1. The document discusses plate girders, which are large flexural members made of welded steel plates used in bridges and buildings.
2. Plate girders are fabricated by welding steel plates to form the web and two flanges.
3. The web resists shear forces while the flanges resist bending moments. Thin, deep webs are prone to buckling under shear forces.
The document discusses composite construction using precast prestressed concrete beams and cast-in-situ concrete. It describes how the two elements act compositely after the in-situ concrete hardens. Composite beams can be constructed as either propped or unpropped. Propped construction involves supporting the precast beam during casting to relieve it of the wet concrete weight, while unpropped construction allows stresses to develop under self-weight. Design and analysis of composite beams involves calculating stresses and deflections considering composite action. Differential shrinkage between precast and in-situ concrete also induces stresses.
This document discusses structural analysis methods for statically indeterminate structures. It defines key terms like degree of static indeterminacy, internal and external redundancy, and methods for analyzing indeterminate structures. Specific methods discussed include the flexibility matrix method, consistent deformation method, and unit load method. Examples of statically indeterminate beams and frames are also provided.
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 provides guidance on the design of lacing and battens for built-up compression members. It discusses the key design considerations and calculations for both single and double lacing systems, including the angle of inclination, slenderness ratio, effective lacing length, bar width and thickness. Similar guidelines are given for battens, covering spacing, thickness, effective depth, transverse shear and overlap. The document also includes an example problem on designing a slab foundation for a column with given load and material properties.
This document discusses various concepts related to structural analysis of arches:
1. An arch is a curved girder supported at its ends, allowing only vertical and horizontal displacements for arch action.
2. The general cable theorem relates the horizontal tension and vertical distance from any cable point to the cable chord moment.
3. Arches are classified based on support conditions (3, 2, or 1 hinged) or shape (curved, parabolic, elliptical, polygonal).
4. Horizontal thrust in arches reduces the bending moment and is calculated differently for various arch types (e.g. parabolic) and loading (e.g. UDL).
This document provides information on industrial buildings, including their components and factors to consider in design. Key points include:
- Industrial buildings are used for manufacturing and storage by industries and include steel plants, warehouses, and factories.
- Site selection considers access, raw materials, utilities, land characteristics, and transportation.
- Major components include the roof, trusses, purlins, girts, bracing, and foundations.
- Design considerations cover roofing/wall materials, bay widths, structural framing, truss configurations, and bracing to resist lateral loads.
The document discusses limit state design of reinforced concrete structures. It introduces limit states as conditions where the structure becomes unfit for use, including limit states of strength and serviceability. Limit state design involves characterizing loads and resistances as random variables and using partial safety factors on loads and resistances to achieve a target reliability. The document outlines the general principles of limit state design according to Indian Standard code IS 800, including defining actions, factors governing strength limits, and serviceability limits related to deflection, vibration and durability.
Static Indeterminacy and Kinematic IndeterminacyDarshil Vekaria
This ppt is more useful for Civil Engineering students.
I have prepared this ppt during my college days as a part of semester evaluation . Hope this will help to current civil students for their ppt presentations and in many more activities as a part of their semester assessments.
I have prepared this ppt as per the syllabus concerned in the particular topic of the subject, so one can directly use it just by editing their names.
The document discusses composite construction using precast prestressed concrete beams and cast-in-situ concrete. It describes how the two elements act compositely after the in-situ concrete hardens. Composite beams can be constructed as either propped or unpropped. Propped construction involves supporting the precast beam during casting to relieve it of the wet concrete weight, while unpropped construction allows stresses to develop under self-weight. Design and analysis of composite beams involves calculating stresses and deflections considering composite action. Differential shrinkage between precast and in-situ concrete also induces stresses.
This document discusses structural analysis methods for statically indeterminate structures. It defines key terms like degree of static indeterminacy, internal and external redundancy, and methods for analyzing indeterminate structures. Specific methods discussed include the flexibility matrix method, consistent deformation method, and unit load method. Examples of statically indeterminate beams and frames are also provided.
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 provides guidance on the design of lacing and battens for built-up compression members. It discusses the key design considerations and calculations for both single and double lacing systems, including the angle of inclination, slenderness ratio, effective lacing length, bar width and thickness. Similar guidelines are given for battens, covering spacing, thickness, effective depth, transverse shear and overlap. The document also includes an example problem on designing a slab foundation for a column with given load and material properties.
This document discusses various concepts related to structural analysis of arches:
1. An arch is a curved girder supported at its ends, allowing only vertical and horizontal displacements for arch action.
2. The general cable theorem relates the horizontal tension and vertical distance from any cable point to the cable chord moment.
3. Arches are classified based on support conditions (3, 2, or 1 hinged) or shape (curved, parabolic, elliptical, polygonal).
4. Horizontal thrust in arches reduces the bending moment and is calculated differently for various arch types (e.g. parabolic) and loading (e.g. UDL).
This document provides information on industrial buildings, including their components and factors to consider in design. Key points include:
- Industrial buildings are used for manufacturing and storage by industries and include steel plants, warehouses, and factories.
- Site selection considers access, raw materials, utilities, land characteristics, and transportation.
- Major components include the roof, trusses, purlins, girts, bracing, and foundations.
- Design considerations cover roofing/wall materials, bay widths, structural framing, truss configurations, and bracing to resist lateral loads.
