanics of materials 10th edition

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P R E FA C E It is intended that this book provide the student with a clear and thorough presentation of the theory and application of the principles of mechanics of materials. To achieve this objective, over the years this work has been shaped by the comments and suggestions of hundreds of reviewers in the teaching profession, as well as many of the author’s students. The tenth edition has been significantly enhanced from the previous edition, and it is hoped that both the instructor and student will benefit greatly from these improvements.

NEW TO THIS EDITION • Updated Material. Many topics in the book have been re-written in order to further enhance clarity and to be more succinct. Also, some of the artwork has been enlarged and improved throughout the book to support these changes.

• New Layout Design. Additional design features have been added to this edition to provide a better display of the material. Almost all the topics are presented on a one or two page spread so that page turning is minimized.

• Improved Preliminary and Fundamental Problems. These problems sets are located just after each group of example problems. They offer students basic applications of the concepts covered in each section, and they help provide the chance to develop their problem-solving skills before attempting to solve any of the standard problems that follow. The problems sets may be considered as extended examples, since in this edition their complete solutions are given in the back of the book. Additionally, when assigned, these problems offer students an excellent means of preparing for exams, and they can be used at a later time as a review when studying for the Fundamentals of Engineering Exam.

• New Photos. The relevance of knowing the subject matter is reflected by the real-world application of the 14 new or updated photos placed throughout the book. These photos generally are used to explain how the principles apply to real-world situations and how materials behave under load.

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• New Problems. There are approximately 30%, or about 430 problems added to this edition, which involve applications to many different fields of engineering.

• New Review Problems. Updated review problems have been placed at the end of each chapter so that instructors can assign them as additional preparation for exams.

CONTENTS The subject matter is organized into 14 chapters. Chapter 1 begins with a review of the important concepts of statics, followed by a formal definition of both normal and shear stress, and a discussion of normal stress in axially loaded members and average shear stress caused by direct shear.

In Chapter 2 normal and shear strain are defined, and in Chapter 3 a discussion of some of the important mechanical properties of materials is given. Separate treatments of axial load, torsion, and bending are presented in Chapters 4, 5, and 6, respectively. In each of these chapters, both linear-elastic and plastic behavior of the material covered in the previous chapters, where the state of stress results from combined loadings. In Chapter 9 the concepts for transforming multiaxial states of stress are presented. In a similar manner, Chapter 10 discusses the methods for strain transformation, including the application of various theories of failure. Chapter 11 provides a means for a further summary and review of previous material by covering design applications of beams and shafts. In Chapter 12 various methods for computing deflections of beams and shafts are covered. Also included is a discussion for finding the reactions on these members if they are statically indeterminate. Chapter 13 provides a discussion of column buckling, and lastly, in Chapter 14 the problem of impact and the application of various energy methods for computing deflections are considered.

Sections of the book that contain more advanced material are indicated by a star (*). Time permitting, some of these topics may be included in the course. Furthermore, this material provides a suitable reference for basic principles when it is covered in other courses, and it can be used as a basis for assigning special projects.

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Alternative Method of Coverage. Some instructors prefer to cover stress and strain transformations first, before discussing specific applications of axial load, torsion, bending, and shear. One possible method for doing this would be first to cover stress and its transformation, Chapter 1 and Chapter 9, followed by strain and its transformation, Chapter 2 and the first part of Chapter 10. The discussion and example problems in these later chapters have been styled so that this is possible. Also, the problem sets have been subdivided so that this material can be covered without prior knowledge of the intervening chapters. Chapters 3 through 8 can then be covered with no loss in continuity.

HALLMARK ELEMENTS Organization and Approach. The contents of each chapter are organized into well-defined sections that contain an explanation of specific topics, illustrative example problems, and a set of homework problems. The topics within each section are placed into subgroups defined by titles. The purpose of this is to present a structured method for introducing each new definition or concept and to make the book convenient for later reference and review.

Chapter Contents. Each chapter begins with a full-page illustration that indicates a broad-range application of the material within the chapter.

The “Chapter Objectives” are then provided to give a general overview of the material that will be covered.

Procedures for Analysis. Found after many of the sections of the book, this unique feature provides the student with a logical and orderly method to follow when applying the theory. The example problems are solved using this outlined method in order to clarify its numerical application. It is to be understood, however, that once the relevant principles have been mastered and enough confidence and judgment have been obtained, the student can then develop his or her own procedures for solving problems.

Photographs. Many photographs are used throughout the book to

enhance conceptual understanding and explain how the principles of mechanics of materials apply to real-world situations.

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Important Points. This feature provides a review or summary of the

most important concepts in a section and highlights the most significant points that should be realized when applying the theory to solve problems.

Example Problems. All the example problems are presented in a concise manner and in a style that is easy to understand.

Homework Problems. Apart from of the preliminary, fundamental, and conceptual problems, there are numerous standard problems in the book that depict realistic situations encountered in engineering practice.

It is hoped that this realism will both stimulate the student’s interest in the subject and provide a means for developing the skill to reduce any such problem from its physical description to a model or a symbolic representation to which principles may be applied. Throughout the book there is an approximate balance of problems using either SI or FPS units. Furthermore, in any set, an attempt has been made to arrange the problems in order of increasing difficulty. The answers to all but every fourth problem are listed in the back of the book. To alert the user to a problem without a reported answer, an asterisk (*) is placed before the problem number. Answers are reported to three significant figures, even though the data for material properties may be known with less accuracy. Although this might appear to be a poor practice, it is done simply to be consistent, and to allow the student a better chance to validate his or her solution.

Appendices. The appendices of the book provide a source for review and a listing of tabular data. Appendix A provides information on the centroid and the moment of inertia of an area. Appendices B and C list tabular data for structural shapes, and the deflection and slopes of various types of beams and shafts.

Accuracy Checking. The Tenth Edition has undergone a rigorous

Triple Accuracy Checking review. In addition to the author’s review of all art pieces and pages, the text was checked by the following individuals: • Scott Hendricks, Virginia Polytechnic University • Karim Nohra, University of South Florida • Kurt Norlin, Bittner Development Group • Kai Beng Yap, Engineering Consultant

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ACKNOWLEDGMENTS Over the years, this text has been shaped by the suggestions and comments of many of my colleagues in the teaching profession. Their encouragement and willingness to provide constructive criticism are very much appreciated and it is hoped that they will accept this anonymous recognition. A note of thanks is given to the reviewers.

S. Apple, Arkansas Tech University A. Bazar, University of California, Fullerton M. Hughes, Auburn University R. Jackson, Auburn University E. Tezak, Alfred State College H. Zhao, Clemson University There are a few people that I feel deserve particular recognition. A longtime friend and associate, Kai Beng Yap, was of great help to me in preparing the problem solutions. A special note of thanks also goes to Kurt Norlin in this regard. During the production process I am thankful for the assistance of Rose Kernan, my production editor for many years, and to my wife, Conny, for her help in proofreading and typing, that was needed to prepare the manuscript for publication.

I would also like to thank all my students who have used the previous edition and have made comments to improve its contents; including all those in the teaching profession who have taken the time to e-mail me their comments, but in particular G. H. Nazari.

