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.
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.
1) The document discusses the analysis of flanged beam sections like T-beams and L-beams. It covers topics like effective flange width, positive and negative moment regions, and ACI code provisions for estimating effective flange width.
2) Examples are provided for analyzing a T-beam and an L-beam section. This includes calculating the effective flange width, checking steel strain, minimum reinforcement requirements, and computing nominal moments.
3) Reinforcement limitations for flange beams are also outlined, covering requirements for flanges in compression and tension.
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 is the sixth edition of the National Structural Code of the Philippines (NSCP) Volume I, which provides requirements for designing buildings, towers, and other vertical structures. It was published in 2010 by the Association of Structural Engineers of the Philippines. The code contains chapters on minimum design loads, materials, and other topics to guide structural design in compliance with the latest standards. The foreword expresses pride in the publication and updates to the code to regulate structural design for safety.
The document discusses different methods of designing concrete structures, focusing on the limit state method. It describes the limit state method's goal of achieving an acceptable probability that a structure will not become unsuitable for its intended use during its lifetime. The document then discusses stress-strain curves for concrete and steel. It covers stress block parameters and equations for calculating the depth of the neutral axis and moment of resistance for singly reinforced concrete beams. The document concludes by providing examples of analyzing an existing beam section and designing a new beam section.
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.
1) The document discusses the analysis of flanged beam sections like T-beams and L-beams. It covers topics like effective flange width, positive and negative moment regions, and ACI code provisions for estimating effective flange width.
2) Examples are provided for analyzing a T-beam and an L-beam section. This includes calculating the effective flange width, checking steel strain, minimum reinforcement requirements, and computing nominal moments.
3) Reinforcement limitations for flange beams are also outlined, covering requirements for flanges in compression and tension.
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 is the sixth edition of the National Structural Code of the Philippines (NSCP) Volume I, which provides requirements for designing buildings, towers, and other vertical structures. It was published in 2010 by the Association of Structural Engineers of the Philippines. The code contains chapters on minimum design loads, materials, and other topics to guide structural design in compliance with the latest standards. The foreword expresses pride in the publication and updates to the code to regulate structural design for safety.
The document discusses different methods of designing concrete structures, focusing on the limit state method. It describes the limit state method's goal of achieving an acceptable probability that a structure will not become unsuitable for its intended use during its lifetime. The document then discusses stress-strain curves for concrete and steel. It covers stress block parameters and equations for calculating the depth of the neutral axis and moment of resistance for singly reinforced concrete beams. The document concludes by providing examples of analyzing an existing beam section and designing a new beam section.
The document provides derivations of design equations for reinforced concrete beams. It begins by deriving the equation for maximum moment capacity of a singly reinforced beam based on concrete strength as M=0.167*fck*b*d^2. It then derives equations for doubly reinforced beams where compression steel is also required. The document further derives equations for design of flanged beams depending on whether the neutral axis lies within the flange or web. It concludes by outlining design procedures for singly and doubly reinforced beams.
Approximate analysis methods make simplifying assumptions to determine preliminary member forces and dimensions for indeterminate structures. Case 1 assumes diagonals cannot carry compression and shares shear between diagonals. Case 2 allows compression in diagonals. Portal and cantilever methods analyze frames by dividing into substructures at assumed hinge locations, solving each sequentially from top to bottom.
This document provides 10 examples of problems related to bearing capacity of foundations. The examples calculate bearing capacity using Terzaghi's analysis for different soil and foundation conditions, including cohesionless and cohesive soils, square and strip footings, and considering the water table depth. One example compares results to field plate load tests. The solutions show calculations for determining soil shear strength parameters, factor of safety, and safe bearing capacity.
A group of 16 square piles extends 12 m into stiff clay soil, underlain by rock at 24 m depth. Pile dimensions are 0.3 m x 0.3 m. Undrained shear strength of clay increases linearly from 50 kPa at surface to 150 kPa at rock. Factor of safety for group capacity is 2.5. Determine group capacity and individual pile capacity.
The group capacity is calculated to be 1600 kN. The individual pile capacity is determined to be 100 kN. The factor of safety of 2.5 is then applied to determine the safe load capacity.
The document provides examples of classifying soils using the AASHTO and USCS soil classification systems. Key steps include determining the particle size distribution, plasticity characteristics (liquid limit, plastic limit, plasticity index), and using this data on classification charts to identify the appropriate soil type symbols. Soils are classified as sand, silt, clay or combinations based on their grain size and plasticity properties.
Name: Sadia Mahajabin
ID : 10.01.03.098
4th year 2nd Semester
Section : B
Department of Civil Engineering
Ahsanullah University of Science and Technology
This document provides information on the structural design of a simply supported reinforced concrete beam. It includes:
- A list of students enrolled in an elementary structural design course.
- Equations and diagrams showing the forces and stresses in a reinforced concrete beam with a singly reinforced bottom section.
- Limits on the maximum depth of the neutral axis according to the grade of steel.
- Examples of analyzing the stresses and determining steel reinforcement for a given beam cross-section.
- A design example calculating the dimensions and steel reinforcement for a rectangular beam with a factored uniform load.
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.
This document discusses structural analysis of cables and arches. It provides examples of determining tensions in cables subjected to concentrated and uniform loads. It also discusses the analysis procedure for cables under uniform loads. Examples are given for calculating tensions at different points of cables supporting bridges. Methods for analyzing fixed and hinged arches are demonstrated through examples finding internal forces at various arch sections.
This document provides an example of designing a rectangular reinforced concrete beam. It includes calculating the loads, bending moment, required tension reinforcement, checking shear capacity and deflection. For a simply supported beam with a uniformly distributed load, the document calculates the steel reinforcement area required using formulas and tables. It then checks that the beam satisfies requirements for shear capacity, minimum and maximum steel ratios, and deflection. The document also provides an example of designing a doubly reinforced beam.
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.
This document outlines a course on principles of reinforced concrete design according to the National Structural Code of the Philippines 2015. The course covers topics like analysis and design of beams, T-beams, columns, slabs, and seismic design provisions. Chapter 1 introduces reinforced concrete components, the advantages and disadvantages of concrete, design codes and notations, concrete properties, and load combinations. It also provides examples of calculating the minimum spacing between reinforcing bars for efficient rectangular beam sections.
The document provides 8 examples of calculating total stress, effective stress, and pore water pressure at different depths for various soil profiles. The examples solve for the stresses and pressures at specific points or depths by considering the layer thicknesses, soil unit weights, depth of water table, and degree of saturation. The effective stress is calculated by subtracting the pore water pressure from the total stress at each point.
This document provides information on reinforced concrete design including:
- Concrete and steel properties such as modulus of elasticity and grades/strengths of reinforcing bars.
- Minimum concrete cover requirements for reinforcement.
- Load factors and combinations for ultimate strength design.
- Flexural design procedures for reinforced concrete beams including assumptions, stress/strain diagrams, and analysis for cases where steel yields or does not yield.
- Requirements for reinforcement spacing, minimum member thicknesses, and ductility.
Solution Manual for Structural Analysis 6th SI by Aslam Kassimaliphysicsbook
https://www.unihelp.xyz/solution-manual-structural-analysis-kassimali/
Solution Manual for Structural Analysis - 6th Edition SI Edition
Author(s): Aslam Kassimali
Solution Manual for 6th SI Edition (above Image) is provided officially. It include all chapters of textbook (chapters 2 to 17) plus appendixes B, C, D.
