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REVIEW OF BASICS IN STRUCTURAL ANALYSIS

FACULTY OF ENGINEERING Prepared by: Dr Dr.. Osama Ibrahim 2015-2016

1

CHAPTER NO CHAPTER NAME INTRODUCTION ONE

PAGE NO 3

TWO

EXTERNAL FORCES

14

THREE

INTERNAL FORCES TRUSS ANALYSIS

28

FIVE

ARCHS and CABLES ANALYSIS

49

SIX

DISPLACEMENTS & DEFLECTIONS OF STRUCTURES

65

SEVEN

ANALYSIS OF R.C BEAMS

100

EIGHT

STEEL TENSION MEMBERS

150

FOUR

2

38

CHAPTER (1) INTRODUCTION

3

Civil engineering is the oldest branch of engineering which is growing right from the Stone Age civilization. American society of Civil Engineering defines civil Engineering as the profession in which a knowledge of the mathematical and physical sciences gained by study, experience and practice is applied with judgment to develop ways to utilize economically the materials and forces of nature for the progressive well-being of man. In this course, scopes of the important field in civil engineering are discussed, structural engineering and construction technology, and how to analyse and design these structures.

1.1 SCOPE OF DIFFERENT FIELDS OF CIVIL ENGINEERING Civil Engineering may be divided into the following fields: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x)

Surveying Building Materials Construction Technology Structural Engineering Geotechnical Engineering Hydraulics Water Resources and Irrigation Engineering Transportation Engineering Environmental Engineering Architecture and Town planning.

1.2 STRUCTURAL ENGINEERING Construction is the major activity of civil engineering which is continuously improving. As land cost is going up there is demand for tall structures in urban areas while in rural areas need is for low constructions. One has to develop technology using locally available materials. Load acting on a structure is ultimately transferred to ground. In doing so, various components of the structure are subjected to internal stresses. For example, in a building, load acting on a slab is transferred by slab to ground through beams, columns and footings. Assessing the internal stresses in the components of a structure is known as structural analysis and finding the suitable size of the structural components is known as design of structure. The structure to be analysed and designed may be of masonry, R.C or steel as shown in figure (1). Upton considerable improvements were seen in classical analysis. With the advent of computers numerical methods emerged and analysis and design packages are becoming popular. A civil engineer has not only to give a safe structure but he has to give economical sections. To get economical section mathematical optimization techniques are used. Frequent earthquakes in the recent years have brought importance of analysis of the structures for earthquake forces. Designing earthquake resistant structures is attracting lot of researches. All these aspects fall under structural engineering field. Design of members and structures of reinforced concrete is a problem distinct from but closely related to analysis. Strictly speaking, it is almost impossible to exactly analyse a concrete structure, and to design exactly is no less difficult. Fortunately, we can make a few fundamental assumptions which make the design of reinforced concrete quite simple, if not easy.

4

Concrete Building

Concrete Tunnel

Steel Structure

Masonry Structure Fig.1 Different Types of Structures 5

A problem unique to the design of reinforced concrete structures is the need to detail each member throughout. Steel structures, in general, require only the detailed design of connections. For concrete structures, we must determine not only the area of longitudinal and lateral reinforcement required in each member, but also the way to best arrange and connect the reinforcement to insure acceptable structural performance. This procedure can be made reasonably simple, if not easy. In this course we will learn to understand the basic performance of concrete and steel as Structural materials, and the behaviour of reinforced concrete members and structures. The overall goal is to be able to design reinforced concrete structures that are: 1- Safe 2- Economical 3- Efficient Reinforced concrete is one of the principal building materials used in engineered structures because: 1- Low cost 2- Weathering and fire resistance 3- Good compressive strength 4- Formability All these criteria make concrete an attractive material for wide range of structural applications such as buildings, dams, reservoirs, tanks, etc.

