The document discusses the behavior and analysis of reinforced concrete beams. It describes three stages that beams undergo as loading increases: 1) the uncracked concrete stage, 2) the cracked-elastic stage, and 3) the ultimate strength stage. It also discusses assumptions made in flexural theory, stress-strain curves for concrete and steel, and methods for calculating stresses in uncracked and cracked beams using the transformed area method. Key points covered include cracking moment, modular ratio, and the three-step transformed area method for cracked sections.
The document discusses properties and testing of concrete. It provides information on the constituents of concrete including cement, coarse aggregate, fine aggregate, and water. It also discusses properties of concrete and reinforcements, including their relatively high compressive strength and lower tensile strength. Various tests performed on concrete are mentioned, including tests on workability, compressive strength, flexural strength, and fresh/hardened concrete. Design philosophies for reinforced concrete include the working stress method, ultimate strength method, and limit state method.
1) The document discusses design considerations for columns according to ACI code, including requirements for different types of columns like tied, spirally reinforced, and composite columns.
2) It provides details on failure modes of tied and spiral columns and code requirements for minimum reinforcement ratios, number of bars, clear spacing, cover, and cross sectional dimensions.
3) Lateral reinforcement requirements are discussed, noting ties help restrain longitudinal bars from buckling while spirals provide additional confinement at ultimate load.
This document discusses the design of compression members in steel structures. It begins by defining compression members as members subjected to compressive stresses, such as columns, struts, and compression flanges. It notes that compression members are more prone to buckling than tension members. The document then discusses factors that influence the buckling strength of compression members, such as the member's length, cross-sectional properties, end conditions, and bracing. It also discusses eccentric loading of columns and the various sections that can be used or built up for compression members.
The balanced cantilever method is used to construct bridges with spans between 50-250m. It involves erecting segments on each side of the pier in a balanced sequence to minimize load imbalance and bending in the piers. This method is advantageous for long spans, marine environments, and where access under the deck is difficult. The cantilever lengths are typically 0.20-0.30 of the main span. Segment construction proceeds until the midspan point is reached, where the balanced pair is closed. The key advantages are single-sided support during construction and uniform construction. However, it is also very expensive and complicated to construct.
This document provides an overview of design in reinforced concrete according to BS 8110. It discusses the basic materials used - concrete and steel reinforcement - and their properties. It describes two limit states for design: ultimate limit state considering failure, and serviceability limit state considering deflection and cracking. Key aspects of beam design are summarized, including types of beams, design for bending and shear resistance, and limiting deflection. Reinforcement detailing rules are also briefly covered.
This chapter discusses structural concrete design. It covers properties of concrete and reinforcing steel, proportioning and mixing concrete, flexural design of beams and slabs, columns under bending and axial load, shear and torsion, development of reinforcement, two-way systems, frames, brackets and corbels, footings, walls, and defining terms. The chapter aims to promote a unified approach to structural concrete design across different design approaches, techniques, and codes worldwide.
The document summarizes the design of a steel exhibition building with a circular plan. It describes the architectural features of the building including the dimensions of the exhibition hall and stalls. It then discusses the structural analysis conducted using STAAD Pro software and consideration of various loads. Next, it details the design of key structural elements like curved beams, trusses, bracings, columns, and base plates. Design calculations are provided for the curved beams. Finally, it provides a bill of materials and concluding remarks on the benefits of the tubular structural design.
Modelling Building Frame with STAAD.Pro & ETABS - Rahul LeslieRahul Leslie
The document discusses modeling a reinforced concrete building frame using STAAD.Pro and ETABS software. It describes how to model the beams, columns, slabs, walls, stairs, and foundations. Initial member sizes are determined based on architectural requirements and design formulas. The building is modeled by framing the beams and columns. Loads like self-weight, floor loads, and wall loads are applied to the frame. Material properties of concrete are also specified. The document provides guidance on modeling the structural elements and applying loads in STAAD.Pro and ETABS to analyze the building frame.
The document discusses properties and testing of concrete. It provides information on the constituents of concrete including cement, coarse aggregate, fine aggregate, and water. It also discusses properties of concrete and reinforcements, including their relatively high compressive strength and lower tensile strength. Various tests performed on concrete are mentioned, including tests on workability, compressive strength, flexural strength, and fresh/hardened concrete. Design philosophies for reinforced concrete include the working stress method, ultimate strength method, and limit state method.
1) The document discusses design considerations for columns according to ACI code, including requirements for different types of columns like tied, spirally reinforced, and composite columns.
2) It provides details on failure modes of tied and spiral columns and code requirements for minimum reinforcement ratios, number of bars, clear spacing, cover, and cross sectional dimensions.
3) Lateral reinforcement requirements are discussed, noting ties help restrain longitudinal bars from buckling while spirals provide additional confinement at ultimate load.
This document discusses the design of compression members in steel structures. It begins by defining compression members as members subjected to compressive stresses, such as columns, struts, and compression flanges. It notes that compression members are more prone to buckling than tension members. The document then discusses factors that influence the buckling strength of compression members, such as the member's length, cross-sectional properties, end conditions, and bracing. It also discusses eccentric loading of columns and the various sections that can be used or built up for compression members.
The balanced cantilever method is used to construct bridges with spans between 50-250m. It involves erecting segments on each side of the pier in a balanced sequence to minimize load imbalance and bending in the piers. This method is advantageous for long spans, marine environments, and where access under the deck is difficult. The cantilever lengths are typically 0.20-0.30 of the main span. Segment construction proceeds until the midspan point is reached, where the balanced pair is closed. The key advantages are single-sided support during construction and uniform construction. However, it is also very expensive and complicated to construct.
This document provides an overview of design in reinforced concrete according to BS 8110. It discusses the basic materials used - concrete and steel reinforcement - and their properties. It describes two limit states for design: ultimate limit state considering failure, and serviceability limit state considering deflection and cracking. Key aspects of beam design are summarized, including types of beams, design for bending and shear resistance, and limiting deflection. Reinforcement detailing rules are also briefly covered.
This chapter discusses structural concrete design. It covers properties of concrete and reinforcing steel, proportioning and mixing concrete, flexural design of beams and slabs, columns under bending and axial load, shear and torsion, development of reinforcement, two-way systems, frames, brackets and corbels, footings, walls, and defining terms. The chapter aims to promote a unified approach to structural concrete design across different design approaches, techniques, and codes worldwide.
The document summarizes the design of a steel exhibition building with a circular plan. It describes the architectural features of the building including the dimensions of the exhibition hall and stalls. It then discusses the structural analysis conducted using STAAD Pro software and consideration of various loads. Next, it details the design of key structural elements like curved beams, trusses, bracings, columns, and base plates. Design calculations are provided for the curved beams. Finally, it provides a bill of materials and concluding remarks on the benefits of the tubular structural design.