This document discusses T-beams, which are more suitable than rectangular beams in reinforced concrete. There are two types of T-beams: monolithic and isolated. It provides notations and code recommendations for T-beams from IS: 456. There are three cases for finding the depth of the neutral axis in a T-beam: when it lies in the flange, in the rib, or at the junction. An example problem is worked through to find the moment of resistance for a given T-beam section using the provided concrete and steel properties.
1. Influence lines represent the variation of reaction, shear, or moment at a specific point on a structural member as a concentrated load moves along the member. They are useful for analyzing the effects of moving loads.
2. To construct an influence line, a unit load is placed at different points along the member and the reaction, shear, or moment is calculated at the point of interest using statics. The values are plotted to show the influence of the load.
3. Influence lines allow engineers to determine the maximum value of a response (reaction, shear, moment) caused by a moving load and locate where on the structure that maximum occurs.
Approximate Analysis of Statically Indeteminate Structures.pdfAshrafZaman33
This document discusses approximate analysis methods for statically indeterminate structures like trusses and frames. It introduces the concept of making a structure statically determinate by assuming certain members carry zero force, then analyzing the simplified structure. Two common assumptions are mentioned: (1) slender diagonals cannot carry compression, and (2) diagonals share the panel load equally between tension and compression. Examples show how to apply these methods to trusses. The document also discusses analyzing building frames by assuming points of inflection act as pins, then using superposition to account for multiple load cases.
This presentation discusses the T-beam design method using the Working Stress Design (WSD) approach for singly and doubly reinforced beams. T-beams have a monolithically cast slab that acts as part of the beam and resists longitudinal compression in positive moment zones. The WSD method designs structures such that all nominal stresses remain in the elastic limit. Singly reinforced beams only have rebar in the tension zone, while doubly reinforced beams require additional rebar in the compression zone to resist the maximum moment. The design procedure for T-beams involves determining the bending moment, section properties, stress limits and distribution, and sizing of reinforcement.
This document discusses the design of singly and doubly reinforced concrete T-beams. It provides definitions of effective flange width for T-beams based on ACI 318 specifications. The document describes how to analyze T-beams as rectangular or T-shaped sections depending on the location of the neutral axis. It presents methods for calculating the nominal moment capacity for T-beams based on whether the neutral axis is within the flange or web. Limitations on reinforcement ratios for flanges are also provided.
The document provides derivations of design equations for reinforced concrete beams. It begins by deriving the equation for maximum moment capacity of a singly reinforced beam based on concrete strength as M=0.167*fck*b*d^2. It then derives equations for doubly reinforced beams where compression steel is also required. The document further derives equations for design of flanged beams depending on whether the neutral axis lies within the flange or web. It concludes by outlining design procedures for singly and doubly reinforced beams.
Approximate analysis methods make simplifying assumptions to determine preliminary member forces and dimensions for indeterminate structures. Case 1 assumes diagonals cannot carry compression and shares shear between diagonals. Case 2 allows compression in diagonals. Portal and cantilever methods analyze frames by dividing into substructures at assumed hinge locations, solving each sequentially from top to bottom.
This document provides 10 examples of problems related to bearing capacity of foundations. The examples calculate bearing capacity using Terzaghi's analysis for different soil and foundation conditions, including cohesionless and cohesive soils, square and strip footings, and considering the water table depth. One example compares results to field plate load tests. The solutions show calculations for determining soil shear strength parameters, factor of safety, and safe bearing capacity.
A group of 16 square piles extends 12 m into stiff clay soil, underlain by rock at 24 m depth. Pile dimensions are 0.3 m x 0.3 m. Undrained shear strength of clay increases linearly from 50 kPa at surface to 150 kPa at rock. Factor of safety for group capacity is 2.5. Determine group capacity and individual pile capacity.
The group capacity is calculated to be 1600 kN. The individual pile capacity is determined to be 100 kN. The factor of safety of 2.5 is then applied to determine the safe load capacity.
The document provides examples of classifying soils using the AASHTO and USCS soil classification systems. Key steps include determining the particle size distribution, plasticity characteristics (liquid limit, plastic limit, plasticity index), and using this data on classification charts to identify the appropriate soil type symbols. Soils are classified as sand, silt, clay or combinations based on their grain size and plasticity properties.
Name: Sadia Mahajabin
ID : 10.01.03.098
4th year 2nd Semester
Section : B
Department of Civil Engineering
Ahsanullah University of Science and Technology
This document provides information on the structural design of a simply supported reinforced concrete beam. It includes:
- A list of students enrolled in an elementary structural design course.
- Equations and diagrams showing the forces and stresses in a reinforced concrete beam with a singly reinforced bottom section.
- Limits on the maximum depth of the neutral axis according to the grade of steel.
- Examples of analyzing the stresses and determining steel reinforcement for a given beam cross-section.
- A design example calculating the dimensions and steel reinforcement for a rectangular beam with a factored uniform load.
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.
This document discusses structural analysis of cables and arches. It provides examples of determining tensions in cables subjected to concentrated and uniform loads. It also discusses the analysis procedure for cables under uniform loads. Examples are given for calculating tensions at different points of cables supporting bridges. Methods for analyzing fixed and hinged arches are demonstrated through examples finding internal forces at various arch sections.
This document provides an example of designing a rectangular reinforced concrete beam. It includes calculating the loads, bending moment, required tension reinforcement, checking shear capacity and deflection. For a simply supported beam with a uniformly distributed load, the document calculates the steel reinforcement area required using formulas and tables. It then checks that the beam satisfies requirements for shear capacity, minimum and maximum steel ratios, and deflection. The document also provides an example of designing a doubly reinforced beam.
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.
This document outlines a course on principles of reinforced concrete design according to the National Structural Code of the Philippines 2015. The course covers topics like analysis and design of beams, T-beams, columns, slabs, and seismic design provisions. Chapter 1 introduces reinforced concrete components, the advantages and disadvantages of concrete, design codes and notations, concrete properties, and load combinations. It also provides examples of calculating the minimum spacing between reinforcing bars for efficient rectangular beam sections.
The document provides 8 examples of calculating total stress, effective stress, and pore water pressure at different depths for various soil profiles. The examples solve for the stresses and pressures at specific points or depths by considering the layer thicknesses, soil unit weights, depth of water table, and degree of saturation. The effective stress is calculated by subtracting the pore water pressure from the total stress at each point.
This document provides information on reinforced concrete design including:
- Concrete and steel properties such as modulus of elasticity and grades/strengths of reinforcing bars.
- Minimum concrete cover requirements for reinforcement.
- Load factors and combinations for ultimate strength design.
- Flexural design procedures for reinforced concrete beams including assumptions, stress/strain diagrams, and analysis for cases where steel yields or does not yield.
- Requirements for reinforcement spacing, minimum member thicknesses, and ductility.
Solution Manual for Structural Analysis 6th SI by Aslam Kassimaliphysicsbook
https://www.unihelp.xyz/solution-manual-structural-analysis-kassimali/
Solution Manual for Structural Analysis - 6th Edition SI Edition
Author(s): Aslam Kassimali
Solution Manual for 6th SI Edition (above Image) is provided officially. It include all chapters of textbook (chapters 2 to 17) plus appendixes B, C, D.