I would greatly appreciate hearing from you if at any time you have any comments or suggestions regarding the contents of this edition.

Russell Charles Hibbeler hibbeler@bellsouth.net

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your work...

your answer specific feedback

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RESOURCES FOR INSTRUCTORS • MasteringEngineering. This online Tutorial Homework program allows you to integrate dynamic homework with automatic grading and adaptive tutoring. MasteringEngineering allows you to easily track the performance of your entire class on an assignment-by-assignment basis, or the detailed work of an individual student.

• Instructor’s Solutions Manual. An instructor’s solutions manual was prepared by the author. The manual includes homework assignment lists and was also checked as part of the accuracy checking program. The Instructor Solutions Manual is available at www.pearsonhighered.com.

• Presentation Resources. All art from the text is available in PowerPoint slide and JPEG format. These files are available for download from the Instructor Resource Center at www.pearsonhighered.com. If you are in need of a login and password for this site, please contact your local Pearson representative.

• Video Solutions. Developed by Professor Edward Berger, Purdue University, video solutions located on the Pearson Engineering Portal offer step-by-step solution walkthroughs of representative homework problems from each section of the text. Make efficient use of class time and office hours by showing students the complete and concise problem solving approaches that they can access anytime and view at their own pace. The videos are designed to be a flexible resource to be used however each instructor and student prefers. A valuable tutorial resource, the videos are also helpful for student self-evaluation as students can pause the videos to check their understanding and work alongside the video.

Access the videos at pearsonhighered.com/engineering-resources/ and follow the links for the Mechanics of Materials text.

RESOURCES FOR STUDENTS • Mastering Engineering. Tutorial homework problems emulate the instrutor’s office-hour environment.

• Engineering Portal—The Pearson Engineering Portal, located at pearsonhighered.com/engineering-resources/includes opportunities for practice and review including: • Video Solutions—Complete, step-by-step solution walkthroughs of representative homework problems from each section of the text. Videos offer fully worked solutions that show every step of the representative homework problems—this helps students make vital connections between concepts.

C O NTE NTS 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7

2 2.1 2.2

3

Stress Chapter Objectives 3 Introduction 3 Equilibrium of a Deformable Body 4 Stress 22 Average Normal Stress in an Axially Loaded Bar 24 Average Shear Stress 32 Allowable Stress Design 46 Limit State Design 48

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 *4.8 *4.9

3.1 3.2 3.3 3.4 3.5 3.6 *3.7

Chapter Objectives 123 Saint-Venant’s Principle 123 Elastic Deformation of an Axially Loaded Member 125 Principle of Superposition 140 Statically Indeterminate Axially Loaded Members 140 The Force Method of Analysis for Axially Loaded Members 147 Thermal Stress 155 Stress Concentrations 162 Inelastic Axial Deformation 165 Residual Stress 167

69

5 3

123

69

Strain Chapter Objectives Deformation 69 Strain 70

Axial Load

Mechanical Properties of Materials Chapter Objectives 85 The Tension and Compression Test 85 The Stress–Strain Diagram 87 Stress–Strain Behavior of Ductile and Brittle Materials 91 Strain Energy 95 Poisson’s Ratio 106 The Shear Stress–Strain Diagram 108 Failure of Materials Due to Creep and Fatigue 111

183

Chapter Objectives 183 Torsional Deformation of a Circular Shaft 183 5.2 The Torsion Formula 186 5.3 Power Transmission 194 5.4 Angle of Twist 206 5.5 Statically Indeterminate Torque-Loaded Members 222 *5.6 Solid Noncircular Shafts 229 *5.7 Thin-Walled Tubes Having Closed Cross Sections 232 5.8 Stress Concentration 242 *5.9 Inelastic Torsion 245 *5.10 Residual Stress 247 5.1

85

Torsion

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6

Contents

Bending

263

Chapter Objectives 263 Shear and Moment Diagrams 263 Graphical Method for Constructing Shear and Moment Diagrams 270 6.3 Bending Deformation of a Straight Member 289 6.4 The Flexure Formula 293 6.5 Unsymmetric Bending 310 *6.6 Composite Beams 320 *6.7 Reinforced Concrete Beams 323 *6.8 Curved Beams 327 6.9 Stress Concentrations 334 *6.10 Inelastic Bending 344 6.1 6.2

9

Chapter Objectives 445 9.1 Plane-Stress Transformation 445 9.2 General Equations of Plane-Stress Transformation 450 9.3 Principal Stresses and Maximum In-Plane Shear Stress 453 9.4 Mohr’s Circle—Plane Stress 469 9.5 Absolute Maximum Shear Stress 481

10 7 7.1 7.2 7.3 7.4 *7.5

Transverse Shear

367

Chapter Objectives 367 Shear in Straight Members 367 The Shear Formula 368 Shear Flow in Built-Up Members 386 Shear Flow in Thin-Walled Members 395 Shear Center for Open Thin-Walled Members 400

8

Combined Loadings

8.1 8.2

Chapter Objectives 413 Thin-Walled Pressure Vessels 413 State of Stress Caused by Combined Loadings 420

413

Stress Transformation 445

Strain Transformation

493

Chapter Objectives 493 10.1 Plane Strain 493 10.2 General Equations of Plane-Strain Transformation 494 *10.3 Mohr’s Circle—Plane Strain 502 *10.4 Absolute Maximum Shear Strain 510 10.5 Strain Rosettes 512 10.6 Material Property Relationships 516 *10.7 Theories of Failure 528

11

Design of Beams and Shafts

Chapter Objectives 545 11.1 Basis for Beam Design 545 11.2 Prismatic Beam Design 548 *11.3 Fully Stressed Beams 562 *11.4 Shaft Design 566

545

C o n t e n t s

12

Deflection of Beams and Shafts

577

Chapter Objectives 577 12.1 The Elastic Curve 577 12.2 Slope and Displacement by Integration 581 *12.3 Discontinuity Functions 599 *12.4 Slope and Displacement by the Moment-Area Method 611 12.5 Method of Superposition 626 12.6 Statically Indeterminate Beams and Shafts 634 12.7 Statically Indeterminate Beams and Shafts—Method of Integration 635 *12.8 Statically Indeterminate Beams and Shafts—Moment-Area Method 640 12.9 Statically Indeterminate Beams and Shafts—Method of Superposition 646

13

Buckling of Columns

Chapter Objectives 665 13.1 Critical Load 665 13.2 Ideal Column with Pin Supports 668 13.3 Columns Having Various Types of Supports 674 *13.4 The Secant Formula 686 *13.5 Inelastic Buckling 692 *13.6 Design of Columns for Concentric Loading 700 *13.7 Design of Columns for Eccentric Loading 710

665

14

Energy Methods

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723

Chapter Objectives 723 14.1 External Work and Strain Energy 723 14.2 Elastic Strain Energy for Various Types of Loading 728 14.3 Conservation of Energy 741 14.4 Impact Loading 748 *14.5 Principle of Virtual Work 759 *14.6 Method of Virtual Forces Applied to Trusses 762 *14.7 Method of Virtual Forces Applied to Beams 770 *14.8 Castigliano’s Theorem 779 *14.9 Castigliano’s Theorem Applied to Trusses 781 *14.10 Castigliano’s Theorem Applied to Beams 784

Appendix A B C

Geometric Properties of an Area 792 Geometric Properties of Structural Shapes 806 Slopes and Deflections of Beams 814

Solutions and Answers for Preliminary Problems 816 Fundamental Problems Partial Solutions and Answers 826 Selected Answers Index

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MECHANICS OF MATERIALS

CHAPTER

1

(© alexskopje/Fotolia) The bolts used for the connections of this steel framework are subjected to stress.