The document discusses limit state design of reinforced concrete structures. It introduces limit states as conditions where the structure becomes unfit for use, including limit states of strength and serviceability. Limit state design involves characterizing loads and resistances as random variables and using partial safety factors on loads and resistances to achieve a target reliability. The document outlines the general principles of limit state design according to Indian Standard code IS 800, including defining actions, factors governing strength limits, and serviceability limits related to deflection, vibration and durability.
Static Indeterminacy and Kinematic IndeterminacyDarshil Vekaria
This ppt is more useful for Civil Engineering students.
I have prepared this ppt during my college days as a part of semester evaluation . Hope this will help to current civil students for their ppt presentations and in many more activities as a part of their semester assessments.
I have prepared this ppt as per the syllabus concerned in the particular topic of the subject, so one can directly use it just by editing their names.
Prestressed concrete is concrete that is placed under compression using tensioned steel strands, cables, or bars. This is done through either pre-tensioning or post-tensioning. In pre-tensioning, the steel components are tensioned before the concrete is poured, while in post-tensioning, the steel components are tensioned after the concrete has hardened. Prestressed concrete provides benefits over reinforced concrete like lower construction costs, thinner structural elements, and longer spans between supports.
This document provides an introduction to prestressed concrete bridge design. It discusses how prestressing concrete induces compression to counteract tensile stresses from loading. Prestressed concrete allows for longer concrete bridge spans through precasting units that are lifted into place. The document covers methods of prestressing including pre-tensioning and post-tensioning. It also summarizes design considerations like serviceability limits, stress limitations, prestress losses, and establishes basic inequalities for prestress force and section properties. Magnel diagrams are introduced as a way to determine appropriate prestress force and eccentricity values.
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
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 provides an overview of the design of compression members (columns) in reinforced concrete structures. It discusses various types of columns based on reinforcement, loading conditions, and slenderness ratio. It describes the classification of columns as short or slender. The document also covers effective length, braced vs unbraced columns, codal provisions for reinforcement, and functions of longitudinal and transverse reinforcement. Key points include types of column reinforcement, minimum reinforcement requirements, cover requirements, and assumptions for the limit state of collapse under compression.
This document discusses Castigliano's theorems for analyzing stresses and strains in structures. It explains that Castigliano's first theorem states that the partial derivative of a structure's strain energy with respect to an applied force equals the displacement at the point of application of that force. Castigliano's second theorem states that the partial derivative of strain energy with respect to a displacement equals the force that produces that displacement. The document provides mathematical expressions to calculate strain energy and uses these theorems to analyze beam deflections under applied loads.
This document provides design requirements for lacing and battening systems used in steel structural elements. It discusses two types of lacing systems - single and double. It outlines 9 design requirements for lacing per Indian code IS 800, including angle of inclination, slenderness ratio, effective length, width/thickness, transverse shear force, strength checks, and end connections. It also discusses 7 design requirements for battening systems, including transverse shear force calculation, slenderness ratio, spacing, thickness, effective depth, overlap for welded connections, and notes battening offers less shear resistance than lacing.
The document provides details on the design procedure for beams. It discusses estimating loads, analyzing beams to determine shear forces and bending moments, and designing beams. The design process involves selecting the beam size and shape, calculating the effective span, determining critical moments and shears, selecting reinforcement, and checking requirements such as shear capacity, deflection limits, and development lengths. An example problem demonstrates designing a singly reinforced concrete beam with a span of 5 meters to support a working live load of 25 kN/m.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins by outlining the basic principles and definitions of the method, including stiffness factors, carry-over factors, and distribution factors. It then provides an example problem, showing the calculation of fixed end moments, establishment of the distribution table through successive approximations, and determination of shear forces and bending moments. Finally, it discusses extensions of the method to structures with non-prismatic members, including using tables to determine necessary values for analysis.
This document discusses the design of compression members subjected to axial load and biaxial bending. It introduces the concept of biaxial eccentricities and explains that columns should be designed considering possible eccentricities in two axes. The document outlines the method suggested by IS 456-2000, which is based on Breslar's load contour approach. It relates the parameter αn to the ratio of Pu/Puz. Finally, it provides a step-by-step process for designing the column section, which involves determining uniaxial moment capacities, computing permissible moment values from charts, and revising the section if needed. It also briefly mentions the simplified method according to BS8110.
Design of Reinforced Concrete Structure (IS 456:2000)MachenLink
This is the 1st Lecture Series on Design Reinforced Cement Concrete (IS 456 -2000).
In this video, you will learn about the objective of structural designing and then basic properties of concrete and steel.
Concrete properties like...
1. Grade of Concrete
2. Modulus of Elasticity
3. Characteristic Strength
4. Tensile Strength
5. Creep and Shrinkage
6. Durability
Reinforced Steel Properties....
1. Grade and types of steel
2. Yield Strength of Mild Steel and HYSD Bars
Static and Kinematic Indeterminacy of Structure.Pritesh Parmar
The document discusses static and kinematic indeterminacy of structures. It defines different types of supports for 2D and 3D structures including fixed support, hinged/pinned support, roller support, and their properties. It also discusses internal joints like internal hinge, internal roller, and internal link. The document explains concepts of static indeterminacy, kinematic indeterminacy, and degree of freedom for different types of structures.
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.
The document discusses code provisions for calculating the effective span of slabs according to IS 456. It describes how to calculate the effective span for simply supported, continuous, and cantilever members. It also discusses load assumptions, reinforcement cover requirements, deflection limits, and provides an overview of one-way slabs, two-way slabs, flat slabs, and flat plates.
