Practical analysis procedures of steel portal frames having different connections rigidities using modified stiffness matrix and end fixity factor concept
The real behaviour of connections in the steel buildings is often underestimated by designers at the structural analysis and design stages, despite their influences on the structural performance, deflection limits and a possibility of achieving a reduction in the material weights, which can significantly reduce the overall cost and amount of energy embodied. This paper, therefore, described systematic and simplified procedures to conduct a first-order elastic structural analysis of the semi-rigid steel portal frames in order to implement a design optimization using a Generalized Reduced Gradient (GRG) algorithm within Solver Add-in tool in Microsoft Excel. The written program used the robustness and efficiency of the Finite Element (FE) method with the versatility of a spreadsheet format in Excel. To simulate the semi-rigid response of the connections, the mathematical representation through End-Fixity Factor and the Modified Stiffness Matrix were used to incorporate such behaviour into structural analysis packages. To validate the written program, a computer-based analysis was conducted using Prokon® software and comparing analysis results with those obtained from the Excel spreadsheet. It demonstrates that Excel's results were perfectly accurate. Consequently, the procedure of establishing spreadsheets as a finite element analysis software for a certain form of frames demonstrates its validity.
Numerical investigations on recycled aggregate rc beamseSAT Journals
Abstract One of the major challenges in our present society is the protection of environment. Some of the important elements in this respect are reduction in the consumption of energy and natural raw materials. The use of recycled aggregates from construction and demolition waste has been receiving increased attention in recent sustainable construction environment. However the question that needs to be addressed is whether one can consistently produce good quality concrete using recycled aggregates (RA). Strength, stiffness and durability characteristics of structural components built using recycled aggregates are required to be studied carefully. Towards this, a static nonlinear finite element analysis is carried out to understand the behavior of reinforced concrete beam made with Recycled aggregate to the extent of 50% replacement of total aggregate volume. Flexural load is applied on the beams. The capacity in terms of strength and stiffness are discussed. Equivalent stress strain graph is derived using moment curvature relationship. Equivalent model is validated by comparing with experimental results available in literature. Response from numerical investigation indicates that Recycled aggregate can be used as structural members. Keywords: - Recycled aggregates, Construction and demolition waste, Reinforced concrete, Moment curvature, Equivalent Stress strain relationship, nonlinear analysis
Comparative Study of Girders for Bridge by Using SoftwareIJERA Editor
According to various research papers, it has been found that composite bridge gives the maximum strength in
comparison to other bridges and the design and analysis of various girders for steel and concrete by using
various software for composite bridge design for girder. In this project, efforts will make to carry outto check
the analysis of girder by using SAP2000 software. Hence, in this project determine three girders which can be
effective to the composite bridges.
Numerical investigations on recycled aggregate rc beamseSAT Journals
Abstract One of the major challenges in our present society is the protection of environment. Some of the important elements in this respect are reduction in the consumption of energy and natural raw materials. The use of recycled aggregates from construction and demolition waste has been receiving increased attention in recent sustainable construction environment. However the question that needs to be addressed is whether one can consistently produce good quality concrete using recycled aggregates (RA). Strength, stiffness and durability characteristics of structural components built using recycled aggregates are required to be studied carefully. Towards this, a static nonlinear finite element analysis is carried out to understand the behavior of reinforced concrete beam made with Recycled aggregate to the extent of 50% replacement of total aggregate volume. Flexural load is applied on the beams. The capacity in terms of strength and stiffness are discussed. Equivalent stress strain graph is derived using moment curvature relationship. Equivalent model is validated by comparing with experimental results available in literature. Response from numerical investigation indicates that Recycled aggregate can be used as structural members. Keywords: - Recycled aggregates, Construction and demolition waste, Reinforced concrete, Moment curvature, Equivalent Stress strain relationship, nonlinear analysis
Comparative Study of Girders for Bridge by Using SoftwareIJERA Editor
According to various research papers, it has been found that composite bridge gives the maximum strength in
comparison to other bridges and the design and analysis of various girders for steel and concrete by using
various software for composite bridge design for girder. In this project, efforts will make to carry outto check
the analysis of girder by using SAP2000 software. Hence, in this project determine three girders which can be
effective to the composite bridges.
