Mechanics of Civil Engineering Structures

By Laszlo P. Kollar and Gabriella Tarjan

Practicing engineers designing civil engineering structures, and advanced students of civil engineering, require foundational knowledge and advanced analytical and empirical tools. Mechanics in Civil Engineering Structures presents the material needed by practicing engineers engaged in the design of civil engineering structures, and students of civil engineering. The book covers the fundamental principles of mechanics needed to understand the responses of structures to different types of load and provides the analytical and empirical tools for design. The title presents the mechanics of relevant structural elements—including columns, beams, frames, plates and shells—and the use of mechanical models for assessing design code application. Eleven chapters cover topics including stresses and strains; elastic beams and columns; inelastic and composite beams and columns; temperature and other kinematic loads; energy principles; stability and second-order effects for beams and columns; basics of vibration; indeterminate elastic-plastic structures; plates and shells. This book is an invaluable guide for civil engineers needing foundational background and advanced analytical and empirical tools for structural design.

• Includes 110 fully worked-out examples of important problems and 130 practice problems with an interaction solution manual (http://hsz121.hsz.bme.hu/solutionmanual)
• Presents the foundational material and advanced theory and method needed by civil engineers for structural design
• Provides the methodological and analytical tools needed to design civil engineering structures
• Details the mechanics of salient structural elements including columns, beams, frames, plates and shells
• Details mechanical models for assessing the applicability of design codes

• Language: English
• Publisher: Elsevier Science
• Release date: Oct 20, 2020
• ISBN: 9780128203224

Laszlo P. Kollar

László P. Kollár is Professor and vice-Head of the Department of Structural Engineering at Budapest University of Technology and Economics, in Hungary. He received his PhD and Habilitation from the Budapest University of Technology and Economics, and his Doctor of Sciences from the Hungarian Academy of Sciences. He was elected to the Academy in 2007, and to the Academia European in 2012, and has been a Visiting Professor in the departments of Aeronautics and Astronautics, and of Civil Engineering, at Stanford University in the USA. He has published over 150 technical papers, as well as three books on the subject of structures, and has served widely as a consultant on the design of steel and reinforced concrete structures.

Book preview

Mechanics of Civil Engineering Structures

First Edition

László P. Kollár

Budapest University of Technology and Economics, Department of Structural Engineering, Budapest, Hungary

Gabriella Tarján

Budapest University of Technology and Economics, Department of Structural Engineering, Budapest, Hungary

Table of Contents

Cover image

Title page

Copyright

Preface

1: Introduction

Abstract

1.1: Design and analysis

1.2: Loads

1.3: Materials

1.4: Modeling

1.5: Structural elements and connections

2: Stresses and strains

Abstract

2.1: Stresses and strains in a plane

2.2: Spatial stresses and strains

2.3: Determination of displacements, strains and stresses

3: Elastic beams and columns

Abstract

3.1: In-plane beam models

3.2: Spatial beam models

3.3: Saint-Venant’s principle

4: Inelastic and composite beams and columns

Abstract

4.1: Composite cross sections made of linearly elastic materials

4.2: Cross sections made of inelastic materials

4.3: *Reinforced concrete cross sections

4.4: *Steel-concrete composites

5: Temperature and other kinematic loads

Abstract

5.1: Beams and columns

5.2: *Shrinkage

5.3: ⁎Prestress

5.4: *Creep

6: Energy principles

Abstract

6.1: Principle of stationary potential energy

6.2: Principle of virtual displacements

6.3: Reciprocal theorems

6.4: *Castigliano’s theorems

6.5: Numerical methods

6.6: *Principle of stationary complementary potential energy

7: Stability and second-order effects of beams and columns

Abstract

7.1: Buckling of discrete systems: Column consisting of rigid bars

7.2: In-plane flexural buckling of (continuous) columns

7.3: Effect of compression and imperfections on displacements and internal forces

7.4: Effect of plasticity on the displacements and on the buckling load

7.6: Analysis of multidegree of freedom discrete systems

7.7: Summation theorems: Application to buckling of multistory buildings

7.8: Energy method

7.9: ⁎Flexural-torsional buckling

7.10: ⁎Lateral-torsional buckling

7.11: ⁎Effect of shear deformations: Buckling of multistory frames

7.12: ⁎Nonlinearity due to change of contact surface

7.13: ⁎Postcritical behavior

7.14: ⁎Vessel stability

8: Basics of vibration

Abstract

8.1: Single degree of freedom systems

8.2: Multidegree of freedom discrete systems

8.3: Continuous systems (beams)

