This chapter discusses the stability and behavior of columns under compressive loads. It begins by introducing columns and their goals of studying stability, critical load, effective length, and the secant formula. It then covers the stability of structures and Euler's formula for pin-ended columns. Subsequent sections extend Euler's formula to other end conditions, discuss eccentric loading and the secant formula, and address design of columns under centric and eccentric loads using empirical equations and allowable stress or load resistance factor approaches.
1) The document discusses the stability and buckling behavior of columns under axial loading. It introduces Euler's formula for determining the critical buckling load of pin-ended beams and describes how this analysis can be extended to columns.
2) Sample problems demonstrate how to design columns for centric and eccentric axial loads using these analytical methods and by considering stress limits. Design approaches vary based on the column's slenderness ratio.
3) The effects of eccentric loading are evaluated using a secant formula approach, where the eccentric load is modeled as a centric load plus a bending moment. Stress limits and interaction equations are provided.
The project consists of designing the structural elements of a two-part school building. The first part has three floors and the second part has one floor. The structural system uses a one-way ribbed slab system. Reinforced concrete beams, columns, footings, and shear walls will be designed according to ultimate strength design philosophy and codes including ACI 318 and IBC 2009. The summary provides an overview of the structural layout and design approach.
This document contains sample problems and information about mechanics of materials and mechanical properties of materials from a course. It includes 3 sample problems involving calculating stresses and strains in loaded structures. It also provides information about stress-strain diagrams and how they characterize different material types as ductile, brittle, elastic or viscoelastic. Key mechanical properties like modulus of elasticity, yield point, proportional limit and ductility are defined. Stress-strain behavior under tension and compression and for different materials like steel, aluminum, rubber, plastics and composites are illustrated with diagrams.
This document summarizes chapter 8 of the textbook "Mechanics of Materials" which discusses determining principle stresses in structural members under combined loading. It provides an introduction to the topic, examples of calculating principle stresses in beams under bending and transverse loads, designing transmission shafts subjected to both torque and bending stresses, and determining stresses at a point due to multiple applied forces. Sample problems are included that walk through applying equations to find principle stresses, shear stresses, and selecting appropriate beam or shaft dimensions based on allowable stress limits.
This document provides an overview of the concepts of stress that will be covered in Chapter 1 of the textbook "Mechanics of Materials". It begins with the objectives of studying mechanics of materials and defining stress and deformation. It then reviews concepts from statics like free body diagrams and force equilibrium. It introduces the different types of stresses - normal stress, shear stress, bearing stress - and provides examples of how to calculate each. It discusses stress under general load conditions and the state of stress. The goal is to analyze and design structures to determine stresses and ensure safety under loads.
3 torsion- Mechanics of Materials - 4th - BeerNhan Tran
This document provides an overview of chapter 3 from the textbook "Mechanics of Materials" which covers the topic of torsion. It begins with an introduction to torsional loads on circular shafts and the concept of net torque due to internal stresses. Key concepts discussed include shear strain, stresses in the elastic range using elastic torsion formulas, and failure modes under torsion. The document provides examples of solving sample problems involving statically indeterminate shafts, determining required shaft diameters, and calculating stresses and angles of twist. It concludes with discussing plastic deformations and torsion of noncircular members.
This document provides an overview of Chapter 2 from the textbook "Mechanics of Materials" which covers stress and strain under axial loading. The chapter discusses key topics like normal strain, stress-strain diagrams for ductile and brittle materials, Hooke's law, elastic vs plastic behavior, fatigue, deformations under axial loading, static indeterminacy, thermal stresses, and Poisson's ratio. It also includes sample problems and examples to demonstrate calculating deformations and reactions for axially loaded members.
1) The document discusses the stability and buckling behavior of columns under axial loading. It introduces Euler's formula for determining the critical buckling load of pin-ended beams and describes how this analysis can be extended to columns.
2) Sample problems demonstrate how to design columns for centric and eccentric axial loads using these analytical methods and by considering stress limits. Design approaches vary based on the column's slenderness ratio.
3) The effects of eccentric loading are evaluated using a secant formula approach, where the eccentric load is modeled as a centric load plus a bending moment. Stress limits and interaction equations are provided.
The project consists of designing the structural elements of a two-part school building. The first part has three floors and the second part has one floor. The structural system uses a one-way ribbed slab system. Reinforced concrete beams, columns, footings, and shear walls will be designed according to ultimate strength design philosophy and codes including ACI 318 and IBC 2009. The summary provides an overview of the structural layout and design approach.
This document contains sample problems and information about mechanics of materials and mechanical properties of materials from a course. It includes 3 sample problems involving calculating stresses and strains in loaded structures. It also provides information about stress-strain diagrams and how they characterize different material types as ductile, brittle, elastic or viscoelastic. Key mechanical properties like modulus of elasticity, yield point, proportional limit and ductility are defined. Stress-strain behavior under tension and compression and for different materials like steel, aluminum, rubber, plastics and composites are illustrated with diagrams.
This document summarizes chapter 8 of the textbook "Mechanics of Materials" which discusses determining principle stresses in structural members under combined loading. It provides an introduction to the topic, examples of calculating principle stresses in beams under bending and transverse loads, designing transmission shafts subjected to both torque and bending stresses, and determining stresses at a point due to multiple applied forces. Sample problems are included that walk through applying equations to find principle stresses, shear stresses, and selecting appropriate beam or shaft dimensions based on allowable stress limits.
This document provides an overview of the concepts of stress that will be covered in Chapter 1 of the textbook "Mechanics of Materials". It begins with the objectives of studying mechanics of materials and defining stress and deformation. It then reviews concepts from statics like free body diagrams and force equilibrium. It introduces the different types of stresses - normal stress, shear stress, bearing stress - and provides examples of how to calculate each. It discusses stress under general load conditions and the state of stress. The goal is to analyze and design structures to determine stresses and ensure safety under loads.