This document discusses T-beams, which are more suitable than rectangular beams in reinforced concrete. There are two types of T-beams: monolithic and isolated. It provides notations and code recommendations for T-beams from IS: 456. There are three cases for finding the depth of the neutral axis in a T-beam: when it lies in the flange, in the rib, or at the junction. An example problem is worked through to find the moment of resistance for a given T-beam section using the provided concrete and steel properties.
1. Influence lines represent the variation of reaction, shear, or moment at a specific point on a structural member as a concentrated load moves along the member. They are useful for analyzing the effects of moving loads.
2. To construct an influence line, a unit load is placed at different points along the member and the reaction, shear, or moment is calculated at the point of interest using statics. The values are plotted to show the influence of the load.
3. Influence lines allow engineers to determine the maximum value of a response (reaction, shear, moment) caused by a moving load and locate where on the structure that maximum occurs.
Approximate Analysis of Statically Indeteminate Structures.pdfAshrafZaman33
This document discusses approximate analysis methods for statically indeterminate structures like trusses and frames. It introduces the concept of making a structure statically determinate by assuming certain members carry zero force, then analyzing the simplified structure. Two common assumptions are mentioned: (1) slender diagonals cannot carry compression, and (2) diagonals share the panel load equally between tension and compression. Examples show how to apply these methods to trusses. The document also discusses analyzing building frames by assuming points of inflection act as pins, then using superposition to account for multiple load cases.
This presentation discusses the T-beam design method using the Working Stress Design (WSD) approach for singly and doubly reinforced beams. T-beams have a monolithically cast slab that acts as part of the beam and resists longitudinal compression in positive moment zones. The WSD method designs structures such that all nominal stresses remain in the elastic limit. Singly reinforced beams only have rebar in the tension zone, while doubly reinforced beams require additional rebar in the compression zone to resist the maximum moment. The design procedure for T-beams involves determining the bending moment, section properties, stress limits and distribution, and sizing of reinforcement.
This document discusses the design of singly and doubly reinforced concrete T-beams. It provides definitions of effective flange width for T-beams based on ACI 318 specifications. The document describes how to analyze T-beams as rectangular or T-shaped sections depending on the location of the neutral axis. It presents methods for calculating the nominal moment capacity for T-beams based on whether the neutral axis is within the flange or web. Limitations on reinforcement ratios for flanges are also provided.
This document provides information about T-beams, which are concrete beams that support slabs integrated monolithically. It discusses how the slab acts as a top flange for the beam. Various T-beam geometries are shown, including methods for analyzing flanged sections. The document outlines provisions for estimating the effective flange width and describes the design of singly and doubly reinforced T-beams using working stress and ultimate strength methods. Reasons for including compression reinforcement are also provided.
As-salamu alaykum
Welcome to the presentation on “T Beam Design: Singly & Doubly by USD method” Presented By -
S. M. Rahat Rahman
ID: 10.01.03.104
1.Contents :
USD (Ultimate Strength Design Method)
T-beam
T - Beam acts Like Singly Reinforced Beam
T – Action vs rectangular Action
Effective Flange width of t-beam
Strength analysis
Nominal moment for t section
2. USD : Based on the ultimate strength of the structure member assuming a failure condition , due to concrete crushing or yielding of steel. Although there is additional strength of steel after yielding (strain hardening zone) which will not be considered in the design.
Actual loads are multiplied by load factor to obtain the ultimate design loads. ACI code emphasizes this method.
3. T Beam : For monolithically casted slabs, a part of a slab act as a part of beam to resist longitudinal compressive force in the moment zone and form a T-Section. This section form the shape of a "T“ . It can resist the longitudinal compression
4. Occurrence and Configuration of T-Beams
• Common construction type
• The slab forms the beam flange, while the part of the beam projecting below the slab forms is what is called web or stem.
5. Singly Reinforced Reinforcement is provided in tension zone only
6. Doubly Reinforced > Concrete can not develop the required compressive force to resist the maximum bending moment
> Reinforcement is provided in both compression and tension zone.
7. T-Beam Act As a Singly Reinforced Beam
8. Continuous T Beam :
When T-shaped sections are subjected to negative bending moments, the flange is located in the tension zone. Since concrete strength in tension is usually neglected in strength design, the sections are treated as rectangular sections.
On the other hand, when sections are subjected to positive bending moments, the flange is located in the compression zone and the section is treated as a T-section.
9. Effective Flange Width
10. Strength analysis of T beam
11. Analysis of T beam
12. T Beam moment calculation
This presentation discusses the design of T beams using the Working Stress Design (WSD) method. It explains that T beams have slabs cast monolithically with beams to act as part of the beam and resist longitudinal compression. The presentation covers designing T beams as singly or doubly reinforced and calculating their moment capacity and steel area based on allowing stresses in concrete and steel to remain in the elastic range.
El documento presenta una clase sobre diseño de estructuras de concreto reforzado. Se discuten temas como el detalle de refuerzo, diseño de losas macizas y vigas T, y losas aligeradas. Se explican conceptos como cortes de fierro, diseño para flexión y corte, y dimensionamiento de vigas T. El documento proporciona ejemplos y fórmulas clave para el análisis y diseño de estas estructuras de concreto armado.
The document is a technical report on structural analysis and load calculations for beams and columns. It was written by Engineer Guido Rodriguez Molina in 2011 in Tacna, Peru as a courtesy to Engineer Jose Acero Martinez. The bulk of the document contains repetitive text on beam calculations and column calculations.
T-Beam Design by USD method-10.01.03.102Sadia Mitu
This document defines and describes T-beams, which are concrete beams with a flange formed by a monolithically cast slab. It provides definitions of T-beams, explaining that the slab acts as a compression flange while the web below resists shear and separates bending forces. The document outlines the ultimate strength design method and effective flange width concept used in T-beam analysis and design. It then presents the design procedure for T-beams, discussing analysis of positive and negative bending moments as well as singly and doubly reinforced beams. Advantages and disadvantages of T-beams are listed at the end.
Análisis y diseño de Vigas de Concreto armadoMiguel Sambrano
Los elementos estructurales sujetos a flexión, son principalmente las vigas y losas. La flexión puede presentarse acompañada de fuerza cortante. Sin embargo, la resistencia a flexión puede estimarse despreciando el efecto de la fuerza cortante.
Para el diseño de secciones a flexión, se usa el Estado Límite de Agotamiento Resistente, donde la resistencia de agotamiento se minora multiplicando por un factor correspondiente; Comparando luego con la demanda o carga real modificada por los factores de mayoración. La norma usada es la COVENIN 1753.
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.
The document discusses the analysis and design of reinforced concrete T-beams and L-beams according to the ACI code. It provides equations to determine the effective flange width of T-beams and L-beams. It then describes the analysis procedure which involves checking code requirements, calculating the depth of the concrete compression block, and determining if the neutral axis falls within the flange or web. The analysis considers the moments contributed by the flange and web portions. Design examples are also provided to demonstrate the process.
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 discusses concepts related to the design of concrete beams including:
1. It introduces concepts like bending, shear, tension and compression as they relate to beam design.
2. It provides formulas for calculating reactions, shear forces, and bending moments in simply supported beams under different loading conditions.