This document discusses the slope-deflection method for analyzing beams and frames. It provides the theory and equations of the slope-deflection method. Examples are included to demonstrate how to use the method to determine support reactions, member end moments, and draw bending moment and shear force diagrams.
Advanced construction equipments and techniquesselva ganesh
This document discusses advanced construction techniques and modern materials. It describes underwater construction methods like caissons and cofferdams. Trenchless technology techniques for installing pipes are also covered, including pipe jacking, auger boring, and microtunneling. Modern materials presented include fly ash bricks, translucent concrete, liquid granite, carbon nanotubes, and solar panel roofing tiles. The document concludes that these advanced techniques and innovative materials can improve properties, recycling, and make construction more efficient.
Prestressed concrete is a structural material that allows for predetermined, engineering stresses to be placed in members to counteract the stresses that occur when they are subject to loading.
The document provides guidance on loads and forces that should be considered when designing bridges, including:
1. Dead loads, live loads, dynamic loads, longitudinal forces, wind loads, centrifugal forces, horizontal water currents, buoyancy, earth pressures, temperature effects, and seismic loads.
2. It describes the various live load models (Class A, B, 70R, AA) and provides details on load intensity, wheel/track configuration, and load combinations.
3. Design recommendations are given for calculating impact factors, braking forces, wind loads, water current pressures, earth pressures, and seismic forces.
determinate and indeterminate structuresvempatishiva
This topic I am uploading here contains some basic topics in structural analysis which includes types of supports, reactions for different support conditions, determinate and indeterminate structures, static and kinematic indeterminacy,external and internal static indeterminacy, kinematic indeterminacy for beams, frames, trusses.
need of finding indeterminacy, different methods available to formulate equations to solve unknowns.
The document discusses using polypropylene fibers in concrete to improve its properties. Various concrete mixtures with 0.15%, 0.20%, 0.25%, and 0.30% polypropylene fiber by volume were tested. Compression tests on cubes at 7 and 28 days found that compressive strength generally increased with fiber content up to 0.20-0.25%, with 0.30% showing little further improvement. Slump tests showed workability decreased slightly with higher fiber amounts. Water absorption decreased with fiber content, indicating reduced permeability. Applications include pavements, runways, bridges and building construction where polypropylene fiber concrete provides improved strength, crack resistance and durability.
Prestressed concrete is concrete that is placed under compression using tensioned steel strands, cables, or bars. This is done through either pre-tensioning or post-tensioning. In pre-tensioning, the steel components are tensioned before the concrete is poured, while in post-tensioning, the steel components are tensioned after the concrete has hardened. Prestressed concrete provides benefits over reinforced concrete like lower construction costs, thinner structural elements, and longer spans between supports.
This document provides an introduction to prestressed concrete bridge design. It discusses how prestressing concrete induces compression to counteract tensile stresses from loading. Prestressed concrete allows for longer concrete bridge spans through precasting units that are lifted into place. The document covers methods of prestressing including pre-tensioning and post-tensioning. It also summarizes design considerations like serviceability limits, stress limitations, prestress losses, and establishes basic inequalities for prestress force and section properties. Magnel diagrams are introduced as a way to determine appropriate prestress force and eccentricity values.
Columns are structural members that experience compression loads. They can buckle if loaded beyond their buckling (or critical) load. Short columns fail through crushing, while long columns fail through lateral buckling. The Euler formula calculates the buckling load of a long column based on its properties and end conditions. The Rankine-Gordon formula provides a more accurate calculation of buckling load that applies to all column types by accounting for both buckling and crushing. Proper design of columns involves ensuring they are loaded below their safe loads, which incorporate factors of safety applied to the theoretical buckling loads.
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 provides an overview of the design of compression members (columns) in reinforced concrete structures. It discusses various types of columns based on reinforcement, loading conditions, and slenderness ratio. It describes the classification of columns as short or slender. The document also covers effective length, braced vs unbraced columns, codal provisions for reinforcement, and functions of longitudinal and transverse reinforcement. Key points include types of column reinforcement, minimum reinforcement requirements, cover requirements, and assumptions for the limit state of collapse under compression.
This document discusses Castigliano's theorems for analyzing stresses and strains in structures. It explains that Castigliano's first theorem states that the partial derivative of a structure's strain energy with respect to an applied force equals the displacement at the point of application of that force. Castigliano's second theorem states that the partial derivative of strain energy with respect to a displacement equals the force that produces that displacement. The document provides mathematical expressions to calculate strain energy and uses these theorems to analyze beam deflections under applied loads.
This document provides design requirements for lacing and battening systems used in steel structural elements. It discusses two types of lacing systems - single and double. It outlines 9 design requirements for lacing per Indian code IS 800, including angle of inclination, slenderness ratio, effective length, width/thickness, transverse shear force, strength checks, and end connections. It also discusses 7 design requirements for battening systems, including transverse shear force calculation, slenderness ratio, spacing, thickness, effective depth, overlap for welded connections, and notes battening offers less shear resistance than lacing.
The document provides details on the design procedure for beams. It discusses estimating loads, analyzing beams to determine shear forces and bending moments, and designing beams. The design process involves selecting the beam size and shape, calculating the effective span, determining critical moments and shears, selecting reinforcement, and checking requirements such as shear capacity, deflection limits, and development lengths. An example problem demonstrates designing a singly reinforced concrete beam with a span of 5 meters to support a working live load of 25 kN/m.