3 torsion- Mechanics of Materials - 4th - BeerNhan Tran
This document provides an overview of chapter 3 from the textbook "Mechanics of Materials" which covers the topic of torsion. It begins with an introduction to torsional loads on circular shafts and the concept of net torque due to internal stresses. Key concepts discussed include shear strain, stresses in the elastic range using elastic torsion formulas, and failure modes under torsion. The document provides examples of solving sample problems involving statically indeterminate shafts, determining required shaft diameters, and calculating stresses and angles of twist. It concludes with discussing plastic deformations and torsion of noncircular members.
This document provides an overview of Chapter 2 from the textbook "Mechanics of Materials" which covers stress and strain under axial loading. The chapter discusses key topics like normal strain, stress-strain diagrams for ductile and brittle materials, Hooke's law, elastic vs plastic behavior, fatigue, deformations under axial loading, static indeterminacy, thermal stresses, and Poisson's ratio. It also includes sample problems and examples to demonstrate calculating deformations and reactions for axially loaded members.
4 pure bending- Mechanics of Materials - 4th - BeerNhan Tran
This chapter discusses pure bending of structural members. Pure bending occurs when equal and opposite couples or bending moments act on a member. The chapter covers bending deformations and strains, stress due to bending calculated using elastic flexure formulas, section properties that influence bending such as moment of inertia, and bending of composite members made of multiple materials. Examples are provided to demonstrate calculating bending stresses in reinforced concrete beams.
Abstract (Dutch)
Samengestelde betonnen liggers vervaardigd van prefab voorgespannen- en/of gewapende elementen zijn zeer populair in de huidige praktijk van de civiele techniek. Twee betonnen, samengestelde delen van de ligger worden gestort op verschillende tijdstippen. Verschillende elasticiteitsmoduli, opeenvolgende belastingaanbrenging, en verschillend krimp en kruip veroorzaken een herverdeling van de normaalspanning en ongelijke rekken en spanningen in twee aansluitende vezels in het aansluitvlak.
Dit seminar richt zich op de berekening volgens de EN 1992-1-1 en EN 1992-2. De aannames met betrekking tot de berekening en de controle van de gewapende en/of voorgespannen samengestelde liggers en doorsnedes zal worden toegelicht.
Ook wordt er ingegaan op:
• De spanning/rek respons van de doorsnede belast door normaalkracht en buigende momenten,
• De principes van het gebruik van de “initiële toestand” in berekeningen van de uiterste grenstoestand en de bruikbaarheidsgrenstoestand,
• De controle van dwarskracht en wringing,
• De interactie tussen alle snedekrachten,
• De principes van de controles van de spanningbeperking,
• De achtergrond van de scheurwijdtecontrole
Speciale aandacht zal er worden gegeven aan de berekening van de schuifspanning in het aansluitvlak, en de beschouwing van de invloed van de verschillende leeftijd van de betonnen delen met betrekking tot de schuifspanningen. Een alternatieve berekeningsmethode ten opzichte van de Eurocode 2 zal worden voorgesteld en worden getest.
De praktische voorbeelden volgens de Eurocode 2 zullen worden uitgevoerd met behulp van de IDEA StatiCa software.
This document provides an overview of the concepts that will be covered in a chapter on stress from the textbook Mechanics of Materials. It includes definitions of key terms like stress, strain, normal stress, shear stress and bearing stress. It also gives examples of how to calculate stresses in different loading conditions like axial, eccentric, shear and bearing loads. The document provides an example problem walking through a static structural analysis and calculation of stresses and factors of safety in the members.
- The document discusses elastoplastic materials and plastic deformations that occur when stresses exceed the yield stress of ductile materials.
- It analyzes plastic deformations by assuming an idealized elastoplastic material model, where deformations are divided into elastic and plastic ranges. Permanent deformations result from loading beyond the yield stress.
- Residual stresses can develop if only part of a structure undergoes plastic deformation, or if different parts experience different plastic strains, even after unloading.
1) The document discusses the concept of stress and its analysis in mechanical engineering.
2) It provides an example of determining internal forces and stresses in a structural system consisting of a boom and rod connected by pins and subjected to a 30 kN load.
3) The analysis involves creating free body diagrams of the overall structure and its components to determine the reaction forces and internal forces, and then calculating the stresses in the members.
This document describes CUMBIA, a set of Matlab codes for analyzing reinforced concrete members. CUMBIA can perform monotonic moment-curvature analysis, force-displacement analysis, and axial load-moment interaction analysis. It uses material models for concrete and steel, including default Mander models for confined and unconfined concrete and King/Raynor models for steel. The section analysis iterates to find the neutral axis position for increasing concrete strains. Member response is obtained using a plastic hinge model, calculating displacements from equivalent plastic hinge length and moment-area relationships.
This document summarizes chapter 6 of the textbook "Mechanics of Materials" which discusses shearing stresses in beams and thin-walled members. It introduces the concepts of shear stress and shear flow distribution in beams. It provides examples of calculating shear stresses and forces in common beam configurations and thin-walled members. It also discusses plastic deformation effects from shear stresses exceeding yield criteria.
(1) The document provides calculations to determine the required base plate thickness for a column base connection according to Eurocode standards. It includes input parameters such as column forces, material properties, bolt sizes and locations.
(2) Three equations are solved simultaneously to determine the maximum pressure under the base plate, tension in the hold down bolts, and active concrete area.
(3) The calculated pressure and bolt tension exceed design values, requiring a redesign of the base plate length/width or use of higher strength concrete.
(4) The minimum required base plate thickness is then calculated based on the design bending moment and material yield strength.
This document provides a preliminary design for a 20-story reinforced concrete building in Los Angeles. It outlines the design process, including establishing seismic loading based on ASCE 7-05 and distributing forces. A dual structural system of concrete shear walls and special moment frames is proposed. Preliminary sizing of structural elements is presented, along with calculations for seismic base shear, story shear distribution, and building overturning moment.
Ch08 10 combined loads & transformationsDario Ch
This document contains a table of contents for a textbook on mechanical engineering. The table of contents lists 14 chapters, beginning with "Stress" and ending with "Buckling of Columns". Each chapter is numbered and given a page range. The document provides an outline of the topics that will be covered in the textbook.
