Beam columns are structural members that experience both bending and axial stresses. They behave similarly to both beams and columns. Many steel building frames have columns that carry significant bending moments in addition to compressive loads. Bending moments in columns are produced by out-of-plumb erection, initial crookedness, eccentric loads, wind loads, and rigid beam-column connections. The interaction of axial loads and bending moments in columns must be considered through an interaction equation to ensure a safe design. Second order effects, or P-Delta effects, produce additional bending moments in columns beyond normal elastic analysis and must be accounted for through moment magnification factors.
The document discusses different types of beams used in structures. It defines a beam as a structural member subjected primarily to bending. Different types of beams discussed include girders, secondary beams, joists, purlins, stringers, floor beams, girts, lintels and spandrels. Beams are classified based on their position, end conditions, fabrication method, and general span ranges. The document also covers beam analysis, including the flexure formula, stability of beam sections, and classification of beam sections as compact, non-compact and slender.
1. The document discusses stresses in solids due to eccentric and combined loading, including bending and direct stresses.
2. It defines the core of a section as the area where a load can be applied without causing tensile stress. For a rectangular section, the core is a rhombus with diagonals of B/3 and D/3.
3. Wind loading on structures like walls and chimneys is also analyzed, calculating bending moments and resultant stresses. Maintaining compressive stresses only is important for structural integrity.
This document discusses compression members and buckling of steel columns. It defines compression members as members subjected to compressive stresses that tend to shorten or squeeze the member. Examples given include struts, columns, truss chords, and beams. It notes that compression members are more prone to buckling than tension members. Buckling occurs when the critical buckling load is reached due to factors like member length, cross-section, end conditions, and imperfections. The effective length factor K is introduced to account for end conditions and sidesway in calculating the critical slenderness ratio.
This document provides an overview of analysis and design methods for concrete slabs, including:
1. Elastic analysis methods like grillage analysis and finite element analysis can be used to determine moments and shear forces in slabs.
2. Yield line theory is an alternative plastic/ultimate limit state approach for determining the ultimate load capacity of ductile concrete slabs. It involves assuming yield line patterns that divide the slab into rigid regions and equating external and internal work.
3. Examples are provided to illustrate yield line analysis for one-way spanning slabs and rectangular two-way slabs. Conventions, assumptions, and calculation procedures are explained.
The document provides steps for designing different structural elements:
1. Design of a beam subjected to torsion including calculation of torsional and bending moments, determination of steel requirements, and detailing.
2. Design of continuous beams involving calculation of bending moments and shears, reinforcement sizing, shear design, deflection check, and detailing including curtailment.
3. Design of circular water tanks with both flexible base and rigid base using approximate and IS code methods. This includes sizing hoop and vertical tension reinforcement, sizing wall thickness, designing cantilever sections and base slabs, and providing detailing diagrams.
A plate girder is a beam composed of welded or riveted steel plates. It consists of two flanges and a web plate. The flanges resist bending moments while the web resists shear forces. Plate girders are commonly used for longer spans than ordinary beams, with spans ranging from 14-40 meters for railroads and 24-46 meters for highways. They have a high depth to thickness ratio for the web, making it slender. Stiffeners are added to the web to prevent buckling. Plate girders are an economical choice for longer spans where their design can be optimized for requirements.
This document provides an overview of the design of steel beams. It discusses various beam types and sections, loads on beams, design considerations for restrained and unrestrained beams. For restrained beams, it covers lateral restraint requirements, section classification, shear capacity, moment capacity under low and high shear, web bearing, buckling, and deflection checks. For unrestrained beams, it discusses lateral torsional buckling, moment and buckling resistance checks. Design procedures and equations for determining effective properties and capacities are also presented.
The document discusses different types of beams used in structures. It defines a beam as a structural member subjected primarily to bending. Different types of beams discussed include girders, secondary beams, joists, purlins, stringers, floor beams, girts, lintels and spandrels. Beams are classified based on their position, end conditions, fabrication method, and general span ranges. The document also covers beam analysis, including the flexure formula, stability of beam sections, and classification of beam sections as compact, non-compact and slender.
1. The document discusses stresses in solids due to eccentric and combined loading, including bending and direct stresses.
2. It defines the core of a section as the area where a load can be applied without causing tensile stress. For a rectangular section, the core is a rhombus with diagonals of B/3 and D/3.
3. Wind loading on structures like walls and chimneys is also analyzed, calculating bending moments and resultant stresses. Maintaining compressive stresses only is important for structural integrity.
This document discusses compression members and buckling of steel columns. It defines compression members as members subjected to compressive stresses that tend to shorten or squeeze the member. Examples given include struts, columns, truss chords, and beams. It notes that compression members are more prone to buckling than tension members. Buckling occurs when the critical buckling load is reached due to factors like member length, cross-section, end conditions, and imperfections. The effective length factor K is introduced to account for end conditions and sidesway in calculating the critical slenderness ratio.
This document provides an overview of analysis and design methods for concrete slabs, including:
1. Elastic analysis methods like grillage analysis and finite element analysis can be used to determine moments and shear forces in slabs.
2. Yield line theory is an alternative plastic/ultimate limit state approach for determining the ultimate load capacity of ductile concrete slabs. It involves assuming yield line patterns that divide the slab into rigid regions and equating external and internal work.
3. Examples are provided to illustrate yield line analysis for one-way spanning slabs and rectangular two-way slabs. Conventions, assumptions, and calculation procedures are explained.
The document provides steps for designing different structural elements:
1. Design of a beam subjected to torsion including calculation of torsional and bending moments, determination of steel requirements, and detailing.
2. Design of continuous beams involving calculation of bending moments and shears, reinforcement sizing, shear design, deflection check, and detailing including curtailment.
3. Design of circular water tanks with both flexible base and rigid base using approximate and IS code methods. This includes sizing hoop and vertical tension reinforcement, sizing wall thickness, designing cantilever sections and base slabs, and providing detailing diagrams.
A plate girder is a beam composed of welded or riveted steel plates. It consists of two flanges and a web plate. The flanges resist bending moments while the web resists shear forces. Plate girders are commonly used for longer spans than ordinary beams, with spans ranging from 14-40 meters for railroads and 24-46 meters for highways. They have a high depth to thickness ratio for the web, making it slender. Stiffeners are added to the web to prevent buckling. Plate girders are an economical choice for longer spans where their design can be optimized for requirements.
This document provides an overview of the design of steel beams. It discusses various beam types and sections, loads on beams, design considerations for restrained and unrestrained beams. For restrained beams, it covers lateral restraint requirements, section classification, shear capacity, moment capacity under low and high shear, web bearing, buckling, and deflection checks. For unrestrained beams, it discusses lateral torsional buckling, moment and buckling resistance checks. Design procedures and equations for determining effective properties and capacities are also presented.
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.
This document provides an overview of shear and torsion behavior in reinforced concrete sections. It discusses several key topics:
1. There is no unified theory to describe shear and torsion behavior, which involves many interactions between forces. Current approaches include truss mechanisms, strut-and-tie models, and compression field theories.
2. Shear stresses are produced by shear forces, torsion, and combinations of these. The origin and distribution of shear stresses is explained.
3. Concrete alone cannot resist much shear or torsion due to its low tensile capacity. Reinforcement is needed to resist forces through truss action after cracking.
4. Design procedures from codes like ACI 318 are summarized
The document discusses various types of structural connections. It begins by defining connections as devices that join structural elements together to safely transfer forces. Connection design is more critical than member design. Failures usually occur at connections and can cause collapse.
The document then discusses different types of connections, including welded, riveted, and bolted connections. Connections are further classified based on the forces transferred, such as truss connections, fully restrained/moment connections, and partially restrained/shear connections. Specific connection types for buildings and frames like moment and shear connections are also explained. Design considerations for various structural connections like weld values, bolt values, and anchor bolts are provided.
The document discusses the design of slender columns. It defines a slender column as having a slenderness ratio (length to least lateral dimension) greater than 12. Slender columns experience appreciable lateral deflection even under axial loads alone. The design of slender columns can be done using three methods - the strength reduction coefficient method, additional moment method, or moment magnification method. The document outlines the step-by-step procedure for designing a slender column using the additional moment method, which involves determining the effective length, initial moments, additional moments, total moments accounting for a reduction coefficient, and redesigning the column for combined axial load and bending.
This document discusses the design of reinforced concrete deep beams. It defines deep beams as having a span/depth ratio less than 2 or a continuous beam ratio less than 2.5. Deep beams behave differently than elementary beam theory due to non-linear stress distributions. Their behavior depends on loading type and cracking typically occurs between one-third to one-half of the ultimate load. Design considerations include checking for minimum thickness, flexural design, shear design, and anchorage of tension reinforcement.