8.4: Summation theorems to calculate the eigenfrequencies

8.5: Effect of normal force and shear deformations on vibration of beams

8.6: Modal analysis

9: Statically indeterminate elastic-plastic structures

Abstract

9.1: Force method of elastic structures

9.2: Analysis of elastic-plastic structures

10: Plates

Abstract

10.1: Material and geometrical equations of thin plates

10.2: *Equilibrium equations of thin plates and the governing equations

10.3: Internal forces in rectangular plates

10.4: Orthotropic plates

10.5: ⁎Composite plates (laminated plates)

10.6: Buckling of plates

10.7: ⁎Local buckling of thin walled beams

10.8: ⁎Plastic analysis

10.9: *Large deflection of plates

10.10: Vibration of floors induced by human activities

10.11: *Ponding

10.12: Plates on elastic foundation

10.13: Plates with arbitrary shapes

10.14: ⁎Shear deformation of plates

11: Shells

Abstract

11.1: Load-bearing mechanism of thin shells

11.2: Elementary differential geometry (basic shell geometries)

11.3: Governing equation of equilibrium in membrane shells

11.4: Membrane theory of shells of revolution subjected to symmetric loads

11.5: General theory of membrane shells

11.6: Bending of shells

11.6.1: Cylindrical shells

11.6.2: Geckeler’s approximation for shells of revolution

11.6.3: Edge disturbance

11.6.4: Sources of bending in shells

11.7: Buckling of shells

Appendix

Elementary linear algebra

Linear equations

Eigenvalue problems

Differential equations

Second-order linear DEs

Higher-order linear DEs with constant coefficients

⁎Linear, homogeneous system of DEs

Practice problems

Section 2

Section 3

Section 4

Section 5

Section 6

Section 7

Section 8

Section 9

Section 10

Section 11

References

Index

Copyright

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Preface

László P. Kollár; Gabriella Tarján

Among topics in this book of special interest to civil engineers are buildings, industrial halls, sport arenas, water containers, and bridges made of a wide range of materials, including reinforced concrete, steel, masonry, fiber-reinforced composites, and plywood. Generally, such structures are designed to codes and standards built on engineering principles, experimental observations, and data, with the corresponding calculations performed by computer algorithms.

While these approaches simplify the engineer’s work, they require that their applications and limitations be known and taken into account. This requires understanding the expected behavior of the structure being designed and the analytical and numerical methods employed in generating the needed design parameters. The book combines physical reasoning, empirical observations, and theoretical analyses to examine, in a logical and unified manner, the mechanics of structures most frequently encountered in civil engineering.

The work is organized into 11 chapters. The first two chapters include an introduction to elasticity Chapters 6–9 are devoted to energy principles, stability, second-order effects, vibrations, and plastic analysis, and in Chapters 3, 4, 10, and 11, the aforementioned concepts are applied to basic structural elements: beams, columns, plates, and shells.

Supporting material, printed in small font, is arranged in two columns. Materials of lesser relevance are denoted by asterisk. For readers not already familiar with them, the basics of differential equations and matrix algebra are presented in the Appendix.

There are sample and practice problems throughout the book. The sample problems provide an outline of the problem and its solution method and illustrate significant information about the physical behavior of the structure. The practice problems offer opportunities for readers to apply their knowledge to problems frequently encountered in civil engineering. An interactive solution manual is developed (see https://www.elsevier.com/books/mechanics-of-civil-engineering-structures/kollar/978-0-12-820321-7), which contains the answers to the practice problems. The manual will provide the results and the necessary explanation to the suggested homework examples. Students can also check their own results and test the level of their understanding through interactive questions. The text, examples, and problems are intended to be useful to upper level engineering students and to those engaged in design.

The book grew out of lectures and accompanying tutorial sessions offered to MSc students at the Budapest University of Technology and Economics. The main text was written by László P. Kollár and the sample and practice problems and the interactive solution manual by Gabriella Tarján with the assistance of the first author.

We acknowledge and thank our teachers and mentors in both our professional and personal lives: the first author's father, Lajos Kollár, our colleagues László Dunai, Zsolt Gáspár, and István Hegedűs at the Budapest University of Technology and Economics and George Springer and Helmut Krawinkler at Stanford University. We owe special thanks to Professor Zsolt Gáspár for reviewing the entire manuscript. Proofreading the manuscript was in the competent hands of Dr. Lili Eszter Hlavicka-Laczák. We obtained valuable remarks from Zsuzsa B. Pap, Orsolya Gáspár, and Dr. Róbert Németh. Flóra Kollár improved the English of the text. We are indebted to our students and teaching assistants, whose feedback helped us in the selection of material and, importantly, showed us where clarifications were needed. Finally, but by no means least, we express our appreciation to our families who endured many long hours of us staring at our computer screens.

January, 2020, Budapest

There is a site for the book that (in addition to the 110 fully worked-out examples in the book) provides access to 130 important practice problems with an interaction solution manual. The website can be found in the book's page at Elsevier.com.http://hsz121.hsz.bme.hu/solutionmanual/.