This document discusses structural stability, statical determinacy, and influence lines. It defines stability as a prerequisite for structures to carry loads, which depends on comparing equations and unknown forces through structural analysis. Statical determinacy determines if a structure remains in equilibrium through static concepts alone. The number of external reactions must exceed the number of equilibrium equations. Influence lines show the variation of reactions, shear, or bending moment due to moving loads and identify their critical positions producing greatest effects.
This document provides an overview of the design of beams and one-way slabs for flexure, shear, and torsion according to IS 456. It discusses key concepts like requirements for flexural reinforcement, minimum and maximum reinforcement limits, clear cover, deflection control, and selection of member sizes. The document also includes a worked example showing the step-by-step design of a rectangular reinforced concrete beam for flexure. Design checks are performed to check for strength and deflection requirements. Modules for the course will cover analysis and design of beams, one-way slabs, and reinforcement detailing in accordance with limit state design principles and code specifications.
Based on the experimental and numerical studies performed by NIST an Alternative Load Path Analysis (ALPA) guideline is being developed by the Disproportionate Collapse Technical Committee to provide design, analysis and modeling methods for engineers in practice to reduce the vulnerability of steel and concrete structures to disproportionate collapse. This article presents a summary of chapter 3 of this guideline, titled Simplified Analysis to Predict Collapse Resistance. Chapter 3 focuses on simplified analysis methods that can be applied in form of spreadsheets or closed form solutions to predict the collapse capacity of steel and concrete buildings.
This document provides an introduction to mechanics of materials, which deals with analyzing how solid objects deform under stress. It discusses key topics like stress and strain analysis, mechanical properties of materials, and static equilibrium as it relates to determining internal forces in loaded structures. The document provides examples of using free body diagrams and equilibrium equations to calculate reactions, shear forces, bending moments, and other internal forces in beams, pipes, and other loaded bodies.
05-Strength of Double Angle Bolted Tension Members (Steel Structural Design &...Hossam Shafiq II
1. The document discusses the limit states and failure modes of bolted double angle tension members, including yielding of the gross section, fracture at the net section, and block shear failure.
2. It provides equations to calculate the effective net area considering shear lag effects, and the block shear strength considering both shear and tensile strengths.
3. An example calculation is shown to determine the tensile resistance of a double unequal angle member bolted at one leg, where fracture at the net section governs with a strength of 393.9 kN.
This document discusses how finite element analysis can be used to optimize composite structures and reduce costs. It provides examples of how FEA was used at various stages of design, including concept design to reduce weight, detailed design to evaluate performance under different loads, laminate optimization to lower material usage, failure analysis to investigate problems, and design verification for quality assurance. One example describes how FEA optimized a helicopter axle by varying fiber orientations between bands, reducing weight from 32kg to 6kg while maintaining safety.
base plate in bending and anchor bolts in tensionabdul latief
This document describes analytical models for predicting the behavior of base plate connections under bending and anchor bolts in tension. It presents models for the stiffness and resistance of the "T-stub" component, which includes the base plate and anchor bolts. The models account for whether contact occurs between the base plate and concrete. The analytical models are verified by tests and finite element analysis. Key factors influencing behavior are the relative stiffness of the base plate and bolts, and the effective length of the T-stub.
IRJETI Comparison of RC Beam-Column Joint with GFRC Beam-Column Joint using S...IRJET Journal
This study compares reinforced concrete (RC) beam-column joints to glass fiber reinforced concrete (GFRC) beam-column joints using finite element analysis in ANSYS. A corner beam-column joint from a two-story building was modeled in SolidWorks and analyzed in ANSYS under 10 N/mm and 20 N/mm loads. The GFRC joint showed significantly less deformation than the RC joint, with deformations reduced by 4-6% when using GFRC. Therefore, using GFRC can improve joint performance by reducing deformations.
This document contains information about flexural analysis and design of beams using ultimate strength design. It includes an example problem of calculating the nominal flexural strength (ΦbMn) for a doubly reinforced concrete beam section under two conditions of concrete compressive strength. It also includes a design example problem of a simply supported rectangular beam with given span, load, depth limitation, width, concrete strength, and steel yield strength to minimize tensile reinforcement.
This document presents the elasticity solution for a cantilever beam with a rectangular cross-section loaded by a concentrated force at its free end. It derives expressions for the stresses, strains, and displacements using the Airy stress function. Three different boundary conditions at the fixed end are considered, resulting in slightly different expressions for the displacements and elastic curve. Shear deformations are found to be more significant for short, deep beams, causing greater transverse deflections than classical beam theory predicts. For beams with span-to-depth ratios over 10, shear effects are negligible and classical theory is accurate.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
4 pure bending- Mechanics of Materials - 4th - BeerNhan Tran
This chapter discusses pure bending of structural members. Pure bending occurs when equal and opposite couples or bending moments act on a member. The chapter covers bending deformations and strains, stress due to bending calculated using elastic flexure formulas, section properties that influence bending such as moment of inertia, and bending of composite members made of multiple materials. Examples are provided to demonstrate calculating bending stresses in reinforced concrete beams.
Abstract (Dutch)
Samengestelde betonnen liggers vervaardigd van prefab voorgespannen- en/of gewapende elementen zijn zeer populair in de huidige praktijk van de civiele techniek. Twee betonnen, samengestelde delen van de ligger worden gestort op verschillende tijdstippen. Verschillende elasticiteitsmoduli, opeenvolgende belastingaanbrenging, en verschillend krimp en kruip veroorzaken een herverdeling van de normaalspanning en ongelijke rekken en spanningen in twee aansluitende vezels in het aansluitvlak.
Dit seminar richt zich op de berekening volgens de EN 1992-1-1 en EN 1992-2. De aannames met betrekking tot de berekening en de controle van de gewapende en/of voorgespannen samengestelde liggers en doorsnedes zal worden toegelicht.