Connections are critical structural elements that join members together to transfer forces safely. Connection design is more important than member design, as connection failures can cause widespread structural collapse. Rigid connections provide strength and ductility to redistribute stresses during events like earthquakes. Common connection types include welded, riveted, and bolted connections, as well as moment connections, shear connections, and splices. Moment connections are particularly important for continuity and resisting lateral loads. Proper connection design is necessary to ensure structural integrity and safety.
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 lap joints, bolted connections, and riveted connections. It provides details on:
- The components and stresses involved in a basic lap joint using a single fastener under tension or compression.
- Requirements for bolted connections including minimum pretension values for high-strength bolts and methods for measuring pretension.
- Types of stresses fasteners experience including shear stresses at the interface of joined parts and bearing stresses transmitted into the surrounding plates.
- Properties and grades of rivets commonly used in structural connections as well as their tensile and shear strengths.
- Methods for calculating the load capacity ("rivet value") of single rivets in lap joints
This document discusses different types of rigid frame knee connections used to join beams and columns. Square knee joints are described, with and without diagonal stiffeners. Other knee types include square knees with brackets, straight haunched knees, and curved haunched knees. Straight haunched knees provide reasonable stiffness and rotation capacity at a lower cost than other options. The document provides design procedures and an example problem for sizing the components of a square knee connection between a W690×140 beam and W360×110 column.
This document provides guidelines for using the structural analysis software ETABS consistently within Atkins Dubai. It covers topics such as modelling procedures, material properties, element definition and sizing, supports, loading, load combinations, and post-analysis checks. The objective is to complement ETABS manuals and comply with codes such as UBC 97, ASCE 7, and BS codes as well as local authority requirements for Dubai projects. The procedures are based on standard practice in Dubai but can be revised based on specific project requirements.
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 summarizes the analysis of reinforced concrete beam cross sections to determine their moment of resistance at the ultimate limit state. It outlines the key assumptions of the strength design method and describes the behavior of beams under small, moderate and ultimate loads. It also discusses balanced, under-reinforced and over-reinforced beam sections, and introduces the concept of the equivalent stress block to simplify calculations. Worked examples are provided to demonstrate how to determine the depth of the neutral axis and moment of resistance for various beam cross sections.
The document discusses the design of compression members in planar trusses. It provides modifications to the slenderness ratio that must be applied when designing single angle compression members to account for potential torsional buckling. It then outlines a design flow chart for selecting compression member sections, including calculating required member capacity and area, selecting a trial section, and performing various checks related to stability, slenderness ratio and member capacity. An alternate method for selecting W-sections or double angle sections using column selection tables is also described.
this slide will clear all the topics and problem related to singly reinforced beam by limit state method, things are explained with diagrams , easy to understand .
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
Peer review presentation for the strut and tie method as an analysis and design approach for the mat on piles foundations of the primary separation cell (vessel).
The document discusses the design of columns in concrete structures. It covers several topics related to column design including: member strength and capacity versus section capacity, moment magnification, issues regarding slenderness effects, P-Delta analysis, and effective design considerations. The key steps in column design are outlined, including determining loads, geometry, materials, checking slenderness, computing design moments and capacities, and iterating the design as needed. Factors that influence column capacity such as slenderness, bracing, and effective length and stiffness are also described.
This document summarizes the design of a raft foundation for a given structure. Key details include:
- The raft is divided into three strips (C-C, B-B, A-A) in the x-direction based on soil pressure.
- Maximum soil pressure is 60.547 kN/m^2 and maximum bending moment is 445.02 kNm.
- The required raft depth is determined to be 860 mm to resist bending and punching shear.
- Longitudinal and transverse reinforcement of 20 mm bars at 200 mm and 220 mm centers respectively are designed.
This document presents an example of analysis design of slab using ETABS. This example examines a simple single story building, which is regular in plan and elevation. It is examining and compares the calculated ultimate moment from CSI ETABS & SAFE with hand calculation. Moment coefficients were used to calculate the ultimate moment. However it is good practice that such hand analysis methods are used to verify the output of more sophisticated methods.
Also, this document contains simple procedure (step-by-step) of how to design solid slab according to Eurocode 2.The process of designing elements will not be revolutionised as a result of using Eurocode 2. Due to time constraints and knowledge, I may not be able to address the whole issues.
This document discusses the properties and design of trusses and purlins. It defines key terms related to trusses like panel loads, which are concentrated loads applied at interior panel points calculated based on the roof load and area contributing to that point. Trusses are analyzed considering unit gravity and wind loads, and the principle of superposition is used. The document provides guidelines for designing purlins, including calculating loads, selecting trial sections, checking stresses and dimensions, and designing sag rods if needed. An example is given to demonstrate the purlin design process for given roof load and truss geometry data.
- Plate girders require stiffeners to prevent buckling of the thin webs under compression. Bearing stiffeners are located at supports and concentrated loads, while intermediate stiffeners are spaced along the web.
- Intermediate stiffeners help develop tension field action after the web buckles, allowing the girder to resist higher shear loads through a truss-like action of the stiffened web.
- The design of intermediate stiffeners involves calculating their required spacing and size based on the web dimensions and shear capacity of the girder considering both the initial buckling strength and additional strength from tension field action.
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.
This document provides an overview of shear and torsion behavior in reinforced concrete sections. It discusses several key topics:
1. There is no unified theory to describe shear and torsion behavior, which involves many interactions between forces. Current approaches include truss mechanisms, strut-and-tie models, and compression field theories.
2. Shear stresses are produced by shear forces, torsion, and combinations of these. The origin and distribution of shear stresses is explained.
3. Concrete alone cannot resist much shear or torsion due to its low tensile capacity. Reinforcement is needed to resist forces through truss action after cracking.
4. Design procedures from codes like ACI 318 are summarized
The document discusses various types of structural connections. It begins by defining connections as devices that join structural elements together to safely transfer forces. Connection design is more critical than member design. Failures usually occur at connections and can cause collapse.
The document then discusses different types of connections, including welded, riveted, and bolted connections. Connections are further classified based on the forces transferred, such as truss connections, fully restrained/moment connections, and partially restrained/shear connections. Specific connection types for buildings and frames like moment and shear connections are also explained. Design considerations for various structural connections like weld values, bolt values, and anchor bolts are provided.
The document discusses the design of slender columns. It defines a slender column as having a slenderness ratio (length to least lateral dimension) greater than 12. Slender columns experience appreciable lateral deflection even under axial loads alone. The design of slender columns can be done using three methods - the strength reduction coefficient method, additional moment method, or moment magnification method. The document outlines the step-by-step procedure for designing a slender column using the additional moment method, which involves determining the effective length, initial moments, additional moments, total moments accounting for a reduction coefficient, and redesigning the column for combined axial load and bending.
This document discusses the design of reinforced concrete deep beams. It defines deep beams as having a span/depth ratio less than 2 or a continuous beam ratio less than 2.5. Deep beams behave differently than elementary beam theory due to non-linear stress distributions. Their behavior depends on loading type and cracking typically occurs between one-third to one-half of the ultimate load. Design considerations include checking for minimum thickness, flexural design, shear design, and anchorage of tension reinforcement.
Connections are critical structural elements that join members together to transfer forces safely. Connection design is more important than member design, as connection failures can cause widespread structural collapse. Rigid connections provide strength and ductility to redistribute stresses during events like earthquakes. Common connection types include welded, riveted, and bolted connections, as well as moment connections, shear connections, and splices. Moment connections are particularly important for continuity and resisting lateral loads. Proper connection design is necessary to ensure structural integrity and safety.
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 lap joints, bolted connections, and riveted connections. It provides details on:
- The components and stresses involved in a basic lap joint using a single fastener under tension or compression.
- Requirements for bolted connections including minimum pretension values for high-strength bolts and methods for measuring pretension.
- Types of stresses fasteners experience including shear stresses at the interface of joined parts and bearing stresses transmitted into the surrounding plates.
- Properties and grades of rivets commonly used in structural connections as well as their tensile and shear strengths.
- Methods for calculating the load capacity ("rivet value") of single rivets in lap joints
This document discusses different types of rigid frame knee connections used to join beams and columns. Square knee joints are described, with and without diagonal stiffeners. Other knee types include square knees with brackets, straight haunched knees, and curved haunched knees. Straight haunched knees provide reasonable stiffness and rotation capacity at a lower cost than other options. The document provides design procedures and an example problem for sizing the components of a square knee connection between a W690×140 beam and W360×110 column.
This document provides guidelines for using the structural analysis software ETABS consistently within Atkins Dubai. It covers topics such as modelling procedures, material properties, element definition and sizing, supports, loading, load combinations, and post-analysis checks. The objective is to complement ETABS manuals and comply with codes such as UBC 97, ASCE 7, and BS codes as well as local authority requirements for Dubai projects. The procedures are based on standard practice in Dubai but can be revised based on specific project requirements.