1.3 DESIGN CODES and SPECIFICATIONS Buildings must be designed and constructed according to the provisions of a building code, which is a legal document containing requirements related to such things as structural safety, fire safety, plumbing, ventilation, and accessibility to the physically disabled. A building code has the force of law and is administered by a governmental entity such as a city, a county, or for some large metropolitan areas, a consolidated government. Building codes do not give design procedures, but specify the design requirements and constraints that must be satisfied. Of particular importance to the structural engineer is the prescription of minimum live loads for buildings. While the engineer is encouraged to investigate the actual loading conditions and attempt to determine realistic values, the structure must be able to support these specified minimum loads. Although some large cities write their own building codes, many municipalities will adopt a “model” building code and modify it to suit their particular needs. There are many examples of codes for structural design specifications as shown in figure (2)

Fig.2 Examples of Codes and Specifications for Structures 6

1.4 DESIGN PROCESS Design of any structure needs the following aspects to be covered:123456-

Requires a fundamental knowledge of material properties and mechanics. Requires knowledge of various types of structural forms and configurations Calculation of loads and load effects acting on the structure as a deflections. Knowledge of structural analysis to calculate design forces. Requires knowledge of designing structural members and connections Ability to evaluate designs and consider other options.

In this course, 1, 2, 3 and 4 will be covered

1.5 CLASSIFICATIONS OF STRUCTURES As an engineer, you must be able to classify structures according to their form and function and you must also be able to recognize various types of elements composing a structure. STRUCTURAL SYSTEM: - Composed of structural members joined together by structural connections. Each structural system may be composed of one or more of the four basic types of structures. The four basic types of structures are:1- Trusses 2- Cables and Arches 3- Frames 4- Surface structures 1.5.1 Trusses Trusses consists of slender members, arranged in a triangular pattern as shown in figure (3)

Fig.3 TRUSSES

7

1.5.2 Cables and Arches Cables and arch are structures used to span long distances as shown in figure (4).

Fig.4 Cables and Arches 1.5.3 Frames Frames are commonly used in building structures. Frames are composed of beams and columns that are connected together as shown in figure (5).

Fig.5 Frames 1.5.4 Surface structure Membrane, plate, or shell type with much less thickness as compared to its other dimensions. It is subjected to tension and compression only as shown in figure (6).

8

Fig.6 Surface Structures

1.6 CLASSIFICATION OF STRUCTURAL MEMEBERS There are five basic types of structural members. These are follows: 1- Tension members and tie rods. 2- Compression members or columns or struts. 3- Flexural members or beams 4- Members subjected to combined loading or beam-columns. Examples of these members can be shown in figure (7), (8)

Fig.7 Tension and compression members

9

Fig.8 Columns and Beams members

1.7 TYPES OF LOADS 1- Dead loads It consists of the weight of the various structural members and weight of any object that permanently attached to the structure as shown in figure (9).

10

Fig.9 Dead loads 2- Live loads It can vary both in their magnitude and location like bridge loads as shown in figure (10)

Fig.10 Live loads 3- Wind loads It is a lateral load subjected to the building as shown in figure (11).

Fig.11 Wind loads 11

4- Snow loads As shown in figure (12)

Fig.12 Snow loads 5- Earthquake Load It is a lateral load. It can be get in consider only in the earthquake regions such as Japan and USA. This load can be shown as figure (13).

Fig.13 Earthquake load 12

1.8 LOAD TRANSFER Loads can be transferred from one element to another as shown in figure (14) from slabs to beams to columns to footings.

Fig.14 Load Transfer through building

13

CHAPTER (2) EXTERNAL FORCES

14

2.1 IDEALIZED STRUCTURES Idealized structure is needed to the engineer to perform a practical force analysis of the whole frame and its member. This is the reason in this section to show different members, member connections and supports and there idealizations. If one know these models may compose idealized model of each real structure after all perform the analysis and design. 2.1.1 Idealized Beams

Fig.15 Idealized Beams 2.1.2 Idealized Frames

Fig.16 Idealized Frames 15

2.1.3 Idealized Trusses

Fig.17 Idealized Trusses 2.1.4 Idealized Loads There are two main loads can be idealized, concentrated loads and distributed loads as shown in figure (18)

Fig.18 Idealized loads 2.1.5 Idealized Connections There are six types of connections as follows showing each type with its idealization:1- Rigid (fixed) connections: This connection carry moment, shear and axial forces between different members. In addition, in this case all members including in such a connection have one and the same rotation and displacements – the nodal rotation and displacements. Typical rigid connections between members in metal and in reinforced concrete constructions and there idealized models are shown in figure (19).