Finite Element Modeling On Behaviour Of Reinforced Concrete Beam Column Joint...IJERA Editor
Recent earthquakes have demonstrated that most of the reinforced concrete structures were severely damaged during earthquakes and they need major repair works. Beam column joints, being the lateral and vertical load resisting members in reinforced concrete structures are particularly vulnerable to failures during earthquakes. The existing reinforced concrete beam column joints which are not designed as per code IS13920:1993 must be strengthened since they do not meet the ductility requirements. The Finite element method (FEM) has become a staple for predicting and simulating the physical behaviour of complex engineering systems. The commercial finite element analysis (FEA) programs have gained common acceptance among engineers in industry and researchers. The details of the finite element analysis of beam column joints retrofitted with carbon fibre reinforced polymer sheets (CFRP) carried out using the package ANSYS are presented in this paper. Three exterior reinforced concrete beam column joint specimens were modelled using ANSYS package. The first specimen is the control specimen. This had reinforcement as per code IS 456:2000. The second specimen which is also the control specimen. This had reinforcement as per code IS 13920:1993. The third specimen had reinforcement as per code IS 456:2000 and was retrofitted with carbon fibre reinforced polymer (CFRP) sheets. During the analysis both the ends of column were hinged. Static load was applied at the free end of the cantilever beam up to a controlled load. The performance of the retrofitted beam column joint was compared with the control specimens and the results are presented in this paper.
Effects of longer span floor system in the constancy of the multistoried stru...eSAT Journals
Abstract The main objective of this study is to investigate the structural integrity, stability and their comparison due to the effects of longer span floor systems considering some constancy in the multi-storied commercial and residential mixed-used structures. In recent times, mixed-use developments and buildings have created an up surging demand in perspective of relatively small area of lands like Bangladesh. But, the commercial developments consisting of underground basement required maximize serviceable column free open-floor spaces for more flexibility, marketability and uninterrupted executive car parking of the end-users. Now, the column free open-plan floor spaces, usually ranges from 18~27 ft, even up to 45 ft. or more [5], offers a bulky change in span length of the slab that results the longer span structure. Again, the longer span structure is directly related with the beam length which promptly affects the thickness of the slab as well as the sizes of beams, columns and the foundations. In this study, two different span lengths of the mixed-use structural Models are considered based on the economical range [18ft-30ft] of the RC floor systems. Then, both of the structural Models are compared based on the following parameters: design aspects, reduction in the number of components, sizes & thickness, weight of steel and volume of concrete. It is found that, longer span structure keeps highest effects on the RC columns and increase in size by 104.3%. This study will also be helpful for a designer to select an appropriate size of the structural components within the economical ranges of these types of particular RC structures in future Keywords: Regular Span Structure, Longer Span Structure, Flat Plate Slab, Flat Slab, Edge Supported Slab Andmat Foundation.