1: Introduction

Abstract

The basic elements of design and analysis of engineering structures are summarized in this Chapter. The possible modeling, which will be referred to several times in the following chapters of the book is discussed in the third subsection.

Keywords

Civil engineering; Structural materials; Structural engineers; Loads; Modelling of structures; Superposition; Nonlinear analysis

The basic elements of design and analysis of engineering structures are summarized in this Chapter. The possible modeling, which will be referred to several times in the following chapters of the book is discussed in the third subsection.

1.1: Design and analysis

Civil engineering structures like buildings, sport arenas, and bridges always satisfy the needs of the people they are constructed for: they are homes, locations of sport activities, and tools for crossing a river. Structural engineers are responsible for designing (and making) their load bearing structures with economy, safety and elegance. This creation is challenging, and it is often called the art of structural engineering. Note, however, that the inverse problem, the analysis and verification of structures subjected to different loads, is also a complicated task (Fig. 1.1). Neither the loads nor the materials are fully known, and there are uncertainties in the behavior of the structure. However, buildings must be designed so they remain safe during their working life.

Fig. 1.1 Design and analysis.

1.2: Loads

The most important loads are listed in Table 1.1. Their calculations are not the task of this book; these are specified in building codes, for example, in Eurocode 1 [7]. Loads may act permanently (dead load) or temporarily (most live loads). Although in design loads are often considered to be static, many of them have severe dynamic effects, which must be taken into account. An introduction to the vibration of structures will be given in Chapter 8; however, the dynamic effects of earthquakes and wind are beyond the scope of this book. Induced effects, such as change in length due to temperature change or shrinkage, are not actually loads, but may cause internal forces similar to the effects of mechanical loads.

Table 1.1

1.3: Materials

The most important structural materials are steel and concrete. Their typical stress-strain curves are given in Fig. 1.2. The tensile strength of concrete is about one-tenth of its compression strength; this is the reason that concrete is used mostly as reinforced concrete (RC), where the tensile force is carried by steel rebars. A typical moment curvature curve of a RC beam element is shown in Fig. 1.2c. It should also be noted that steel is about 10 times more rigid and 10 times stronger than concrete in compression. Creep plays a significant role in the deformation of concrete, when a concrete column is loaded in compression and it deforms instantaneously ɛe; the final deformation (several months after the loading of the structure) will be about 3ɛe, i.e., due to creep the final displacement will be threefold of the elastic displacement.

Fig. 1.2 Stress-strain curves of steel (a), concrete (b), and moment-curvature curve of reinforced concrete beams (c).

Both steel and RC show large deformations before failure (Fig. 1.2a and c); this is called ductility. Structures made of ductile materials are favorable over those of brittle materials (Fig. 1.3), their load resistances (for identical strength) are usually higher, and before collapse they show large deformations.

Fig. 1.3 Typical stress-strain curves of brittle and ductile materials.

In designing concrete, the possible brittleness in tension and the large deformations which occur over time must be taken into account, and for both steel and RC structures, plastic design, which takes into account ductility, is beneficial.

The typical elastic moduli, strengths, and specific weights of building materials are listed in Table 1.2.

•Wood has about the same strength as concrete; it has significant creep, and wood is brittle in tension.

•Glass is also brittle, and for high stress it shows significant creep.

•Brick (masonry) is brittle, and has negligible tensile strength; however, masonry in shear shows ductility.

•Fiber reinforced plastics (FRP), which are most commonly made of glass or carbon fibers (GFRP, CFRP), are also brittle materials. GFRP can have roughly the strength of steel, while CFRP can be significantly stronger.

Table 1.2

Fibrous materials, both wood and FRP, are highly orthotropic, which means that they show significantly higher stiffness and strength in the fiber direction than perpendicular to it.

Next to the last column of Table 1.2 the strength divided by the specific weight is given, normalized by that of steel. The values show that to obtain the same load bearing, the weight of concrete would be about four times of that of steel, which is why large span structures are usually made of steel and not concrete. In the last column very rough data are given on the price of the strength, i.e., on the price of a stocky column capable of carrying a given compression load, made of different materials, compared to that of steel. Note that the price varies with location and also changes with time. These numbers serve only as an orientation, to give an indication of cost. The data clearly show that although CFRP is beneficial due to its high strength-to-weight ratio, one must pay a price for it.

1.4: Modeling

To analyze a structure, we must first build a model. The possible steps of modeling (and the steps of analysis) are shown in Fig. 1.4.

Fig. 1.4 Steps of modeling.

A structure is given with its geometry and materials together with its loads (self-weight, furniture, jumping people, etc.).