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 summarizes the analysis of reinforced concrete beam cross sections to determine their moment of resistance at the ultimate limit state. It outlines the key assumptions of the strength design method and describes the behavior of beams under small, moderate and ultimate loads. It also discusses balanced, under-reinforced and over-reinforced beam sections, and introduces the concept of the equivalent stress block to simplify calculations. Worked examples are provided to demonstrate how to determine the depth of the neutral axis and moment of resistance for various beam cross sections.
The document discusses the design of compression members in planar trusses. It provides modifications to the slenderness ratio that must be applied when designing single angle compression members to account for potential torsional buckling. It then outlines a design flow chart for selecting compression member sections, including calculating required member capacity and area, selecting a trial section, and performing various checks related to stability, slenderness ratio and member capacity. An alternate method for selecting W-sections or double angle sections using column selection tables is also described.
this slide will clear all the topics and problem related to singly reinforced beam by limit state method, things are explained with diagrams , easy to understand .
The document discusses buckling and its theories. It defines buckling as the failure of a slender structural member subjected to compressive loads. It provides examples of structures that can experience buckling. It explains Euler's theory of buckling which derived an equation for the critical buckling load of a long column based on its bending stress. The assumptions of Euler's theory are listed. Four cases of long column buckling based on end conditions are examined: both ends pinned, both ends fixed, one end fixed and one end pinned, one end fixed and one end free. Effective lengths are defined for each case and the corresponding critical buckling loads given. Limitations of Euler's theory are noted. Rankine's empirical formula for calculating ultimate
Peer review presentation for the strut and tie method as an analysis and design approach for the mat on piles foundations of the primary separation cell (vessel).
The document discusses the design of columns in concrete structures. It covers several topics related to column design including: member strength and capacity versus section capacity, moment magnification, issues regarding slenderness effects, P-Delta analysis, and effective design considerations. The key steps in column design are outlined, including determining loads, geometry, materials, checking slenderness, computing design moments and capacities, and iterating the design as needed. Factors that influence column capacity such as slenderness, bracing, and effective length and stiffness are also described.
This document summarizes the design of a raft foundation for a given structure. Key details include:
- The raft is divided into three strips (C-C, B-B, A-A) in the x-direction based on soil pressure.
- Maximum soil pressure is 60.547 kN/m^2 and maximum bending moment is 445.02 kNm.
- The required raft depth is determined to be 860 mm to resist bending and punching shear.
- Longitudinal and transverse reinforcement of 20 mm bars at 200 mm and 220 mm centers respectively are designed.
This document presents an example of analysis design of slab using ETABS. This example examines a simple single story building, which is regular in plan and elevation. It is examining and compares the calculated ultimate moment from CSI ETABS & SAFE with hand calculation. Moment coefficients were used to calculate the ultimate moment. However it is good practice that such hand analysis methods are used to verify the output of more sophisticated methods.
Also, this document contains simple procedure (step-by-step) of how to design solid slab according to Eurocode 2.The process of designing elements will not be revolutionised as a result of using Eurocode 2. Due to time constraints and knowledge, I may not be able to address the whole issues.
This document discusses the properties and design of trusses and purlins. It defines key terms related to trusses like panel loads, which are concentrated loads applied at interior panel points calculated based on the roof load and area contributing to that point. Trusses are analyzed considering unit gravity and wind loads, and the principle of superposition is used. The document provides guidelines for designing purlins, including calculating loads, selecting trial sections, checking stresses and dimensions, and designing sag rods if needed. An example is given to demonstrate the purlin design process for given roof load and truss geometry data.
- Plate girders require stiffeners to prevent buckling of the thin webs under compression. Bearing stiffeners are located at supports and concentrated loads, while intermediate stiffeners are spaced along the web.
- Intermediate stiffeners help develop tension field action after the web buckles, allowing the girder to resist higher shear loads through a truss-like action of the stiffened web.
- The design of intermediate stiffeners involves calculating their required spacing and size based on the web dimensions and shear capacity of the girder considering both the initial buckling strength and additional strength from tension field action.
This document discusses design considerations for steel beams, including:
1. Deflection limits for buildings, bridges, and delicate machinery are discussed, ranging from L/360 to L/2000.
2. Initial beam selection can be done by limiting the span-to-depth ratio (L/d) based on the member type to indirectly control deflections.
3. Explicit span-to-depth ratio limits are provided for various member types, such as L/d ≤ 5500/Fy for buildings and L/d ≤ 20 for bridges.
4. Formulas are provided for calculating beam deflections under different loading conditions like uniform and point loads.
This document discusses tension members and how to calculate their net cross-sectional area. Tension members experience axial tensile forces that cause elongation. Built-up members may be needed if a single shape lacks sufficient capacity, rigidity, or requires unusual connection details. Net area calculation accounts for holes from fasteners by subtracting their total area from the gross area. Inclined portions of the failure plane add area. Shear lag reduces the effective net area based on connection efficiency. Pitch, gage, and stagger refer to fastener spacing.
This document provides an introduction to beams and beam mechanics. It discusses different types of beams and supports, how to calculate beam reactions and internal forces like shear force and bending moment, shear force and bending moment diagrams, theories of bending and deflection, and methods for analyzing statically determinate beams including the direct method, moment area method, and Macaulay's method. The key objectives are determining the internal forces in beams, establishing procedures to calculate shear force and bending moment, and analyzing beam deflection.
This document discusses the design of compression members. It defines compression members as members subjected to compressive loads that tend to shorten or squeeze them. Common examples include columns, struts, and members with bending and compressive loads. The strength of compression members is reduced compared to tension members due to their tendency to buckle when loaded. Longer columns have a greater risk of buckling. Other factors like load eccentricity, imperfections, and residual stresses also influence the buckling load. The document discusses various structural sections used for columns and considerations for local and overall buckling stability.
This document provides an overview of stresses in beams due to bending. It defines key terms like beam, bending moment, neutral axis, and radius of curvature. It then derives the bending formula that relates stress, bending moment, moment of inertia, and distance from the neutral axis. Several examples are worked through to demonstrate calculating stress in standard beam cross sections given bending moment values. The document concludes with self-assessment exercises for the reader.
This document provides an overview of helical spring design and analysis. It discusses stresses in helical springs, deflection calculations, compression spring types, stability, materials selection, static design considerations, critical frequencies, and examples. The chapters cover key topics like curvature effects, stresses from torsion and shear, deflection-force relationships, buckling analysis, material properties, recommended design ranges, and the wave equation for vibrational frequency.
The document discusses the analysis and design of beams subjected to bending. It provides examples of how to:
1) Determine shear and bending moment diagrams by drawing free body diagrams and applying equilibrium equations.
2) Calculate maximum shear forces and bending moments.
3) Relate loads, shears, and bending moments.
4) Select beam cross sections based on required section modulus to limit normal stresses to below allowable values.
The document discusses the analysis and design of beams subjected to bending. It provides examples of how to:
1) Determine shear and bending moment diagrams by drawing free body diagrams and applying equilibrium equations.
2) Calculate maximum shear forces and bending moments.
3) Relate loads, shears, and bending moments.
4) Select beam cross sections based on required section modulus to limit normal stresses to below allowable values.
Shaft design Erdi Karaçal Mechanical Engineer University of GaziantepErdi Karaçal
This document discusses the design of an industrial railway car shaft that is subjected to various loading conditions including bending, torsion, axial loading, and shear. The author performs both static failure analysis and fatigue failure analysis to size the shaft diameter. For fatigue analysis, the author calculates stress concentration factors and endurance limits. An initial diameter of 37.63mm is obtained from static analysis, which is then checked against fatigue analysis criteria. The final recommended diameter is 58mm, providing a safety factor of 1.55 when accounting for torsional loads in addition to bending. Deflection analysis is also performed to evaluate the shaft deformation.
The document discusses the benefits of exercise for mental health. Regular physical activity can help reduce anxiety and depression and improve mood and cognitive functioning. Exercise causes chemical changes in the brain that may help protect against mental illness and improve symptoms.
The document discusses various masonry design codes and philosophies. It explains that limit states design (LSD) is considered the most rational as it considers both safety under ultimate loads and serviceability under service loads, unlike working stress method (WSM) and ultimate load method (ULM). LSD has been adopted by most modern codes like ACI, IBC, and Eurocode. The document then discusses provisions for axial load, shear, and other limit states in different codes based on LSD and allowable stress design (ASD) formats.
Basic concept for structures with damping devicesF.R. Khan
1. Damping devices are intended to consume seismic energy and reduce structural response through added equivalent viscous damping. Figures 1 and 2 show how added damping and stiffness from damping devices can reduce displacement response but have different effects on acceleration response.