Fig.19 Idealized Rigid connection 16

2- Hinged (pin) connections: This connection carry shear and axial forces but not moment between different members. Hinged connection allow to the jointed members to have different rotations but the same displacements. Typical hinged connections between members in metal and in reinforced concrete constructions and there idealized models are shown in figure (20)

Fig.20 Idealized hinged connection 3- Fixed Support: This support carry moment, shear and axial forces between different members. This kind of support doesn’t allow any displacements of the support point. So if the displacement along the x axis is u, the displacement along y axis is y and the rotation is called ϕ then we can say that: uA = 0; VA = 0 and ϕA = 0.

Fig.21 Idealized Fixed Support 4- Hinged (pin) support: This support carry shear and axial forces but not moment between different members. The hinged support allows rotation of the support point but the two displacement are equal zero or: uA = 0; VA = 0 and ϕA ≠ 0.

Fig.22 Idealized Hinged Support

17

5- Roller support: This support carry only shear forces between jointed members. The roller support allows rotation and one displacement of the support point: uA ≠ 0; VA = 0 and ϕA ≠ 0.

Fig.23 Idealized Roller Support 6- Spring supports: These supports are like the previous but with the difference that they are not ideally rigid but with some real stiffness. The spring has a stiffness constant c equals to the force caused by displacement d = 1.

Fig.24 Idealized Spring Support

Example No.1 Draw the idealization model of the following figures to do the mathematical calculation of analysis and design.

Solution

18

Solution

2.2 Equivalent system of Distributed Loads

a/2

w0a

w0 a 2a/3

w0a/2

w1a

(w2- w1)a/2

a/2

a/3

w0 a

w2

w1 a

2.3 Solving Reactions for Beams It is possible to determine the forces and stresses in beams by utilizing the equations of equilibrium, that is

19

For a two-dimensional beam, there are at most three equilibrium equations for each part, so that if there is a total n parts and r reactions, we have r = 3n

statistically determinate

r > 3n

statistically indeterminate

Example No.2 Classify each of the beams shown as statically determinate or statistically indeterminate

20

In this course, we will focus on determine structures only. To get reaction at any support, you should follow the following instructions: 1- Draw free body diagram (i.e. replace supports with forces and moments as applicable) 2- Replace any uniform distributed load (UDL) with a single force acting at the midpoint of the UDL. 3- Find the right hand reaction by taking moments about the point of action of the left hand reaction. 4- Find the left hand reaction by adding all the forces together for the x and y directions and putting these equal to zero.

Example No.3 Calculate the reactions for the following beam Solution -

Use the sum of the moments must be zero (Σ M = 0) equation to calculate the magnitude of the reactions Sign convention: clockwise moments positive (+ve), anti-clockwise moments negative (-ve). There are to unknowns (RL and RR) and one must be eliminated to be able to calculate the reaction. We select the rotation point at RR because reaction RR times distance is then zero. To find RL take the moment about RR (now there is only one unknown in our equation).

If this formula is used, you must be certain that RL is correctly calculated. At this stage, it is better to calculate RR as well

21

Example No.4 Determination of the reactions of a simple beam with a uniformly distributed load Solution

Example No.5 Calculate the reactions at the supports.

Solution -

For equilibrium (∑M=0) Sum of forces in X and Y directions = 0

∑Ma = 0 (clockwise positive) 30 *1 + (15*4)*(1+4/2) + 45*6 - Rb*8 = 0 Rb = 60 KN ∑ Fy = 0 (positive upwards) Ra – 30 – (15*4) – 45 – 60 = 0 Ra = 75 KN 22

Example No.6 Determine the reactions on the beam shown.