The mechanical model includes the following: (i) structural idealization (simply supported beam, frame, etc.); (ii) load idealization (distributed load instead of moving people, horizontal triangular load to represent earthquakes, etc.); and (iii) material idealization, often called material law (e.g., Hooke’s law).

The mathematical model and its solution contain typically three (sets of) equations: (i) material equations; (ii) geometrical equations; and (iii) equilibrium equations. As a result of their solution we obtain reaction forces, internal forces (or stress resultants), displacements, stresses, and strains.

Requirements may be that a structure must always fulfil a particular role, the bridge carries the traffic, or a building serves the needs of its residents. For a structural engineer, there are usually two kinds of requirements: to carry the loads with proper reliability, and to satisfy the serviceability requirements (displacement limits, vibration control, crack width for concrete, etc.).

Mathematical modeling gives the relationship between the:

•loads,

•displacements,

•internal forces (or stresses, stress resultants), and

•deformations (strains).

As an example, a spring fixed at one end and loaded by force P at the other end is considered. The end-displacement is denoted by u, the internal force is the spring-force, denoted by N, and the deformation is the elongation of the spring, denoted by Δ. The three equations are given in the second column of Table 1.3. The material equation connects the deformation and the internal force, the geometrical equation the displacement and the deformation, while the equilibrium equation the load and the internal force. (Another SDOF example, shown in Fig. 1.8b is given in footnote a of Chapter 7, page 229.)

Table 1.3

a Derivation of the equilibrium equation: when Δx is small, and the distributed load is considered to be uniform, the equilibrium is: − σ(x) + Δxp + σ(x + Δx) = 0, which results in: p =  − (σ(x + Δx) − σ(x))/Δx. When Δx tends to zero, according to the definition of the derivative: p =  − σ′ (see Eq. 2.94).

When the structure is subjected to several forces and there are several independent displacements, the equations connect the load vector (p), the displacement vector (u), the vector of internal forces (N), and the deformation vector (Δ). The relationships are given by matrix equations. An example is shown in the Appendix for a two spring system (Example L.2 of Appanedix Linear Algebra, page 499). The equations for a truss are given in the third column of Table 1.3. There are two load-components at each nodal point, hence there are 12 elements in the load vector. Similarly there are 12 displacements, horizontal and vertical motions of each node. The internal forces are the bar forces, while the deformations are the elongations of the bars. Both Δ and N have 15 elements.

A continuum can be described by continuous functions. For example, in the case of a beam in bending, we have the load function: p(x) and the displacement function: v(x). The deformation function is the curvature: κ(x) and the internal force is the bending moment: M(x). For this case the relationships are given by (differential) operators. For the beam bending problem, these are given in the fourth column of Table 1.3. (Note that for the uniqueness of the solution, the boundary conditions must also be given.)

Finally, in the last column of Table 1.3 the equations of a bar in tension are given. In this case the internal force is the normal stress (σ) and the deformation is the axial strain (ɛ), which are connected by Hooke’s law, E is the modulus of elasticity, and p is the axial load per unit volume.

In the following, regardless of the problem (beam, plate, shell, truss, etc.), the unknowns will be called displacements, internal forces, and deformations.

In structural analyses the above three sets of equations must be presented and solved (Fig. 1.5). The most common way of solution is that the deformations and internal forces are eliminated, and we obtain one equation which gives the relationship between the loads and the displacements, as shown in the last row of Table 1.3 (P = N → P = kΔ → P = ku) and by a thick arrow in Fig. 1.5.

Fig. 1.5 Relationships between loads, internal forces, deformations, and displacements.

The three sets of equations can be written with the following unified notations:

   (1.1)

Here we use the following notations:

will be discussed in Section 6.1. Eliminating the internal forces and deformations gives:

   (1.2)

There are structures—called statically determinate structures—where the internal forces (σ) can be directly calculated from the equilibrium equations. In this case the three sets of equations can be solved one after the other. We will see that certain shell structures can be modeled by neglecting the bending stiffness, and then the internal forces can be calculated from the equilibrium equations directly. This is the membrane theory of shells (Section 11.5).

When all three equations (Eq. 1.1) are linear then the analysis of the structure is linear as well, which has the very important consequence that superposition can be used: the effects (stresses, displacements) of two loads can be calculated independently and then they can be added together. This is illustrated in Fig. 1.6.

Fig. 1.6 Superposition in case of linear analysis: the effects of two loads can be calculated independently, and then added together.

There are three possible sources of nonlinearity due to the three equations.a

Material nonlinearity (Table 1.4). For small strains, most of the materials (e.g., steel) behave in a linearly elastic manner; however, for larger strains the relationship between stress and strain is nonlinear (Fig. 1.2a). Its simplest model is an elastic-plastic stress-strain curve (Fig. 1.7a). Another source of material nonlinearity is that some materials behave differently under tension and compression, and in the modeling tensile strength is often neglected (Fig. 1.7b).