2. Cyclic loading tests of SMCD dampers show stable restoring force characteristics within a deformation range of ±10mm and allow over 20 cycles of deformation. SMCD dampers provide sufficient energy dissipation but may need replacement after severe earthquakes.
The document discusses limit state design of reinforced concrete structures. It defines two main limit states - limit state of collapse and limit state of serviceability. Limit state of collapse deals with strength and stability under maximum loads, while serviceability deals with deflection, cracking and durability under service loads. Characteristic loads have a 95% probability of not being exceeded and are factored up using partial safety factors for design loads. Material strengths are reduced using partial safety factors to calculate design strengths. The document also derives design coefficients for moment of resistance, depth of neutral axis and percentage steel reinforcement for rectangular sections under flexure.
Chapter 6-influence lines for statically determinate structuresISET NABEUL
Influence lines provide a systematic way to determine how forces in a structure vary with the position of a moving load. To construct influence lines for statically determinate structures:
1) Place a unit load at various positions along the member and use static analysis to determine the reaction, shear, or moment at the point of interest.
2) The influence line is drawn by plotting the value of the function versus load position.
3) Influence lines for beams consist of straight line segments, and the maximum shear or moment can be found using the area under the influence line curve.
Shear Strenth Of Reinforced Concrete Beams Per ACI-318-02Engr Kamran Khan
This document provides a 4 PDH course on the shear strength of reinforced concrete beams per ACI 318-02. It covers topics such as the different modes of failure for beams without shear reinforcement, the shear strength criteria, and calculations for the shear strength provided by concrete. The course content includes introductions to shear stresses in beams, Mohr's circle analysis, beam classifications, and equations for determining nominal shear strength based on the concrete strength and web reinforcement.
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Introduction to transportation engineeringCivil Zone
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Capacity & level of service (transportation engineering)Civil Zone
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This document discusses the alignment of highways, including horizontal and vertical elements. It covers topics such as grade line, horizontal and vertical curves, sight distance requirements, and super elevation. The key points are:
- Highway alignment consists of horizontal and vertical elements, including tangents and curves. Curves can be simple, compound, spiral, or reverse.
- Grade line refers to the longitudinal slope/rise of the highway. Factors in selecting a grade line include earthwork, terrain, sight distance, flood levels, and groundwater.
- Horizontal alignment deals with tangents and circular curves that connect changes in direction. Vertical alignment includes highway grades and parabolic curves.
- Proper design of curves
- Hydraulics engineering is the application of fluid mechanics principles to water-related structures like canals, rivers, dams and reservoirs. It is a branch of civil engineering concerned with water flow and conveyance.
- Ancient Egyptians, Mesopotamians, and Armenians made important early contributions to hydraulics engineering, developing irrigation systems using canals and qanats.
- Notable hydraulic structures through history include one of the world's oldest dams built in Egypt between 2950-2690 BC, and ship locks that raised or lowered boats between different water levels.
This document provides an introduction to hydropower engineering. It discusses how hydropower works by capturing the kinetic energy of falling water through turbines connected to generators. The amount of electricity generated depends on water flow rate and head (drop height). It also categorizes different types of hydropower developments including run-of-river, diversion canal, storage, and pumped storage plants. Site selection factors for hydropower include available water resources, water storage capacity, water head, and accessibility of the site.
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This document discusses the properties and analysis of trusses. It defines a truss as a frame structure where all members experience axial forces. Trusses are analyzed as pin-jointed frames if the joints intersect at a single point and loads are only applied at panel points. The document compares trusses to rigid frames and outlines various truss types including common roof trusses like the Howe, Pratt, Fink and Warren trusses. It also defines related terms like pitch, rise, purlins and loads on truss roofs.
1. The document designs bearing and end bearing stiffeners for a plate girder. For the bearing stiffener, a 200 x 15 mm stiffener plate is required on both sides under the web crippling limit state.
2. For the end bearing stiffener, a 240 x 18 mm stiffener plate is required on both sides due to the web crippling and bearing stiffener requirements at unframed ends.
3. Both designs satisfy all other limit states checked such as web local yielding, web sidesway buckling, and have sufficient weld strength.
This document provides a flow chart for designing built-up compression members and summarizes the design of a sample column consisting of two back-to-back channels with flat lacing. The key steps are: 1) Select a built-up section that satisfies strength and stability requirements, 2) Design flat lacing to resist shear forces using bars of appropriate size, length, and strength, 3) Satisfy design checks for lacing bar geometry and capacity. For the example, two 330x74mm channels are selected and flat lacing with 50x10mm bars 425mm long is designed to resist the shear force.
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Literature Review Basics and Understanding Reference Management.pptxDr Ramhari Poudyal
Three-day training on academic research focuses on analytical tools at United Technical College, supported by the University Grant Commission, Nepal. 24-26 May 2024
CHINA’S GEO-ECONOMIC OUTREACH IN CENTRAL ASIAN COUNTRIES AND FUTURE PROSPECTjpsjournal1
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china’s geo economic outreach in central Asian countries and its future prospect. China is thriving in trade,
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We have compiled the most important slides from each speaker's presentation. This year’s compilation, available for free, captures the key insights and contributions shared during the DfMAy 2024 conference.
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Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Using recycled concrete aggregates (RCA) for pavements is crucial to achieving sustainability. Implementing RCA for new pavement can minimize carbon footprint, conserve natural resources, reduce harmful emissions, and lower life cycle costs. Compared to natural aggregate (NA), RCA pavement has fewer comprehensive studies and sustainability assessments.
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Steel strucure lec # (19)
1. Prof. Dr. Zahid Ahmad Siddiqi
BEAM COLUMNS
Beam columns are structural members that are
subjected to a combination of bending and axial
stresses.
The structural behaviour resembles
simultaneously to that of a beam and a column.
Majority of the steel building frames have
columns that carry sizable bending moments in
addition to the usual compressive loads.
2. Prof. Dr. Zahid Ahmad Siddiqi
The sources of this bending moment are shown in
Figure 5.1 and explained below:
M=P´e
e
P
e
P
e
P
a) Out-Of-Plumb b) Initial Crookedness c) Eccentric Load
Figure 5.1. Sources of Eccentricity in Columns.
3. Prof. Dr. Zahid Ahmad Siddiqi
It is almost impossible to erect the columns
perfectly vertical and centre loads exactly on
columns.
Columns may be initially crooked or have other
flaws with the result that lateral bending is
produced.
In some cases, crane beams parallel to columns-
line and other perpendicular beams rest on brackets
projecting out from columns. This produces high
values of bending moments.
4. Prof. Dr. Zahid Ahmad Siddiqi
Wind and other lateral loads act within the
column height and produce bending.
The bending moments from the beams are
transferred to columns if the connections are
rigid.
CONTROLLING DESIGN FACTOR:
SECOND ORDER EFFECTS
The elastic analysis carried out to calculate
deflections and member forces for the given
loads is called 1st order and analysis.
5. Prof. Dr. Zahid Ahmad Siddiqi
The high axial load present in the column
combined with this elastic deflection produces
extra bending moment in the column, as is clear
from Figure 5.2.
The analysis of structure including this extra
moment is called 2nd order analysis.
Similarly, other higher order analysis may also be
performed.
In practice, usually 2nd order analysis is
sufficiently accurate with the high order results of
much lesser numerical value.
6. Prof. Dr. Zahid Ahmad Siddiqi
d
Maximum lateral
deflection due to
bending moment
(M)
P
M
M
P
Extra moment = P´d,
which produces more
deflections
Deflected shape
or elastic curve
due to applied
bending moment
(M)
Figure 5.2. Eccentricity Due to First Order Deflections.
7. Prof. Dr. Zahid Ahmad Siddiqi
The phenomenon in which the moments are
automatically increased in a column beyond the
usual analysis for loads is called moment
magnification or 2nd order effects.
The moment magnification depends on many
factors but, in some cases, it may be higher
enough to double the 1st order moments or even
more.
In majority of practical cases, this magnification
is appreciable and must always be considered for
a safe design.
8. Prof. Dr. Zahid Ahmad Siddiqi
1st order deflection produced within a member
(d) usually has a smaller 2nd order effect called P-
d effect, whereas magnification due to sides-way
(D) is much larger denoted by P-D effect (refer to
Figure 5.3).
P-Delta effect is defined as the secondary effect
of column axial loads and lateral deflections on
the moments in members.
The calculations for actual 2nd order analysis are
usually lengthy and can only be performed on
computers.
9. Prof. Dr. Zahid Ahmad Siddiqi
For manual calculations, empirical methods are
used to approximately cater for these effects in
design.
2nd order effects are more pronounced when loads
closer to buckling loads are applied and hence the
empirical moment magnification formula contains
a ratio of applied load to elastic buckling load.
The factored applied load should, in all cases, be
lesser than 75% of the elastic critical buckling load
but is usually kept much lesser than this limiting
value.