Modelling Building Frame with STAAD.Pro & ETABS - Rahul LeslieRahul Leslie
The document discusses modeling a reinforced concrete building frame using STAAD.Pro and ETABS software. It describes how to model the beams, columns, slabs, walls, stairs, and foundations. Initial member sizes are determined based on architectural requirements and design formulas. The building is modeled by framing the beams and columns. Loads like self-weight, floor loads, and wall loads are applied to the frame. Material properties of concrete are also specified. The document provides guidance on modeling the structural elements and applying loads in STAAD.Pro and ETABS to analyze the building frame.
This document discusses the working stress method for designing reinforced concrete structures. It defines key terms like neutral axis, lever arm, and moment of resistance. It describes the assumptions and steps of the working stress method, including designing for under-reinforced, balanced, and over-reinforced beam sections. The document also discusses limitations of the working stress method and introduces the limit state method as a more modern approach.
ETABS is structural analysis software used to analyze and design buildings. It was developed in 1975 by students and later released commercially in 1985 by Computers and Structures Inc. The Burj Khalifa in Dubai was one of the first major projects analyzed using ETABS.
To model a structure in ETABS, materials like concrete and steel must first be defined along with their properties. Frame sections for beams, columns, walls and slabs are then created. The grid is drawn representing the building plan. Beams, columns, walls and slabs can then be drawn by connecting nodes on the grid. Modeling tools allow modifying the structural model by merging joints, aligning elements, and editing frames.
The document discusses bar bending schedules (BBS), which provide essential information for bending and placing reinforcement bars during construction. A BBS includes the location, type, size, length, number, and bending details of each bar. It allows bars to be pre-bent in a factory and transported to the construction site, reducing time. A BBS also improves quality control and provides better estimates of steel requirements.
This document is the Indian Railway Standard Code of Practice for plain, reinforced and prestressed concrete for general bridge construction from 1997. It provides definitions for terminology used, specifies materials and workmanship for concrete, reinforcement and prestressing tendons. It defines loads, load combinations and requirements for limit state design. It provides recommendations for the design and construction of plain concrete, reinforced concrete and prestressed concrete bridges. The document covers topics such as concrete mix design, formwork, reinforcement, transportation and curing of concrete, prestressing, precast construction, and load testing.
Design of column base plates anchor boltKhaled Eid
This document discusses the design of column base plates and steel anchorage to concrete. It covers base plate materials and design for different load cases including axial, moment, and shear loads. It also discusses anchor rod types, materials, and design for tension and shear loading based on calculations of the steel and concrete breakout strengths according to building codes.
Prepared by madam rafia firdous. She is a lecturer and instructor in subject of Plain and Reinforcement concrete at University of South Asia LAHORE,PAKISTAN.
This document summarizes the key aspects of loadbearing masonry construction. It discusses the advantages of masonry, including its ability to provide structure, insulation, and fire protection simultaneously. It also describes the development of modern codes of practice, which have expanded the use of loadbearing masonry beyond empirical rules to the rational design of multi-storey buildings. The document outlines basic design considerations for loadbearing masonry, such as compatible building typologies, and provides a high-level classification of masonry wall systems.
A continuous beam has more than one span carried by multiple supports. It is commonly used in bridge construction since simple beams cannot support large spans without requiring greater strength and stiffness. Continuous prestressed concrete beams provide adequate strength and stiffness while allowing for redistribution of moments, resulting in higher load capacity, reduced deflections, and more evenly distributed bending moments compared to equivalent simple beams. Analysis of continuous beams requires determining primary moments from prestressing, secondary moments induced by support reactions, and the combined resultant moments.
This document discusses the design of floor slabs including one-way spanning slabs, two-way spanning slabs, continuous slabs, cantilever slabs, and restrained slabs. It covers slab types based on span ratios, bending moment coefficients, determining design load, reinforcement requirements, shear and deflection checks, crack control, and reinforcement curtailment details for different slab conditions. The document is authored by Eng. S. Kartheepan and is related to the design of floor slabs for a civil engineering project.
Lec11 Continuous Beams and One Way Slabs(1) (Reinforced Concrete Design I & P...Hossam Shafiq II
The document discusses reinforced concrete continuity and analysis methods for continuous beams and one-way slabs. It describes how steel reinforcement must extend through members to provide structural continuity. The ACI/SBC coefficient method of analysis is summarized, which uses coefficient tables to determine maximum shear forces and bending moments for continuous beams and one-way slabs under various loading conditions in a simplified manner compared to elastic analysis. Requirements for applying the coefficient method include having multiple spans with ratios less than 1.2, prismatic member sections, and live loads less than 3 times dead loads.
Workshop under the Capacity Building Programme of the Southern Road Connectivity Project / Expressway Connectivity Improvement Plan Project, March 2016
This document discusses different types and classifications of columns. It defines a column as a vertical structural member primarily designed to carry axial compression loads. Columns can be classified based on their shape, reinforcement, and type of loading. Common shapes include square, rectangular, circular, L-shaped, and T-shaped sections. Reinforcement types include tied columns with ties, spiral columns with helical reinforcement, and composite columns with encased steel. Columns are either concentrically loaded with forces through the centroid, or eccentrically loaded off-center. The document also covers column capacity calculations, resistance factors, and provides an example problem.
This is just an overview about the Reinforced Concrete Deck Girder Bridge
(RCDG Bridge)
the Presentation includes:
Materials for Construction,
Parts of a typical RCDG bridge,
The Forces Acting on the bridge, etc.
This document provides information about the course "Design & Detailing of RC Structures 10CV321" taught by Dr. G.S. Suresh at NIE Mysore. It lists several reference books for the course and provides the evaluation pattern for both theory and drawing components. It also outlines the course content which includes limit state design method, stress-strain behavior of materials, assumptions in limit state design, behavior of reinforced concrete beams, stress block parameters, and calculation of ultimate flexural strength.
Trusses Analysis Of Statically DeterminateAmr Hamed
The document discusses the analysis of statically determinate trusses. It describes the characteristics of determinate trusses, including their slender members, pinned/bolted/welded joints, and loads acting at joints with members in tension or compression. It also discusses terminology and selection criteria for different types of trusses used in roofs and bridges. The document outlines the assumptions and methods for analyzing trusses, including the method of joints and method of sections.
This document discusses reinforced concrete columns. It begins by defining columns and different column types, including based on shape, reinforcement, loading conditions, and slenderness ratio. Short columns fail due to material strength while slender columns are at risk of buckling. The document covers column design considerations like unsupported length and effective length. It provides examples of single storey building column design and discusses minimum longitudinal reinforcement requirements in columns.
Calulation of deflection and crack width according to is 456 2000Vikas Mehta
This document discusses the calculation of crack width in reinforced concrete flexural members. It provides information on:
1) Crack width is calculated to satisfy serviceability limits and is only relevant for Type 3 pre-stressed concrete members that crack under service loads.
2) Crack width depends on factors like amount of pre-stress, tensile stress in bars, concrete cover thickness, bar diameter and spacing, member depth and location of neutral axis, bond strength, and concrete tensile strength.