In this chapter we will discuss how engineers design these connections and their fasteners.

STRESS

CHAPTER OBJECTIVES n

In this chapter we will review some of the important principles of statics and show how they are used to determine the internal resultant loadings in a body. Afterwards the concepts of normal and shear stress will be introduced, and specific applications of the analysis and design of members subjected to an axial load or direct shear will be discussed.

1.1

INTRODUCTION

Mechanics of materials is a branch of mechanics that studies the internal effects of stress and strain in a solid body. Stress is associated with the strength of the material from which the body is made, while strain is a measure of the deformation of the body. A thorough understanding of the fundamentals of this subject is of vital importance for the design of any machine or structure, because many of the formulas and rules of design cited in engineering codes are based upon the principles of this subject.

3

4

C h a p t e r 1 S t r e ss

Historical Development.

1

The origin of mechanics of materials dates back to the beginning of the seventeenth century, when Galileo Galilei performed experiments to study the effects of loads on rods and beams made of various materials. However, it was not until the beginning of the nineteenth century when experimental methods for testing materials were vastly improved. At that time many experimental and theoretical studies in this subject were undertaken, primarily in France, by such notables as Saint-Venant, Poisson, Lamé, and Navier.

Through the years, after many fundamental problems had been solved, it became necessary to use advanced mathematical and computer techniques to solve more complex problems. As a result, mechanics of materials has expanded into other areas of mechanics, such as the theory of elasticity and the theory of plasticity.

1.2 EQUILIBRIUM OF A DEFORMABLE BODY Since statics plays an important role in both the development and application of mechanics of materials, it is very important to have a good grasp of its fundamentals. For this reason we will now review some of the main principles of statics that will be used throughout the text.

Loads. A body can be subjected to both surface loads and body forces. Surface loads that act on a small area of contact are reported by concentrated forces, while distributed loadings act over a larger surface area of the body. When the loading is coplanar, as in Fig. 1–1a, then a resultant force FR of a distributed loading is equal to the area under the distributed loading diagram, and this resultant acts through the geometric center or centroid of this area.

700 N

FR 400 N 200 N/m

B

A 1m

1m

1m (a) Fig. 1–1

1.5 m

1.2 Equilibrium of a Deformable Body

A body force is developed when one body exerts a force on another body without direct physical contact between the bodies. Examples include the effects caused by the earth’s gravitation or its electromagnetic field. Although these forces affect all the particles composing the body, they are normally represented by a single concentrated force acting on the body. In the case of gravitation, this force is called the weight W of the body and acts through the body’s center of gravity.

1

Support Reactions. For bodies subjected to coplanar force systems, the supports most commonly encountered are shown in Table 1–1. As a general rule, if the support prevents translation in a given direction, then a force must be developed on the member in that direction.

Likewise, if rotation is prevented, a couple moment must be exerted on the member. For example, the roller support only prevents translation perpendicular or normal to the surface. Hence, the roller exerts a normal force F on the member at its point of contact. Since the member can freely rotate about the roller, a couple moment cannot be developed on the member.

Many machine elements are pin connected in order to enable free rotation at their connections. These supports exert a force on a member, but no moment.

TABLE 1–1 Type of connection

u

Reaction

F

Cable

Type of connection

Reaction Fy

u Fx One unknown: F

Two unknowns: Fx, Fy

External pin

Fy Fx

Roller

F One unknown: F

Internal pin

Two unknowns: Fx, Fy M

Fy

Fx F u Smooth support

One unknown: F

Three unknowns: Fx, Fy, M

Fixed support Fx

Fy

F Journal bearing

One unknown: F

Thrust bearing

5

Two unknowns: Fx, Fy

6

C h a p t e r 1 S t r e ss

Equations of Equilibrium. Equilibrium of a body requires both a balance of forces, to prevent the body from translating or having accelerated motion along a straight or curved path, and a balance of moments, to prevent the body from rotating. These conditions are expressed mathematically as the equations of equilibrium:

1

ΣF = 0 ΣMO = 0

In order to design the members of this building frame, it is first necessary to find the internal loadings at various points along their length.

(1–1)

Here, Σ F represents the sum of all the forces acting on the body, and Σ MO is the sum of the moments of all the forces about any point O either on or off the body.

If an x, y, z coordinate system is established with the origin at point O, the force and moment vectors can be resolved into components along each coordinate axis, and the above two equations can be written in scalar form as six equations, namely,

ΣFx = 0 ΣMx = 0

ΣFz = 0 ΣMz = 0

(1–2)

700 N

FR 400 N 200 N/m

Often in engineering practice the loading on a body can be represented as a system of coplanar forces in the x–y plane. In this case equilibrium of the body can be specified with only three scalar equilibrium equations, that is,

B

A 1m

ΣFy = 0 ΣMy = 0

1m

1m

1.5 m

(a)

ΣFx = 0 ΣFy = 0 ΣMO = 0 700 N

400 N

1m Ay

(1–3)

2m (b)

Fig. 1–1

1.5 m

Bx By

Successful application of the equations of equilibrium must include all the known and unknown forces that act on the body, and the best way to account for these loadings is to draw the body’s free-body diagram before applying the equations of equilibrium. For example, the free-body diagram of the beam in Fig. 1–1a is shown in Fig. 1–1b. Here each force is identified by its magnitude and direction, and the body’s dimensions are included in order to sum the moments of the forces.

1.2 Equilibrium of a Deformable Body MRO

F4

F3

7

FR O

section

F1

F1

F2 (a)

F1

F2 (b)

Fig. 1–2

Internal Resultant Loadings. In mechanics of materials, statics is primarily used to determine the resultant loadings that act within a body. This is done using the method of sections. For example, consider the body shown in Fig. 1–2a, which is held in equilibrium by the four external forces.* In order to obtain the internal loadings acting on a specific region within the body, it is necessary to pass an imaginary section or “cut” through the region where the internal loadings are to be determined. The two parts of the body are then separated, and a free-body diagram of one of the parts is drawn. When this is done, there will be a distribution of internal force acting on the “exposed” area of the section, Fig. 1–2b. These forces actually represent the effects of the material of the top section of the body acting on the bottom section.

Although the exact distribution of this internal loading may be unknown, its resultants FR and MRO, Fig. 1–2c, are determined by applying the equations of equilibrium to the segment shown in Fig. 1–2c. Here these loadings act at point O; however, this point is often chosen at the centroid of the sectioned area.

*The body’s weight is not shown, since it is assumed to be quite small, and therefore negligible compared with the other loads.