10. Prof. Dr. Zahid Ahmad Siddiqi
INTERACTION EQUATION AND
INTERACTION DIAGRAM
P D
P
Extra Moment
M = P´D
M
Figure 5.3.
A Deflected Beam-Column.
The combined stress at any
point in a member subjected to
bending and direct stress, as in
Figure 5.3, is obtained by the
formula:
f = ± ±
A
P
x
x
I
yM
y
y
I
xM
11. Prof. Dr. Zahid Ahmad Siddiqi
For a safe design, the maximum compressive
stress (f) must not exceed the allowable material
stress (Fall) as follows:
f = ± ± £ FallA
P
x
x
I
yM
y
y
I
xM
+ + £ 1
allAF
P
allx
x
FS
M
ally
y
FS
M
+ + £ 1
maxP
P
max,x
x
M
M
max,y
y
M
M
This equation is called interaction equation
showing interaction of axial force and bending
moment in an easy way.
12. Prof. Dr. Zahid Ahmad Siddiqi
If this equation is plotted against the various terms
selected on different axis, we get an interaction
curve or an interaction surface depending on
whether there are two or three terms in the
equation, respectively.
1.0
1.0
0,0
Figure 5.4. A Typical Interaction Curve.
13. Prof. Dr. Zahid Ahmad Siddiqi
Pr = required axial compressive strength
(Pu in LRFD)
Pc = available axial compressive
strength
= fcPn, fc = 0.90 (LRFD)
= Pn / Wc, Wc = 1.67 (ASD)
Mr= required flexural strength (Mu in
LRFD)
Mc = available flexural strength
= fbMn, fb = 0.90 (LRFD)
= Mn / Wb, Wb = 1.67 (ASD)
14. Prof. Dr. Zahid Ahmad Siddiqi
AISC INTERACTION EQUATIONS
The following interaction equations are
applicable for doubly and singly symmetric
members:
If ³ 0.2, axial load is considerable, and
following equation is to be satisfied:
c
r
P
P
£ 1.0
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
15. Prof. Dr. Zahid Ahmad Siddiqi
If < 0.2, axial load is lesser, beam
action is dominant, and the applicable
equation is:
c
r
P
P
£ 1.0÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
2
MOMENT ADJUSTMENT FACTOR
(Cmx or Cmy)
Moment adjustment factor (Cm) is based on
the rotational restraint at the member ends
and on the moment gradient in the members.
It is only defined for no-sway cases.
16. Prof. Dr. Zahid Ahmad Siddiqi
1. For restrained compression members in
frames braced against joint translation (no
sidesway) and not subjected to transverse loading
between their supports in the plane of bending:
Cm = 0.6 – 0.4
2
1
M
M
where M1 is the smaller end moment and M2 is
the larger end moment.
is positive when member is bent in
reverse curvature and it it is negative when
member is bent in single curvature (Figure 5.5b).
21 / MM
17. Prof. Dr. Zahid Ahmad Siddiqi
P
P
M2
M1
a) Reverse Curvature
P
P
M2
M1
b) Single Curvature
Figure 5.5. Columns Bent in Reverse and Single Curvatures.
When transverse load is applied between the
supports but or sway is prevented,
for members with restrained ends Cm = 0.85
for members with unrestrained ends Cm = 1.0
18. Prof. Dr. Zahid Ahmad Siddiqi
K-VALUES FOR FRAME BEAM-COLUMNS
K-values for frame columns with partially fixed
ends should be evaluated using alignment charts
given in Reference-1.
However, if details of adjoining members are not
given, following approximate estimate may be
used:
K = 1.2 – 1.5 if sidesway is permitted with
partially fixed ends
K = 1 if sidesway is prevented but end
conditions are not mentioned
19. Prof. Dr. Zahid Ahmad Siddiqi
MOMENT MAGNIFICATION FACTORS
Moment magnification factors (B1 and B2) are
used to empirically estimate the magnification
produced in the column moments due to 2nd order
effects.
These are separately calculated for sway or lateral
translation case (lt-case) and for no-sway or no
translation case (nt-case).
Accordingly, the frame is to be separately
analysed for loads producing sway and not
producing sway.
20. Prof. Dr. Zahid Ahmad Siddiqi
Mlt = moment due to lateral loads producing
appreciable lateral translation.
B2 = moment magnification factor to take
care of PuD effects for sway and
deflections due to lateral loads.
Mnt = the moment resulting from gravity
loads, not producing appreciable lateral
translation.
B1 = moment magnification factor to take
care of Pud effects for no translation
loads.
21. Prof. Dr. Zahid Ahmad Siddiqi
Mr = required magnified flexural strength
for second order effects
= B1 Mnt + B2 Mlt
Pr = required magnified axial strength
= Pnt + B2 Plt
No-Sway Magnification
B1 = ³ 1.0
11 er
m
PP
C
a-
22. Prof. Dr. Zahid Ahmad Siddiqi
where
a = 1.0 (LRFD) and 1.60 (ASD)
Pe1 = Euler buckling strength for
braced frame
= p2 EI / (K1 L)2
K1= effective length factor in the
plane of bending for no lateral
translation, equal to 1.0 or a
smaller value by detailed analysis
23. Prof. Dr. Zahid Ahmad Siddiqi
Sway Magnification
The sway magnification factor, B2, can be
determined from one of the following formulas:
B2 =
2
1
1
e
nt
P
P
å
å
-
a
where,
a = 1.0 (LRFD) and 1.60 (ASD)
SPnt = total vertical load supported by
the story, kN, including gravity loads
24. Prof. Dr. Zahid Ahmad Siddiqi
SPe2 = elastic critical buckling
resistance for the story
determined by sidesway
buckling analysis
= Sp2 EI / (K2 L)2
where I and K2 is calculated in the plane of
bending for the unbraced conditions
25. Prof. Dr. Zahid Ahmad Siddiqi
SELECTION OF TRIAL BEAM-
COLUMN SECTION
The only way by which interaction of axial
compression and bending moment can be
considered, is to satisfy the interaction equation.
However, in order to satisfy these equations, a
trial section is needed.
For this trial section, maximum axial compressive
strength and bending strengths may be
determined.
26. Prof. Dr. Zahid Ahmad Siddiqi
The difficulty in selection of a trial section for a
beam column is that whether it is selected based
on area of cross-section or the section modulus.
No direct method is available to calculate the
required values of the area and the section
modulus in such cases.
For selection of trial section, the beam-column
is temporarily changed into a pure column by
approximately converting the effect of bending
moments into an equivalent axial load.
27. Prof. Dr. Zahid Ahmad Siddiqi
Peq = equivalent or effective axial load
= Pr + Mrx mx + Mry my
mx (for first trial) = 8.5 - 0.7K1xLx
my (for first trial) = 17 - 1.4K1yLy
mx = 10 - 14(d / 1000)2 - 0.7K1xLx
my = 20 - 28(d / 1000)2 - 1.4K1yLy
28. Prof. Dr. Zahid Ahmad Siddiqi
The above equation is evaluated for Peq and a
column section is selected from the
concentrically loaded column tables for that
load.
The equation for Peq is solved again using a
revised value of m.
Another section is selected and checks are then
applied for this trial section.
29. Prof. Dr. Zahid Ahmad Siddiqi
WEB LOCAL STABILITY
For stiffened webs in combined flexural and axial
compression:
If £ 0.125 lp =
yb
u
P
P
f ÷
÷
ø
ö
ç
ç
è
æ
-
yb
u
y P
P
F
E
f
75.2
176.3
For A36 steel, lp = ÷
÷
ø
ö
ç
ç
è
æ
-
yb
u
P
P
f
75.2
17.106
If > 0.125 lp =
yb
u
P
P
f yyb
u
y F
E
P
P
F
E
49.133.212.1 ³
÷
÷
ø
ö
ç
ç
è
æ
-
f
3.4233.28.31 ³
÷
÷
ø
ö
ç
ç
è
æ
-
yb
u
P
P
f
For A36 steel, lp =
where l = h / tw and Py = Fy Ag
30. Prof. Dr. Zahid Ahmad Siddiqi
FLOW CHART FOR DESIGN OF
BEAM-COLUMNS
Known Data: Pu, Mntx, Mltx , Mnty, Mlty, KxLx, KyLy
Mr = Mu = Mnt + Mlt for the first trial
Calculate Mr both in the x and y directions
Peq = Pr + Mrx(mx) + Mry(my)
Assume an approximate magnification of
15% for the moments only.