3) The method of calculation involves determining the shortest distance from the surface to a bar and using equations involving member depth, neutral axis depth, average strain at the surface level. Permissible crack widths are specified depending on exposure
The document discusses the design of footings for structures. It begins by explaining that footings are needed to transfer structural loads from members made of materials like steel and concrete to the underlying soil. It then describes different types of shallow and deep foundations, including spread, strap, combined, and raft footings. The document provides details on designing isolated and combined footings to resist vertical loads and moments based on provisions in IS 456. It also discusses wall footings and combined footings that support multiple columns. In summary, the document covers the purpose of footings, various footing types, and design of isolated and combined footings.
The document discusses the behavior and analysis of reinforced concrete beams. It describes the three stages a beam undergoes when loaded: uncracked, cracked-elastic, and ultimate strength. The transformed area method is presented for calculating stresses in cracked beams. An example problem demonstrates using this method to find bending stresses in a beam section. The allowable resisting moment is also determined based on specified material stresses.
Effect of creep on composite steel concrete sectionKamel Farid
Creep and Shrinkage are inelastic and time-varying strains.
For Steel-Concrete Composite beam creep and shrinkage are highly associated with concrete.
Simple approach depending on modular ratio has been adopted to compute the elastic section properties instead of the theoretically complex calculations of creep.
This document discusses the working stress method for designing reinforced concrete structures. It defines key terms like neutral axis, lever arm, and moment of resistance. It describes the assumptions and steps of the working stress method, including designing for under-reinforced, balanced, and over-reinforced beam sections. The document also discusses limitations of the working stress method and introduces the limit state method as a more modern approach.
ETABS is structural analysis software used to analyze and design buildings. It was developed in 1975 by students and later released commercially in 1985 by Computers and Structures Inc. The Burj Khalifa in Dubai was one of the first major projects analyzed using ETABS.
To model a structure in ETABS, materials like concrete and steel must first be defined along with their properties. Frame sections for beams, columns, walls and slabs are then created. The grid is drawn representing the building plan. Beams, columns, walls and slabs can then be drawn by connecting nodes on the grid. Modeling tools allow modifying the structural model by merging joints, aligning elements, and editing frames.
The document discusses bar bending schedules (BBS), which provide essential information for bending and placing reinforcement bars during construction. A BBS includes the location, type, size, length, number, and bending details of each bar. It allows bars to be pre-bent in a factory and transported to the construction site, reducing time. A BBS also improves quality control and provides better estimates of steel requirements.
This document is the Indian Railway Standard Code of Practice for plain, reinforced and prestressed concrete for general bridge construction from 1997. It provides definitions for terminology used, specifies materials and workmanship for concrete, reinforcement and prestressing tendons. It defines loads, load combinations and requirements for limit state design. It provides recommendations for the design and construction of plain concrete, reinforced concrete and prestressed concrete bridges. The document covers topics such as concrete mix design, formwork, reinforcement, transportation and curing of concrete, prestressing, precast construction, and load testing.
Design of column base plates anchor boltKhaled Eid
This document discusses the design of column base plates and steel anchorage to concrete. It covers base plate materials and design for different load cases including axial, moment, and shear loads. It also discusses anchor rod types, materials, and design for tension and shear loading based on calculations of the steel and concrete breakout strengths according to building codes.
Prepared by madam rafia firdous. She is a lecturer and instructor in subject of Plain and Reinforcement concrete at University of South Asia LAHORE,PAKISTAN.
This document summarizes the key aspects of loadbearing masonry construction. It discusses the advantages of masonry, including its ability to provide structure, insulation, and fire protection simultaneously. It also describes the development of modern codes of practice, which have expanded the use of loadbearing masonry beyond empirical rules to the rational design of multi-storey buildings. The document outlines basic design considerations for loadbearing masonry, such as compatible building typologies, and provides a high-level classification of masonry wall systems.
A continuous beam has more than one span carried by multiple supports. It is commonly used in bridge construction since simple beams cannot support large spans without requiring greater strength and stiffness. Continuous prestressed concrete beams provide adequate strength and stiffness while allowing for redistribution of moments, resulting in higher load capacity, reduced deflections, and more evenly distributed bending moments compared to equivalent simple beams. Analysis of continuous beams requires determining primary moments from prestressing, secondary moments induced by support reactions, and the combined resultant moments.
This document discusses the design of floor slabs including one-way spanning slabs, two-way spanning slabs, continuous slabs, cantilever slabs, and restrained slabs. It covers slab types based on span ratios, bending moment coefficients, determining design load, reinforcement requirements, shear and deflection checks, crack control, and reinforcement curtailment details for different slab conditions. The document is authored by Eng. S. Kartheepan and is related to the design of floor slabs for a civil engineering project.
Lec11 Continuous Beams and One Way Slabs(1) (Reinforced Concrete Design I & P...Hossam Shafiq II
The document discusses reinforced concrete continuity and analysis methods for continuous beams and one-way slabs. It describes how steel reinforcement must extend through members to provide structural continuity. The ACI/SBC coefficient method of analysis is summarized, which uses coefficient tables to determine maximum shear forces and bending moments for continuous beams and one-way slabs under various loading conditions in a simplified manner compared to elastic analysis. Requirements for applying the coefficient method include having multiple spans with ratios less than 1.2, prismatic member sections, and live loads less than 3 times dead loads.
Workshop under the Capacity Building Programme of the Southern Road Connectivity Project / Expressway Connectivity Improvement Plan Project, March 2016
This document discusses different types and classifications of columns. It defines a column as a vertical structural member primarily designed to carry axial compression loads. Columns can be classified based on their shape, reinforcement, and type of loading. Common shapes include square, rectangular, circular, L-shaped, and T-shaped sections. Reinforcement types include tied columns with ties, spiral columns with helical reinforcement, and composite columns with encased steel. Columns are either concentrically loaded with forces through the centroid, or eccentrically loaded off-center. The document also covers column capacity calculations, resistance factors, and provides an example problem.
This is just an overview about the Reinforced Concrete Deck Girder Bridge
(RCDG Bridge)
the Presentation includes:
Materials for Construction,
Parts of a typical RCDG bridge,
The Forces Acting on the bridge, etc.
This document provides information about the course "Design & Detailing of RC Structures 10CV321" taught by Dr. G.S. Suresh at NIE Mysore. It lists several reference books for the course and provides the evaluation pattern for both theory and drawing components. It also outlines the course content which includes limit state design method, stress-strain behavior of materials, assumptions in limit state design, behavior of reinforced concrete beams, stress block parameters, and calculation of ultimate flexural strength.
Trusses Analysis Of Statically DeterminateAmr Hamed
The document discusses the analysis of statically determinate trusses. It describes the characteristics of determinate trusses, including their slender members, pinned/bolted/welded joints, and loads acting at joints with members in tension or compression. It also discusses terminology and selection criteria for different types of trusses used in roofs and bridges. The document outlines the assumptions and methods for analyzing trusses, including the method of joints and method of sections.