1

F2 (c)

8

C h a p t e r 1 S t r e ss Torsional Moment T

MRO

MRO Normal N Force

FR

FR

1 O

F1

Bending M Moment

F2 (c)

O V Shear Force

F1

F2 (d)

Fig. 1–2 (cont.)

Three Dimensions.

For later application of the formulas for mechanics of materials, we will consider the components of FR and MRO acting both normal and tangent to the sectioned area, Fig. 1–2d. Four different types of resultant loadings can then be defined as follows:

Normal force, N. This force acts perpendicular to the area. It is

developed whenever the external loads tend to push or pull on the two segments of the body.

Shear force, V. The shear force lies in the plane of the area, and it is the body to slide over one another.

Torsional moment or torque, T. This effect is developed when the

external loads tend to twist one segment of the body with respect to the other about an axis perpendicular to the area.

The weight of this sign and the wind loadings acting on it will cause normal and shear forces and bending and torsional moments in the supporting column.

Bending moment, M. The bending moment is caused by the external loads that tend to bend the body about an axis lying within the plane of the area.

Notice that graphical representation of a moment or torque is shown in three dimensions as a vector (arrow) with an associated curl around it. By the right-hand rule, the thumb gives the arrowhead sense of this vector and the fingers or curl indicate the tendency for rotation (twisting or bending).

1.2 Equilibrium of a Deformable Body F2

section

F3

y F2

O

F1

F4 (a)

F1

(b)

Fig. 1–3

Coplanar Loadings. If the body is subjected to a coplanar system of

forces, Fig. 1–3a, then only normal-force, shear-force, and bendingmoment components will exist at the section, Fig. 1–3b. If we use the x, y, z coordinate axes, as shown on the left segment, then N can be obtained by applying ΣFx = 0, and V can be obtained from ΣFy = 0. Finally, the bending moment MO can be determined by summing moments about point O (the z axis), ΣMO = 0, in order to eliminate the moments caused by the unknowns N and V.

I MPO RTAN T POINTS • Mechanics of materials is a study of the relationship between the external loads applied to a body and the stress and strain caused by the internal loads within the body.

• External forces can be applied to a body as distributed or concentrated surface loadings, or as body forces that act throughout the volume of the body.

• Linear distributed loadings produce a resultant force having a magnitude equal to the area under the load diagram, and having a location that passes through the centroid of this area.

• A support produces a force in a particular direction on its attached member if it prevents translation of the member in that direction, and it produces a couple moment on the member if it prevents rotation.

• The equations of equilibrium ΣF = 0 and ΣM = 0 must be satisfied in order to prevent a body from translating with accelerated motion and from rotating.

• The method of sections is used to determine the internal

resultant loadings acting on the surface of a sectioned body. In general, these resultants consist of a normal force, shear force, torsional moment, and bending moment.

Shear Force V MO Bending Moment x N Normal Force

9

1

10

C h a p t e r 1 S t r e ss

PRO C E DU RE F O R A NA LYSIS

1

The resultant internal loadings at a point located on the section of a body can be obtained using the method of sections. This requires the following steps.

Support Reactions.

• When the body is sectioned, decide which segment of the body

is to be considered. If the segment has a support or connection to another body, then before the body is sectioned, it will be necessary to determine the reactions acting on the chosen segment. To do this, draw the free-body diagram of the entire body and then apply the necessary equations of equilibrium to obtain these reactions.

Free-Body Diagram.

• Keep all external distributed loadings, couple moments, torques, and forces in their exact locations, before passing the section through the body at the point where the resultant internal loadings are to be determined.

• Draw a free-body diagram of one of the “cut” segments and

indicate the unknown resultants N, V, M, and T at the section.

These resultants are normally placed at the point representing the geometric center or centroid of the sectioned area.

• If the member is subjected to a coplanar system of forces, only N, V, and M act at the centroid.

• Establish the x, y, z coordinate axes with origin at the centroid and show the resultant internal loadings acting along the axes.

Equations of Equilibrium.

• Moments should be summed at the section, about each of the

coordinate axes where the resultants act. Doing this eliminates the unknown forces N and V and allows a direct solution for M and T.

• If the solution of the equilibrium equations yields a negative

value for a resultant, the directional sense of the resultant is opposite to that shown on the free-body diagram.

The following examples illustrate this procedure numerically and also provide a review of some of the important principles of statics.

11

1.2 Equilibrium of a Deformable Body

EXAMPLE

1.1

Determine the resultant internal loadings acting on the cross section at C of the cantilevered beam shown in Fig. 1–4a.

1 270 N/m

A

B C 3m

6m (a)

Fig. 1–4

SOLUTION Support Reactions. The support reactions at A do not have to be determined if segment CB is considered.

Free-Body Diagram. The free-body diagram of segment CB is shown in Fig. 1–4b. It is important to keep the distributed loading on the segment until after the section is made. Only then should this loading be replaced by a single resultant force. Notice that the intensity of the distributed loading at C is found by proportion, i.e., from Fig. 1–4a, w>6 m = (270 N>m)>9 m, w = 180 N>m. The magnitude of the resultant of the distributed load is equal to the area under the loading curve (triangle) and acts through the centroid of this area. Thus, F = 12(180 N>m)(6 m) = 540 N, which acts 13(6 m) = 2 m from C as shown in Fig. 1–4b.

540 N 180 N/m MC NC

C VC

B 2m

4m (b)

Equations of Equilibrium. Applying the equations of equilibrium we have + ΣFx = 0; S + c ΣFy = 0; a+ ΣMC = 0;

-NC = 0 NC = 0 VC - 540 N = 0 VC = 540 N -MC - 540 N(2 m) = 0 MC = -1080 N # m

Ans. Ans. Ans.

The negative sign indicates that MC acts in the opposite direction to that shown on the free-body diagram. Try solving this problem using segment AC, by first checking the support reactions at A, which are given in Fig. 1–4c.

135 N

540 N

90 N/m

180 N/m MC

1215 N

3645 N m

A

C

1m

1.5 m VC 0.5 m (c)

NC

12

C h a p t e r 1 S t r e

EXAMPLE

ss

1.2 The 500-kg engine is suspended from the crane boom in Fig. 1–5a.

Determine the resultant internal loadings acting on the cross section of the boom at point E.

D

1

1.5 m C A 1m

E 1m

B 1m

SOLUTION Support Reactions. We will consider segment AE of the boom, so we must first determine the pin reactions at A. Since member CD is a two-force member, it acts like a cable, and therefore exerts a force FCD having a known direction. The free-body diagram of the boom is shown in Fig. 1–5b. Applying the equations of equilibrium,

(a)

a+ ΣMA = 0; FCD 1 35 2 (2 m) - [500(9.81) N](3 m) = 0

FCD

FCD = 12 262.5 N

5

3

Ax

4

+ ΣF = 0; S x

A 2m

1m

Ax = 9810 N

Ay 500(9.81) N (b)

Ax - (12 262.5 N) 1 45 2 = 0

+ c ΣFy = 0;

-Ay + (12 262.5 N) 1 35 2 - 500(9.81) N = 0 Ay = 2452.5 N

Free-Body Diagram. in Fig. 1–5c.