31. Prof. Dr. Zahid Ahmad Siddiqi
Select section as a simple column depending
upon the following criteria:
1. Asel » Areq
2. Minimum weight
3. Connecting leg width b > bmin
4. Depth of W-section £ 360 mm
mx (for first trial) = 8.5 - 0.7K1xLx
my (for first trial) = 17 - 1.4K1yLy
mx = 10 - 14(d / 1000)2 - 0.7K1xLx
my = 20 - 28(d / 1000)2 - 1.4K1yLy
32. Prof. Dr. Zahid Ahmad Siddiqi
The column selection tables may also be
employed to select the section using the values of
Peq and KyLy.
See rx/ry from column selection table for selected
section
Calculate (KyLy)eq =
yx
xx
rr
LK
Re-enter the table for greater of KyLy and (KyLy)eq
and revise to obtain suitable section for the load
Peq.
33. Prof. Dr. Zahid Ahmad Siddiqi
Find new values of m for subsequent trials.
Select a new section and repeat until values
of load capacities, Peq, and m are stabilized.
Peq = Pr + Mrx (mx) + Mry (my)
Select a new section and repeat until values of
load capacities, Peq and m are stabilized.
34. Prof. Dr. Zahid Ahmad Siddiqi
Calculate Cmx and Cmy for no sway conditions
Calculate , , and
x
xx
r
LK1
y
yy
r
LK1
R = maximum of the above values
Check for maximum slenderness ratio: R £ 200
x
xx
r
LK2
y
yy
r
LK2
35. Prof. Dr. Zahid Ahmad Siddiqi
Axial strength of trial section:
Calculate fcFcr corresponding to the R-value or
directly read it from the table in Reference-1 and
evaluate the compression capacity by multiplying
with the area of cross-section.
Pc = fcPn = fcFcr Ag / 1000
Calculate Euler buckling strength (Pe1)x, (Pe1)y
, (Pe2)x and (Pe2)y for both lt and nt cases.
Pe1 = p2 EI / (K1 L)2 / 1000 (kN)
36. Prof. Dr. Zahid Ahmad Siddiqi
Calculate no-sway moment magnification factors
B1x = ³ 1.0 : B1y = ³ 1.0
Note: Pr in the above formulas is the actual
factored axial load and not Peq.
Calculate B2x and B2y.
B2 =
( )xe
r
mx
P
P
C
1
1 a-
( )ye
r
my
P
P
C
1
1 a-
where a = 1.0 for LRFD procedure.
2
1
1
e
nt
P
P
å
å
-
a
37. Prof. Dr. Zahid Ahmad Siddiqi
Calculate design moments
Mrx = Mux = B1x Mntx + B2x Mltx
Mry = Muy = B1y Mnty + B2y Mlty
Bending strength of the trial section:
fbMny = fb Fy Zy / 106 (kN-m)
There are no chances of lateral buckling because
the lateral direction for y-axis bending is the
stronger direction.
38. Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section as a beam.
Find Lp and Lr from column table and check
against Lbx.
Calculate fbMnx as for a beam using Lbx, Lp, Lr
and beam selection tables. Use Cb = 1.0 in the
expressions.
Calculate to see which interaction
equation is applicable.
c
r
P
P
39. Prof. Dr. Zahid Ahmad Siddiqi
Check interaction equations:
£ 1.0For ³ 0.2
£ 1.0For < 0.2
Get the value of Left Side of equation (LS) up to
2nd decimal place, truncating the 3rd decimal digit,
which should not be more than 1.00.
This means that LS can be as high as 1.0099 but
not 1.01.
c
r
P
P
c
r
P
P
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
2
40. Prof. Dr. Zahid Ahmad Siddiqi
Values of LS between 0.9 and 1.0 ® Very economical design
Values of LS between 0.8 and 0.9 ® Economical design
Values of LS between 0.7 and 0.8 ® May be acceptable,
but better to try
an economical section
Values of LS lesser than 0.7 ® Revise by selecting
a lighter section
Values of LS greater than 1.0099 ® Select a stronger section
Check shear strength, which may usually be
omitted in hot rolled W sections because of very
high available strength.
Write the final solution using standard designation.
41. Prof. Dr. Zahid Ahmad Siddiqi
Example 5.1: Design the columns in a single-
bay multi-storey unbraced frame shown in Figure
5.6, where P is the load from the top stories.
Ratio of moment of inertia of beams with respect
to columns may be assumed as shown in the
figure. Approximate analyses results are also
provided in Figures 5.7 and 5.8. Assume that
sway is not allowed in the y-direction.
Solution:
Total Factored Loads
1. Load Combination 1, Gravity Load
Combination (1.2 D + 1.6 L)
42. Prof. Dr. Zahid Ahmad Siddiqi
8.5 m
II
PP
wI I
H
6.0 m
6.0 m
1.4 I
Figure 5.6. Frame And Loading For Example 5.1.
43. Prof. Dr. Zahid Ahmad Siddiqi
Pu = 1.2(1025) + 1.6(410) = 1886 kN
wu = 1.2(7.3) + 1.6(22.0) = 43.96 kN/m
P = 1025 kN dead load
= 410 kN live load
w = 7.3 kN/m dead load
= 22.0 kN/m live load
H = 345 kN wind load
44. Prof. Dr. Zahid Ahmad Siddiqi
1886kN1886kN
43.96kN/m
75.8 kN-m
37.9 kN-m
227 kN-m
Figure 5.7. Partial Gravity Load Analysis Results.
45. Prof. Dr. Zahid Ahmad Siddiqi
2. Load Combination 2, Wind Load
Combination (1.2D+0.5L+1.3W)
Pu = 1.2(1025) + 0.5(410) = 1435 kN
Hu = 1.3(345) = 448.5 kN
wu = 1.2(7.3)+0.5(22.0) = 19.76 kN/m
Value of Kx
Gtop = = = 2.02
( )
( ) beamsforLI
columnsforLI
å
å ( )
5.84.1
62
I
I
Gbotton = 1.0 for sway columns
46. Prof. Dr. Zahid Ahmad Siddiqi
1435kN1435kN
19.76kN/m
51.2 kN-m
25.6 kN-m
( No Sway Part )
93.7 kN-m
138 kN
138 kN
448.5kN
586 kN-m
759 kN-m
Doh
( Sway Part )
586 kN-m
759 kN-m
Figure 5.8. Partial Lateral Load Analysis Results.
47. Prof. Dr. Zahid Ahmad Siddiqi
Kx = 1.0 for braced frame
Kx = 1.45 for unbraced conditions
Value of Ky:
No data of connected elements is given for y-
direction and hence the approximate value may
conservatively be assumed for no sway in this
direction.
Ky = 1.0
48. Prof. Dr. Zahid Ahmad Siddiqi
Here, design is made for the wind combination and
check is then made for the gravity combination.
Design for Combination 2:
Pr = 1435 + 138 + 19.76 ´ 8.5/2
= 1656.7 kN
According to AISC, max. moments for different
types of loading (nt or lt case), acting at different
locations or of different signs, are to be added
magnitude-wise in any combination.
The Right column is critical for the axial load.
49. Prof. Dr. Zahid Ahmad Siddiqi
Mntx = 51.2 kN-m
Mltx = 759 kN-m
Mnty = Mlty = 0
K2xLx= 1.45 ´ 6 = 8.7 m for lt-case
K1xLx = 1.00 ´ 6 = 6.00 m for nt-case
K1yLy = 1.00 ´ 6 = 6.00 m
Peq = Pu + 1.15 Mux (m)
= 1656.7 + 1.15 (51.2 + 759.0) (4.3)
= 5663 kN
mx (for first trial) = 8.5 - 0.7K1xLx
= 8.5 - 0.7 ´ 6 = 4.3
Assume 15% magnification of moments.