Ook wordt er ingegaan op:
• De spanning/rek respons van de doorsnede belast door normaalkracht en buigende momenten,
• De principes van het gebruik van de “initiële toestand” in berekeningen van de uiterste grenstoestand en de bruikbaarheidsgrenstoestand,
• De controle van dwarskracht en wringing,
• De interactie tussen alle snedekrachten,
• De principes van de controles van de spanningbeperking,
• De achtergrond van de scheurwijdtecontrole
Speciale aandacht zal er worden gegeven aan de berekening van de schuifspanning in het aansluitvlak, en de beschouwing van de invloed van de verschillende leeftijd van de betonnen delen met betrekking tot de schuifspanningen. Een alternatieve berekeningsmethode ten opzichte van de Eurocode 2 zal worden voorgesteld en worden getest.
De praktische voorbeelden volgens de Eurocode 2 zullen worden uitgevoerd met behulp van de IDEA StatiCa software.
This document provides an overview of the concepts that will be covered in a chapter on stress from the textbook Mechanics of Materials. It includes definitions of key terms like stress, strain, normal stress, shear stress and bearing stress. It also gives examples of how to calculate stresses in different loading conditions like axial, eccentric, shear and bearing loads. The document provides an example problem walking through a static structural analysis and calculation of stresses and factors of safety in the members.
- The document discusses elastoplastic materials and plastic deformations that occur when stresses exceed the yield stress of ductile materials.
- It analyzes plastic deformations by assuming an idealized elastoplastic material model, where deformations are divided into elastic and plastic ranges. Permanent deformations result from loading beyond the yield stress.
- Residual stresses can develop if only part of a structure undergoes plastic deformation, or if different parts experience different plastic strains, even after unloading.
1) The document discusses the concept of stress and its analysis in mechanical engineering.
2) It provides an example of determining internal forces and stresses in a structural system consisting of a boom and rod connected by pins and subjected to a 30 kN load.
3) The analysis involves creating free body diagrams of the overall structure and its components to determine the reaction forces and internal forces, and then calculating the stresses in the members.
This document describes CUMBIA, a set of Matlab codes for analyzing reinforced concrete members. CUMBIA can perform monotonic moment-curvature analysis, force-displacement analysis, and axial load-moment interaction analysis. It uses material models for concrete and steel, including default Mander models for confined and unconfined concrete and King/Raynor models for steel. The section analysis iterates to find the neutral axis position for increasing concrete strains. Member response is obtained using a plastic hinge model, calculating displacements from equivalent plastic hinge length and moment-area relationships.
This document summarizes chapter 6 of the textbook "Mechanics of Materials" which discusses shearing stresses in beams and thin-walled members. It introduces the concepts of shear stress and shear flow distribution in beams. It provides examples of calculating shear stresses and forces in common beam configurations and thin-walled members. It also discusses plastic deformation effects from shear stresses exceeding yield criteria.
(1) The document provides calculations to determine the required base plate thickness for a column base connection according to Eurocode standards. It includes input parameters such as column forces, material properties, bolt sizes and locations.
(2) Three equations are solved simultaneously to determine the maximum pressure under the base plate, tension in the hold down bolts, and active concrete area.
(3) The calculated pressure and bolt tension exceed design values, requiring a redesign of the base plate length/width or use of higher strength concrete.
(4) The minimum required base plate thickness is then calculated based on the design bending moment and material yield strength.
This document provides a preliminary design for a 20-story reinforced concrete building in Los Angeles. It outlines the design process, including establishing seismic loading based on ASCE 7-05 and distributing forces. A dual structural system of concrete shear walls and special moment frames is proposed. Preliminary sizing of structural elements is presented, along with calculations for seismic base shear, story shear distribution, and building overturning moment.
Ch08 10 combined loads & transformationsDario Ch
This document contains a table of contents for a textbook on mechanical engineering. The table of contents lists 14 chapters, beginning with "Stress" and ending with "Buckling of Columns". Each chapter is numbered and given a page range. The document provides an outline of the topics that will be covered in the textbook.
This document discusses structural stability, statical determinacy, and influence lines. It defines stability as a prerequisite for structures to carry loads, which depends on comparing equations and unknown forces through structural analysis. Statical determinacy determines if a structure remains in equilibrium through static concepts alone. The number of external reactions must exceed the number of equilibrium equations. Influence lines show the variation of reactions, shear, or bending moment due to moving loads and identify their critical positions producing greatest effects.
This document provides an overview of the design of beams and one-way slabs for flexure, shear, and torsion according to IS 456. It discusses key concepts like requirements for flexural reinforcement, minimum and maximum reinforcement limits, clear cover, deflection control, and selection of member sizes. The document also includes a worked example showing the step-by-step design of a rectangular reinforced concrete beam for flexure. Design checks are performed to check for strength and deflection requirements. Modules for the course will cover analysis and design of beams, one-way slabs, and reinforcement detailing in accordance with limit state design principles and code specifications.
Based on the experimental and numerical studies performed by NIST an Alternative Load Path Analysis (ALPA) guideline is being developed by the Disproportionate Collapse Technical Committee to provide design, analysis and modeling methods for engineers in practice to reduce the vulnerability of steel and concrete structures to disproportionate collapse. This article presents a summary of chapter 3 of this guideline, titled Simplified Analysis to Predict Collapse Resistance. Chapter 3 focuses on simplified analysis methods that can be applied in form of spreadsheets or closed form solutions to predict the collapse capacity of steel and concrete buildings.
This document provides an introduction to mechanics of materials, which deals with analyzing how solid objects deform under stress. It discusses key topics like stress and strain analysis, mechanical properties of materials, and static equilibrium as it relates to determining internal forces in loaded structures. The document provides examples of using free body diagrams and equilibrium equations to calculate reactions, shear forces, bending moments, and other internal forces in beams, pipes, and other loaded bodies.
05-Strength of Double Angle Bolted Tension Members (Steel Structural Design &...Hossam Shafiq II
1. The document discusses the limit states and failure modes of bolted double angle tension members, including yielding of the gross section, fracture at the net section, and block shear failure.
2. It provides equations to calculate the effective net area considering shear lag effects, and the block shear strength considering both shear and tensile strengths.
3. An example calculation is shown to determine the tensile resistance of a double unequal angle member bolted at one leg, where fracture at the net section governs with a strength of 393.9 kN.