Performance of single layer steel barrel vault under bucklingeSAT Journals
Abstract Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behavior of the part beyond 'buckling' is not evident from the normal arguments of static. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones. If a structure is subject to compressive loads, then a buckling analysis may be necessary. The study presented in this paper is intended to help designers of steel braced barrel vaults by identifying the significant differences in determining which configuration(s) would be best in different conditions of use. The study presented is of parametric type and covers several other important parameters like rise to span ratio, different boundary conditions, such that barrel vault acts as an arch, as a beam or as a shell, The buckling strength of a three different configuration of a double layer braced barrel vaults are presented in this paper for rise/span ratio varying from 0.2-0.7 and having four different types of boundary conditions. Through consideration of these parameters, the paper presents an assessment of the effect of the vault configuration on the overall buckling strength. Keywords: Buckling, Steel Barrel Vault, Single Layer
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Non Linear Analysis of Composite Beam Slab Junction with Shear Connectors usi...inventionjournals
Frame finite-element models permit obtaining, at moderate computational cost, significant information on the dynamic response behavior of steel–concrete composite beam frame structures. As an extension of conventional monolithic beam models, composite beams with deformable shear connection were specifically introduced and adopted for the analysis of composite beams, in which the flexible shear connection allows development of partial composite action influencing structural deformation and distribution of stresses. The use of beams with deformable shear connection in the analysis of frame structures raises very specific modeling issues, such as the characterization of the cyclic behavior of the deformable shear connection and the assembly of composite beam elements with conventional beam–column elements. In addition, the effects on the dynamic response of composite beam frame structures of various factors, such as the shear connection boundary conditions and the mass distribution between the two components of the composite beam, are still not clear and deserve more investigation. The object of this paper is to provide deeper insight into the natural vibration properties and nonlinear seismic response behavior of composite beam frame structures and how they are influenced by various modeling assumptions. For this purpose, a materially nonlinear-only finite-element formulation is used for static and dynamic response analyses of steel–concrete frame structures using composite beam elements with deformable shear connection. Realistic uniaxial cyclic constitutive laws are adopted for the steel and concrete materials of the beams and columns and for the shear connection. The resulting finite-element model for a benchmark problem is validated using experimental test results from the literature review
A Review: Production of Third Generation Advance High Strength Steelsijsrd.com
To fulfill the requirement for steel with higher strength while retaining its formability, advanced high strength steels (AHSS) was developed. Several authors worked on the first generation AHSS steels with Dual phase structure and successfully achieved high strengths as compared to conventional counter-parts but could not achieve good formability. Several authors worked to overcome this problem and substantially improved the ductility and strength by forming second generation AHSS. However, DP steels were not formed (in second generation). Second generation steels (included ASS, TWIP steels etc.) were dependent on expensive alloy additions and were not cost effective. Authors became successful in achieving good strength and ductility but because of alloying, the weldability suffered. Finally, a few authors have reported their work on third generation AHSS, where first generation dual phase steels have been modified through suitable alloy additions to improve formability, strength and toughness. Work has been reported where carbon has been reduced to improve weldability. There is extremely limited work has been done for production of dual phase multipurpose steels with controlled cooling.
Finite element analysis and parametric study of curved concrete box girder us...eSAT Journals
Abstract The horizontally curved bridges are becoming the norm of highway interchanges and urban expressways as a result of complicated geometrics, limited rights of way, and traffic mitigation. This type of superstructure has gained popularity because it addresses the needs of transportation engineering. A study of box girder curved in plan with rectangular cross-section has been carried out in the present investigation. The finite element software ABAQUS is used to carryout analysis of these box girders. The analysis is carried out for the dead load, super imposed dead load and live load of IRC Class A loading. The paper presents a parametric study of curved box girders by varying span and radius of curvature and by keeping the span to depth ratio constant. The parametric investigations performed on curved box girders helps to evaluate the effects of different parameters on the behavior of the girder. This study would enable bridge engineers to better understand the behavior of curved concrete box girders. Key Words: Curved concrete box girder, ABAQUS
Beam column joints in concrete framed structure have been identified as critical member for transferring forces and bending moments between beams and columns. The change of moments in beam and columns across the joint region, under loadings, induces high shear force and stresses as compared with other adjacent members. The shear failure caused is often brittle in nature which is not an acceptable structural performance. Retrofitting enhances the moment carrying capacity of joint. Often beam column joints need to be strengthened. Author proposes use of ferrocement for retrofitting as wrapping technique, cost effective alternative to costly FRP wrapping technique. In this present research study, modelling & comparison of Beam-Column joint with and without ferrocement jacket is carried out by finite element method using software ANSYS APDL. The comparison shows enhanced performance of the jacketed model over Non jacketed in terms of stresses, ultimate load carrying capacity.