Example No.7 Determine the reactions on the beam shown. Assume A is a pin and the support at B is a roller (smooth surface).

23

Example No.8 The compound beam in figure below is fixed at A. Determine the reactions at A, B, and C. Assume that the connection at pin and C is a roller.

Solution

24

2.4 SOLVING REACTIONs FOR FRAMES Example No.9 Find the reactions for the shown figure

Solution

25

Example No.10 From the figure below, determine the horizontal and vertical components of reaction at the pin connections A, B, and C of the supporting gable frame.

Solution

26

CHAPTER (3) INTERNAL FORCES

27

3.1 INTRODUCTION Recall from mechanics of materials that the internal forces P (generic axial), V (shear) and M (moment) represent resultants of the stress distribution acting on the cross section of the beam. Internal Axial Force (P) ≡ equal in magnitude but opposite in direction to the algebraic sum (resultant) of the components in the direction parallel to the axis of the beam of all external loads and support reactions acting on either side of the section being considered. T ≡ Tension C ≡ Compression Internal Shear Force (V) ≡ equal in magnitude but opposite in direction to the algebraic sum (resultant) of the components in the direction perpendicular to the axis of the beam of all external loads and support reactions acting on either side of the section being considered. Internal Bending Moment (M) ≡ equal in magnitude but opposite in direction to the algebraic sum of the moments about (the centroid of the cross section of the beam) the section of all external loads and support reactions acting on either side of the section being considered. Positive Sign Conventions: Tension axial force on the section, Shears that produces clockwise moments, and Bending moments that produce compression in the top fibres and tension in the bottom fibres of the beam as shown in figure (25) and (26). Shear and bending moment diagrams depict the variation of these quantities along the length of the member. Proceeding from one end of the member to the other, sections are passed. After each successive change in loading along the length of the member, a FBD (Free Body Diagram) is drawn to determine the equations expressing the shear and bending moment in terms of the distance from a convenient origin. Plotting these equations produces the shear and bending moment diagrams.

Fig.25 Beam Sign Convention for Shear and Moment

28

Fig.26 Sign Convention for Shear and Moment on a Portal Frame

3.2 FINDING SHEAR FORCE AT ANY SECTION 1- Draw Free Body Diagram: - replace supports with forces and moments as applicable. 2- Find Reactions: - Take moments about one of the supports to determine one reaction, sum forces in x and y directions to determine the other reaction, draw deflected shape. 3-Find Shear Forces:- Determine SFs by taking cuts just before and just after each applied load and reaction, Draw the SF diagram remembering +ve shear convention. 29

3.3 FINDING BENDING MOMENTS There are two methods in finding BM at any section Theoretical method and Simplified method a) Theoretical Method: - Determine BMs by taking cuts at each applied load. In civil engineering, a positive BM is usually drawn on the tension side of the beam. b) Simplified Method: - BM is the area under the shear force diagram (SF is the gradient of the BM diagram). SF = 0 when the BM is a maximum or minimum. BM = 0 at points of contra flexure.

3.4 COMMON RESULTS OF BEAM ANALYSIS

3.5 SOLVED PROBLEMS Example No.1 Determine shear forces and BMs at points B, C, D and E in the following Beam

123-

Reaction at point A = 4.4 KN. Determination of shear forces from point A Cut just left of B: SFB- + 4.4 = -4.4 KN Cut just right of B: SFB+ + 4.4KN – 12KN = +7.6 KN Cut just right of C: SFC + 4.4KN -12 + 17 + -9.4 KN Cut just right of D: SFC + 4.4 -12 + 17 -8 = -1.4 KN Cut just right of E: SFC + 4.4 -12 + 17 -8 -12 + +10.6 KN Determination moments at sections from left: At B: MB = -4.4 * 2 = -8.8 KN.m At C: MC = -8.8 KN.m + 7.6 * 3 = + 14 KN.m At D: MD = 14 KN.m + 3...