Table 1.4

a In theory, we may apply large displacements, with equilibrium equations written on the original geometry; however it does not make sense (the geometry changes considerably), thus this combination is not shown in the table.

Fig. 1.7 Material models which show nonlinearity due to plasticity (a) and neglected tensile strength (b).

The equilibrium equations may be written on the original (undeformed) geometry, which results in a linear internal force-load relationship. When the change in geometry is taken into account (which is called second-order analysis), the equilibrium equations are nonlinear (Chapter 7).

Finally, the relationship between the displacements and the strains is linear when the displacements are small; for higher displacements the relationship is nonlinear. (See footnotes i of Chapter 2 and a of Chapter 7 on pages 25 and 229.)

These last two sources of nonlinearity are called geometrical nonlinearity (Table 1.4), which may affect the behavior considerably. For example, in the structure shown in Fig. 1.8 the bar forces depend on the inclinations of the bars: the smaller the angle (with respect to the horizontal line), the higher the forces. Higher forces mean higher strains and higher displacements, which lead to nonlinear behavior. When the equilibrium equations are written on the original geometry and the displacements are small, the analysis is called first-order analysis.b

Fig. 1.8 Effect of change in geometry.

The effect of geometrical nonlinearity is especially important for compressed structures (Fig. 1.8a), where the change in geometry results in the reduction of the stiffness of the structure, and in the increase of stresses. These effects will be discussed in Chapter 7. The change in geometry may also cause the loss of stability of the structure.

1.5: Structural elements and connections

In the analysis of complex structures, engineers usually consider simpler structural forms and elements: frames, trusses, plates, shells, walls, arches, beams, and columns, and their seizing is often based on their cross sections (Fig. 1.9). Engineers must decide whether the structure as a whole, the structural elements, or the cross sections are considered in the analysis.

Fig. 1.9 Structures, basic structural forms and elements, and cross sections.

A relevant part of design is the analysis of connections; the beams are resting on beams or columns, columns are attached to the foundations, and the moment-resistant beam-column connections of frames must be properly designed. Although these questions are important, designs of welded and bolted steel joints as well as monolith or prefabricated RC connections are challenging, and many structural failures occur at the connections; their analysis is out of the scope of this book.

References☆

[7] Eurocode. EC 0: Basis of structural design (EN 1990); EC 1: Actions on structures (EN 1991); EC 2: Design of concrete structures (EN 1992); EC 3: Design of steel structures (EN 1993); EC 4: Design of composite steel and concrete structures (EN 1994); EC 5: Design of timber structures (EN 1995); EC 6: Design of masonry structures (EN 1996); EC 7: Geotechnical design (EN 1997); EC 8: Design of structures for earthquake resistance (EN 1998); EC 9: Design of aluminium structures (EN 1999). European Committee for Standardization (CEN); 2010.

---

a There is an additional source of nonlinearity due to the change in the contact surface (Section 7.12).

b Sometimes neglecting of geometrical nonlinearity is called the assumptions of small displacements. In the case of buckling or bifurcation, however (see Chapter 7), even a very small change in geometry may change the behavior considerably. When we are away from the buckling geometry (e.g., the structure shown in Fig. 1.8), a direct consequence of the assumption of small displacements is that the equilibrium equations are linear.

☆ To view the full reference list for the book, click here

2: Stresses and strains

Abstract

In this chapter, stresses and strains are discussed in both 2-D and 3-D, and then a few solutions will be shown under plane stress (or plane strain) condition, which have practical importance.

Keywords

Material equations; Constitutive equations; Airy stress function; Failure criterion; Yield criterion; Plane stress; Plane strain; Stress concentration

In this chapter, stresses and strains will be discussed in both 2-D and 3-D, and then a few solutions will be shown under plane stress (or plane strain) condition, which have practical importance.

2.1: Stresses and strains in a plane

2.1.1: Stresses, their transformation, and principal stresses

Let us consider a pressure vessel subjected to internal pressure (Fig. 2.1). Due to the pressure, internal forces arise among the particles of the wall. We cut out a small rectangular element from the wall, where the edges are parallel to the axial and the hoop direction (Fig. 2.1a), and to ensure equilibrium the forces among the particles are replaced by distributed loads (dimension N/m²). These loads are called stresses (Fig. 2.1a)a. We denote the hoop stress by σ1 and the axial stress by σ2. It can be shown that the hoop stress is twice the axial stressb. The stress is defined always at a cut: in Fig. 2.1a the stresses are given along two perpendicular cuts, one is parallel to the axis of the cylinder. Let us make a cut now, which is in an arbitrary direction. The direction of the cut is denoted by y and its normal by x, the angle of which is α from the axis of the pressure vessel. The forces among the particles are replaced by a distributed load (Fig. 2.1c), its specific value is denoted by ρx (Fig. 2.1d). Its normal and tangential components are the normal stress and the shear stress, which are denoted by σx and τxy, respectively (Fig. 2.1e). The signs of the stresses are defined in the following way: normal stress is positive for tension. A face is positive, if one of the coordinate axes (in Fig. 2.1e: x) points outward of the face. On a positive face the shear stress is positive in the positive coordinate direction (in Fig. 2.1e: y).