The document discusses the moment distribution method for analyzing statically indeterminate structures. It begins by outlining the basic principles and definitions of the method, including stiffness factors, carry-over factors, and distribution factors. It then provides an example problem, showing the calculation of fixed end moments, establishment of the distribution table through successive approximations, and determination of shear forces and bending moments. Finally, it discusses extensions of the method to structures with non-prismatic members, including using tables to determine necessary values for analysis.
This document discusses the design of compression members subjected to axial load and biaxial bending. It introduces the concept of biaxial eccentricities and explains that columns should be designed considering possible eccentricities in two axes. The document outlines the method suggested by IS 456-2000, which is based on Breslar's load contour approach. It relates the parameter αn to the ratio of Pu/Puz. Finally, it provides a step-by-step process for designing the column section, which involves determining uniaxial moment capacities, computing permissible moment values from charts, and revising the section if needed. It also briefly mentions the simplified method according to BS8110.
Design of Reinforced Concrete Structure (IS 456:2000)MachenLink
This is the 1st Lecture Series on Design Reinforced Cement Concrete (IS 456 -2000).
In this video, you will learn about the objective of structural designing and then basic properties of concrete and steel.
Concrete properties like...
1. Grade of Concrete
2. Modulus of Elasticity
3. Characteristic Strength
4. Tensile Strength
5. Creep and Shrinkage
6. Durability
Reinforced Steel Properties....
1. Grade and types of steel
2. Yield Strength of Mild Steel and HYSD Bars
Static and Kinematic Indeterminacy of Structure.Pritesh Parmar
The document discusses static and kinematic indeterminacy of structures. It defines different types of supports for 2D and 3D structures including fixed support, hinged/pinned support, roller support, and their properties. It also discusses internal joints like internal hinge, internal roller, and internal link. The document explains concepts of static indeterminacy, kinematic indeterminacy, and degree of freedom for different types of structures.
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.
The document discusses code provisions for calculating the effective span of slabs according to IS 456. It describes how to calculate the effective span for simply supported, continuous, and cantilever members. It also discusses load assumptions, reinforcement cover requirements, deflection limits, and provides an overview of one-way slabs, two-way slabs, flat slabs, and flat plates.
This document discusses the slope-deflection method for analyzing beams and frames. It provides the theory and equations of the slope-deflection method. Examples are included to demonstrate how to use the method to determine support reactions, member end moments, and draw bending moment and shear force diagrams.
Advanced construction equipments and techniquesselva ganesh
This document discusses advanced construction techniques and modern materials. It describes underwater construction methods like caissons and cofferdams. Trenchless technology techniques for installing pipes are also covered, including pipe jacking, auger boring, and microtunneling. Modern materials presented include fly ash bricks, translucent concrete, liquid granite, carbon nanotubes, and solar panel roofing tiles. The document concludes that these advanced techniques and innovative materials can improve properties, recycling, and make construction more efficient.
Prestressed concrete is a structural material that allows for predetermined, engineering stresses to be placed in members to counteract the stresses that occur when they are subject to loading.
The document provides guidance on loads and forces that should be considered when designing bridges, including:
1. Dead loads, live loads, dynamic loads, longitudinal forces, wind loads, centrifugal forces, horizontal water currents, buoyancy, earth pressures, temperature effects, and seismic loads.
2. It describes the various live load models (Class A, B, 70R, AA) and provides details on load intensity, wheel/track configuration, and load combinations.
3. Design recommendations are given for calculating impact factors, braking forces, wind loads, water current pressures, earth pressures, and seismic forces.
determinate and indeterminate structuresvempatishiva
This topic I am uploading here contains some basic topics in structural analysis which includes types of supports, reactions for different support conditions, determinate and indeterminate structures, static and kinematic indeterminacy,external and internal static indeterminacy, kinematic indeterminacy for beams, frames, trusses.
need of finding indeterminacy, different methods available to formulate equations to solve unknowns.
The document discusses using polypropylene fibers in concrete to improve its properties. Various concrete mixtures with 0.15%, 0.20%, 0.25%, and 0.30% polypropylene fiber by volume were tested. Compression tests on cubes at 7 and 28 days found that compressive strength generally increased with fiber content up to 0.20-0.25%, with 0.30% showing little further improvement. Slump tests showed workability decreased slightly with higher fiber amounts. Water absorption decreased with fiber content, indicating reduced permeability. Applications include pavements, runways, bridges and building construction where polypropylene fiber concrete provides improved strength, crack resistance and durability.
This document discusses aluminium and its alloys. Key points include:
- Aluminium is a lightweight metal that is abundant, corrosion resistant, and highly conductive. It is extracted from bauxite via electrolysis.
- Aluminium alloys include heat treatable alloys like duralumin that can be strengthened via precipitation hardening as well as non heat-treatable alloys.
- Common fabrication methods for aluminium include casting, rolling, extrusion, and welding. Various heat treatments can further influence the properties of aluminium alloys.
- Applications of aluminium alloys span transportation, infrastructure, packaging, and more due to its combination of properties like strength, conductivity, and corrosion resistance
Polypropylene is a thermoplastic polymer derived from the monomer propylene. It was first synthesized in the 1950s and is now the second most widely produced plastic globally. There are two main methods for producing polypropylene - gas phase and liquid phase polymerization processes, which both use Ziegler-Natta catalysts. As a fiber, polypropylene has good strength, abrasion resistance, and chemical resistance. It finds major uses in nonwovens like diapers and filters, as well as clothing, ropes, furniture and medical devices.