This document discusses reinforced concrete columns. It begins by defining columns and different column types, including based on shape, reinforcement, loading conditions, and slenderness ratio. Short columns fail due to material strength while slender columns are at risk of buckling. The document covers column design considerations like unsupported length and effective length. It provides examples of single storey building column design and discusses minimum longitudinal reinforcement requirements in columns.
Calulation of deflection and crack width according to is 456 2000Vikas Mehta
This document discusses the calculation of crack width in reinforced concrete flexural members. It provides information on:
1) Crack width is calculated to satisfy serviceability limits and is only relevant for Type 3 pre-stressed concrete members that crack under service loads.
2) Crack width depends on factors like amount of pre-stress, tensile stress in bars, concrete cover thickness, bar diameter and spacing, member depth and location of neutral axis, bond strength, and concrete tensile strength.
3) The method of calculation involves determining the shortest distance from the surface to a bar and using equations involving member depth, neutral axis depth, average strain at the surface level. Permissible crack widths are specified depending on exposure
The document discusses the design of footings for structures. It begins by explaining that footings are needed to transfer structural loads from members made of materials like steel and concrete to the underlying soil. It then describes different types of shallow and deep foundations, including spread, strap, combined, and raft footings. The document provides details on designing isolated and combined footings to resist vertical loads and moments based on provisions in IS 456. It also discusses wall footings and combined footings that support multiple columns. In summary, the document covers the purpose of footings, various footing types, and design of isolated and combined footings.
The document discusses the behavior and analysis of reinforced concrete beams. It describes the three stages a beam undergoes when loaded: uncracked, cracked-elastic, and ultimate strength. The transformed area method is presented for calculating stresses in cracked beams. An example problem demonstrates using this method to find bending stresses in a beam section. The allowable resisting moment is also determined based on specified material stresses.
Effect of creep on composite steel concrete sectionKamel Farid
Creep and Shrinkage are inelastic and time-varying strains.
For Steel-Concrete Composite beam creep and shrinkage are highly associated with concrete.
Simple approach depending on modular ratio has been adopted to compute the elastic section properties instead of the theoretically complex calculations of creep.
OUTLINE:
Introduction
Shoring Process
Effective Beam Flange Width
Shear Transfer
Strength Of Steel Anchors
Partially Composite Beams
Moment Capacity Of Composite Sections
Deflection
Design Of Composite Sections
This document discusses shear failure in reinforced concrete beams. It begins by explaining that beams require shear reinforcement to prevent dangerous shear failure before flexural failure under overloading. Shear failure is difficult to predict due to diagonal tension cracks forming within the beam. The document then examines shear stresses and diagonal tension stresses in homogeneous beams and how flexural cracks and diagonal cracks form in reinforced concrete beams without shear reinforcement. It notes that shear reinforcement in the form of stirrups is required to control diagonal cracking and prevent diagonal tension failure.
CE 72.52 - Lecture 7 - Strut and Tie ModelsFawad Najam
The document discusses the strut-and-tie approach for analyzing concrete structures. It begins with background concepts such as Bernoulli's hypothesis, St. Venant's principle, and the lower bound theorem of plasticity. It then discusses how axial stresses, shear stresses, and the interaction of stresses affect concrete sections. The document outlines the ACI approach to shear-torsion design and provides equations from ACI 318 for calculating the concrete shear capacity. It introduces the concept of modeling concrete as a truss system and compares this to flexural behavior in beams. The strut-and-tie method is presented as a unified approach for considering all load effects. Guidelines are provided for developing an appropriate strut-and-tie model and
CE 72.52 Lecture 4 - Ductility of Cross-sectionsFawad Najam
This document provides information on ductility of concrete structures. It discusses how ductility is key to good seismic performance of structures. Ductility is defined and different levels of ductility are described, from the material level to the structural level. Factors that affect ductility include confinement of concrete, reinforcement, cross-section shape, and applied loads. Moment-curvature relationships are used to compute ductility at the cross-section level. Confinement improves concrete ductility by modifying its stress-strain behavior. Spiral reinforcement increases concrete strength under triaxial compression. Moment-curvature curves can indicate yield points and failure mechanisms for different types of sections.
This document discusses prestressed concrete, including:
- The basic concepts of prestressing including using metal bands, pre-tensioned spokes, and introducing stresses to counteract external loads.
- Design concepts like losses in prestressing structures from elastic shortening, creep, shrinkage, relaxation, friction, and anchorage slip.
- Provisions for prestressing in the Indian Road Congress Bridge Code and Indian Standard Code.
- Construction aspects like casting of girders, post-tensioning work, and load testing of structures.
The document provides an overview of prestressed concrete structures including:
- Definitions of prestressing where internal stresses counteract external loads.
- The key terminology used including tendons, anchorage, pretensioning vs post-tensioning.
- The materials used including cement, concrete, and steel types.
- The stages of loading and advantages of prestressing over reinforced concrete.
- Details of pretensioning and post-tensioning systems including equipment, processes, and differences between the two methods.
This document discusses the working stress method for designing reinforced concrete structures. It defines key terms like neutral axis, lever arm, and moment of resistance. It describes the assumptions and steps of the working stress method, including designing for under-reinforced, balanced, and over-reinforced beam sections. The document also discusses limitations of the working stress method and introduces the limit state method as a more modern approach.
Deflections in PT elements pt structure for all pt slabs in civil industry.pdfvijayvijay327286
The document discusses factors that influence deflections in prestressed concrete members and methods for predicting deflections. It covers:
- Short term deflections of unracked members which can be estimated using Mohr's theorem.
- How the tendon profile affects deflections, providing formulas for straight, trapezoidal, parabolic, and other tendon types.
- Downward deflections due to self-weight and imposed loads that can be calculated using formulas provided.
- Estimation of long-term deflections accounting for creep and shrinkage effects, discussing various methods like those of Busemann, McHenry, and Neville.
Lec10 Bond and Development Length (Reinforced Concrete Design I & Prof. Abdel...Hossam Shafiq II
This document discusses bond and development length in reinforced concrete. It defines bond as the adhesion between concrete and steel reinforcement, which is necessary to develop their composite action. Bond is achieved through chemical adhesion, friction from deformed bar ribs, and bearing. Development length refers to the minimum embedment length of a reinforcement bar needed to develop its yield strength by bonding to the surrounding concrete. The development length depends on factors like bar size, concrete strength, bar location, and transverse reinforcement. It also provides equations from design codes to calculate the development length for tension bars, compression bars, bundled bars, and welded wire fabric. Hooked bars can be used when full development length is not available, and the document discusses requirements for standard hook geome
Prestress loss occurs as prestress reduces over time from its initial applied value. There are two types of prestress loss - immediate losses during prestressing/transfer and long-term time-dependent losses. Immediate losses include elastic shortening, anchorage slip, and friction. Long-term losses include creep and shrinkage of concrete and relaxation of prestressing steel. The quantification of losses is based on strain compatibility between concrete and steel. For a pre-tensioned concrete sleeper, the percentage loss due to elastic shortening was calculated to be approximately 2.83% based on the stress in concrete at the level of the tendons.