This document discusses how finite element analysis can be used to optimize composite structures and reduce costs. It provides examples of how FEA was used at various stages of design, including concept design to reduce weight, detailed design to evaluate performance under different loads, laminate optimization to lower material usage, failure analysis to investigate problems, and design verification for quality assurance. One example describes how FEA optimized a helicopter axle by varying fiber orientations between bands, reducing weight from 32kg to 6kg while maintaining safety.
base plate in bending and anchor bolts in tensionabdul latief
This document describes analytical models for predicting the behavior of base plate connections under bending and anchor bolts in tension. It presents models for the stiffness and resistance of the "T-stub" component, which includes the base plate and anchor bolts. The models account for whether contact occurs between the base plate and concrete. The analytical models are verified by tests and finite element analysis. Key factors influencing behavior are the relative stiffness of the base plate and bolts, and the effective length of the T-stub.
IRJETI Comparison of RC Beam-Column Joint with GFRC Beam-Column Joint using S...IRJET Journal
This study compares reinforced concrete (RC) beam-column joints to glass fiber reinforced concrete (GFRC) beam-column joints using finite element analysis in ANSYS. A corner beam-column joint from a two-story building was modeled in SolidWorks and analyzed in ANSYS under 10 N/mm and 20 N/mm loads. The GFRC joint showed significantly less deformation than the RC joint, with deformations reduced by 4-6% when using GFRC. Therefore, using GFRC can improve joint performance by reducing deformations.
This document contains information about flexural analysis and design of beams using ultimate strength design. It includes an example problem of calculating the nominal flexural strength (ΦbMn) for a doubly reinforced concrete beam section under two conditions of concrete compressive strength. It also includes a design example problem of a simply supported rectangular beam with given span, load, depth limitation, width, concrete strength, and steel yield strength to minimize tensile reinforcement.
This document presents the elasticity solution for a cantilever beam with a rectangular cross-section loaded by a concentrated force at its free end. It derives expressions for the stresses, strains, and displacements using the Airy stress function. Three different boundary conditions at the fixed end are considered, resulting in slightly different expressions for the displacements and elastic curve. Shear deformations are found to be more significant for short, deep beams, causing greater transverse deflections than classical beam theory predicts. For beams with span-to-depth ratios over 10, shear effects are negligible and classical theory is accurate.
Kittel c. introduction to solid state physics 8 th edition - solution manualamnahnura
1. The document discusses crystallographic planes and directions in a cube, the Miller indices of planes with respect to primitive axes, and the spacing between dots projected onto different planes of a crystal structure.
2. Key concepts from crystallography such as Miller indices, primitive lattice vectors, reciprocal lattice vectors, and the first Brillouin zone are defined. Calculations of interplanar spacing and lattice parameters are shown for simple cubic and face-centered cubic lattices.
3. Binding energies, cohesive energies, and equilibrium properties are calculated and compared for body-centered cubic and face-centered cubic crystal structures. Approximations made in describing crystal binding using Madelung energies and pair potentials are
This document provides solutions to problems involving the natural frequency and motion of single degree of freedom oscillatory systems. It analyzes problems related to beams, frames and pendulums undergoing harmonic motion. The solutions involve calculating the system stiffness and natural frequency, and determining displacement, velocity and acceleration as a function of time using the equations of motion for an undamped harmonic oscillator. Key parameters like mass, stiffness, initial conditions and system geometry are provided to find the desired responses.
This paper defines a class of functions represented by generalized Dirichlet series whose coefficients satisfy certain conditions. The region of convergence depends on the fixed Dirichlet series. This class of functions forms a complete normed linear space called Ω(u,p) with an inner product. It is proven that Ω(u,p) is a Banach space for 1≤p<∞ but not a Hilbert space unless p=2. A Schauder basis is obtained for Ω(u,p) as the set of functions {±kkesλk} where λk are the exponents of the fixed Dirichlet series.
Gate 2017 mechanical engineering evening session kulkarni Academy
This document contains a summary of a mechanical engineering evening session held on 4 February 2017 at Kulkarni's Academy of Mechanical Engineering. It includes 27 multiple choice questions covering topics like thermodynamics, fluid mechanics, heat transfer, strength of materials, machine design and other mechanical engineering subjects. The questions have answers ranging from simple calculations to conceptual understanding. This session was aimed at providing practice on various mechanical engineering topics for students.
1. The document contains solutions to problems from Chapter 17 of the physics textbook "Solutions to Physics: Principles with Applications, 5/E, by Giancoli". The problems involve calculating work, electric potential, electric field, and other concepts related to electric charge and forces.
2. The solutions show calculations using equations for work, kinetic energy, electric potential energy, and other physics equations. Values are calculated for quantities like voltage, electric field strength, speed, and more.
3. Diagrams are included with some of the solutions, such as vector diagrams for electric field calculations and diagrams of charge configurations. Twenty-seven unique problems are solved with varying levels of calculation complexity.
APPLICATION OF HIGHER ORDER DIFFERENTIAL EQUATIONSAYESHA JAVED
1) The document discusses modeling and applications of second order differential equations. It provides examples of second order differential equations that model vibrating springs and electric current circuits.
2) Solving second order differential equations involves finding the complementary function and particular integral. The type of roots in the auxiliary equation determines the form of the complementary function.
3) An example solves a second order differential equation modeling a vibrating spring to find the position of a mass attached to the spring at any time.
- The document provides the solution to problem 25.72 from the textbook, which asks to derive an expression for the total electric potential energy of a solid sphere with uniform charge density.
- The sphere is modeled as being built up of concentric spherical shells, with each shell carrying a small charge dq.
- The expression derived for the total potential energy is Ue = (3/5)kεQ2/R, where Q is the total charge on the sphere, R is the radius, and kε is the Coulomb's constant.
- The derivation involves integrating the electric potential energy dUe = Vdq over the volume of the sphere, where V is the potential and dq is the charge
This document summarizes methods for determining the deflection curve of beams subjected to different loading conditions. It presents the differential equations that relate the bending moment, shear force, and load to the deflection curve. Several examples are provided to demonstrate integrating these equations to obtain the slope and deflection of beams with various end conditions and load distributions. Methods like superposition are also described to handle cases with multiple loads.