3. It explains concepts like the neutral axis, stress blocks, and strain diagrams that are important to beam design.
4. It discusses factors that influence the strength of beams like the moment of inertia and reinforcement ratio.
5. It compares working stress and limit state methods of design.
This document summarizes lecture content on the flexural analysis and design of reinforced concrete T-beams and L-beams. It discusses the effective width of flanges, flexural behavior cases where the flange is in tension or compression, and analysis approaches depending on whether the neutral axis lies within or outside the flange. Procedures are provided for checking tension-controlled failure, calculating flexural capacity from the compression and tension sides, and designing beams by selecting trial dimensions, calculating reinforcement ratios, and detailing. An example is given to design an interior T-beam under given loading and moment capacity conditions.
Lec 20 21-22 -flexural analysis and design of beams-2007-rCivil Zone
This document summarizes lecture content on the flexural analysis and design of reinforced concrete T-beams and L-beams. It discusses the effective width of flanges, flexural behavior cases where the flange is in tension or compression, and analysis approaches depending on whether the neutral axis lies within or outside the flange. Formulas are provided for calculating forces, moments, and steel ratios. The design process involves selecting beam dimensions, determining the effective width, checking the neutral axis position, and iterating as needed to satisfy strength and serviceability requirements. An example problem demonstrates the full T-beam design process.
This document provides information on analysis and design of reinforced concrete beams. It discusses key concepts such as modular ratio, neutral axis, stress diagrams, and types of reinforcement. It also defines under-reinforced, balanced, and over-reinforced beam sections. Several examples are provided to illustrate determination of neutral axis depth, moment of resistance, steel percentage, and stresses in concrete and steel reinforcement. Design aspects like maximum load capacity are also explained through examples.
Lec 10-flexural analysis and design of beamnsCivil Zone
The document discusses the concepts of balanced steel ratio, tension controlled sections, transition sections, compression controlled sections, strength reduction factors, maximum steel ratio, and minimum reinforcement for flexural members in reinforced concrete beams. The balanced steel ratio corresponds to the amount of steel that yields at the same time as the concrete crushes. Tension controlled sections have a steel strain over 0.005 when concrete strain is 0.003. Transition sections have steel strain between yield and 0.005 when concrete is at 0.003. Compression controlled sections have steel strain under yield when concrete is at 0.003.
The document discusses the concepts of balanced steel ratio, tension controlled sections, transition sections, compression controlled sections, strength reduction factors, maximum steel ratio, and minimum reinforcement for flexural members in reinforced concrete beams. The balanced steel ratio corresponds to the amount of steel that yields at the same time as the concrete crushes. Tension controlled sections have a steel strain over 0.005 when concrete strain is 0.003. Transition sections have steel strain between yield and 0.005 when concrete is at 0.003. Compression controlled sections have steel strain under yield when concrete is at 0.003.
This document discusses composite construction, specifically composite steel and concrete beams. It provides definitions and examples of composite construction, explaining that it aims to make each material perform the function it is best suited for. It then describes the differences between non-composite and composite beam behavior. The document goes on to discuss elements of composite construction like decking and shear studs. It also summarizes the design process for composite beams, covering moment capacity, shear capacity, shear connector capacity, and longitudinal shear capacity calculations.
Here are the key steps to solve this problem:
1) Check if a < t condition is satisfied. Here a = c = d - As/bw*fy/0.85fc = 300 - As/200*414/0.85*20.7
2) Use the formula for balanced steel ratio: ρb = 0.85*fc/(d-a/2)*bw/fy
3) Solve for As = ρb*bw*d
4) Check strain compatibility
5) Report required As
The required steel area is As = 145 cm^2
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.
rectangular and section analysis in bending and shearqueripan
The document discusses the design of reinforced concrete beams for bending and shear. It covers the analysis of singly and doubly reinforced rectangular beam sections. Key points covered include the concept of neutral axis, under-reinforced and over-reinforced sections, design of bending reinforcement, design of shear reinforcement including link spacing, and deflection criteria. Worked examples are provided to demonstrate the design of bending and shear reinforcement for rectangular beams.
The document discusses bolted connections. It begins by explaining that all components of a bolted connection must be verified, including shear resistance of the beam web, compression resistance of the beam web, tension resistance of the beam web, bending resistance of the beam flanges, bending resistance of the cover plate, and compression resistance of the beam flange and web. It then describes the different potential failure modes for bolted connections, including failure due to crushing of the plate material in the bolt holes, shear failure, tension failure of the connected plates, tension failure of the bolt, and combined shear and tension failure. Finally, it provides tables summarizing classification of bolted connection categories according to how the bolts are loaded, as well as
Design of Beam- RCC Singly Reinforced BeamSHAZEBALIKHAN1
Concrete beams are an essential part of civil structures. Learn the design basis, calculations for sizing, tension reinforcement, and shear reinforcement for a concrete beam.
Lecture 5 s.s.iii Design of Steel Structures - Faculty of Civil Engineering IaşiUrsachi Răzvan
1) The document discusses various types of column designs for industrial buildings, including columns with constant or variable cross-sections, built-up or compound cross-sections, and stiffening elements.
2) It provides details on column base designs like hinged bases, fixed bases, and bases with gusset plates. Hold-down bolts, shear lugs, and resistance to combined forces are also examined.
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- Strain and stress diagrams are presented, showing cracked concrete and steel stresses and strains.
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- Minimum beam depths are provided to avoid requiring deflection calculations for preliminary design.
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This document provides information on the flexural analysis and design of reinforced concrete beams based on the ultimate strength design method. It discusses under-reinforced and over-reinforced failure modes. For under-reinforced beams, it describes the three stages of loading: uncracked stage, cracked stage, and steel yielding stage. Equations are derived for calculating the reinforcement ratio ρ. For over-reinforced beams, it discusses failure when the concrete reaches its strain limit before steel yields. The document also provides guidelines for determining if a section is under-reinforced or over-reinforced, and criteria for selection of the strength reduction factor Φ. In the end, it lists the data and outputs required for capacity analysis of a
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Lec06 Analysis and Design of T Beams (Reinforced Concrete Design I & Prof. Abdelhamid Charif)
1. 25-Feb-13
CE370: Prof. A. Charif 1
CE 370
REINFORCED CONCRETE-I
Prof. Abdelhamid Charif
Analysis and Design of T-Beams
• T-shaped beams are frequently used in structures
• There are two types of T-beams :
Beams directly cast and delivered as isolated
T-beams (especially in bridges)
T-shaped beams resulting from interaction of slabs
with beams (building slabs)
• The obvious advantage of T-beams is to reduce useless
tensile concrete quantity
T-Beams
2
2. 25-Feb-13
CE370: Prof. A. Charif 2
Isolated T-beams
T-Beams
Slab interaction
T-beams
3
A beam is considered as such even if it has a larger base (heel) to
accommodate tension steel.
The shape and size of concrete on the tension side, assumed to be
cracked, has no effect on the theoretical resisting moments.
T-Beams
Precast T-Beams for bridges
4
3. 25-Feb-13
CE370: Prof. A. Charif 3
T-Beams
RC floors normally consist of slabs and beams that are cast
monolithically. The two act together to resist loads and because of
this interaction, the effective section of the beam is a T or L section.
T-section for
interior beams
L-section for
exterior beams
5
T-Beams
The effective flange width resulting from slab interaction
depends on the slab type (one-way or two-way slab).
The present course focuses on one-way slabs.