Propene undergoes addition polymerization to produce polypropylene, one of the most versatile thermoplastics. Polypropylene has properties including low density, heat resistance, and chemical inertness that make it suitable for wide applications like packaging film, fibers, and injection molded products. It is mainly produced using Ziegler-Natta catalysts in bulk or gas phase polymerization, and newer metallocene catalysts allow production of isotactic or syndiotactic polypropylene structures.
The document discusses the fabrication of composite welded steel plate girder bridges being constructed by Northern Railway in Ambala Division on two level crossings at Barnala and Budhwala. Key steps in the fabrication process are outlined, including testing of steel plates, cutting plates to shape, drilling holes, welding plates together to form girders, adding stiffeners and splice plates, and specifying bearings. The bridges involve multi-span plate girders with concrete decks and will eliminate hazards posed by the level crossings once completed.
This document provides an overview of aluminum alloys, including their chemistry, classification system, applications, manufacturing processes, heat treatments, and common defects. It discusses the major alloying elements used in aluminum like copper, manganese, silicon, magnesium, and zinc. It also summarizes the various production methods for wrought aluminum alloys like extrusion and heat treating processes like annealing, solution heat treatment, and precipitation hardening. Finally, it outlines typical casting, extrusion, forging, and heat treatment defects seen in aluminum alloys.
This document discusses the properties and applications of aluminum and its alloys. It outlines that aluminum is lightweight, corrosion resistant, and electrically and thermally conductive. However, in its pure form aluminum is soft and has a low melting point. The document then discusses how aluminum is commonly alloyed with other metals like copper, magnesium, and manganese to increase its strength and maximum operating temperature. These aluminum alloys have many applications in transportation, infrastructure, consumer goods, and oil and gas due to their high strength to weight ratio and corrosion resistance.
This document discusses various applications of aluminium alloys including electrical conductors, transport, packaging, building and architecture, and miscellaneous uses. It then provides more detailed information on specific aluminium alloys commonly used in marine applications such as 5052, 5059, 5083, 5086, 6061, and 6063 aluminium alloys and their material properties and typical uses. The advantages of aluminium alloys over steel for vessels include lighter weight and better corrosion resistance, while disadvantages are more difficult welding and lower strength.
This document discusses various methods for measuring stream flow. There are direct and indirect methods. Direct methods like area-velocity measure discharge by determining the cross-sectional area and average velocity. Indirect methods relate discharge to easily measured water level/stage using structures or the slope-area method with Manning's equation. Accurate stage measurements are important for estimating discharge from stage-discharge curves developed through direct measurements.
This document discusses surface runoff, stream flow, hydrographs, and unit hydrographs. It begins by defining surface runoff and stream flow, explaining that surface runoff occurs when precipitation is unable to infiltrate the ground and flows overland into streams, rivers, and other bodies of water. It then discusses measuring stream flow through various methods like current meters and weirs to determine discharge. The document introduces the concept of hydrographs, which plot discharge over time, and unit hydrographs, which represent the hydrograph resulting from 1 unit of excess precipitation. It provides examples of using unit hydrographs and the S-curve method to develop hydrographs of different durations.
This document discusses points and crossings in railway tracks. It defines points and crossings as arrangements that guide trains during directional changes. Points divert vehicles, while crossings provide gaps in rails for flanged wheels to roll over. The key components of a point and crossing assembly are the point, lead, and crossing element. Crossings introduce gaps where tracks cross to allow wheels to pass between tracks. Points and crossings are necessary because railway vehicles have inside wheel flanges and require special arrangements to navigate the rails and change tracks or directions.
Geometric design of tracks aims to provide smooth and safe running of trains at maximum speed while carrying heavy loads. This involves proper design of gradients, curvature, and super elevation (cant).
There are different types of gradients - ruling gradient which is the maximum gradient permitted, momentum gradient which is steeper and uses train momentum, and pusher gradient requiring extra locomotives. Gradients are designed considering train performance and load. Curvature introduces greater resistance requiring grade compensation of ruling gradients.
Super elevation (cant) involves raising the outer rail on curves to counteract centrifugal forces. Equilibrium cant provides equal wheel load distribution. Higher speeds result in cant deficiency which must be limited for passenger safety. Contrary flexures like
These are the arrangements provided in a railway track to divert a train from the main line to a branch line, such as turnouts that use pairs of points and crossings. The document defines various railway track components like the tongue rail, stock rails, stretcher bar, and switch. It describes different types of turnouts like left and right hand turnouts. The document also discusses railway signaling, explaining different types of signals based on their operation, function, and location. Signaling facilitates efficient train movement, safety between trains, and maximum track utilization.
This document is a project report on the geometric design of railway tracks submitted by Mohit M. Jain to Gujarat Technological University in India. It introduces the topic of geometric design and its importance for ensuring safe and efficient train operation. The following chapters discuss geometric cross sections, gradients including different types, curves, superelevation, and gauge widening on curves. The report provides information on the key design considerations for railway tracks.
This document discusses precipitation measurement and estimation. It begins by defining precipitation and its different types. It then discusses various methods for measuring precipitation, including manual and automatic rain gauges. It emphasizes the importance of establishing a raingauge network according to WMO recommendations to spatially average precipitation measurements over a region. The document concludes by outlining different methods that can be used to estimate missing precipitation data, including station averaging, normal ratio, inverse distance weighting, and regression techniques.