The free-body diagram of segment AE is shown

Equations of Equilibrium.

ME

9810 N A

E 1m

2452.5 N

VE

+ ΣF = 0; N + 9810 N = 0 S x E

NE

NE = -9810 N = -9.81 kN + c ΣFy = 0;

Fig. 1–5

-VE - 2452.5 N = 0 VE = -2452.5 N = -2.45 kN

(c)

a+ ΣME = 0;

Ans.

Ans.

ME + (2452.5 N)( 1 m) = 0 ME = -2452.5 N # m = -2.45 kN # m

Ans.

13

1.2 Equilibrium of a Deformable Body

EXAMPLE

1.3

Determine the resultant internal loadings acting on the cross section at C of the beam shown in Fig. 1–6a.

900 lb 8 ft

300 lb/ft 2 ft

B

1

2 ft

2 ft C

Ax

30

A 6 ft

30

A

FBD

Ay

D (a)

(b)

Fig. 1–6

SOLUTION Support Reactions. Here we will consider segment BC, but first we must find the force components at pin A. The free-body diagram of the entire beam is shown in Fig. 1–6b. Since member BD is a two-force member, like member CD in Example 1.2, the force at B has a known direction, Fig. 1–6b. We have a+ ΣMA = 0; (900 lb)(2 ft) - (FBD sin 30°) 10 ft = 0

FBD = 360 lb

Free-Body Diagram. Using this result, the free-body diagram of segment BC is shown in Fig. 1–6c.

Equations of Equilibrium.

2 ft

+ ΣF = 0; N - (360 lb) cos 30° = 0 S x C NC = 312 lb

Ans.

+ c ΣFy = 0; (360 lb) sin 30° - VC = 0 VC = 180 lb

Ans.

a+ ΣMC = 0; MC - (360 lb) sin 30°(2 ft) = 0 MC = 360 lb # ft

Ans.

B

MC 30

VC 360 lb (c)

NC

14

C h a p t e r 1 S t r e

EXAMPLE

ss

1.4 Determine the resultant internal loadings acting on the cross section at B of the pipe shown in Fig. 1–7a. End A is subjected to a vertical force of 50 N, a horizontal force of 30 N, and a couple moment of 70 N # m. Neglect the pipe’s mass.

1 C 0.75 m 0.5 m D

B

50 N 1.25 m 70 N m

The problem can be solved by considering segment AB, so we do not need to calculate the support reactions at C. Free-Body Diagram. The free-body diagram of segment AB is shown in Fig. 1–7b, where the x, y, z axes are established at B. The resultant force and moment components at the section are assumed to act in the positive coordinate directions and to pass through the centroid of the cross-sectional area at B.

30 N

A

SOLUTION

(a)

Equations of Equilibrium. Applying the six scalar equations of equilibrium, we have* z (FB)z

(FB)y (MB)z (MB)y (MB)x (FB)x

B

0.5 m y

50 N 1.25 m

x 70 N·m

30 N

A (b)

Fig. 1–7

ΣFx = 0; (FB)x = 0

Ans.

ΣFy = 0;

(FB)y + 30 N = 0

(FB)y = -30 N

Ans.

ΣFz = 0;

(FB)z - 50 N = 0

(FB)z = 50 N

Ans.

Σ(MB)x = 0;

(MB)x + 70 N # m - 50 N (0.5 m) = 0

(MB)x = -45 N # m Σ(MB)y = 0;

Ans.

(MB)y + 50 N (1.25 m) = 0

(MB)y = -62.5 N # m

Ans.

Σ(MB)z = 0;

Ans.

(MB)z + (30 N)(1.25) = 0 (MB)z = -37.5 N # m

NOTE: What do the negative signs for (FB)y, (MB)x, (MB)y, and (MB)z

indicate? The normal force NB = (FB)y = 30 N, whereas the shear force is VB = 2(0)2 + (50)2 = 50 N. Also, the torsional moment is TB = (MB)y = 62.5 N # m, and the bending moment is MB = 2(45)2 + (37.5)2 = 58.6 N # m. *The magnitude of each moment about the x, y, or z axis is equal to the magnitude of each force times the perpendicular distance from the axis to the line of action of the force.

The direction of each moment is determined using the right-hand rule, with positive moments (thumb) directed along the positive coordinate axes.

15

1.2 Equilibrium of a Deformable Body

It is suggested that you test yourself on the solutions to these examples, by covering them over and then trying to think about which equilibrium equations must be used and how they are applied in order to determine the unknowns. Then before solving any of the problems, build your skills by first trying to solve the Preliminary Problems, which actually require little or no calculations, and then do some of the Fundamental Problems given on the following pages. The solutions and answers to all these problems are given in the back of the book. Doing this throughout the book will help immensely in understanding how to apply the theory, and thereby develop 1 your problem-solving skills.

P R E LIMIN ARY PRO B L EM S P1–1. In each case, explain how to find the resultant internal loading acting on the cross section at point A. Draw all necessary free-body diagrams, and indicate the relevant equations of equilibrium. Do not calculate values. The lettered dimensions, angles, and loads are assumed to be known.

B A r

D

f O u 2a

a

P

(d)

A

B

u

C

P

a

C

P

a A

(a)

a

w

u

B

(e)

a

a

a

P

C

A

2a

a

a C

B

A

3a

(b)

a

a D

M B

A a/2

C a/2

a (c)

a

u

B

P P

(f)

16

C h a p t e r 1 S t r e ss

F UN DAMEN TAL PRO B L EM S 1

F1–1. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

F1–4. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

10 kN/m 10 kN 60 kN m

C

A

A 2m

3m

B 1m

B

C

1m

3m

2m

F1–4

F1–1 F1–5. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

F1–2. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

300 lb/ft

A

200 N/m

100 N/m

C 3 ft

A 1.5 m

3 ft

B

C

B 3 ft

F1–5

1.5 m

F1–2

F1–6. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

F1–3. Determine the resultant internal normal force, shear force, and bending moment at point C in the beam.

5 kN/m

C

A 20 kN/m

B

3m C B

A 2m

2m

F1–3

2m

D

2m

2m

F1–6

2m

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DANCE ON STILTS AT THE GIRLS’ UNYAGO, NIUCHI

Newala, too, suffers from the distance of its water-supply—at least the Newala of to-day does; there was once another Newala in a lovely valley at the foot of the plateau. I visited it and found scarcely a trace of houses, only a Christian cemetery, with the graves of several missionaries and their converts, remaining as a monument of its former glories. But the surroundings are wonderfully beautiful. A thick grove of splendid mango-trees closes in the weather-worn crosses and headstones; behind them, combining the useful and the agreeable, is a whole plantation of lemon-trees covered with ripe fruit; not the small African kind, but a much larger and also juicier imported variety, which drops into the hands of the passing traveller, without calling for any exertion on his part. Old Newala is now under the jurisdiction of the native pastor, Daudi, at Chingulungulu, who, as I am on very friendly terms with him, allows me, as a matter of course, the use of this lemon-grove during my stay at Newala.