6
4. 25-Feb-13
CE370: Prof. A. Charif 4
Exterior beam – Minimum of:
Beam tributary width
Web width plus six times slab
thickness
Web width plus one-twelfth
of beam span
Interior Beam – Minimum of:
Beam tributary width
Web width plus 16 times slab
thickness
One-quarter of beam span
Effective flange width in one way slabs
7
Beam tributary width
Beam tributary width is the transverse width of the slab
transmitting load to the beam (supported by the beam)
It is determined using mid-lines between beam lines
For exterior beams, offsets are included
Figure shows tributary widths for beams A and B
A
B
C
Lt
Lt
8
5. 25-Feb-13
CE370: Prof. A. Charif 5
Flange Dimensions
(Isolated T-Beams)
For an isolated T-beam:
The flange thickness may
not be less than one-half
the web width
The effective flange
width may not be larger
than four times the web
width
wf
w
f bb
b
h 4
2
9
Location of neutral axis and
Rectangular stress block depth a
• Neutral axis (N.A.) for T-beams can
fall either in the flange or in the web
• At ultimate state, it is the depth of
the rectangular stress block that
counts.
If the stress block falls in the flange, as is very frequent for
positive moments, the rectangular beam formulas apply.
In this case tensile concrete is assumed to be cracked, and its
shape has no effect (other than weight).
The section is analyzed as a rectangular one using the flange
width b = bf
10
6. 25-Feb-13
CE370: Prof. A. Charif 6
Location of rectangular
stress block (Contd.)
If the stress block falls in the
web, the compression area
does not consist of a single
rectangle, and the rectangular
beam design procedure does
not apply.
The section is in this case
divided in two parts as will be
seen later (decomposition
method)
11
Case of a negative moment
• If the T-section is subjected to a negative moment,
then the flange will be in tension
• The section can thus be analyzed as an inverted
rectangular one using the web width b = bw
12
7. 25-Feb-13
CE370: Prof. A. Charif 7
Minimum Steel for T-Beams
• Minimum steel quantity is imposed by codes for the
same reasons as in rectangular beams
db
ff
f
A w
yy
c
s
4.1
,
4
Max
'
min,
Exception: For a statically determinate T-beam subjected
to a negative moment (cantilever), the minimum steel
area is:
According to SBC/ACI:
db
f
f
A w
y
c
s
2
'
min,
13
CE 370
REINFORCED CONCRETE-I
Prof. Abdelhamid Charif
Analysis of T-Beams
8. 25-Feb-13
CE370: Prof. A. Charif 8
15
Analysis of T-Beams
Yielding of Tension Steel
• With a larger compression area from the flange, a T-
beam is usually under reinforced and tension steel
should yield before failure.
• We therefore first present the method of analysis in
case of steel yielding.
• Case of steel not yielding is presented later.
methodiondecompositusein web,Block:If
continueflange,inBlockOK:If
85.0
85.0:mequilibriuForce '
'
f
f
fc
ys
ysfc
ha
ha
bf
fA
afAabf
16
Location of rectangular stress block
in Analysis Case
• Assume stress block in
flange
• Express force equilibrium
• Deduce value of a and
check.
2
a
d
wb
sA
fb
fh
fcabf '
85.0
a
ys fA
9. 25-Feb-13
CE370: Prof. A. Charif 9
Analysis steps for T-Beams
nn
f
f
f
fc
ys
s
MM
c
cd
β
a
c
a
ha
b
ha
bf
fA
a
A
and,Calculate6.
caseyieldingnogotoyieldingnotsteelIf.5
003.0strainsteelanddepthaxisneutralCalculate.4
ofvaluenewfindtomethodiondecompositUse
webreachesblockStressIf
4goto,widthflangeusingsectionrrectangulaaas
analysiscontinueflange,inblockStressIf
thicknessflangedepth withblockCompare.3
85.0
depthitsdetermineandflangeinblockstressAssume2.
areasteelminimumCheck1.
s
1
'
min
17
Decomposition Method (1)
• When the stress block exceeds the flange, the section is divided:
T-section = W-section + F-section
• W-section = Compression part of the web (unknown depth a)
• F-section = Overhanging parts of the flange (known depth hf)
18
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
Total steel area and nominal moment are decomposed:
nwnfnswsfs MMMAAA
10. 25-Feb-13
CE370: Prof. A. Charif 10
Decomposition Method (2)
19
sfssw
y
fwc
sfysffwcf AAA
f
hbbf
AfAhbbfC
and
)(85.0
)(85.0
'
'
Force equilibrium in the flange part gives the two steel areas :
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
Force equilibrium in the web gives the compression block depth :
w
'
c
ysw
yswwwcw
bf.
fA
afATabfC
850
85.0 '
22
22
a
dfA
h
dfAMMM
a
dfAM
h
dfAM
ysw
f
ysfnwnfn
yswnw
f
ysfnf
Decomposition Method (3)
20
Using appropriate lever arms, the nominal moments are thus :
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
11. 25-Feb-13
CE370: Prof. A. Charif 11
Problem 1
m1.5iswidthtributaryBeamthick.mm100isthatslabfloorawith
integrallycastisspanm9withbeamTheMPa.420andMPa30
figure.in theshownbeamTinteriorofmomentdesignDetermine
y
'
c ff
21
600d
250
286
widthEffective
100
mmd
mmd
t 655
545min
Solution 1
mm1500
mm22504/90004Span /
mm18501001625016
1500widthTributary
Min
:widthFlangeEffective
bhb
mml
fw
t
OKAmm5.369428
4
6and
mm0.50060025000333.0,00326.0Max
600250
420
4.1
,
4204
30
Max
4.1
,
4
Max
Checking
mins
22
2
minmin
'
min
min
ss
ss
w
yy
c
s
s
AA
AA
db
ff
f
A
A
22
12. 25-Feb-13
CE370: Prof. A. Charif 12
:flangeinblockstressAssume
0.90controlTension005.0005.00313.0
72.47
72.47545
003.0003.0:checkStrain
mm72.47
85.0
56.40
flangeinisblockstressOKmm100a
mm56.40
15003085.0
4205.2694
85.0
min
min
min
1
'
t
f
fc
ys
εε
c
cd
ε
β
a
c
h
bf
fA
a
Solution 1 – Cont.
23
600d
250
286
mm1500widthEffective
100
mmd
mmd
t 655
545min
layerbottomatcontrol-nfor tensiocheckmustThen we
:005.0butIf:Note minmin y
mkNM
mkNmmNM
M
a
dfAM
n
n
n
ysn
.6.80955.8999.0:momentDesign
.55.899.1055.899
2
56.40
6004205.3694
2
:momentNominal
6
Solution 1 – Cont.
The T-beam can therefore resist any ultimate moment
equal to or less than 809.6 kN.m
24
600d
250
286
mm1500widthEffective
100
mmd
mmd
t 655
545min
13. 25-Feb-13
CE370: Prof. A. Charif 13
Problem 2
Analysis of a T-section with six 20-mm
bars in two layers as shown.
Net layer spacing Sl = 30 mm
Stirrup diameter = 10 mm
MPafMPaf yc 42020'
Steel depths :
mmddmmdd
mm
dd
AA
AdAd
d
mmdSdd
mmd
d
hdd
t
ss
ss
bl
s
b
t
490540
515
2
49050540)2030(540)(
54060600)10
2
20
40(600)
2
cover(
2min1
21
21
2211
12
1
600
75
525
300
Total steel area is : 2
2
96.1884
4
20
6 mmAs
It is greater than the minimum steel area:
2
min
'
min
515515300
420
4.1
515300
420
4.1
,515300
4204
20
Max
4.1
,
4
Max
mm
x
A
db
f
db
f
f
A
s
w
y
w
y
c
s
Solution 2
First assume compression block in the flange (a ≤ hf ) and
if true analyze as a rectangular section (bf , h).