This document provides information about the hydrological cycle and water budget. It begins with the objectives of understanding water sources and the hydrological cycle components of evaporation, precipitation, infiltration, runoff and subsurface flow. It then discusses the global water resources and usage, including increasing population growth. The bulk of the document defines and explains the various components of the hydrological cycle, including evaporation, condensation, precipitation types, interception, infiltration, subsurface flow, runoff and storage. It provides an example water balance equation and long-term water balance calculation. Finally, it briefly discusses the global water cycle and a typical hydrological cycle for the UAE.
This document provides an introduction to groundwater hydrology. It discusses key topics such as the hydrologic cycle, aquifers, water tables, wells, Darcy's law, and groundwater management. Shallow groundwater recharges more quickly than deep groundwater but is more prone to contamination. Aquifers are underground units that store and transmit water, and can be unconfined, confined, or semi-confined. The water table and piezometric surface indicate the level of groundwater in unconfined and confined aquifers respectively. Darcy's law describes the flow of water through porous media. Groundwater management aims to balance withdrawal and recharge to prevent water levels from dropping. Artificial recharge
This lecture discusses the equilibrium of particles and free-body diagrams. It defines particle equilibrium as being at rest if originally at rest or having constant velocity if in motion. The condition for particle equilibrium is that the sum of all forces acting on the particle is zero. Examples of connections involving springs and cables/pulleys are provided. The procedure for drawing free-body diagrams involves outlining the particle, showing all known and unknown forces, and labeling the forces. Example problems are solved to demonstrate applications of these concepts.
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.
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1. N.W.F.P. University of Engineering and
Technology P h
T h l Peshawar
Lecture 13: Plate Girder
By: Prof Dr. Akhtar Naeem Khan
chairciv@nwfpuet.edu.pk
1
2. Plate Girders
A girder is a flexural member which is required
to carr hea loads on relati el long spans
carry heavy relatively
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 2
4. Plate Girder
Plate girders are typically used as long-span
g yp y g p
floor girders in buildings, as bridge girders, and
as crane girders in industrial structures.
g
Commonly term girder refers to a flexural x-
section made up of a number of elements
elements.
They are generally considerably deeper than the
y g y y p
deepest rolled sections and usually have webs
thinner than rolled sections.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 4
5. Plate Girder
Modern plate girders are normally fabricated
by welding together two flanges and a web
p
plate.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 5
6. Plate Girder
Plate girders are at their most impressive in
modern b id construction where main spans of
d bridge t ti h i f
well over 200m are feasible, with corresponding
cross-section d th h
ti depths, haunched over th
h d the
supports, in the range of 5-10m.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 6
7. Plate Girder
Because plate girders are fabricated
separately, each may be designed
p y, y g
individually to resist the applied
actions using proportions that ensure
low self-weight and high load
resistance.
resistance
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 7
8. Plate Girder
Changes in X-Section
There is also considerable scope for variation
of cross-section in the longitudinal direction.
Ad i
designer may choose t reduce th fl
h to d the flange
thickness (or breadth) in a zone of low
applied moment
moment.
Equally, in a zone of high shear, the designer
might choose to thicken the web plate.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 8
9. Plate Girder
Changes in Material
Alternatively, higher g
y g grade steel might be
g
employed for zones of high applied moment
and shear, while standard grade would be
used elsewhere. S
d l h So-called "h b id" girders
ll d "hybrid" i d
with different strength material in the flanges
and the web offer another possible means of
more closely matching resistance to
requirements.
requirements
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 9
12. Plate Girder
Any cross-section of a plate girder is normally
subjected to a combination of shear force and
bending moment.
The primary function of the top and bottom
flange plates of the girder is to resist the axial
compressive and tensile forces arising from
the applied bending moment.
moment
The primary function of the web plate is to
resist the applied shear force.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 12
13. Plate Girder
Plate girders are normally designed to support
heavy loads over long spans in situations where it
is necessary to produce an efficient design by
providing girders of high strength to weight ratio.
To produce the lowest axial flange force for a
given bending moment, the web depth (d) must be
made as large as possible. To reduce the self
weight, the web thickness (tw) must be reduced to
a minimum
minimum.
As a consequence, in many instances the web
plate is of slender proportions and i th f
l t i f l d ti d is therefore
prone to buckling at relatively low values of
applied shear
shear.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 13
14. Plate Girder
For efficient design it is usual to choose a
relatively deep girder, thus minimizing the
required area of flanges for a given applied
moment, Msd.
t
This obviously entails a deep web whose
y p
area will be minimized by reducing its
thickness to the minimum required to carry
the applied shear, Vsd.
h li d h
Such a web may be quite slender (
y q (i.e. a high
g
d/tw ratio) and may be prone to local buckling
and shear buckling.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 14
15. Plate Girder
Web buckling does not determine the
ultimate strength of a plate girder.
lti t t th f l t id
Plate elements do not collapse when they
p y
buckle; they can possess a substantial post-
buckling reserve of resistance.
For an efficient design, any calculation
relating to the ultimate limit state should take
g
the post-buckling action into account.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 15
16. Design Criteria
Criteria for design of plate girder may be
based on
Elastic bend-buckling strength
Elastic h
El ti shear-buckling strength
b kli t th
Post-bend-buckling
Post bend buckling strength
Post shear buckling(Tension
Post-shear-buckling(Tension field)strength
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 16
17. Design Criteria
The designer has the choice of following four
combinations
1. Elastic bend buckling + Elastic shear buckling
g g
(conventional flexural behavior)
2.