Fig. 2.1 Stresses in a pressure vessel.

If σ1 and σ2 are known, stresses σx and τxy can be calculated unambiguously from the equilibrium of a triangular element (Fig. 2.2b). Let the length of the hypotenuse be Δs (Fig. 2.2c). The equilibrium equation in the x direction is as follows:

   (2.1)

where h is the wall thickness. Equilibrium equation in the y direction is as follows:

   (2.2)

Fig. 2.2 Stresses in a triangular element (shear stress is zero along the legs).

From these equations, we obtain the following:

   (2.3)

   (2.4)

We may observe (Eq. 2.4) that the following equality holdsc

   (2.5)

As a consequence the shear stresses on the two sides of a rectangular element have identical intensity (Fig. 2.3a). (Observe the sign of the shear stress: the two upper faces are positive since y and x point outward of the faces, and the positive shear stresses are in the x and y directions.) The stress in the y direction, σy, is obtained from Eq. (2.3) by replacing α with α + 90°.

Fig. 2.3 Stresses in a rectangular and in a triangular element.

Now, we assume that the stresses are known perpendicular to the x and y cuts (Fig. 2.3a) and we wish to determine the stresses if the rectangle is rotated by β and the new coordinate system is denoted by x′ and y′. β is positive when x is rotated toward y (Fig. 2.3b). The equilibrium of the triangle shown in Fig. 2.3c gives

   (2.6)

   (2.7)

σy′ (shown also in Fig. 2.3b) can be obtained from Eq. (2.6) by replacing β with β + 90°:

   (2.8)

Eqs. (2.6)–(2.8) can be given also in matrix form:

   (2.9)

where matrix Tσ (in squared brackets) is the transformation matrix of stresses. We obtained that if the stresses are known at two perpendicular cuts (two normal stresses and one shear stress), the stresses can be calculated unambiguously at an arbitrary direction (Example 2.1).

Example 2.1

Stresses of a butt-welded joint

Steel plate given in Fig. (a) is joined by an inclined butt weld. The plate is subjected to tensile stresses in both x and y directions, σx = 54 N/mm² and σy = 38 N/mm². Determine the normal and shear stresses of the butt weld.

Solution. Stresses of the butt weld are referred to the weld throat plane shown in Fig. (b). Stresses of the weld are obtained by transforming the stresses of the plate into the x ′ − y ′ − z′ coordinate system attached to the weld throat plane (β = − atan3/4 = − 36.87°). Transformation matrix of the stresses is (cosβ = 4/5, sinβ = − 3/5):

Normal and shear stresses of the weld are determined by Eq. (2.9):

The question arises in which direction the normal and the shear stress will be maximum. By differentiating Eq. (2.6) with respect to β, we may observe that the derivative of σx′ is equal to 2τxy′ (Eq. 2.7). Since at an extreme value of a function its derivative is zero, the maximum and minimum normal stress occur when τxy′ = 0. This equality gives (Eq. 2.7)

   (2.10)

It has two solutions between 0° and 180°, which are perpendicular to each other. The corresponding (minimum and maximum) stresses are called principal stresses, and they are denoted by subscripts 1 and 2. Introducing βo into Eq. (2.6), we obtain

   (2.11)

For an arbitrary β inequality σ2 ≤ σ(β) ≤ σ1 holds. It can be shown that the maximum shear stress occurs at βo + 45°, and its value is

   (2.12)

(For example, for the pressure vessel [Fig. 2.2] the principal stresses are in the axial and in the hoop direction, while the maximum shear stress is in the 45° direction.)

From Eqs. (2.11), (2.12), we have

   (2.13)

Pure shear. We investigate the case when the normal stresses are zero in the x-y coordinate system (σx = σy = 0), while the shear stress is not zero τxy ≠ 0 (Fig. 2.4a).

Fig. 2.4 Pure shear and the equivalent compression-tension.

This stress state is called pure shear. (With a good approximation, this is the case at the middle of the web of an I-beam subjected to transverse loads.) The directions of the principal stresses are obtained from Eq. (2.10), the principal stresses from Eq. (2.11) (βo = 45°), and the maximum shear stress from Eq. (2.12):

   (2.14)

It means that pure shear is equivalent to the sum of a pure compression and a pure tension in the ± 45° directions (Fig. 2.4b).