Deep beams are structural elements where a significant portion of the load is carried to the supports by compression forces combining the load and reaction. As a result, the strain distribution is nonlinear and shear deformations are significant compared to pure flexure. Examples include floor slabs under horizontal loads, short span beams carrying heavy loads, and transfer girders. The behavior of deep beams is two-dimensional rather than one-dimensional, and plane sections may not remain plane. Analysis requires a two-dimensional stress approach.
All reinforced concrete beams crack, generally starting at loads well below service level, and possibly even prior to loading due to restrained shrinkage. Flexural cracking due to loads is not only inevitable, but actually necessary for the reinforcement to be used effectively. Prior to the formation of flexural cracks, the steel stress is no more than n times the stress in the adjacent concrete, where n is the modular ratio E5/Ec. For materials common in current practice, n is approximately 8.
This document discusses various types of losses in prestressing force that occur in pre-tensioned and post-tensioned concrete members. It defines key terms like prestressing force, pre-tensioning and post-tensioning. It explains different types of losses - elastic shortening, anchorage slip, friction, creep, shrinkage and relaxation. Methods to calculate losses in prestressing force due to elastic shortening are presented for different member types like axial and bending members. Friction loss occurring uniquely in post-tensioning is also explained.
This document discusses rock slope stability analysis and engineering. It describes a rock slope in Hong Kong supported with tensioned rock anchors and shotcrete. The principles of rock slope design concern the orientation and characteristics of discontinuities like joints and faults. Slope stability analysis requires determining the friction angle and cohesion of potential sliding surfaces. Shear strength testing is used to define the cohesion and friction angle parameters in the Coulomb failure criterion for analyzing rock slope stability.
Prestressed concrete is concrete in which internal stresses are introduced to counteract external loads. Tendons are stretched elements that impart prestress, and anchorage devices enable the tendons to impart and maintain prestress. There are two main methods - pretensioning, where tendons are tensioned before concrete is cast, and post-tensioning, where tendons are tensioned against hardened concrete. Prestressed concrete uses high-strength materials like cement, concrete, and steel tendons or strands to achieve its compressive strength and durability advantages over reinforced concrete.
This document discusses torsion force and its importance in pre-stressed concrete lab courses. It defines torsion force as a twisting force applied to an object by twisting one end. The document outlines objectives about defining torsion force, explaining formulas, and discussing its importance in pre-stressing. Key concepts covered include the torsion formula for solid and hollow circular cross-sections, limitations of the equations, origins of torsion theory, reasons for understanding torsion in pre-stressed concrete, analyzing torsion, and defining pure torsion.
Similar to Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif) (20)
Ch8 Truss Bridges (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. Metwally ...Hossam Shafiq II
This chapter discusses truss bridges. It begins by defining a truss as a triangulated assembly of straight members that can be used to replace girders. The main advantages of truss bridges are that primary member forces are axial loads and the open web system allows for greater depth.
The chapter then describes the typical components of a through truss bridge and the most common truss forms including Pratt, Warren, curved chord, subdivided, and K-trusses. Design considerations like truss depth, economic spans, cross section shapes, and wind bracing are covered. The chapter concludes with sections on determining member forces, design principles, and specific design procedures.
Ch7 Box Girder Bridges (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. Metw...Hossam Shafiq II
1. Box girder bridges have two key advantages over plate girder bridges: they possess torsional stiffness and can have much wider flanges.
2. For medium span bridges between 45-100 meters, box girder bridges offer an attractive form of construction as they maintain simplicity while allowing larger span-to-depth ratios compared to plate girders.
3. Advances in welding and cutting techniques have expanded the structural possibilities for box girders, allowing for more economical designs of large welded units.
Ch5 Plate Girder Bridges (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. Me...Hossam Shafiq II
Plate girders are commonly used as main girders for short and medium span bridges. They are fabricated by welding together steel plates to form an I-shape cross-section, unlike hot-rolled I-beams. Plate girders offer more design flexibility than rolled sections as the plates can be optimized for strength and economy. However, their thin plates are more susceptible to various buckling modes which control the design. Buckling considerations of the compression flange, web in shear and bending must be evaluated to determine the plate girder's load capacity.
Ch4 Bridge Floors (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. Metwally ...Hossam Shafiq II
This chapter discusses bridge floors for roadway and railway bridges. It describes three main types of structural systems for roadway bridge floors: slab, beam-slab, and orthotropic plate. For railway bridges, the two main types are open timber floors and ballasted floors. The chapter then covers design considerations for allowable stresses, stringer and cross girder cross sections, and provides an example design for the floor of a roadway bridge with I-beam stringers and cross girders.
Ch3 Design Considerations (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. M...Hossam Shafiq II
This chapter discusses design considerations for steel bridges. It outlines two main design philosophies: working stress design and limit states design. The chapter then focuses on the working stress design method, which is based on the Egyptian Code of Practice for Steel Constructions and Bridges. It provides allowable stress values for various steel grades and loading conditions, including stresses due to axial, shear, bending, compression and tension loads. Design of sections is classified based on compact and slender criteria. The chapter also addresses stresses from repeated, erection and secondary loads.
Ch2 Design Loads on Bridges (Steel Bridges تصميم الكباري المعدنية & Prof. Dr....Hossam Shafiq II
This document discusses design loads on bridges. It describes various types of loads that bridges must be designed to resist, including dead loads from the bridge structure itself, live loads from traffic, and environmental loads such as wind, temperature, and earthquakes. It provides specifics on how to calculate loads from road and rail traffic according to Egyptian design codes, including truck and train configurations, impact factors, braking and centrifugal forces, and load distributions. Other loads like wind, thermal effects, and concrete shrinkage are also summarized.
Ch1 Introduction (Steel Bridges تصميم الكباري المعدنية & Prof. Dr. Metwally A...Hossam Shafiq II
This document provides an introduction to steel bridges, including:
1. It discusses the history and evolution of bridge engineering and the key components of bridge structures.
2. It describes different classifications of bridges according to materials, usage, position, and structural forms. The structural forms include beam bridges, frame bridges, arch bridges, cable-stayed bridges, and suspension bridges.
3. It provides examples of different types of bridges and explains the basic structural systems used in bridges, including simply supported, cantilever, and continuous beams as well as rigid frames.
Lec09 Shear in RC Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif)Hossam Shafiq II
This document discusses shear in reinforced concrete beams. It covers shear stress and failure modes, shear strength provided by concrete and steel stirrups, design according to code provisions, and critical shear sections. Key points include: transverse loads induce shear stress perpendicular to bending stresses; shear failure is brittle and must be designed to exceed flexural strength; nominal shear strength comes from concrete and steel stirrups according to code equations; design requires checking section adequacy and providing minimum steel area and maximum stirrup spacing. Critical shear sections for design are located a distance d from supports.