If not, use decomposition method.
14. 25-Feb-13
CE370: Prof. A. Charif 14
Force equilibrium C = T gives the compression block depth is :
mm
bf
fA
a
fc
ys
616.77
6002085.0
42096.1884
85.0 '
This value is greater than the flange thickness a > hf
Compression block is thus in the web
Use decomposition method:
T-section = W-section + F-section
nfnwn
sfsws
MMM
AAA
2
2
'
246.974
714.910
420
75)300600(2085.0
and
85.0
mmAAA
mmA
AAA
f
hbbf
A
sfssw
sf
sfssw
y
fwfc
sf
Solution 2 – Cont.
mm
a
c
hamma
bf
fA
afAabf
f
wc
ysw
yswwc
39.94
85.0
232.80
)(232.80
3002085.0
420246.974
85.0
85.0
1
'
'
New value of compression
block depth is obtained
from the force equilibrium
in the web part :
control-Tension005.0005.0
0021.001257.0
39.94
39.94490
003.0003.0
min
min
min
OK
OK
c
cd
t
y
Steel strain check at minimum depth:
Solution 2 – Cont.
No need to calculate strain at bottom layer.
15. 25-Feb-13
CE370: Prof. A. Charif 15
Nominal moment:
mkNM
mkNMMM
mkNmmNM
a
dfAM
mkNmmNM
h
dfAM
n
nfnwn
nw
yswnw
nf
f
ysfnf
.26.339958.37690.0
.958.376
.315.194.194314612
2
232.80
515420246.974
2
.364.182.182643693
2
75
515420714.910
2
Solution 2 – Cont.
Problem 3
MPa420MPa30 y
'
c ff
30
750d
350
328
750widthEffective
100
mmd
mmd
t 860
640min
Compute the design moment for the shown T-beam with all
dimensions in mm.
OKA
mm0.875
75035000333.0,00326.0Max
750350
420
4.1
,
4204
30
Max
4.1
,
4
Max
mm0.643432
4
8
mins
2
min
min
'
min
22
s
s
s
w
yy
c
s
s
A
A
A
db
ff
f
A
A
16. 25-Feb-13
CE370: Prof. A. Charif 16
2
2
'
43.400557.24280.6434
57.2428
420
)100350750(3085.0)(85.0
:With
mmAAA
mm
f
hbbf
A
sfssw
y
fwc
sf
mm
a
c
mmhmm
bf.
fA
a f
w
'
c
ysw
75.221
85.0
49.188
:isdepthaxisNeutral
)100(49.188
3503085.0
42043.4005
850
:isdepthblocknCompressio
1
Solution 3
31
nfnwnsfsws
f
fc
ys
MMMAAA
mmhmm
bf
fA
a
:iondecompositusingAnalyzein webliesblockStress
1003.141
7503085.0
4206434
85.0
:flangeinblockstressAssume
'
mkNM
mkNMMM
mmN
a
dfAM
mmN
h
dfAM
c
cd
n
nwnfn
yswnw
f
ysfnf
y
t
.48.16352.181790.0:momentDesign
.2.1817:momentnominalTotal
.102.1103
2
49.188
75042043.4005
2
.100.714
2
100
75042057.2428
2
:aremomentsNominal
layerbottomatcontrol-nfor tensiocheckmustThen we
:005.0butIf:Note
0.90controlTensionOK005.0005.0
00566.0
75.221
75.221640
003.0003.0:checkStrain
6
6
minmin
min
min
min
Solution 3 – Cont.
32
17. 25-Feb-13
CE370: Prof. A. Charif 17
33
Analysis of T-beams with tension steel not yielding
2
:check003.0
:continueif1
4
1
285.0
600
formula.rectangleUse:)(flangein thefirstitassumestillweweb,
in thelikelymoreisblockncompressiotheyielding,steelnoithAlthough w
6000030
003.0:ThenIf
1
1
'
a
dfAMεEf
c
cd
haca
P
dP
c
bf
A
P
ha
c
cd
A
c
cd
.EAT
c
cd
EAfAT
ssnsssyss
f
fc
s
f
sss
ssssssys
Situations where steel does not yield at failure in T-beams are very
rare. The case of one tension steel layer is treated here. With many
layers, non yielding must be solved using strain compatibility
method (as was done with rectangular beams).
34
Analysis of T-beams with tension steel not yielding
Decomposition (more likely)
sssyss
w
fwf
wc
s
sfwfcwcwf
wcwcwfwfcf
wfsf
εEf
c
cd
ca
QP
PdQP
c
b
hbb
Q
bf
A
PPdcQPc
c
cd
AhbbfcbfCCT
cbfabfChbbfC
CCC
c
cd
ATha
:check003.0and:Deduce
1
)(
4
1
2
)(
:issolutionPositive
)(
85.0
600
with0)(
60085.085.0
85.085.085.0
600:methodiondecomposituseif
1
2
11
'
2
'
1
'
1
'''
18. 25-Feb-13
CE370: Prof. A. Charif 18
35
Analysis of T-beams with tension steel not yielding
Decomposition - Continued
sfssw
s
fwfc
sf
sswnw
f
ssfnf
nwnfn
sssyss
AAA
f
hbbf
A
a
dfAM
h
dfAM
MMM
εEf
c
cd
ca
and
)(85.0
22
:check003.0
'
1
36
600d
100
750
350
328
Problem 4
MPa420MPa20 y
'
c ff
Compute the design moment for the shown T-beam with all
dimensions in mm. The large base (heel) allows many bars in a
single layer and does not change the T-section behavior.
OKA
mm0.700
60035000333.0,00266.0Max
600350
420
4.1
,
4204
20
Max
4.1
,
4
Max
mm0.643432
4
8
mins
2
min
min
'
min
22
s
s
s
w
yy
c
s
s
A
A
A
db
ff
f
A
A
19. 25-Feb-13
CE370: Prof. A. Charif 19
2
2
'
95.481405.16190.6434
05.1619
420
)100350750(2085.0)(85.0
:With
mmAAA
mm
f
hbbf
A
sfssw
y
fwc
sf
mm
a
c
mmhmm
bf.
fA
a f
w
'
c
ysw
86.399
85.0
88.339
:isdepthaxisNeutral
)100(88.339
3502085.0
42095.4814
850
:isdepthblocknCompressio
1
Solution 4
37
nfnwnsfsws
f
fc
ys
MMMAAA
mmhmm
bf
fA
a
:iondecompositusingAnalyzein webliesblockStress
10094.211
7502085.0
4206434
85.0
:flangeinblockstressandyieldingsteelAssume
'
mmc
QP
b
hbb
Q
bf
A
P
QP
PdQP
c
c
cd
w
fwf
wc
s
y
s
20.3631
)45.13430.763(
60030.7634
1
2
)45.13430.763(
45.134
85.0350
100)350750(
30.763
85.03502085.0
6434600
)(
85.0
600
with1
)(
4
1
2
)(
:iondecomposittheuseandwebin theisithatdirectly tassumeratherWe
checkandflangein theblockncompressiofirst theassumeagaincanWe
web)in theisblockstresstlikely thastill(but:yieldingNot
00150.0
86.399
86.399600
003.0003.0:strainSteel
2
11
'2
s
Solution 4 – Cont.