2d beam element with combined loading bending axial and torsionrro7560
The document discusses beam theory and finite element modeling of beams and frames. It provides information on modeling beams using one-dimensional beam elements with cubic shape functions. The formulation describes defining the element stiffness matrix and calculating the element's contribution to the global structural stiffness matrix and force vector based on applied loads. Boundary conditions and sample problems are presented to demonstrate the element modeling approach.
This document discusses improving the continuous operation of Vasilescu-Karpen's concentration pile, which produces electricity while lowering the temperature of the electrolyte and environment. It presents the historical context of electrochemistry and outlines a two-dimensional mathematical model and equation transformations for numerically analyzing the pile's electric, magnetic, temperature, velocity, and density fields. The model considers heat transfer effects and variable liquid density. Specific boundary conditions are defined for steady-state operation. Stable and unstable operating regimes are identified based on the pile's heating and cooling energy functions.
The document discusses recursive algorithms and recurrence relations. It provides examples of solving recurrence relations for different algorithms like Towers of Hanoi, selection sort, and merge sort. Recurrence relations define algorithms recursively in terms of smaller inputs. They are solved to find closed-form formulas for the running time of algorithms.
(i) The document discusses a computational approach using finite element method to analyze the dynamic stability of pile structures subjected to periodic loads.
(ii) It develops the governing Mathieu-Hill type eigenvalue equation to determine stability and instability regions for different ranges of static and dynamic load factors.
(iii) Key steps involve discretizing the pile into finite elements, developing element stiffness, mass and stability matrices, and assembling them to solve the eigenvalue problem and analyze dynamic stability conditions for the pile structure.
This presentation is intended for undergraduate students in physics and engineering.
Please send comments to solo.hermelin@gmail.com.
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This presentation is in the Physics folder.
This document discusses power flow analysis and the Newton-Raphson power flow method. It provides details on setting up the power flow problem, including defining the power balance equations in terms of real and reactive power. It also describes calculating the Jacobian matrix and differentiating the power flow equations to populate the matrix. An example power flow case is presented on a two bus system to illustrate applying the Newton-Raphson method through multiple iterations to solve for the voltage magnitude and angle.
This document discusses cables and arches used in structural engineering. It provides information on how cables carry loads primarily in tension, while arches carry loads in compression. The document outlines various assumptions made in analyzing cables, including that cables are flexible, inextensible, and resist only axial forces. It also discusses analyzing cables under concentrated and uniformly distributed loads. Additional considerations are provided for cable-supported structures, including wind forces that must be considered in design.
Structural engineering involves relating physical forces to structural elements that resist them. Analysis determines forces in each element of a defined structure, while design configures elements to resist known forces. The process iterates between analysis and design until complete. Structures resist vertical and horizontal loads, and include large items like bridges as well as everyday items. Structural design requires data on the structure type, site conditions like soil properties, and loading conditions from dead and live loads, wind, and earthquakes as defined by codes. Design methods are selected based on local practices.
This document discusses the design of beams and columns in buildings. It explains that beams are used in floors and roofs to safely support loads, and are primarily subjected to bending and shear. Beams must be sized to limit deflection based on the length to avoid issues like ceiling cracks. Columns carry primary axial loads and are designed for both compression and any additional bending from other forces. The document also discusses different types of beams, forces and supports, and references sources for further information.
This document provides an excerpt from the textbook "Mechanics of Materials" by Ferdinand P. Beer, E. Russell Johnston, Jr., and John T. DeWolf. The excerpt discusses columns and their stability. It begins by explaining how column design considers both stress and deformation. It then introduces Euler's formula for calculating the critical buckling load of columns. Subsequent sections extend the formula to other column end conditions, provide sample problems, and discuss design of columns under centric and eccentric loading. Key concepts covered include pin-ended beams, the secant formula for eccentric loading, and design approaches for steel and aluminum alloy columns.
1) The moment coefficient method is used to analyze statically indeterminate structures by providing coefficients that account for the internal forces and moments based on the structure's loading and geometry.
2) The ACI code provides tables of moment coefficients for various structural conditions that are based on elastic analysis but also consider inelastic redistribution of forces.
3) The moment coefficient method expresses the moments as coefficients multiplied by the total factored load and the clear span length. This allows for a more precise analysis compared to assuming uniform moments.
This document discusses the design of slender columns and two-way slabs. It provides an example problem for designing a long rectangular braced column. The steps include computing factored loads, determining the k value, checking for slenderness effects, calculating the required moment strength, and designing the reinforcement. It also compares one-way and two-way slab behavior, discussing different slab systems like flat plates, waffle slabs, and ribbed slabs.
This document is a course material document on structures II that covers:
- Column analysis including failure modes, end conditions, and analysis of wood and steel columns
- Column design including design of wood and steel columns using various equations and methods
- It provides examples of analyzing and designing columns and references resources on copyright and terms of use.
The document discusses bar bending schedules (BBS), which provide details of reinforcing bars used in concrete structures. It explains that a BBS includes the member identification, bar mark, steel type, diameter, length, number of bars, and bending dimensions. It then provides examples of BBS for beams, slabs, columns and walls. Measurement techniques for bar lengths are also outlined, along with best practices. The document concludes by presenting a sample BBS calculation for a beam and listing relevant codes, specifications and online BBS software.
- A beam-and-slab system uses a one-way slab supported by beams cast compositely with the slab, forming an efficient floor system.
- The slab is designed as a continuous slab using continuous beam theory or a simplified coefficient method. Beams act as L-beams on edges and T-beams interiorly.
- T-beams and L-beams save weight while providing a greater lever arm for strength and stiffness over conventional beams. Ensuring the beam and slab are securely connected requires attention to vertical and horizontal shear forces.