Lec06 Analysis and Design of T Beams (Reinforced Concrete Design I & Prof. Ab...Hossam Shafiq II
1) T-beams are commonly used structural elements that can take two forms: isolated precast T-beams or T-beams formed by the interaction of slabs and beams in buildings.
2) The analysis and design of T-beams considers the effective flange width provided by slab interaction or the dimensions of an isolated precast flange.
3) Two methods are used to analyze T-beams: assuming the stress block is in the flange and using rectangular beam theory, or using a decomposition method if the stress block extends into the web.
Lec05 Design of Rectangular Beams with Tension Steel only (Reinforced Concret...Hossam Shafiq II
The document discusses design considerations for rectangular reinforced concrete beams with tension steel only. It covers topics such as beam proportions, deflection control, selection of reinforcing bars, concrete cover, bar spacing, effective steel depth, minimum beam width, and number of bars. Beam proportions should have a depth to width ratio of 1.5-2 for normal spans and up to 4 for longer spans. Minimum concrete cover and bar spacings are specified to protect the steel. Effective steel depth is the distance from the extreme compression fiber to the steel centroid. Design assumptions must be checked against the final design.
Lec04 Analysis of Rectangular RC Beams (Reinforced Concrete Design I & Prof. ...Hossam Shafiq II
This document discusses the ultimate flexural analysis of reinforced concrete beams according to building codes. It covers topics such as concrete stress-strain relationships, stress distributions at failure, nominal and design flexural strength, moments in beams, tension steel ratios, minimum steel requirements, ductile and brittle failure modes, and calculations for balanced and maximum steel ratios. Diagrams illustrate key concepts regarding stress blocks, strain distributions, and section types. Formulas are presented for determining balanced steel ratio, maximum steel ratio, and checking neutral axis depth.
An In-Depth Exploration of Natural Language Processing: Evolution, Applicatio...DharmaBanothu
Natural language processing (NLP) has
recently garnered significant interest for the
computational representation and analysis of human
language. Its applications span multiple domains such
as machine translation, email spam detection,
information extraction, summarization, healthcare,
and question answering. This paper first delineates
four phases by examining various levels of NLP and
components of Natural Language Generation,
followed by a review of the history and progression of
NLP. Subsequently, we delve into the current state of
the art by presenting diverse NLP applications,
contemporary trends, and challenges. Finally, we
discuss some available datasets, models, and
evaluation metrics in NLP.
Impartiality as per ISO /IEC 17025:2017 StandardMuhammadJazib15
This document provides basic guidelines for imparitallity requirement of ISO 17025. It defines in detial how it is met and wiudhwdih jdhsjdhwudjwkdbjwkdddddddddddkkkkkkkkkkkkkkkkkkkkkkkwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwwioiiiiiiiiiiiii uwwwwwwwwwwwwwwwwhe wiqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq gbbbbbbbbbbbbb owdjjjjjjjjjjjjjjjjjjjj widhi owqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq uwdhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhwqiiiiiiiiiiiiiiiiiiiiiiiiiiiiw0pooooojjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjjj whhhhhhhhhhh wheeeeeeee wihieiiiiii wihe
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Data Communication and Computer Networks Management System Project Report.pdfKamal Acharya
Networking is a telecommunications network that allows computers to exchange data. In
computer networks, networked computing devices pass data to each other along data
connections. Data is transferred in the form of packets. The connections between nodes are
established using either cable media or wireless media.
Sri Guru Hargobind Ji - Bandi Chor Guru.pdfBalvir Singh
Sri Guru Hargobind Ji (19 June 1595 - 3 March 1644) is revered as the Sixth Nanak.
• On 25 May 1606 Guru Arjan nominated his son Sri Hargobind Ji as his successor. Shortly
afterwards, Guru Arjan was arrested, tortured and killed by order of the Mogul Emperor
Jahangir.
• Guru Hargobind's succession ceremony took place on 24 June 1606. He was barely
eleven years old when he became 6th Guru.
• As ordered by Guru Arjan Dev Ji, he put on two swords, one indicated his spiritual
authority (PIRI) and the other, his temporal authority (MIRI). He thus for the first time
initiated military tradition in the Sikh faith to resist religious persecution, protect
people’s freedom and independence to practice religion by choice. He transformed
Sikhs to be Saints and Soldier.
• He had a long tenure as Guru, lasting 37 years, 9 months and 3 days
Cricket management system ptoject report.pdfKamal Acharya
The aim of this project is to provide the complete information of the National and
International statistics. The information is available country wise and player wise. By
entering the data of eachmatch, we can get all type of reports instantly, which will be
useful to call back history of each player. Also the team performance in each match can
be obtained. We can get a report on number of matches, wins and lost.
Lec03 Flexural Behavior of RC Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif)
1. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 1
CE370
REINFORCED CONCRETE-I
Prof. Abdelhamid CHARIF
Flexural Behavior and
Analysis of RC Beams
Behavior stages in a RC beam
• A small transverse load placed on a simply supported RC beam is
gradually increased until beam failure.
• This loading causes positive bending moment (tension, and steel
in bottom) and shear force
• Perfect bond is assumed between steel bars and concrete
• The beam will go through three distinct stages before collapse
occurs. These are:
• The uncracked concrete stage
• The concrete cracked – elastic strength stage
• The ultimate – strength stage.
2
2. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 2
3
Simply supported beam under uniform load
4
Basic Assumptions in Flexure theory of RC Members
1. Plane sections before bending remain plane after bending.
This assumption has been experimentally verified for all normal
beams except deep beams (length to thickness ratio less than 3).
2. The strain in reinforcement equals to the strain in concrete at the
same level (perfect bond, no slip).
1 & 2 imply that strain in reinforcement and concrete is
directly proportional to the distance from N.A.
3. Tensile strength of concrete is neglected in flexural strength
calculations (after cracking).
4. Stresses in concrete and reinforcement are calculated from
strains using material stress-strain curves.
Other assumptions will be made at ultimate state
3. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 3
5
Recall Concrete Stress-Strain Curve
Compression: For a stress less or equal to 50% of fc’ ,
the curve is almost linear.
Tension: Strength very small and curve assumed linear.
6
Recall Steel Stress-Strain Curve
(Large plastic deformations after yielding)
ysy
ysss
s
f
E
f
when
when
gradesallfor
200000MPaEs
4. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 4
Uncracked Concrete Stage
• At small loads when the tensile stresses are less than the
modulus of rupture, the entire cross section of the beam resists
bending.
• Standard bending theory can be used.
• Compression develops on top side and tension on bottom (for
positive moments.
7
• Neutral axis passes through
centroid
• Moment of inertia of gross
section is used
8
Uncracked Concrete Stage
Strains and stresses
5. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 5
Concrete-Cracked – Elastic Stage
• As load is increased, the modulus of rupture of the concrete is
exceeded, and cracks begin to develop in bottom of the beam.
• Cracking Moment: The moment at which tensile stress in the
bottom of the beam equals the modulus of rupture (i.e. when
cracks begin to form) is referred to as the cracking moment, Mcr
• As load is further increased, cracks quickly spread closer to the
neutral axis, which then begins to move upward.