Finite Element Modeling On Behaviour Of Reinforced Concrete Beam Column Joint...IJERA Editor
Recent earthquakes have demonstrated that most of the reinforced concrete structures were severely damaged during earthquakes and they need major repair works. Beam column joints, being the lateral and vertical load resisting members in reinforced concrete structures are particularly vulnerable to failures during earthquakes. The existing reinforced concrete beam column joints which are not designed as per code IS13920:1993 must be strengthened since they do not meet the ductility requirements. The Finite element method (FEM) has become a staple for predicting and simulating the physical behaviour of complex engineering systems. The commercial finite element analysis (FEA) programs have gained common acceptance among engineers in industry and researchers. The details of the finite element analysis of beam column joints retrofitted with carbon fibre reinforced polymer sheets (CFRP) carried out using the package ANSYS are presented in this paper. Three exterior reinforced concrete beam column joint specimens were modelled using ANSYS package. The first specimen is the control specimen. This had reinforcement as per code IS 456:2000. The second specimen which is also the control specimen. This had reinforcement as per code IS 13920:1993. The third specimen had reinforcement as per code IS 456:2000 and was retrofitted with carbon fibre reinforced polymer (CFRP) sheets. During the analysis both the ends of column were hinged. Static load was applied at the free end of the cantilever beam up to a controlled load. The performance of the retrofitted beam column joint was compared with the control specimens and the results are presented in this paper.
Effects of longer span floor system in the constancy of the multistoried stru...eSAT Journals
Abstract The main objective of this study is to investigate the structural integrity, stability and their comparison due to the effects of longer span floor systems considering some constancy in the multi-storied commercial and residential mixed-used structures. In recent times, mixed-use developments and buildings have created an up surging demand in perspective of relatively small area of lands like Bangladesh. But, the commercial developments consisting of underground basement required maximize serviceable column free open-floor spaces for more flexibility, marketability and uninterrupted executive car parking of the end-users. Now, the column free open-plan floor spaces, usually ranges from 18~27 ft, even up to 45 ft. or more [5], offers a bulky change in span length of the slab that results the longer span structure. Again, the longer span structure is directly related with the beam length which promptly affects the thickness of the slab as well as the sizes of beams, columns and the foundations. In this study, two different span lengths of the mixed-use structural Models are considered based on the economical range [18ft-30ft] of the RC floor systems. Then, both of the structural Models are compared based on the following parameters: design aspects, reduction in the number of components, sizes & thickness, weight of steel and volume of concrete. It is found that, longer span structure keeps highest effects on the RC columns and increase in size by 104.3%. This study will also be helpful for a designer to select an appropriate size of the structural components within the economical ranges of these types of particular RC structures in future Keywords: Regular Span Structure, Longer Span Structure, Flat Plate Slab, Flat Slab, Edge Supported Slab Andmat Foundation.
Performance of single layer steel barrel vault under bucklingeSAT Journals
Abstract Buckling is a critical state of stress and deformation, at which a slight disturbance causes a gross additional deformation, or perhaps a total structural failure of the part. Structural behavior of the part beyond 'buckling' is not evident from the normal arguments of static. Buckling failures do not depend on the strength of the material, but are a function of the component dimensions & modulus of elasticity. Therefore, materials with a high strength will buckle just as quickly as low strength ones. If a structure is subject to compressive loads, then a buckling analysis may be necessary. The study presented in this paper is intended to help designers of steel braced barrel vaults by identifying the significant differences in determining which configuration(s) would be best in different conditions of use. The study presented is of parametric type and covers several other important parameters like rise to span ratio, different boundary conditions, such that barrel vault acts as an arch, as a beam or as a shell, The buckling strength of a three different configuration of a double layer braced barrel vaults are presented in this paper for rise/span ratio varying from 0.2-0.7 and having four different types of boundary conditions. Through consideration of these parameters, the paper presents an assessment of the effect of the vault configuration on the overall buckling strength. Keywords: Buckling, Steel Barrel Vault, Single Layer
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Non Linear Analysis of Composite Beam Slab Junction with Shear Connectors usi...inventionjournals