2 Elastic bend buckling + Post shear buckling
3. Post bend buckling + Elastic shear buckling
g g
4. Post bend buckling + Post shear buckling
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 17
18. Elastic Bend Buckling
Strength
The extreme f fiber bending stress at which a
perfectly flat web buckles is given by
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 18
19. Elastic Bend Buckling
Strength
Using a FOS of 1.25 w.r.t service load bending
stress fb gives an eqnuation which is AASHTO
slenderness limit for plat girders webs
Using AASHTO allowable stress fb=0.55Fy
“ h/t=165 f A36 steel “
h/t 165 for t l
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 19
20. Elastic Bend Buckling
Strength
The bend buckling resistance of beam webs can be
increased considerably by reinforcing the slender webs
with Longitudinal stiffeners.
Means webs thinner than those given by the equation can be
used.
used
A typical longitudinally stiffened girder is shown after failure
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 20
21. Web Stiffeners
They usually consists of rectangular
bars to welded to web.
Transverse stiffeners may be in pairs,
one on each side of web, or they may
placed on one side of web.
Longitudinal stiffeners are usually
placed on one side of web.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 21
24. Web Stiffeners
The main function of the longitudinal stiffeners is
to increase the buckling resistance of the web
with respect o bot s ea a d be d g loads. An
t espect of both shear and bending oads
effective stiffener will remain straight, thereby
sub-dividing the web p
g panel and limiting the
g
buckling to the smaller sub-panels. The resulting
increase in the ultimate resistance of the girder
g
can be significant.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 24
25. Web Stiffeners
Efficiency of stiffener is a function of its location
in the compression zone
The optimum location for a longitudinal stiffener
has been determined to be at least h/5 from
compression edge.
In this case k=129. The corresponding allowable web
k 129.
slenderness is h/t=330 as compare to 165
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 25
26. Web Stiffeners
Stiffener acts as a beam supported at the ends
where a vertical stiffener holds the web in line
line.
Stiffener acts as a beam column and hence must
be
b proportioned i t
ti d in terms of x-sectional area and
f ti l d
moment of inertia.
AASHTO specifies Is as
Stiffener acts as a beam supported at the ends
where a vertical stiffener holds the web in line.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 26
27. Web Stiffeners
The stiffeners must also be proportioned to
resist local buckling.
buckling
For plates supported on one longitudinal
p pp g
edge AASHTO require b/t<1625/√fb
Multiple longitudinal stiffeners are used for
large depth webs.
As longitudinal stiffener is also acting as a
column so it must be satisfied for critical
stress (Fcrs>0.6Fcrf)
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 27
28. Post buckling bending
strength
If bending strain increases after Fcr, the upper
g , pp
edge of panels shortens and bottom edge
lengthens.
If web were to remain flat there will be increase in
stress.
Because the web has buckled, the increase in
stress is non linear
non-linear.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 28
29. Post buckling bending
strength
As
A variation i post-buckled state i not k
i ti in t b kl d t t is t known,
simplify assumptions are made.
Non-linear compression is replaced with linear
distribution acting on effective depth be.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 29
30. Post buckling bending
strength
Point A gives point that enables a girder to reach its full
yield moment(925 /√Fy=154).
If stiffeners at h/5 is provided gives point B
B.
Considering the
post buckling A B
strength, the 0.94
0.82
0 82
point where M/My
reduction in web 0.4
effectiveness 0.18
0 18
begins s taken to
154 315 360
be 980/√Fy=170.
980/√Fy 170.
h/t
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 30
31. Post buckling bending
strength
Equation connecting the revised point A
with points corresponding t h/t 360 i
ith i t di to h/t=360 is
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 31
32. Post buckling bending
strength
LRFD
Where
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 32
34. Compression Flange Vertical
buckling
If plate girder web is too slender the compression
plate-girder slender,
flange may buckle in vertical plane at stress less
than yield stress
stress.
The compression flange is a beam-column
p g
continuous over vertical stiffener as supports
Its stability depends on stiffener spacing and
relative stiffness of the flange and the web. Fcr is
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 34
35. Compression Flange Vertical
buckling
Slenderness of webs with vertical stiffeners is taken conservatively
AISC ASD/LRFD li it the h/t by the given equation with
limits th b th i ti ith
Aw/Af =0.5
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 35
36. Shear buckling of beam webs
Shear buckling is seldom a determining
factor i d i
f t in design of rolled section b t
f ll d ti but
plate girders have much larger h/t so it
must be considered.
tb id d
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 36
37. Shear buckling of beam webs
Transverse stiffeners are used to
increase th b kli strength b
i the buckling t th by
increasing factor k through a reduction
in aspect ratio a/h.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 37
38. Transverse Stiffeners
Transverse stiffeners play an important role in
allowing the full ultimate load resistance of a
plate girder to be achieved.
In the first place they increase the buckling
resistance of the web;
Secondly they must continue to remain effective
after the web buckles, to provide anchorage for
p g
the tension field;
finally they must prevent any tendency for the
flanges to move towards one another.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 38
39. Transverse Stiffeners
The satisfactory performance of a
transverse stiffener can best be illustrated
by comparing the girders shown, after
shown
testing.
Figure 2
g
Figure 1
Fi
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 39
40. Transverse Stiffeners
In Figure 1 the stiffeners have remained straight.
g g
In Figure 2 the stiffener has failed and has been
unable to limit the buckling to the adjacent sub-
sub
panels of the girder; instead, the buckle has run
through the stiffener p
g position extending over
g
both panels. Consequently, significant reduction
in the failure load of the girder occurred.