• Note: The cracks occur at all places along the beam where the
actual moment is greater than the cracking moment.
9
Concrete-Cracked – Elastic Stage (contd.)
• As concrete in the cracked zone cannot resist tensile stresses –
the steel must do it. Perfect bond is still maintained.
• This stage will continue as long as the compression stress in the
top fibers is less than about one-half of the concrete compressive
strength fc’ and as long as steel stress is less than its yield stress
• In this stage the compressive stresses vary linearly with the
distance from the neutral axis (straight line).
10
ratioModular
with
:pointsameat thestrainsEqual
n
E
E
nnff
E
E
ff
E
f
E
f
c
s
cs
c
s
cs
s
s
c
c
sc
6. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 6
• Service or Working Loads: Loads that actually occur when a
structure is in normal service use.
• Under working loads, moments develop which are considerably
larger than the cracking moments. Obviously the tensile side of
the beam will be cracked.
• Under working loads, concrete compression stress is less than
50% of its strength and linear stress-strain variation is assumed.
• Service loads and serviceability limit state correspond thus to
the cracked elastic stage
• To compute concrete and steel stresses in this range, the
transformed-area method is used.
11
Cracked – Elastic Stage and Service Loads
Beam Failure- Ultimate Strength Stage
• As load is increased further, compressive stresses are greater than
0.5fc’, cracks and neutral axis move further upward .
• Concrete compressive stresses begin to change appreciably from
linear to curved, and reinforcing bars yield.
12
7. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 7
Moment Curvature Diagram
• From the measured compressive strain in concrete and tensile
strain in steel, the relationship between the bending moment
and the curvature can be tracked up to failure.
13
dy
y
sc
ntaanglessmallVery
NAfromdistancesomeatfiberbeamainstrain
curvatureorsectionbeamtheofchangeAngle
y
s
c
d
• Bending moments cause
curvatures (rotations).
• Curvature more objective
than deflection as
bending deformation.
14
Typical Moment Curvature Diagram
for a Reinforced Concrete Beam
There are three
different stages
in the curve :
OC – CY - YF
8. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 8
15
Moment Curvature Diagram
1. First stage is for moments less than the cracking moment Mcr .
The entire beam cross section is available to resist bending. The
relationship is quasi-linear.
2. When the moment is increased beyond Mcr, the slope of the
curve decreases as the cracked beam is not as stiff as in the
initial stage. The diagram will still follow a straight line from Mcr
to the point where steel is stressed to its yield point.
15
3. After steel yields, the beam has
very little additional moment
capacity, and only a small
additional load is required to
substantially increase rotations
and deflections. The slope of the
diagram is now almost flat.
Bending stress in an Uncracked Beam
• Steel area is small (usually 2% or less), and its effect on beam
properties is negligible as long as the beam is uncracked
• The bending stress in such a beam can be obtained based on
the gross properties of the beam’s cross section.
• The neutral axis coincides with the centroid
• The stress in the concrete at a distance y from the neutral axis
of the cross section can be determined from the following
flexure formula:
16
section.crossgrosstheofinertiaofMoment
sectiontheofmomentBending
g
cr
g
I
MM
I
My
f
9. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 9
Cracking Moment
17
section.crosstheofinertiaofmomentGross
momentBending
g
g
I
M
I
My
f
t
gr
cr
g
tcr
r
cr
t
y
If
M
I
yM
f
fff
yy
:Thus
9.5.2.3)sectionSBC/(ACI0.7ruptureofModulus
fibertensionextremetoaxiscentroidalfromdistance
Substitute
'
Example 1: Uncracked section
• Assuming concrete is uncracked, compute bending stresses in
extreme fibers of the section below for a bending moment of
34 kN.m. fc’ = 30 MPa.
• To check that the section is uncracked, first determine the
cracking moment of the section.
18
375
450
300
283
All dimensions in mm
10. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 10
19
Solution of Example 1
uncrackedissectionthekN.m,34.0At
kN.m8.38
N.mm108.38
225
1028.283.3
:)(MomentCracking
mm1028.2450300
12
1
12
1
MPa83.3307.07.0
:ruptureofModulus
6
9
4933
'
M
M
y
If
M
M
bhI
ff
cr
t
gr
cr
cr
g
cr
19
375
450
300
283
Solution of Example 1 (continued)
20
uncracked.wellissectionThe
MPa83.3307.07.0ruptureofModulus
MPa35.3
1028.2
225)1034(
225
2
:fibersextremetwoAt the
mm1028.2450300
12
1
12
1
:StressesBending
'
9
6
4933
r
cr
g
g
g
ff
ff
I
My
f
mm
h
y
bhI
I
My
f
An uncracked beam is assumed to be homogeneous with neutral
axis passing through the centroid of the beam section.
Bending stress is given by the standard flexural equation.
375
450
300
283
11. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 11
Calculation of elastics stresses
in cracked sections
• When bending moment is sufficiently large to cause tensile stress
in the extreme fibers , greater than the modulus of rupture, it is
assumed that all of the concrete on the tensile side of the beam is
cracked and must be neglected in the flexure calculations.
• Cracking of the beam does not necessarily mean that the beam is
going to fail. The reinforcing bars on the tension side begin to pick
up the tension caused by the applied moment.
• Assumption of perfect bond is made between reinforcing bars
and concrete. Thus the strain in the concrete and in the steel will
be the same at equal distances from the neutral axis.
• Although the strains in the two materials at a particular point are
the same, their stresses can not be the same as they have
different values of elastic modulus.
21
Modular Ratio (n)
22
steelofthattimesbewillconcreterequiredofArea
required?bewilltwotheofareaswhatconcreteorsteelbyresistedbetoisforceIf
with
nnAA
n
A
A
A
F
n
A
F
nff
F
E
E
nnfff
E
E
f
E
f
E
f
sc
s
c
cs
cs
c
s
csc
c
s
s
s
s
c
c
sc
Definition: The ratio of the steel modulus to the concrete modulus
is called the modular ratio.
If the modular ratio for a beam is 10, stress in steel will be 10 times
the stress in concrete at the same distance from the neutral axis.
Otherwise: when n = 10, 500 mm2 of steel will carry the same total
force as 5000 mm2 of concrete.
At the same point, steel and concrete strains are equal and the
stresses are proportional to their modulus of elasticity.
12. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 12
Transformed Area Method
• When the steel area As is replaced with an equivalent area of
fictitious concrete (nAs), which supposedly can resist tension,
the area is referred to as the transformed area.
• As the transformed area is of concrete only, it can be analyzed
by the usual methods for elastic homogeneous beams.
23
Tensile concrete is
ignored
Tensile steel area As
replaced by equivalent
concrete area nAs
N.A.
cf
sA
n
fs
snA
Steps in transformed area method
1. Locate the neutral axis:
– Assume it is located a distance x from the compression
surface of the beam.
– Equate the first moment of the compression area of the beam
cross section about the neutral axis to the first moment of
the tensile area about the neutral axis.