38
20. 25-Feb-13
CE370: Prof. A. Charif 20
2
2
'
1
76.469524.17380.6434
24.1738
2.391
001)350750(2085.0)(85.0
2.391001956.0200000
:confirmedyieldingNo
001956.0
2.363
2.363600
003.0003.0
assumedaswebin theblocknCompressio
72.30820.36385.020.363
mmAAA
mm
f
hbbf
A
MPaεEf
c
cd
mmcammc
sfssw
s
fwfc
sf
sss
ys
s
Solution 4 – Cont.
39
mkNM
mkNMMM
mmNM
a
dfAM
mmNM
h
dfAM
n
nwnfn
nw
sswnw
nf
f
ssfnf
.19.7756.119265.0yielding)No(65.0
.6.1192
.106.818
2
72.308
6002.39176.4695
2
.100.374
2
100
6002.39124.1738
2
6
6
Solution 4 – Cont.
40
21. 25-Feb-13
CE370: Prof. A. Charif 21
CE 370
REINFORCED CONCRETE-I
Prof. A. Charif
Design of T Beams
Design of T-Beams
The same minimum thickness for deflection control is used for
rectangular or T-beams. Design steps are :
• Determine all dimensions of T-section
• Estimate steel depth according to expected number of layers
• Determine location of compression block
• If stress block is in flange, design as a rectangular section using
the flange width
• If stress block is in web, use decomposition method for design
• Check minimum steel
• Perform strain checks using actual provided steel
• Location of compression block determined by comparing
design moment capacity of full flange with ultimate moment.
42CE 370 : Prof. Abdelhamid Charif
22. 25-Feb-13
CE370: Prof. A. Charif 22
2
fh
d
wb
sA
fb
fh ffc hbf '
85.0
Moment capacity of full flange
nff
f
ffcnff
M
h
dhbfM
flangefullofmomentDesign
2
85.0:flangefullofmomentNominal '
43CE 370 : Prof. Abdelhamid Charif
Location of rectangular stress block
• The location of the stress block (flange or web) is determined
with the following steps:
1. Compute design moment capacity of full flange
2. Compare this moment with given ultimate moment
methodiondecompositeUs
)(webinblockStress:If
hwith widtbeamrrectangulaaasDesign
)(flangeinblockStress:If
2
85.0 '
funff
f
funff
f
ffcnff
haMM
b
haMM
h
dhbfM
44CE 370 : Prof. Abdelhamid Charif
23. 25-Feb-13
CE370: Prof. A. Charif 23
Decomposition Method in Design (1)
• When the stress block exceeds the flange, the section is divided:
T-section = W-section + F-section
• W-section = Compression part of the web (unknown depth a)
• F-section = Overhanging parts of the flange (known depth hf)
45CE 370 : Prof. Abdelhamid Charif
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
Decomposition Method in Design (2)
nffnf
f
ysfnf
y
fwc
sfysffwc
sfswsfs
MM
h
dfAM
f
hbbf
AfAhbbf
AAAA
:Note
2
:partflangeofmomentNominal
)(85.0
)(85.0
:givespartflangeofmEquilibriu
'
'
46CE 370 : Prof. Abdelhamid Charif
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
24. 25-Feb-13
CE370: Prof. A. Charif 24
Decomposition Method in Design (3)
nfuwu
w
nfu
f
ysfnf
MMM
b
MM
h
dfAM
momentultimatereducedatosubjectedwhen
th widthsection wirrectangulaaasdesignedthereforeispartwebThe
part.webby thetakenismomentultimateremainingThe
2
toequalmomentultimateanresistspartFlange
47CE 370 : Prof. Abdelhamid Charif
wb
sA
fbb
fh
a
d
2
a
d
wb
swA
a
wb
sfA
2/)( wbb
fh
2
fh
d
Decomposition Method in Design (4)
Steel area component Asw is the solution of quadratic equation :
22'
'
with
7.1
4
11
85.0
db
MM
db
M
R
f
R
f
dbf
A
w
nfu
w
wu
wn
c
wn
y
wc
sw
Total steel area Asf + Asw must then be compared to the minimum
value Asmin .
The new value of the stress block depth (to be used for strain check)
is obtained from force equilibrium in the web using the actual
provided steel area.
wc
ypsw
sfpspsw
bf
fA
aAAA '
,
,,
85.0
48CE 370 : Prof. Abdelhamid Charif
25. 25-Feb-13
CE370: Prof. A. Charif 25
Design Problem-1
MPafMPaf
mmdb
kN.m
y
'
c
w
42030:Take
expected).layer(onelyrespective450and300asgivenareand
m6isspanBeam300.0ofmomentultimatean
tosubjectedwhenbelowshownsystemfloorfor thebeamTDesign the
mm300wb
mm100fh
mm450d
m3
sA sAsA
m3 m3 m3
49CE 370 : Prof. Abdelhamid Charif
Solution 1
mmb
mm
mmhb
mmm
ffw 1500
15004/6000Span/4
19001001630016
30000.3
2
0.3
2
3.0
widthTributary
ofLesser:widthFlangeEffective
mmb
haMM
mkNM
mkNmmNM
h
dhbfM
f
funff
nff
nff
f
ffcnff
1500th widthsection wirrectangulaaasDesign
)(flangeinblockStress
.0.1377153090.0momentDesign
.0.1530.101530
2
100
45010015003085.0
2
85.0
:flangefullofmomentNominal
6
'
50CE 370 : Prof. Abdelhamid Charif
26. 25-Feb-13
CE370: Prof. A. Charif 26
Required steel area As is given by :
OKislayerOne)5.1963(254useWe
4.1803
307.1
974.0.14
11
420
45015003085.0
0974.1
450150090.0
10300
with
7.1
4
11
85.0
2
,
2
2
6
2'
'
mmA
mmA
R
db
M
R
f
R
f
dbf
A
ps
s
n
f
u
n
c
n
y
fc
s
Solution 1 – Cont.
51CE 370 : Prof. Abdelhamid Charif
Designed section
OKA450
450300
420
4.1
,
4204
30
Max
4.1
,
4
Max
Checking
mins
2
min
min,
'
min,
min
ss
s
w
yy
c
s
s
AmmA
A
db
ff
f
A
A
450
300
254
1500widthEffective
100
0.90controlTensionOK005.00502.0
003.0
365.25
365.25450
003.0
365.25
85.0
56.21
depthaxisNeutral
)100(56.21
15003085.0
4205.1963
85.0
depthblockStress
1
'
,
t
t
f
fc
yps
ε
c
cd
ε
mm
β
a
c
mmhmm
bf
fA
a
52CE 370 : Prof. Abdelhamid Charif
27. 25-Feb-13
CE370: Prof. A. Charif 27
Design Problem-2
mm375wb
mm75fh
mm700h
m.81
sA sAsA
m.81 m.81 m.81
mmhd
MPafMPaf
mmhb
kN.m
y
'
c
w
61090:asdepthsteeltheestimatewelayers,2Expecting
42022
lyrespective700and375asgivenareand
m5.4isspanBeam1250.0ofmomentultimatean
tosubjectedwhenbelowshownsystemfloorfor thebeamTDesign the
53CE 370 : Prof. Abdelhamid Charif
Solution 2
mmb
mm
mmhb
mmm
ffw 1350
13504/5400Span/4
1575751637516
18008.1
2
8.1
2
1.8
widthTributary
ofLesser:widthFlangeEffective
methodiondecompositusingDesign
)(in webblockStress).1250(
.0.6.975108490.0momentDesign
.0.1084.101084
2
75
6107513502285.0
2
85.0
:flangefullofmomentNominal
6
'
funff
nff
nff
f
ffcnff
hamkNMM
mkNM
mkNmmNM
h
dhbfM
54CE 370 : Prof. Abdelhamid Charif
28. 25-Feb-13
CE370: Prof. A. Charif 28
mkNmmNM
h
dfAM
mmA
f
hbbf
AfAhbbf
A
nf
f
ysfnf
sf
y
fwfc
sfysffwfc
sf
.86.782.1086.782
2
75
6104208.3255
2
:partflangeofcapacityNominal
8.3255
420
)753751350(2285.0
)(85.0
)(85.0
:steelrequiredgivespartflangein themEquilibriu
6
2
'
'
Solution 2 – Cont.