FEET MUTILATED BY THE RAVAGES OF THE “JIGGER” (Sarcopsylla penetrans)

The water-supply of New Newala is in the bottom of the valley, some 1,600 feet lower down. The way is not only long and fatiguing, but the water, when we get it, is thoroughly bad. We are suffering not only from this, but from the fact that the arrangements at Newala are nothing short of luxurious. We have a separate kitchen—a hut built against the boma palisade on the right of the baraza, the interior of which is not visible from our usual position. Our two cooks were not long in finding this out, and they consequently do—or rather neglect to do—what they please. In any case they do not seem to be very particular about the boiling of our drinking-water—at least I can attribute to no other cause certain attacks of a dysenteric nature, from which both Knudsen and I have suffered for some time. If a man like Omari has to be left unwatched for a moment, he is capable of anything. Besides this complaint, we are inconvenienced by the state of our nails, which have become as hard as glass, and crack on the slightest provocation, and I have the additional infliction of pimples all over me. As if all this were not enough, we have also, for the last week been waging war against the jigger, who has found his Eldorado in the hot sand of the Makonde plateau. Our men are seen all day long—whenever their chronic colds and the dysentery likewise raging among them permit—occupied in removing this scourge of Africa from their feet and trying to prevent the disastrous consequences of its presence. It is quite common to see natives of this place with one or two toes missing; many have lost all their toes,

or even the whole front part of the foot, so that a well-formed leg ends in a shapeless stump. These ravages are caused by the female of Sarcopsylla penetrans, which bores its way under the skin and there develops an egg-sac the size of a pea. In all books on the subject, it is stated that one’s attention is called to the presence of this parasite by an intolerable itching. This agrees very well with my experience, so far as the softer parts of the sole, the spaces between and under the toes, and the side of the foot are concerned, but if the creature penetrates through the harder parts of the heel or ball of the foot, it may escape even the most careful search till it has reached maturity.

Then there is no time to be lost, if the horrible ulceration, of which we see cases by the dozen every day, is to be prevented. It is much easier, by the way, to discover the insect on the white skin of a European than on that of a native, on which the dark speck scarcely shows. The four or five jiggers which, in spite of the fact that I constantly wore high laced boots, chose my feet to settle in, were taken out for me by the all-accomplished Knudsen, after which I thought it advisable to wash out the cavities with corrosive sublimate. The natives have a different sort of disinfectant—they fill the hole with scraped roots. In a tiny Makua village on the slope of the plateau south of Newala, we saw an old woman who had filled all the spaces under her toe-nails with powdered roots by way of prophylactic treatment. What will be the result, if any, who can say?

The rest of the many trifling ills which trouble our existence are really more comic than serious. In the absence of anything else to smoke, Knudsen and I at last opened a box of cigars procured from the Indian store-keeper at Lindi, and tried them, with the most distressing results. Whether they contain opium or some other narcotic, neither of us can say, but after the tenth puff we were both “off,” three-quarters stupefied and unspeakably wretched. Slowly we recovered—and what happened next? Half-an-hour later we were once more smoking these poisonous concoctions—so insatiable is the craving for tobacco in the tropics.

Even my present attacks of fever scarcely deserve to be taken seriously. I have had no less than three here at Newala, all of which have run their course in an incredibly short time. In the early afternoon, I am busy with my old natives, asking questions and making notes. The strong midday coffee has stimulated my spirits to

an extraordinary degree, the brain is active and vigorous, and work progresses rapidly, while a pleasant warmth pervades the whole body. Suddenly this gives place to a violent chill, forcing me to put on my overcoat, though it is only half-past three and the afternoon sun is at its hottest. Now the brain no longer works with such acuteness and logical precision; more especially does it fail me in trying to establish the syntax of the difficult Makua language on which I have ventured, as if I had not enough to do without it. Under the circumstances it seems advisable to take my temperature, and I do so, to save trouble, without leaving my seat, and while going on with my work. On examination, I find it to be 101·48°. My tutors are abruptly dismissed and my bed set up in the baraza; a few minutes later I am in it and treating myself internally with hot water and lemon-juice.

Three hours later, the thermometer marks nearly 104°, and I make them carry me back into the tent, bed and all, as I am now perspiring heavily, and exposure to the cold wind just beginning to blow might mean a fatal chill. I lie still for a little while, and then find, to my great relief, that the temperature is not rising, but rather falling. This is about 7.30 p.m. At 8 p.m. I find, to my unbounded astonishment, that it has fallen below 98·6°, and I feel perfectly well. I read for an hour or two, and could very well enjoy a smoke, if I had the wherewithal—Indian cigars being out of the question.

Having no medical training, I am at a loss to account for this state of things. It is impossible that these transitory attacks of high fever should be malarial; it seems more probable that they are due to a kind of sunstroke. On consulting my note-book, I become more and more inclined to think this is the case, for these attacks regularly follow extreme fatigue and long exposure to strong sunshine. They at least have the advantage of being only short interruptions to my work, as on the following morning I am always quite fresh and fit.

My treasure of a cook is suffering from an enormous hydrocele which makes it difficult for him to get up, and Moritz is obliged to keep in the dark on account of his inflamed eyes. Knudsen’s cook, a raw boy from somewhere in the bush, knows still less of cooking than Omari; consequently Nils Knudsen himself has been promoted to the vacant post. Finding that we had come to the end of our supplies, he began by sending to Chingulungulu for the four sucking-pigs which we had

bought from Matola and temporarily left in his charge; and when they came up, neatly packed in a large crate, he callously slaughtered the biggest of them. The first joint we were thoughtless enough to entrust for roasting to Knudsen’s mshenzi cook, and it was consequently uneatable; but we made the rest of the animal into a jelly which we ate with great relish after weeks of underfeeding, consuming incredible helpings of it at both midday and evening meals. The only drawback is a certain want of variety in the tinned vegetables. Dr. Jäger, to whom the Geographical Commission entrusted the provisioning of the expeditions—mine as well as his own—because he had more time on his hands than the rest of us, seems to have laid in a huge stock of Teltow turnips,[46] an article of food which is all very well for occasional use, but which quickly palls when set before one every day; and we seem to have no other tins left. There is no help for it—we must put up with the turnips; but I am certain that, once I am home again, I shall not touch them for ten years to come.

Amid all these minor evils, which, after all, go to make up the genuine flavour of Africa, there is at least one cheering touch: Knudsen has, with the dexterity of a skilled mechanic, repaired my 9 × 12 cm. camera, at least so far that I can use it with a little care.

How, in the absence of finger-nails, he was able to accomplish such a ticklish piece of work, having no tool but a clumsy screw-driver for taking to pieces and putting together again the complicated mechanism of the instantaneous shutter, is still a mystery to me; but he did it successfully. The loss of his finger-nails shows him in a light contrasting curiously enough with the intelligence evinced by the above operation; though, after all, it is scarcely surprising after his ten years’ residence in the bush. One day, at Lindi, he had occasion to wash a dog, which must have been in need of very thorough cleansing, for the bottle handed to our friend for the purpose had an extremely strong smell. Having performed his task in the most conscientious manner, he perceived with some surprise that the dog did not appear much the better for it, and was further surprised by finding his own nails ulcerating away in the course of the next few days. “How was I to know that carbolic acid has to be diluted?” he mutters indignantly, from time to time, with a troubled gaze at his mutilated finger-tips.