– Solve the resulting quadratic equation.
2. Calculate the moment of inertia of the transformed section
wit respect to the neutral axis.
3. Compute the stresses in the concrete and the steel with the
flexure formula ( f = My/I ).
24
13. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 13
Example 2: Transformed Area Method
• (a) Calculate the bending stresses in the beam shown below
using the transformed area method
• fc’ = 20 MPa, n = 9 and M = 95 kN.m
• (b) Determine the allowable resisting moment of the beam, if
the allowable stresses are : fac = 10 MPa and fas = 140 MPa
25
425
500
300
283
All dimensions in mm
26
Solution:
425
500
300
)1846(
283
sA
All dimensions in mm
300
425
x
)425( x
2
mm16614snA
49
2323
2
22
22
mm10571.1
168.5)(425(16614)168.5300
3
1
)(425)(168.5300
3
1
:axis)neutral(aboutInertiaofMoment
mm5.168issolutionPositive
84.44778.200559470734)76.110(
04707376.1100706095016614150
166147060950150)425(16614150)425(
2
)(300
:)isfibertopfromdepthaxis(neutralaxisneutralaboutmomentsfirstTaking
I
xnAI
x
xxxx
xxxxxnA
x
x
x
s
s
14. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 14
27
MPa6.139
10571.1
5.256)1095(
9
:mm)256.5168.5-425(levelSteel
MPa2.10
10571.1
5.168)1095(
:mm)168.5(ncompressioExtreme
:StressesBending
9
6
9
6
I
My
nf
I
My
n
f
fy
I
My
fy
I
My
f
s
s
c
c
kN.m2.95N.mm102.95
5.2569
10571.1140)/(
mm256.5168.5-425:levelsteelAt the
:steelintensionallowabletoingcorrespondmomentAllowable
kN.m2.93N.mm102.93
5.168
10571.110
:concreteinncompressioallowabletoingcorrespondmomentAllowable
:momentresistingAllowable(b)
6
9
6
9
ny
If
y
Inf
M
y
y
If
M
y
If
M
I
yM
f
asas
as
ac
ac
allowable
allowable
allowable
allowable
Solution – Cont.
28
Concrete and steel will reach their permissible stresses at these
moments respectively.
Discussion:
Concrete and steel will not reach their maximum allowable stresses
at exactly the same bending moments.
The resisting moment of the section is 93.2 kNm (the smallest)
because if that value is exceeded, concrete becomes overstressed
even though the steel stress is less than its allowable stress.
kN.m2.95
kN.m2.93
:momentsresistingAllowable
as
ac
M
M
Solution – Cont.
15. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 15
2929
d
h
b
sA
b
h
x
)( xd
snA
I
xdM
nfxdy
I
Mx
fxy
xnA
bx
Ix
xnAdnA
bx
xdnA
bx
xdnA
x
x
x
s
c
s
ssss
)(
:)(stresstensionSteel
:)(stressncompressioconcreteTop
)(d)(
3
:NA)(aboutinertiaofMomentforSolve
0
2
)(
2
)(
2
)(b
:)isfibertopfromdepthaxis(neutralaxisneutralaboutmomentsFirst
2
3
22
Steps for a General Rectangular Section
Ultimate State and Nominal Flexural Moment
• After concrete compression stresses exceed about 50%
of fc’, they no longer vary linearly. The variation is
rather nonlinear as shown in Figure (b)
30
d
yf
b
sA
c
(a) Beam
(b) Actual compression
stress variation
Neutral axis depth
is now noted c
16. 24/2/2013
CE 370 : Prof. Abdelhamid Charif 16
31
The ultimate flexural moment capacity is called “Nominal”
moment and is noted Mn. It is derived using this assumption:
The curved compression diagram can be replaced by a
rectangular one with a constant stress of 0.85 fc’ (Whitney).
The Whitney rectangular stress block of depth “a” is assumed to
have the same area and same centre of gravity as the curved
diagram (generating the same force and the same moment).
d
yf
b
sA
c ca 1
abfC c
'
85.0
'
85.0 cf
(a) Beam
(b) Actual block of
compression stress
(c) Whitney block of
compression stress
ys fAT
Ultimate State and Nominal Flexural Moment
32
Apart from the previous standard assumptions (sections remain
plane after bending, perfect bond between concrete and steel bars,
concrete tensile strength ignored), two other assumptions are added
in the ultimate stage:
• Concrete is assumed to fail (crush) when its compressive strain
at the extreme fiber reaches its ultimate value εcu = 0.003
• The nonlinear compressive stress block for concrete is replaced
by an equivalent rectangular one with constant stress of 0.85 fc’
and a depth a = 1 c
Additional assumptions at ultimate state
MPafforf
MPaffor
cc
c
3065.0,008.009.1Max
3085.0
1
1 shall not be smaller than 0.65
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CE 370 : Prof. Abdelhamid Charif 17
33
Factor 1 to account for skewed shape
of compression stress block
Question: Why does 1 vary (reduced for fc’ > 30 MPa) ?
MPafforf
MPaffor
cc
c
3065.0,008.009.1Max
3085.0
1
34
Variation of coefficient 1
Observe skewed shape of compression
stress-strain curves for higher strength
The shapes of the stress-
strain curves for high
strength concrete are
skewed. This shifts the
centroid of the area
(below the curve) closer
to the origin, and thus
reduces the length of the
equivalent rectangle.
Thus 1 must be smaller
for high strength
concrete.
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CE 370 : Prof. Abdelhamid Charif 18
35
There are three types of flexural failure of a structural
member:
1. Steel reaches its yield strength before concrete reaches
its ultimate strain of 0.003 (Under-reinforced section)
2. Steel reaches yield limit at the same time as concrete
reaches its ultimate strain (Balanced section).
3. Concrete fails before the yield of steel, in the case of
presence of a high percentage of steel in the section
(Over-reinforced section).
Types of ultimate flexural failure
36
Steel reaches its yield strength before concrete reaches its ultimate
strain. Large deformations occur before concrete crushing. Failure
is ductile. After yielding, the steel tensile force remains constant.
To maintain an equal equilibrating concrete compression force, the
neutral axis is shifted up. Cracks will also progress upwards.
Under-reinforced section
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CE 370 : Prof. Abdelhamid Charif 19
37
Steel reaches its yield limit at the same time as concrete
reaches the ultimate strain. Failure is sudden and brittle
Balanced section
38
Concrete crushes before the yield of steel, due to the
presence of a high percentage of steel in the section.
Failure is very sudden and brittle
Over-reinforced section
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CE 370 : Prof. Abdelhamid Charif 20
39
Moment-Curvature for various steel areas
(Increasing steel area reduces ductility)
Types of flexural failure
• Brittle failures are very dangerous and must be
avoided.
• All codes of practice (ACI, SBC, …) require beams to
be designed as under-reinforced to ensure ductile
behavior.
• In fact SBC and ACI codes require more ductility than
just under reinforced.
40