mkNM
MMM
b
wu
nfuwu
w
.426.54586.7829.01250
momentultimatereducedaunder
th widthsection wirrectangulaaasdesignedpartWeb
55CE 370 : Prof. Abdelhamid Charif
Steel area component Asw given by :
2
2
6
2'
'
7.2731
227.1
343.44
11
420
6103752285.0
343.4
61037590.0
10426.545
with
7.1
4
11
85.0
mmA
R
db
M
R
f
R
f
dbf
A
sw
wn
w
wu
wn
c
wn
y
wc
sw
Solution 2 – Cont.
2
5.59877.27318.3255:areasteelTotal mmAAA swsfs
56CE 370 : Prof. Abdelhamid Charif
29. 25-Feb-13
CE370: Prof. A. Charif 29
OKA5.762
610375
420
4.1
,
4204
22
Max
4.1
,
4
Max
Checking
mins
2
min
min,
'
min,
min
ss
s
w
yy
c
s
s
AmmA
A
db
ff
f
A
A
Solution 2 – Cont.
)5.6157(2810requiresThis
5.5987:areasteelrequiredTotal
2
,
2
mmA
mm
ps
375
2810
3501widthEffective
75
610
57CE 370 : Prof. Abdelhamid Charif
required.isdesign-reifseecheck tomomentPerform
mm)(610valueAssumed0.607
2
5783028
63664700)141040(
OKbarsfor tenlayersTwo5
05.5
3028
0422810630375
cover26
:spacinglayerandspacingbarformm30Assuming
:layeronein28barsofnumberMaximum
21
11min2
1
max
mm
dd
d
mmdSdddd
mmhdd
n
n
Sd
ddSb
n
lb
t
bb
bsb
Solution 2 – Cont.
58CE 370 : Prof. Abdelhamid Charif
30. 25-Feb-13
CE370: Prof. A. Charif 30
Solution 2 – Cont.
layerbottomatcontrol-nfor tensiocheckmustThen we
:005.0butIf:Note
0.90controlTensionOK005.000548.0
46.204
46.204578
003.0003.0
46.204
85.0
79.173
depthaxisNeutral
)75(79.173
3752285.0
4207.2901
85.0
:isdepthblockStress
7.29018.32555.6157:partwebsteelActual
entreinforcemprovidedactualtheusingperformedbemustChecks
minmin
min
min
min
1
'
,
2
,,
y
f
wc
ypsw
sfpspsw
ε
c
cd
ε
mm
β
a
c
mmhmma
bf
fA
a
mmAAA
59CE 370 : Prof. Abdelhamid Charif
Moment check
60
375
3501fb
75
607
2810
79.173
a
The moment check is necessary as
the final steel depth (607) is less than
the initially assumed value (610).
requireddesign-reNoOK).1250(
.35.127161.141290.0:momentDesign
.61.1412:momentnominalTotal
.1086.633
2
79.173
6074207.2901
2
.1075.778
2
75
6074208.3255
2
:aremomentsNominal
6
,
6
mkNMM
mkNM
mkNMMM
mmN
a
dfAM
mmN
h
dfAM
un
n
nwnfn
ypswnw
f
ysfnf
31. 25-Feb-13
CE370: Prof. A. Charif 31
Design Problem 3
Design the shown T-section for an
ultimate bending moment of 440 kN.m
MPafMPaf yc 42025'
600
75
525
300
Expecting two tension steel layers, and with
25-mm layer spacing, the effective steel depth
at the centroid is estimated as :
d = h – 90 = 600 – 90 = 510 mm
The full flange moment capacity is:
mkNMmkNM
mkNmmNM
h
dhbfM
unff
nff
f
ffcnff
.0.440.65.40683.4519.0
.83.451.1083.451
2
75
510756002585.0
2
85.0
6
'
Compression block is thus in the web.
Decompose as follows: T-section = W-section + F-section
mkNmmNM
h
dfAM
mm
f
hbbf
A
AAAMMM
nf
f
ysfnf
y
fwfc
sf
sfswsnfnwn
.914.225.10914.225
2
75
51042039.1138
2
39.1138
420
753006002585.085.0
6
2
'
funff hamkNMmkNM .0.440.65.406
Solution 3 – Cont.
32. 25-Feb-13
CE370: Prof. A. Charif 32
The web is designed as rectangular section for an ultimate moment:
mkNMMM nfuwu .68.236914.2259.0440
The steel area component Asw is the solution of a quadratic
equation given by:
2
2
2
6
2'
'
9.248239.11385.1344:isareasteelTotal
5.1344
257.1
3702.34
11
420
5103002585.0
3702.3
51030090.0
1068.236
7.1
4
11
85.0
mmAAA
mm
db
M
R
f
R
f
dbf
A
sfsws
w
wu
wn
c
wn
y
wc
sw
Solution 3 – Cont.
CE 370 : Prof. Abdelhamid Charif 64
mm
dd
d
mmddd
mmdd t
5.517
2
:iscemtroidatdepthsteelEffective
49520255402025
540101040600
21
1min2
1
2
mins
2
min
'
min,
min
9.2482OKA0.510
510300
420
4.1
,
4204
25
Max
4.1
,
4
Max
Checking
mmAAmmA
db
ff
f
A
A
sss
w
yy
c
s
s
Use eight 20-mm bars in two layers = 2513.27 mm2.
The steel depths are :
Solution 3 – Cont.
The final steel depth is just greater than the assumed value : OK
Moment check is not necessary.
33. 25-Feb-13
CE370: Prof. A. Charif 33
Compression block and neutral axis depths are computed using
the actual steel area :
Strain check:
ControlTensionOK005.00109.0
0021.00109.0
565.106
565.106495
003.0003.0
min
min
min
OK
c
cd
y
mm
a
cmm
bf
fA
a
mmA
AAA
mmA
wc
ypsw
psw
sfpspsw
ps
565.106
85.0
58.90
58.90
3002585.0
42088.1374
85.0
88.137439.113827.2513
:ispartin webareasteelActual
27.2513
4
20
8:isareasteelActual
1
'
,
2
,
,,
2
2
,
Solution 3 – Cont.
Thank you
66CE 370 : Prof. Abdelhamid Charif