Since we came to Newala we have been making excursions in all directions through the surrounding country, in accordance with old habit, and also because the akida Sefu did not get together the tribal elders from whom I wanted information so speedily as he had promised. There is, however, no harm done, as, even if seen only from the outside, the country and people are interesting enough.

The Makonde plateau is like a large rectangular table rounded off at the corners. Measured from the Indian Ocean to Newala, it is about seventy-five miles long, and between the Rovuma and the Lukuledi it averages fifty miles in breadth, so that its superficial area is about two-thirds of that of the kingdom of Saxony. The surface, however, is not level, but uniformly inclined from its south-western edge to the ocean. From the upper edge, on which Newala lies, the eye ranges for many miles east and north-east, without encountering any obstacle, over the Makonde bush. It is a green sea, from which here and there thick clouds of smoke rise, to show that it, too, is inhabited by men who carry on their tillage like so many other primitive peoples, by cutting down and burning the bush, and manuring with the ashes. Even in the radiant light of a tropical day such a fire is a grand sight.

Much less effective is the impression produced just now by the great western plain as seen from the edge of the plateau. As often as time permits, I stroll along this edge, sometimes in one direction, sometimes in another, in the hope of finding the air clear enough to let me enjoy the view; but I have always been disappointed.

Wherever one looks, clouds of smoke rise from the burning bush, and the air is full of smoke and vapour. It is a pity, for under more favourable circumstances the panorama of the whole country up to the distant Majeje hills must be truly magnificent. It is of little use taking photographs now, and an outline sketch gives a very poor idea of the scenery. In one of these excursions I went out of my way to make a personal attempt on the Makonde bush. The present edge of the plateau is the result of a far-reaching process of destruction through erosion and denudation. The Makonde strata are everywhere cut into by ravines, which, though short, are hundreds of yards in depth. In consequence of the loose stratification of these beds, not only are the walls of these ravines nearly vertical, but their upper end is closed by an equally steep escarpment, so that the

western edge of the Makonde plateau is hemmed in by a series of deep, basin-like valleys. In order to get from one side of such a ravine to the other, I cut my way through the bush with a dozen of my men.

It was a very open part, with more grass than scrub, but even so the short stretch of less than two hundred yards was very hard work; at the end of it the men’s calicoes were in rags and they themselves bleeding from hundreds of scratches, while even our strong khaki suits had not escaped scatheless.

NATIVE PATH THROUGH THE MAKONDE BUSH, NEAR MAHUTA

I see increasing reason to believe that the view formed some time back as to the origin of the Makonde bush is the correct one. I have no doubt that it is not a natural product, but the result of human occupation. Those parts of the high country where man—as a very slight amount of practice enables the eye to perceive at once—has not yet penetrated with axe and hoe, are still occupied by a splendid timber forest quite able to sustain a comparison with our mixed forests in Germany. But wherever man has once built his hut or tilled his field, this horrible bush springs up. Every phase of this process

may be seen in the course of a couple of hours’ walk along the main road. From the bush to right or left, one hears the sound of the axe— not from one spot only, but from several directions at once. A few steps further on, we can see what is taking place. The brush has been cut down and piled up in heaps to the height of a yard or more, between which the trunks of the large trees stand up like the last pillars of a magnificent ruined building. These, too, present a melancholy spectacle: the destructive Makonde have ringed them— cut a broad strip of bark all round to ensure their dying off—and also piled up pyramids of brush round them. Father and son, mother and son-in-law, are chopping away perseveringly in the background—too busy, almost, to look round at the white stranger, who usually excites so much interest. If you pass by the same place a week later, the piles of brushwood have disappeared and a thick layer of ashes has taken the place of the green forest. The large trees stretch their smouldering trunks and branches in dumb accusation to heaven—if they have not already fallen and been more or less reduced to ashes, perhaps only showing as a white stripe on the dark ground.

This work of destruction is carried out by the Makonde alike on the virgin forest and on the bush which has sprung up on sites already cultivated and deserted. In the second case they are saved the trouble of burning the large trees, these being entirely absent in the secondary bush.

After burning this piece of forest ground and loosening it with the hoe, the native sows his corn and plants his vegetables. All over the country, he goes in for bed-culture, which requires, and, in fact, receives, the most careful attention. Weeds are nowhere tolerated in the south of German East Africa. The crops may fail on the plains, where droughts are frequent, but never on the plateau with its abundant rains and heavy dews. Its fortunate inhabitants even have the satisfaction of seeing the proud Wayao and Wamakua working for them as labourers, driven by hunger to serve where they were accustomed to rule.

But the light, sandy soil is soon exhausted, and would yield no harvest the second year if cultivated twice running. This fact has been familiar to the native for ages; consequently he provides in time, and, while his crop is growing, prepares the next plot with axe and firebrand. Next year he plants this with his various crops and

lets the first piece lie fallow. For a short time it remains waste and desolate; then nature steps in to repair the destruction wrought by man; a thousand new growths spring out of the exhausted soil, and even the old stumps put forth fresh shoots. Next year the new growth is up to one’s knees, and in a few years more it is that terrible, impenetrable bush, which maintains its position till the black occupier of the land has made the round of all the available sites and come back to his starting point.

The Makonde are, body and soul, so to speak, one with this bush.

According to my Yao informants, indeed, their name means nothing else but “bush people.” Their own tradition says that they have been settled up here for a very long time, but to my surprise they laid great stress on an original immigration. Their old homes were in the south-east, near Mikindani and the mouth of the Rovuma, whence their peaceful forefathers were driven by the continual raids of the Sakalavas from Madagascar and the warlike Shirazis[47] of the coast, to take refuge on the almost inaccessible plateau. I have studied African ethnology for twenty years, but the fact that changes of population in this apparently quiet and peaceable corner of the earth could have been occasioned by outside enterprises taking place on the high seas, was completely new to me. It is, no doubt, however, correct.

The charming tribal legend of the Makonde—besides informing us of other interesting matters—explains why they have to live in the thickest of the bush and a long way from the edge of the plateau, instead of making their permanent homes beside the purling brooks and springs of the low country.

“The place where the tribe originated is Mahuta, on the southern side of the plateau towards the Rovuma, where of old time there was nothing but thick bush. Out of this bush came a man who never washed himself or shaved his head, and who ate and drank but little.

He went out and made a human figure from the wood of a tree growing in the open country, which he took home to his abode in the bush and there set it upright. In the night this image came to life and was a woman. The man and woman went down together to the Rovuma to wash themselves. Here the woman gave birth to a stillborn child. They left that place and passed over the high land into the valley of the Mbemkuru, where the woman had another child, which

was also born dead. Then they returned to the high bush country of Mahuta, where the third child was born, which lived and grew up. In course of time, the couple had many more children, and called themselves Wamatanda. These were the ancestral stock of the Makonde, also called Wamakonde,[48] i.e., aborigines. Their forefather, the man from the bush, gave his children the command to bury their dead upright, in memory of the mother of their race who was cut out of wood and awoke to life when standing upright. He also warned them against settling in the valleys and near large streams, for sickness and death dwelt there. They were to make it a rule to have their huts at least an hour’s walk from the nearest wateringplace; then their children would thrive and escape illness.”