50. Prof. Dr. Zahid Ahmad Siddiqi
Using column tables of Reference 1 for this Peq
and KyLy = 6.0 m;
Trial Section = W360 ´ 262
Peq = Pu + 1.15 Mux (m)
= 1656.7 + 1.15 (51.2 + 759.0) (3.99)
= 5374 kN
Revised mx = 10 - 14(d / 1000)2 - 0.7K1xLx
= 10 - 14 ´ 0.362 - 0.7 ´ 6
= 3.99
51. Prof. Dr. Zahid Ahmad Siddiqi
Trial Section-1: W360 ´ 237
A = 30,100 mm2
rx = 162 mm, ry = 102 mm
rx/ry = 1.60
Ix = 79,100 ´ 104 mm4
M1 / M2 is positive because of reverse curvature
Cmx = 0.6 – 0.4
2
1
M
M
= 0.6 – 0.4 = 0.4÷
ø
ö
ç
è
æ
2.51
6.25
= = 37.04 ( for nt case)
x
xx
r
LK1
162
10000.6 ´
yx
xx
rr
LK
/
2
60.1
7.8
(KyLy)eq = = = 5.44 m (not critical)
52. Prof. Dr. Zahid Ahmad Siddiqi
= = 53.70 (for lt-case)
x
xx
r
LK2
162
10007.8 ´
= = 58.82
y
yy
r
LK1
102
10000.6 ´
R » 59 < 200 OK
fcFcr = 187.09 MPa
Pc = fcFcr Ag = = 5,631 kN
1000
100,3009.187 ´
for nt-case
Pe1x = p2 EI / (K1x L)2
=
10006000
10100,79000,200
2
42
´
´´´p
= 43,371 kN
54. Prof. Dr. Zahid Ahmad Siddiqi
B2x = = = 1.08
Mrx = B1x Mntx + B2x Mltx
= 1.0 (51.20) + 1.08(759.00)
= 870.9 kN-m
From column selection table:
Lp = 5.06 m, Lr = 25.43 m
xe
nt
P
P
,2
1
1
å
å
-
a
256,41
30380.1
1
1
´
-
Pr = Pnt + B2 Plt
= 1518.98 + 1.08(138)
= 1668.02kN
55. Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section:
= 6.5 < lp = 10.8 OK
f
f
t
b
2
= = 0.245
yb
u
P
P
f ( ) 100,301000/2509.0
7.1656
´´
For web, lp = 3.4233.28.31 ³
÷
÷
ø
ö
ç
ç
è
æ
-
yb
u
P
P
f
= 66.3 for A36 steel
= 15.3 < lp OK
wt
h
56. Prof. Dr. Zahid Ahmad Siddiqi
Lb = 6.00m > Lp, bending strength is to be
calculated using the inelastic LTB formula.
Mp = 250 ´ 4700 ´ 103 / 106 = 1175.0 kN-m
fbMp = 0.9 ´ 1175 = 1057.5 kN-m
Mr = 0.7 ´ Fy ´ Sx / 106
= 0.7 ´ 250 ´ 4160 ´ 103 / 106
= 728.0 kN-m
BF = = 21.94 kN÷
ø
ö
ç
è
æ
-
-
=
-
-
06.543.25
7281175
pr
rp
LL
MM
57. Prof. Dr. Zahid Ahmad Siddiqi
Mcx = fb[Mp - BF(Lb - Lp)]
= 0.9 [1175 - 21.94(6.0 - 5.06)]
= 1038.9 kN-m
Check Interaction Equation:
c
r
P
P
631,5
02.1668
= = 0.296 > 0.2
÷÷
ø
ö
çç
è
æ
+
cx
rx
c
r
M
M
P
P
9
8
= 0.296 + ÷
ø
ö
ç
è
æ
9.1038
92.870
9
8
= 1.041 > 1.00 NG
58. Prof. Dr. Zahid Ahmad Siddiqi
Trial Section-2: W360 ´ 262
A = 33,400 mm2
rx = 163 mm, ry = 102 mm
rx/ry = 1.60
Ix = 89,100 ´ 104 mm4
(KyLy)eq =
yx
xx
rr
LK
/
2
=
60.1
7.8
= 5.44 m (not critical)
59. Prof. Dr. Zahid Ahmad Siddiqi
Cmx = 0.4 (as before)
x
xx
r
LK1
163
10000.6 ´
= = 36.81 ( for nt case)
x
xx
r
LK2
163
10007.8 ´
= = 53.37 ( for lt case)
y
yy
r
LK1
102
10000.6 ´
= = 58.82
R » 59 < 200 OK
fcFcr = 187.09 MPa
60. Prof. Dr. Zahid Ahmad Siddiqi
Pc = fcPn = fcFcr Ag
=
1000
400,3309.187 ´
= 6,248 kN
Pe1x = p2 EI / (K1x L)2
=
10006000
10100,89000,200
2
42
´
´´´p
= 48,854 kN for nt-case
Pe2x = p2 EI / (K2x L)2
=
10008700
10100,89000,200
2
42
´
´´´p
= 23,236 kN for lt-case
61. Prof. Dr. Zahid Ahmad Siddiqi
xer
mx
PP
C
,11 a-
854,487.165611
4.0
´-
B1x =
= = 0.41 B1x = 1.0
SPnt = 1435 ´ 2 + 19.75 ´ 8.5
= 3038 kN
SPe2,x = 2 ´ 23,236 = 46,472 kN
B2x =
xe
nt
P
P
,2
1
1
å
å
-
a =
472,46
30380.1
1
1
´
-
= 1.07
62. Prof. Dr. Zahid Ahmad Siddiqi
Mrx = B1x Mntx + B2x Mltx
= 1.0 (51.20) + 1.07(759.00)
= 863.33 kN-m
Pr = Pnt + B2 Plt
= 1518.98 + 1.07(138)
= 1666.64 kN
From column selection table:
Lp = 5.08 m, Lr = 30.44 m
63. Prof. Dr. Zahid Ahmad Siddiqi
Check conditions of compact section:
f
f
t
b
2
= 6.0 < lp = 10.8 OK
yb
u
P
P
f ( ) 400,331000/2509.0
7.1656
´´
= = 0.220
For web,
lp = 3.4233.28.31 ³
÷
÷
ø
ö
ç
ç
è
æ
-
yb
u
P
P
f
for A36 steel
= 67.1
wt
h
= 13.7 < lp OK
64. Prof. Dr. Zahid Ahmad Siddiqi
Lb = 6.00m > Lp, bending strength is to be
calculated using the inelastic LTB formula.
Mp = 250 ´ 5240 ´ 103 / 106
= 1310.0 kN-m
fbMp = 0.9 ´ 1310 = 1179 kN-m
Mr = 0.7 ´ Fy ´ Sx / 106
= 0.7 ´ 250 ´ 4600 ´ 103 / 106
= 805.0 kN-m
65. Prof. Dr. Zahid Ahmad Siddiqi
BF = ÷
ø
ö
ç
è
æ
-
-
=
-
-
08.544.30
8051310
pr
rp
LL
MM
= 19.91 kN
Mcx = fb[Mp - BF(Lb - Lp)]
= 0.9 [1310 - 19.91(6.0 - 5.08)]
= 1162.5 kN-m
c
r
P
P
248,6
64.1666
= = 0.267 > 0.2
66. Prof. Dr. Zahid Ahmad Siddiqi
Check Interaction Equation:
÷÷
ø
ö
çç
è
æ
+
cx
rx
c
r
M
M
P
P
9
8
÷
ø
ö
ç
è
æ
5.1162
33.863
9
8
= 0.267 +
= 0.927 < 1.00 OK
Section Selected For Wind
Combination: W360 ´ 262
Check for Combination 1:
Pr = Pu = 1886 + 43.96 ´ 8.5/2
= 2073 kN
Mntx = 75.8 kN-m
67. Prof. Dr. Zahid Ahmad Siddiqi
Cmx = 0.4 same as before
xer
mx
PP
C
,11 a- 854,48207311
4.0
´-
B1x = =
= 0.42 B1x = 1.0
Mrx = B1x ´ Mntx = 75.8 kN-m
c
r
P
P
248,6
2073
= = 0.332 > 0.2
68. Prof. Dr. Zahid Ahmad Siddiqi
Check Interaction Equation:
÷÷
ø
ö
çç
è
æ
+
cx
rx
c
r
M
M
P
P
9
8
÷
ø
ö
ç
è
æ
5.1162
8.75
9
8
= 0.332 +
= 0.39 < 1.00 OK
Final Selection: W360 ´ 262
69. Prof. Dr. Zahid Ahmad Siddiqi
Example 5.2: Design the column for the
following data:
1.Braced frame
2.Pu = 1750 kN
3.Mntx = 330 kN-m
4.Mltx = 0
5.Mnty = 105 kN-m
6.K1xLx = K1yLy = 7.3 m
7.Lb = 7.3 m
8.Cm = 0.85
9.Fy = 250 MPa
70. Prof. Dr. Zahid Ahmad Siddiqi
Solution:
Peq = Pu + Mux mx + Muy my
For first trial: mx = 8.5 - 0.7 K1xLx
= 8.5 - 0.7 ´ 7.3 = 3.39
my = 17 - 1.4 K1yLy
= 17 - 1.4 ´ 7.3 = 6.78
Assume 15% magnification.