The document provides information on column design according to BS 8110-1:1997, including general recommendations, classifications of columns, effective length and minimum eccentricity, design moments, and design. Short columns have a length to height or breadth ratio less than 15 for braced or 10 for unbraced. Braced columns have lateral stability from walls or bracing. Additional moments are considered for slender or unbraced columns based on deflection. Design moments are calculated considering axial load and biaxial bending for different column classifications. Shear design also considers axial load and reinforcement is required if shear exceeds the shear capacity. The interaction diagram is constructed based on equilibrium equations relating stresses on a column cross section to axial load and bending
This document discusses different types and classifications of columns. It defines a column as a vertical structural member primarily designed to carry axial compression loads. Columns can be classified based on their shape, reinforcement, and type of loading. Common shapes include square, rectangular, circular, L-shaped, and T-shaped sections. Reinforcement types include tied columns with ties, spiral columns with helical reinforcement, and composite columns with encased steel. Columns are either concentrically loaded with forces through the centroid, or eccentrically loaded off-center. The document also covers column capacity calculations, resistance factors, and provides an example problem.
This document discusses thin-walled column design considering local, distortional, and Euler buckling. It introduces elastic buckling prediction methods including numerical and hand calculation approaches. Experimental data on over 100 cold-formed steel columns demonstrates failure mechanisms associated with local and distortional buckling. Design methods considered are the AISI effective width specification and direct strength methods using hand solutions or finite strip analysis. Performance assessments show direct strength methods provide accurate strength predictions for a variety of cross-sections.
The document discusses different failure modes of columns under eccentric loading. It defines equations for determining the load carrying capacity (Pn) and moment capacity (Mn) of columns. Three failure cases are analyzed: 1) pure axial load/crushing failure where only compression steel contributes, 2) balanced failure where steel yields in tension and compression, and 3) pure flexural failure where the column acts like a beam with no axial load. Equations are derived for Pn and Mn for each case.
This document summarizes a presentation on beams, slabs, and columns. It lists the group members presenting and defines beams, slabs, and columns. It describes different types of beams, slabs, reinforcement details, formwork types, construction process including casting, compaction and curing. Construction examples are provided and materials used are outlined.
This document discusses the design of two-way slabs. It begins by defining two-way slabs as slabs that span in two directions when the ratio of long to short spans is less than 2. It describes the main types of two-way slabs as flat slabs with drop panels and slabs with beams. The document outlines the basic design steps, including choosing the slab type and thickness, selecting a design method, calculating moments, determining reinforcement, and checking shear strength. It provides details on determining maximum bending moments and reinforcement spacing and requirements. Finally, it compares the direct design method and equivalent frame method for analyzing two-way slab systems.
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This document provides an overview of design in reinforced concrete according to BS 8110. It discusses the basic materials used - concrete and steel reinforcement - and their properties. It describes two limit states for design: ultimate limit state considering failure, and serviceability limit state considering deflection and cracking. Key aspects of beam design are summarized, including types of beams, design for bending and shear resistance, and limiting deflection. Reinforcement detailing rules are also briefly covered. Design examples are provided to illustrate bending and shear design of beams.
The document discusses reinforced concrete columns, including their functions, failure modes, classifications, and design considerations. Columns primarily resist axial compression but may also experience bending moments. They can fail due to compression, buckling, or a combination. Design depends on whether the column is short or slender, braced or unbraced. Reinforcement is determined based on the loads applied, including axial load only, symmetrical beam loading, or loading in one or two bending directions. Links are included to prevent bar buckling. Examples show how to design column longitudinal reinforcement and links for different load cases.
This document provides three thumb rules for column placement in structures:
1. The minimum column size should not be less than 9"x9" for a single story structure, and 12"x9" for a 1.5 story structure, using appropriate concrete grades. Larger column sizes are needed for greater distances or heights.
2. The distance between column centers should not exceed 4m for 9"x9" columns, and larger column sizes are needed for greater distances.
3. Columns should be arranged in a rectangular grid or circular pattern, not zigzag, to avoid structural issues in load transfer, wall construction, and beam placement. Following these thumb rules can help prevent mistakes in structural design.
This presentation summarizes the key aspects of one-way slab design:
1) One-way slabs have an aspect ratio of 2:1 or greater, where bending occurs primarily along the long axis. They can be solid, hollow, or ribbed.
2) Design and analysis treats a unit strip of the slab as a rectangular beam of unit width and the slab thickness as the depth.
3) The ACI code specifies minimum slab thickness, concrete cover, span length, bar spacing, reinforcement ratios, and other design requirements.
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5) One-
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2. 10.1 Introduction
Column = vertical prismatic members subjected
to compressive forces
Goals of this chapter:
1. Study the stability of elastic
columns
2. Determine the critical load Pcr
3. The effective length
4. Secant formula
3. Previous chapters:
-- concerning about
(1) the strength and
(2) excessive deformation (e.g. yielding)
This chapter:
-- concerning about
(1) stability of the structure (e.g. bucking)
4. 10.2 Stability of Structures
allow
P
A
σ σ= <
cr
PL
AE
δ δ= <
2
2
( )sin ( )cr
LP Kθ θ∆ = ∆ (10.1)
sin θ θ∆ ≈ ∆ (10.2)
Concerns before:
New concern:
Stable?
Unstable?
5. 2
2
( ) ( )cr
LP Kθ θ∆ = ∆
sin θ θ∆ ≈ ∆
4 /crP K L= (10.2)
The system is stable, if
The system is unstable if
Since
4 /crP K L<
4 /crP K L>
A new equilibrium state may be
established
6. 2
2
( )sin ( )LP M Kθ θ= =
4 sin
PL
K
θ
θ
=
The new equilibrium position is:
4sin
PL
K
θ
θ
=
(10.3)
or
7. After the load P is applied, there are
three possibilities:
1. P < Pcr – equilibrium & θ = 0 -- stable
2. P > Pcr – equilibrium & θ = θ -- stable
3. P > Pcr – unstable – the structure
collapses, θ = 90o
8. 10.3 Euler’s Formula for Pin-Ended Columns
Determination of Pcr for the configuration in Fig. 10.1
ceases to be stable
2
2
d y M P
y
dx EI EI
= = −
2
2
0
d y P
y
dx EI
+ =
Assume it is a beam subjected to
bending moment:
(10.5)
(10.4)
9. 2 P
p
EI
=
2
2
2
0
d y
p y
dx
+ =
sin cosy A px B px= +
Defining:
The general solution to this harmonic function is:
(10.7)
(10.6)
(10.8)
B.C.s:
@ x = 0, y = 0 → B = 0
@ x = L, y = 0
Eq. (10.8) reduces to
0sinA pL = (10.9)
10. 0sinA pL =
2 2
2
2
n
p
L
π
=
2 P
p
EI
=
(10.9)