In-plane hydrostatic stress state. Now, we investigate the special case (Fig. 2.5), when the shear stress is zero and the normal stresses are identical (σx = σy = σ, τxy = 0). For this case Eq. (2.11) simplifies to

   (2.15)

Fig. 2.5 In-plane hydrostatic stress state.

Eq. (2.10) contains a fraction of zero/zero, and hence βo cannot be determined. This stress state is called in-plane hydrostatic stress state. In this case the normal stress is identical in any direction of cut, and the shear stress is zero. We can say that every direction is principal direction.

For the sake of visualization, Eqs. (2.6), (2.8) are shown in the polar (σx′(β) and σy′(β)) coordinate system (Fig. 2.6). If axis x is rotated by 90°, it will overlap with the y axis, and hence σx′(β + 90°) = σy′(β).

Fig. 2.6 Transformation of normal stresses ( σ x ′ and σ y ′) in polar coordinate systems.

Mohr-circle

To visualize the stresses the Mohr circle is often applied (Fig. 2.7). The direction of the first principal stress is denoted by βo (Fig. 2.7b). Let us define β′ as

   (2.16)

Fig. 2.7 Illustration of stresses with Mohr circle.

Eq. (2.9) results in the following expression (with τxy = 0):

   (2.17)

which can be rearranged as

   (2.18)

This is the equation of a circle in the σx′, τxy′ coordinate system:

   (2.19)

The origin and the radius are

   (2.20)

   (2.21)

The circle is given in Fig. 2.7. We can use the Mohr circle in the following way:

Drawing of the Mohr circle. Two points of the Mohr circle are given by (σx, τxy) and (σy,− τxy). Connecting them, we obtain the diameter, and then the circle can be drawn (Fig. 2.7d).

Principal stresses and principal directions. The angle between the diameter x–y and the horizontal axis is equal to twice the angle of principal directions. The intersections of the circle and the horizontal axis give the principal stresses. The radius of the circle is equal to the maximum shear stress.

Stresses in an arbitrary direction. We wish to know the stresses in the direction that has an angle β′ from the first principal direction (Fig. 2.7c). We draw the diameter at angle 2β′. The coordinates on the circle give stresses σx′, σy′, and τxy′ (Fig. 2.7e).

Typical stress states. The advantage of using the Mohr circle is not the fact that it makes hand calculation simple, rather that it makes visualizing the stresses possible. This is illustrated for four cases in Fig. 2.8.

Fig. 2.8 The four typical stress states and their Mohr circles.

In case of pure tension, one of the principal stresses is equal to the tensile stress, while the other one is zero. We may observe that in every direction the normal stress will be tension (the full circle is on the right side of the axis τ).

In the case of pure compression, one of the principal stresses is equal to the compression stress, while the other one is zero, and in every direction the normal stress will be compression (the Mohr circle is on the left side of axis τ).

In the case of pure shear, the two principal stresses are equal and opposite, and their direction is ± 45°.

For in-plane hydrostatic stress state, the principal stresses are identical, and there is no special direction. The Mohr circle becomes a point.

2.1.2: Failure criteria

We distinguish between stress (σ) and strength (f), the latter one is the resistance of the material, which can be loaded up to σ ≤ f. The strength can be determined by unidirectional tests. The question arises how we can decide whether it can resist a stress combination when there are three stresses in the material (Fig. 2.9a).

Fig. 2.9 Stresses on a square element (a), maximal normal stresses (b), and maximal shear stresses (c).

Rankine failure criterion (or Coulomb failure criterion, theory of maximum normal stresses). It seems a reasonable decision that the maximum normal stresses (i.e., the principal stresses) are determined and we compare them to the strength of the material:

   (2.22)

where the vertical lines mean absolute value. This criterion was introduced by Rankine (~ 1850), and it is shown graphically in Fig. 2.10a. The Rankine failure criterion gives reasonable results for brittle materials.

Fig. 2.10 The Rankine (a), the Tresca (b), and the von Mises (c) failure (yield) criterion. When the calculated stresses σ 1 and σ 2 are within the curve, the material can carry the load.

Tresca failure criterion (theory of maximum shear stresses). We add a further condition to Eq. (2.22), which is an upper bound for the maximum shear stress. Assuming that the limit, that is, the shear strength, is half of the normal strength, we have

   (2.23)

This criterion was developed by Tresca (1868), and it is shown in Fig. 2.10b. (Taking Eq. (2.12) into account, the last expression in Eq. (2.23) can be given in the following form: | σ1 − σ2 | ≤ f.) This criterion can be used for ductile materials; hence, it is also called Tresca yield criterion.