Peq = 1750 + 1.15 ´ 330 ´ 3.39 + 1.15
´ 105 ´ 6.78
= 3855 kN
71. Prof. Dr. Zahid Ahmad Siddiqi
KyLy = 7.3 m
From column load table, the trial section is:
W360 ´ 196
mx = 10 - 14 (d/1000)2 - 0.7 K1xLx
= 10 - 14 (0.36)2 - 0.7 ´ 7.3 = 3.08
my = 20 - 28 (d/1000)2 - 1.4 K1yLy
= 20 - 28 (0.36)2 - 1.4 ´ 7.3 = 6.15
Peq = 1750 + 1.15 ´ 330 ´ 3.08
+ 1.15 ´ 105 ´ 6.15
= 3661 kN
72. Prof. Dr. Zahid Ahmad Siddiqi
From column load table, the trial section is:
W360 ´ 179
rx/ry = 1.67
K1xLx / 1.67 = 4.37 < KyLy
KyLy is critical
Trial Section No. 1: W360 ´ 179
73. Prof. Dr. Zahid Ahmad Siddiqi
Ag = 22,800 mm2
rx = 158 mm
ry = 95.0 mm
Lp = 4.73 m
Lr = 21.20 m
Ix = 57,400 ´ 104 mm4
Iy = 20,600 ´ 104 mm4
Zx = 3,474 ´ 103 mm3
74. Prof. Dr. Zahid Ahmad Siddiqi
Zy = 1,671 ´ 103 mm3
Sx = 3,110 ´ 103 mm3
Mp = 868.5 kN-m
Mr = 544.25 kN-m
BF = 19.69 kN
Mcx = fbMnx = 736.11 kN-m
Cm = 0.85 (given)
75. Prof. Dr. Zahid Ahmad Siddiqi
158
10003.71 ´
=
x
xx
r
LK
0.95
10003.71 ´
=
y
yy
r
LK
= 46.20
R » 77 < 200 OK
fcFcr = 164.32 MPa
= 76.84
Pc = fcFcrAg =
1000
32.164
´ 22,800 = 4108 kN
76. Prof. Dr. Zahid Ahmad Siddiqi
10007300
10400,57000,200
2
42
´
´´´p
10007300
10600,20000,200
2
42
´
´´´p
Pe1,x =
= 21,262 kN
Pe1,y =
= 7,630 kN
B1x =
262,21
17501
1
85.0
1
,1
´
-
=
-
xe
nt
m
P
P
C
a
= 0.93
77. Prof. Dr. Zahid Ahmad Siddiqi
B1x = 1.0
B1y =
630,7
17501
1
85.0
´
-
= 1.10
Pr is not magnified as Plt = 0.
Mux = B1x ´ Mntx = 330 kN-m
Muy = B1y ´ Mnty = 1.10 ´ 105 = 115.5 kN-m
78. Prof. Dr. Zahid Ahmad Siddiqi
6
3
10
10671,12509.0 ´´´
c
r
P
P
3746
1750
Mcy = fbMpy
=
= 375.98 kN-m
bf / 2tf = 7.8 < 10.8 OK
h / tw = 19.3 < 42.3 (worst case) OK
= = 0.467 > 0.2
79. Prof. Dr. Zahid Ahmad Siddiqi
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
÷
ø
ö
ç
è
æ
+
98.375
5.115
11.736
330
9
8
Interaction Equation:
= 0.467 +
= 1.139 > 1.0 NG
Trial Section No. 2: W360 ´ 196
Ag = 25,000 mm2
rx = 160 mm
ry = 95.5 mm
81. Prof. Dr. Zahid Ahmad Siddiqi
Pc = 4108 kN
Pe1,x = 23,595 kN
Pe1,y = 8,445 kN
B1x = 1.0
B1y = 1.07
Mux = 330 kN-m
Muy = 112.5 kN-m
Mcy = fbMpy = 416.7 kN-m
82. Prof. Dr. Zahid Ahmad Siddiqi
c
r
P
P
4108
1750
= = 0.426 > 0.2
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
÷
ø
ö
ç
è
æ
+
7.416
5.112
57.817
330
9
8
Interaction Equation:
= 0.426 +
= 1.025 > 1.0 NG
Trial Section No. 3: W360 ´ 216
83. Prof. Dr. Zahid Ahmad Siddiqi
Ag = 27,500 mm2
rx = 161 mm
ry = 101 mm
Lp = 5.03 m
Lr = 25.43 m
Ix = 71,200 ´ 104 mm4
Iy = 28,200 ´ 104 mm4
Zx = 4,260 ´ 103 mm3
84. Prof. Dr. Zahid Ahmad Siddiqi
Zy = 2,180 ´ 103 mm3
Sx = 3,800 ´ 103 mm3
Mp = 1065 kN-m
Mr = 665 kN-m
BF = 19.61 kN
Mcx = fbMnx = 490.5 kN-m
c
r
P
P
4665
1750
= = 0.375 > 0.2
85. Prof. Dr. Zahid Ahmad Siddiqi
÷
÷
ø
ö
ç
ç
è
æ
++
cy
ry
cx
rx
c
r
M
M
M
M
P
P
9
8
÷
ø
ö
ç
è
æ
+
5.490
3.107
44.918
330
9
8
Interaction Equation:
= 0.375 +
= 0.889 > 1.0 OK
Final Selection: W360 ´ 216
86. Prof. Dr. Zahid Ahmad Siddiqi
Example 5.3: Design
the beam column
shown in Figure 5.9, if
sidesway is allowed
along weak axis but is
prevented along strong
axis. The moments
shown are factored
and are due to lateral
loads. The column
ends are partially fixed.
Pu =290 kN
Pu =290 kN
5.2m
220
kN-m
320 kN-m
320 kN-m
Figure 5.9. Column of Example 5.3.
87. Prof. Dr. Zahid Ahmad Siddiqi
Solution:
Although lateral load is present, sway is not
allowed along strong axis. Hence, the
moments may be considered to be of nt-case.
Pu = 290 kN; Mntx = 320 kN-m; Mltx = 0 kN-m
Due to unavailability of the connection data, use
approximate values of Kx and Ky as follows:
K1x = 1.0 ; K1y = 1.0 ; K2y = 1.2
K1xLx = 5.2 m ; K1yLy = 5.2 m
K2yLy = 6.24 m
88. Prof. Dr. Zahid Ahmad Siddiqi
For first trial: mx = 8.5 - 0.7 K1xLx
= 8.5 - 0.7 ´ 5.2 = 4.86
Assume 15% magnification.
Peq = Pu + 1.15 Mux (mx)
= 290 + 1.15 ´ 320(4.86)
= 2079 kN
W360 ´ 110 is uneconomical.
From column selection table, the trial section
is: W310 ´ 97
89. Prof. Dr. Zahid Ahmad Siddiqi
mx = 10 - 14 (d/1000)2 - 0.7 K1xLx
= 10 - 14 (0.31)2 - 0.7 ´ 5.2 = 5.01
Peq = 290 + 1.15 ´ 320 ´ 5.01
= 2134 kN
From column load table, the trial section is:
W310 ´ 97
rx/ry = 1.75
K1xLx / 1.75 = 2.97 < KyLy
KyLy is critical
90. Prof. Dr. Zahid Ahmad Siddiqi
Trial Section No.1: W310 ´ 97
Ag = 12,300 mm2
rx = 134 mm
ry = 76.7 mm
Lp = 3.82 m
Lr = 13.90 m
Ix = 22,700 ´ 104 mm4
Iy = 7,240 ´ 104 mm4
91. Prof. Dr. Zahid Ahmad Siddiqi
Zx = 1,586 ´ 103 mm3 ; Zy = 723 ´ 103 mm3
Sx = 1,440 ´ 103 mm3 ; Mp= 396.5 kN-m
Mr= 252 kN-m ; BF= 14.34 kN
Mcx = fbMnx= 339.04 kN-m
Check for local stability:
Þ Compact Section
bf/2tf = 9.9 < lp
h/tw = 24.9 < lp 42.3 for the worst case
92. Prof. Dr. Zahid Ahmad Siddiqi
Cmx = 1.0 (Consider member with
unrestrained ends to be on conservative side.)
K1xLx/rx = 5200/134 = 38.81
K1yLy/ry = 5200/76.7 = 67.80
K2yLy/ry = 6240/76.7 = 81.36
R = 82 < 200 OK
fcFcr = 157.54 MPa
Pc = fcFcrAg / 1000
= 157.54 ´ 12,300/1000 = 1938 kN
93. Prof. Dr. Zahid Ahmad Siddiqi
10005200
10200,22000,200
2
42
´
´´´p
xe
nt
mx
P
P
C
,1
1
a
-
206,16
2901
1
1
´
-
Pe1,x =
= 16,206 kN
B1x = ³ 1.0 =
= 1.018
Mrx = B1x Mntx
= 1.018 ´ 320 = 325.76 kN-m
94. Prof. Dr. Zahid Ahmad Siddiqi
c
r
P
P
1938
290
= = 0.150 < 0.2
÷÷
ø
ö
çç
è
æ
+
cx
rx
c
r
M
M
P
P
2 04.339
76.325
2
150.0
+=
= 1.036 > 1.0 NG
Trial Section No.2: W310 ´ 107
95. Prof. Dr. Zahid Ahmad Siddiqi
÷÷
ø
ö
çç
è
æ
+
cx
rx
c
r
M
M
P
P
2
77.379
12.325
2
134.0
+=
= 0.923 < 1.0 OK
Final Selection: W310 ´ 107