Therefore,
1. A = 0 → y = 0 → the column is straight!
2. sin pL = 0 → pL = nπ → p = nπ /L →
Since (10.6)
We have
2 2
2
n EI
P
L
π
=
For n = 1
2
2cr
EI
P
L
π
=
(10.10)
-- Euler’s formula (10.11)
11. Substituting Eq. (10.11) into Eq. (10.6),
2
2cr
EI
P
L
π
=
2 P
p
EI
=
2 2
2
2 2
crP EI
p
EI L EI L
π π
= = =
(10.6)
(10.11)
Therefore,
Hence p
L
π
=
Equation (10.8) becomes sin
x
y A
L
π
= (10.12)
This is the elastic curve after the beam is buckled.
12. If P < Pcr → sin pL ≠ 0
0sinA pL =
1. A = 0 → y = 0 → the column is straight!
2. sin pL = 0 → pL = nπ →
2
2cr
EI
P
L
π
=
(10.9)
Hence, A = 0 and y = 0 → straight configuration
13. Critical Stress:
Introducing
2
2
( )
cr
E
L
r
π
σ =
2
2
cr
cr
P EI
A L A
π
σ = =
2
I Ar=
Where r = radius of gyration
, ,y zx
x y z
I II
r r r
A A A
= = =
Where r = radius of gyration (10.13)
L/r = Slenderness ratio
14. 10.4 Extension of Euler’s Formula to columns with
Other End Conditions
2
2cr
e
EI
P
L
π
=
2
2
( )
cr
e
E
L
r
π
σ =
(10.11')
(10.13')
Case A: One Fixed End, One Free End
Le = 2L
15. Case B: Both Ends Fixed
At Point C
RCx = 0
Q = 0 → 0
VQ
It
π = =
Point D = inflection point → M = 0 → AD and DC are symmetric
Hence, Le = L/2
16. 2
2
d y M P V
y x
dx EI EI EI
= = − −
2 P
p
EI
=
2
2
2
d y V
p y x
dx EI
+ = −
Case C: One Fixed End, One Pinned End
M = -Py - Vx
Since
Therefore,
The general solution:
The particular solution:
sin cosy A px B px= +
2
V
y x
p EI
= −
17. 2 P
p
EI
=
V
y x
P
= −
Substituting into the particular solution, it follows
As a consequence, the complete solution is
sin cos
V
y A px B px x
P
= + − (10.16)
18. sin
V
A pL L
P
=
sin
V
y A px x
P
= −
(10.16)
B.C.s:
@ x = 0, y = 0 → B = 0
@ x = L, y = 0 →
(10.17)
Eq. (10.16) now takes the new form
sin cos
V
y A px B px x
P
= + −
19. Taking derivative of the question,
cos
dy V
Ap px
dx P
= −
cos
V
Ap pL
P
=
sin
V
A pL L
P
=
B.C.s: @ x = L, dy/dx = θ = 0
(10.18)
(10.17)
10 17
10 18
( . )
( . )
tan pL pL= (10.19)
20. Solving Eq. (10.19) by trial and error,
2 2
4 4934 4 4934 20 19064. . / . /pL p L p L= ⇒ = ⇒ =
2 2
2
20 19.P EI
p P p EI
EI L
= ⇒ = =
2
2 2
20 19.
cr
e
EI EI
P
L L
π
= =
Since
Therefore,
Case C
Solving for Le
Le = 0.699L ≈ 0.7 L
26. 10.6 Design of Columns under a Centric Load
Assumptions in the preceding sections:
-- A column is straight
-- Load is applied at the center of the column
-- σ < σy
Reality: may violate these assumptions
-- use empirical equations and rely lab data
27. Facts:
1. Long Columns: obey Euler’s Equation
2. Short Columns: dominated by σy
3. Intermediate Columns: mixed behavior
Test Data:
29. Real Case Design using Empirical Equations:
Two Approaches:
1. Allowable Stress Design
2. Load & Resistance Factor Design
30. 1. For L/r ≥ Cc [long columns]:
[Euler’s eq.]
Structural Steel – Allowable Stress Design
Approach I -- w/o Considering F.S.
2
2
( / )cr
E
L r
π
σ =
2
2
1
2
( / )
[ ]cr Y
c
L r
C
σ σ= −
2. For L/r ≤ Cc [short & interm. columns]:
where
2
2 2
c
Y
E
C
π
σ
=
31. 1. L/r ≥ Cc :
(10.43)
Approach II -- Considering F.S.
2
2
1 92. . ( / )
cr
all
E
F S L r
σ π
σ = =
2. L/r ≤ Cc :
21
1
2
/
[ ( ) ]
. . .
cr Y
all
c
L c
F S F S C
σ σ
σ = = − (10.45)
32. 10.7 Design of Columns under an Eccentric Load
centric bendingσ σ σ= +
max
P Mc
A I
σ = +
(10.56)
(10.57)
(I) Allowable Stress Method
(II) Interaction Method
Two Approaches:
1. The section is far from the ends
2. σ < σy
33. I. Allowable-Stress Method
all
P Mc
A I
σ+ ≤ (10.58)
-- σall is obtained from Section 10.6.
-- The results may be too conservative.
34. II. Interaction Method
1
/ /
all all
P A Mc I
σ σ
+ ≤
1
/ /
( ) ( )all centric all bending
P A Mc I
σ σ
+ ≤
(10.59)
(10.60)(Interaction Formula)
( )all centric
σ -- determined using the largest Le
Case A: If P is applied in a plane of
symmetry:
35. 1max max/ //
( ) ( ) ( )
x x x x
all centric all bending all bending
M z I M x IP A
σ σ σ
+ + ≤ (10.61)
Case B: If P is NOT Applied in a Plane of Symmetry:
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