Von Mises failure (or yield) criteriond (theory of maximum distortion strain energy). The aforementioned criterion has sharp corners; however, at least for ductile materials, smooth functions should be used. Quadratic criterions can satisfy this condition, where the polynomial of stresses is used up to the second degree. Its most common form is

   (2.24)

which coincides with the Tresca yield criterion at its six corner points. The graphical interpretation (an ellipse) is shown in Fig. 2.10c.e This condition can be used for ductile materials, typically this is used for steel (Examples 2.2 and 2.3).

Example 2.2

Web of a thin-walled beam

Consider a thin walled beam subjected to bending and shear. In the wall of the beam, axial normal stresses (σx) and shear stresses (τxy) arise, and normal stresses perpendicular to the axis are negligible (σy = 0) (Fig. a). The strength of the material is f, accordingly in case of uniaxial normal stress material failure occurs when σx = f. How does the ultimate normal stress change if the shear stress is τxy = 0.25f? Use the Tresca or von Mises failure criterion.

Solution. Principal stresses are calculated from Eq. (2.11), the maximum shear stress is given by Eq. (2.12):

Taking these expressions into consideration, the Tresca failure criterion (Eq. 2.23) can be written in the following form:

If the second inequality holds, then the first is satisfied automatically. Thus it is sufficient to consider only the second equation:

Substituting the principal stresses into the von Mises criterion (Eq. 2.24), the following expression is obtained:

The two yield criteria are represented graphically in Figs. (b) and (c).

Considering shear stress τxy = 0.25f σx becomes

Thus the presence of shear stress—according to both failure criteria—reduces the resistance by 13% and 10%, respectively. (As it is written earlier, in the case of steel material, von Mises criterion is generally used.)

Example 2.3

Stresses of transverse fillet-welded joint

The lap joint of a tie rod given in Fig. (a) is subjected to a tensile force, F = 120 kN. Determine the stresses in the throat plane and check the weld if its strength is f = 200 MPa using the von Mises criterion. Length and throat thickness of the weld are l = 322 mm and a = 4 mm (horizontal and vertical leg lengths are equal).

Solution. Stresses of a fillet weld are referred to the weld throat plane given in Fig. (b). As an approximation, uniform stress distribution is considered. From the tensile force, stress arises in the y direction, and its value referred to the throat plane is

Stress of the throat plane at point A is given in Fig. (c). Stress in y direction has two components, shear stress parallel to the throat plane and normal stress perpendicular to the throat plane (Fig. c):

(for the multiplier 3, see the previous example), which gives

Thus the fillet weld safely resists the given tensile force.

Note that the failure criterion for welding in general is not identical to the von Mises yield criterion, where the normal stresses parallel to the axis of the weld also must be taken into account, while for checking the weld it is neglected. (In this example the normal stress parallel to the axis of the weld is zero.)

2.1.3: Strains and their transformation

Materials subjected to stresses deform. For example, the pressure vessel shown in Fig. 2.1 elongates both in the hoop and in the longitudinal direction. First the strain is defined for a simple bar in tension. When the bar is loaded, its length will change by ΔL as shown in Fig. 2.11b. This change in length is the sum of the relative motions among the particles (molecules). We may assume that the change in length of the half bar will be the half of ΔL, or the L/k part of the bar elongates ΔL/k. It is convenient to introduce the specific elongation, which is called (normal) strain and denoted by ɛ:

   (2.25)

Fig. 2.11 Normal strain in a bar in tension.

When the elongation is not uniform, this expression gives the average strain. The strain at a point is obtained in such a way that a very short length L is chosen, where the strain can be considered to be uniform. More precisely the length L tends to zerof:

   (2.26)

Let us consider the pressure vessel again (Fig. 2.1). The wall elongates both in the hoop and in the axial direction. The normal strain in the direction of the x coordinate is defined in the following way. On the unloaded and undeformed structure, we mark a short straight line that is parallel to x, and its length is Δx (Fig. 2.12), and we measure its length after the deformations take place (Δx′). The normal strain in the x direction is denoted by ɛx and defined as

   (2.27)

Fig. 2.12 Normal strain in the x ( ɛ x ) and y ( ɛ y ) directions.

Similarly the normal strain in the y direction is (Fig. 2.12)

   (2.28)

Let us now mark two short straight lines, which are perpendicular to each other (Fig. 2.13). After the deformations occur, the angle between the lines also changes. This change is called angular (or shear) strain and denoted by γxy:

   (2.29)

Fig. 2.13 Illustration of angular strain ( γ xy ).

The aforementioned strains (ɛx, ɛy, γxy) are called engineering strains. They belong to the x,y coordinate system; for another (rotated) coordinate system, their values will be different. They are denoted by ɛx′, ɛy′, and γxy′ (Fig. 2.14). It will be shown that these strains can be unambiguously calculated from the previous ones as

   (2.30)

where matrix Tɛ (in squared brackets) is the transformation matrix of strains. We derive this matrix in two different ways.