In Figure 1 One can also see the effect of aspect
ratio,i.e greater a/ less k a d s a Fcr.
at o, e g eate a/h ess and small c
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 40
41. Transverse Stiffeners
The stiffener must be of adequate
rigidity in th di ti perpendicular t
i idit i the direction di l to
the plane of the web to prevent web
buckling. This condition is satisfied
p
provided the stiffener has a second
moment of area Is that satisfies the
following empirical formulae:
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 41
42. Transverse Stiffeners
AISC/LRFD Moment of Inertia of
stiffener is:
tiff i
where
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 42
43. Transverse Stiffeners
Transverse stiffeners spacing can be
determined from the following
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 43
44. Tension Field Action
The resulting shear stresses on an
element of a web are equivalent t
l t f b i l t to
principal stresses, one Tensile and one
Compressive, at 45 to the shear stress.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 44
45. Tension Field Action
Once a web panel has buckled in shear, it
p ,
loses its resistance to carry additional
compressive stresses.
stresses
On the other hand tensile principal stress
p p
continues to increase in strain in the
diagonal direction.
direction
Such a panel has a considerable p buckling strength,
p post g g ,
since increase in tension is limited only by yield stress.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 45
46. Tension Field Action
In this post b ckling range a ne load carr ing
post-buckling range, new load-carrying
mechanism is developed, whereby any additional
shear load is carried by an inclined tensile
membrane stress field. This tension field anchors
against the top and bottom flanges and against the
transverse stiffeners on either side of the web
panel. The load-carrying action of the plate girder
than becomes similar to that of the N-truss
In the post-buckling range, the resistance offered by
the web plates is analogous to that of the diagonal
tie bars in the truss.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 46
47. Tension Field Action
Phases of behavior up to collapse of a typical panel in shear
Prior to Buckling Post Buckling Collapse
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 47
48. Tension Field Action
The load-carrying action of the plate girder
load carrying
than becomes similar to that of the N-truss
In the post-buckling range, the resistance
offered by the web plates is analogous to
that of the diagonal tie bars in the truss.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 48
51. Tension Field Action
ft V
V
Vt=Tsinφ
Vt = ft ht cosφ sinφ T=ft ht cosφ
Vt = (1/2)ft ht sin2φ φ
Vt =(1/2) ft ht φ=45
Vty=(1/2) Fy ht………….(1)
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 51
55. Tension Field Action
(1)
Taking inelastic and strain hardening range
(2)
(3)
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 55
56. Tension Field Action
Codal equations are derived from
eqn;(1),(2),(3)
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 56
57. Tension Field Action
AISC/LRFD
k
a/h
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 57
58. Combined Bending & Shear
of Webs
Interaction diagram is based on Tension-
field f
fi ld of webs
b
If the web is completely yielded in
shear,any accompanying moment must
be
b resisted entirely b fl
i t d ti l by flanges.
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 58
59. Combined Bending & Shear
Bending & shear Interaction Curve
B B C D
V vyAw)
V/(F A
E 1/√3
0.75 0.83 1.0 1.07 1.12 M/My
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 59
60. Combined Bending & Shear
1.0
Mu/φMn
0.8
0.6
LRFD I t
Interaction Curve
ti C
0.4
0.2
0.2 0.4 0.6 0.8 1.0 Vu/φVn
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 60
61. Web Proportioning
Notations
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 61
62. Web Proportioning
Depth of girder is influenced by many
factors:
Headroom
Clearance for high water in deck bridges
Traffic passing beneath the bridge
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 62
63. Web Proportioning
Depth: Overall girder depth, h, will
usually be in the range
Lo/12 ≤ h ≤ Lo/8
/8,
occasionally lighter loads may be
accommodated with Lo/20 /20.
Flange:
g
The breadth, b, will usually be in the range
h/5 ≤ b ≤ h/3,
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 63
64. Design Procedure
1. Maximum Moment & Shear for Factored Load
2. Web D i
2 W b Design
1. p girder L/12 ≤ h ≤ L/8
Assume depth of g
2. Depth of Web hw=h-2tf
3. Web slenderness
1. For a/h <5 …………….
2. and for a/h > 5 ……………………
3.
3 hw/tw= 970/√Fy
4. Select optimum tw
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 64
65. Design Procedure
4. Flange Design
1. Find Af
2. Select suitable tf and bf
3.
3 Flange slenderness
1. bf/ 2tf < 65/√Fy …………….Compact
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 65
66. Design Procedure
5.
5 Check trial girder section
1. Web local buckling limit state
1. hw/tw< 640/√Fy…………………..Compact
2. 640/√Fy< hw/tw < 970/√Fy……Non-Compact
3. hw/tw > 970/√Fy…………………..Slender
2. Flange local buckling limit state
1. bf/ 2tf < 65/√Fy …………….Compact
3. Lateral Torsional Buckling
g
1. Calculate Iy
2. A=Af+Aw/6
3. ry= √Iy/A
4. Find Lb/ry
5. λp= 300/√Fy ………….. λ< λp ______Compact
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 66
67. Design Procedure
6. Bending strength
1. Calculate
C l l t Ix
2. Calculate Sxt
3. .
4. .
5. φMn≥ Mu
CE-409: Lecture 13 Prof. Dr Akhtar Naeem Khan 67
68. Procedure for Design
6. Bending strength
1. Calculate Ix
2.
2 Calculate Sxt
3. .
4. .
5.
CE-409: Lecture 13
φMn≥ Mu Prof. Dr Akhtar Naeem Khan