The explanation of the name Makonde given by my informants is somewhat different from that contained in the above legend, which I extract from a little book (small, but packed with information), by Pater Adams, entitled Lindi und sein Hinterland. Otherwise, my results agree exactly with the statements of the legend. Washing?

Hapana—there is no such thing. Why should they do so? As it is, the supply of water scarcely suffices for cooking and drinking; other people do not wash, so why should the Makonde distinguish himself by such needless eccentricity? As for shaving the head, the short, woolly crop scarcely needs it,[49] so the second ancestral precept is likewise easy enough to follow. Beyond this, however, there is nothing ridiculous in the ancestor’s advice. I have obtained from various local artists a fairly large number of figures carved in wood, ranging from fifteen to twenty-three inches in height, and representing women belonging to the great group of the Mavia, Makonde, and Matambwe tribes. The carving is remarkably well done and renders the female type with great accuracy, especially the keloid ornamentation, to be described later on. As to the object and meaning of their works the sculptors either could or (more probably) would tell me nothing, and I was forced to content myself with the scanty information vouchsafed by one man, who said that the figures were merely intended to represent the nembo—the artificial deformations of pelele, ear-discs, and keloids. The legend recorded

by Pater Adams places these figures in a new light. They must surely be more than mere dolls; and we may even venture to assume that they are—though the majority of present-day Makonde are probably unaware of the fact—representations of the tribal ancestress.

The references in the legend to the descent from Mahuta to the Rovuma, and to a journey across the highlands into the Mbekuru valley, undoubtedly indicate the previous history of the tribe, the travels of the ancestral pair typifying the migrations of their descendants. The descent to the neighbouring Rovuma valley, with its extraordinary fertility and great abundance of game, is intelligible at a glance—but the crossing of the Lukuledi depression, the ascent to the Rondo Plateau and the descent to the Mbemkuru, also lie within the bounds of probability, for all these districts have exactly the same character as the extreme south. Now, however, comes a point of especial interest for our bacteriological age. The primitive Makonde did not enjoy their lives in the marshy river-valleys.

Disease raged among them, and many died. It was only after they had returned to their original home near Mahuta, that the health conditions of these people improved. We are very apt to think of the African as a stupid person whose ignorance of nature is only equalled by his fear of it, and who looks on all mishaps as caused by evil spirits and malignant natural powers. It is much more correct to assume in this case that the people very early learnt to distinguish districts infested with malaria from those where it is absent.

This knowledge is crystallized in the ancestral warning against settling in the valleys and near the great waters, the dwelling-places of disease and death. At the same time, for security against the hostile Mavia south of the Rovuma, it was enacted that every settlement must be not less than a certain distance from the southern edge of the plateau. Such in fact is their mode of life at the present day. It is not such a bad one, and certainly they are both safer and more comfortable than the Makua, the recent intruders from the south, who have made USUAL METHOD OF good their footing on the western edge of the CLOSING HUT-DOOR plateau, extending over a fairly wide belt of country. Neither Makua nor Makonde show in their dwellings anything of the size and comeliness of the Yao houses in the plain, especially at Masasi, Chingulungulu and Zuza’s. Jumbe Chauro, a Makonde hamlet not far from Newala, on the road to Mahuta, is the

most important settlement of the tribe I have yet seen, and has fairly spacious huts. But how slovenly is their construction compared with the palatial residences of the elephant-hunters living in the plain.

The roofs are still more untidy than in the general run of huts during the dry season, the walls show here and there the scanty beginnings or the lamentable remains of the mud plastering, and the interior is a veritable dog-kennel; dirt, dust and disorder everywhere. A few huts only show any attempt at division into rooms, and this consists merely of very roughly-made bamboo partitions. In one point alone have I noticed any indication of progress—in the method of fastening the door. Houses all over the south are secured in a simple but ingenious manner. The door consists of a set of stout pieces of wood or bamboo, tied with bark-string to two cross-pieces, and moving in two grooves round one of the door-posts, so as to open inwards. If the owner wishes to leave home, he takes two logs as thick as a man’s upper arm and about a yard long. One of these is placed obliquely against the middle of the door from the inside, so as to form an angle of from 60° to 75° with the ground. He then places the second piece horizontally across the first, pressing it downward with all his might.

It is kept in place by two strong posts planted in the ground a few inches inside the door. This fastening is absolutely safe, but of course cannot be applied to both doors at once, otherwise how could the owner leave or enter his house? I have not yet succeeded in finding out how the back door is fastened.

MAKONDE LOCK AND KEY AT JUMBE CHAURO

This is the general way of closing a house. The Makonde at Jumbe Chauro, however, have a much more complicated, solid and original one. Here, too, the door is as already described, except that there is only one post on the inside, standing by itself about six inches from one side of the doorway. Opposite this post is a hole in the wall just large enough to admit a man’s arm. The door is closed inside by a large wooden bolt passing through a hole in this post and pressing with its free end against the door. The other end has three holes into which fit three pegs running in vertical grooves inside the post. The door is opened with a wooden key about a foot long, somewhat curved and sloped off at the butt; the other end has three pegs corresponding to the holes, in the bolt, so that, when it is thrust through the hole in the wall and inserted into the rectangular opening in the post, the pegs can be lifted and the bolt drawn out.[50]

MODE OF INSERTING THE KEY

With no small pride first one householder and then a second showed me on the spot the action of this greatest invention of the Makonde Highlands. To both with an admiring exclamation of “Vizuri sana!” (“Very fine!”). I expressed the wish to take back these marvels with me to Ulaya, to show the Wazungu what clever fellows the Makonde are. Scarcely five minutes after my return to camp at Newala, the two men came up sweating under the weight of two heavy logs which they laid down at my feet, handing over at the same time the keys of the fallen fortress. Arguing, logically enough, that if the key was wanted, the lock would be wanted with it, they had taken their axes and chopped down the posts—as it never occurred to them to dig them out of the ground and so bring them intact. Thus I have

two badly damaged specimens, and the owners, instead of praise, come in for a blowing-up.

The Makua huts in the environs of Newala are especially miserable; their more than slovenly construction reminds one of the temporary erections of the Makua at Hatia’s, though the people here have not been concerned in a war. It must therefore be due to congenital idleness, or else to the absence of a powerful chief. Even the baraza at Mlipa’s, a short hour’s walk south-east of Newala, shares in this general neglect. While public buildings in this country are usually looked after more or less carefully, this is in evident danger of being blown over by the first strong easterly gale. The only attractive object in this whole district is the grave of the late chief Mlipa. I visited it in the morning, while the sun was still trying with partial success to break through the rolling mists, and the circular grove of tall euphorbias, which, with a broken pot, is all that marks the old king’s resting-place, impressed one with a touch of pathos.

Even my very materially-minded carriers seemed to feel something of the sort, for instead of their usual ribald songs, they chanted solemnly, as we marched on through the dense green of the Makonde bush:—