Frame finite-element models permit obtaining, at moderate computational cost, significant information on the dynamic response behavior of steel–concrete composite beam frame structures. As an extension of conventional monolithic beam models, composite beams with deformable shear connection were specifically introduced and adopted for the analysis of composite beams, in which the flexible shear connection allows development of partial composite action influencing structural deformation and distribution of stresses. The use of beams with deformable shear connection in the analysis of frame structures raises very specific modeling issues, such as the characterization of the cyclic behavior of the deformable shear connection and the assembly of composite beam elements with conventional beam–column elements. In addition, the effects on the dynamic response of composite beam frame structures of various factors, such as the shear connection boundary conditions and the mass distribution between the two components of the composite beam, are still not clear and deserve more investigation. The object of this paper is to provide deeper insight into the natural vibration properties and nonlinear seismic response behavior of composite beam frame structures and how they are influenced by various modeling assumptions. For this purpose, a materially nonlinear-only finite-element formulation is used for static and dynamic response analyses of steel–concrete frame structures using composite beam elements with deformable shear connection. Realistic uniaxial cyclic constitutive laws are adopted for the steel and concrete materials of the beams and columns and for the shear connection. The resulting finite-element model for a benchmark problem is validated using experimental test results from the literature review
A Review: Production of Third Generation Advance High Strength Steelsijsrd.com
To fulfill the requirement for steel with higher strength while retaining its formability, advanced high strength steels (AHSS) was developed. Several authors worked on the first generation AHSS steels with Dual phase structure and successfully achieved high strengths as compared to conventional counter-parts but could not achieve good formability. Several authors worked to overcome this problem and substantially improved the ductility and strength by forming second generation AHSS. However, DP steels were not formed (in second generation). Second generation steels (included ASS, TWIP steels etc.) were dependent on expensive alloy additions and were not cost effective. Authors became successful in achieving good strength and ductility but because of alloying, the weldability suffered. Finally, a few authors have reported their work on third generation AHSS, where first generation dual phase steels have been modified through suitable alloy additions to improve formability, strength and toughness. Work has been reported where carbon has been reduced to improve weldability. There is extremely limited work has been done for production of dual phase multipurpose steels with controlled cooling.
Finite element analysis and parametric study of curved concrete box girder us...eSAT Journals
Abstract The horizontally curved bridges are becoming the norm of highway interchanges and urban expressways as a result of complicated geometrics, limited rights of way, and traffic mitigation. This type of superstructure has gained popularity because it addresses the needs of transportation engineering. A study of box girder curved in plan with rectangular cross-section has been carried out in the present investigation. The finite element software ABAQUS is used to carryout analysis of these box girders. The analysis is carried out for the dead load, super imposed dead load and live load of IRC Class A loading. The paper presents a parametric study of curved box girders by varying span and radius of curvature and by keeping the span to depth ratio constant. The parametric investigations performed on curved box girders helps to evaluate the effects of different parameters on the behavior of the girder. This study would enable bridge engineers to better understand the behavior of curved concrete box girders. Key Words: Curved concrete box girder, ABAQUS
Beam column joints in concrete framed structure have been identified as critical member for transferring forces and bending moments between beams and columns. The change of moments in beam and columns across the joint region, under loadings, induces high shear force and stresses as compared with other adjacent members. The shear failure caused is often brittle in nature which is not an acceptable structural performance. Retrofitting enhances the moment carrying capacity of joint. Often beam column joints need to be strengthened. Author proposes use of ferrocement for retrofitting as wrapping technique, cost effective alternative to costly FRP wrapping technique. In this present research study, modelling & comparison of Beam-Column joint with and without ferrocement jacket is carried out by finite element method using software ANSYS APDL. The comparison shows enhanced performance of the jacketed model over Non jacketed in terms of stresses, ultimate load carrying capacity.
Seismic rehabilitation of beam column joint using gfrp sheets-2002
Similar to Practical analysis procedures of steel portal frames having different connections rigidities using modified stiffness matrix and end fixity factor concept
The influence of cable sag on the dynamic behaviour of cable stayed suspensio...eSAT Journals
Abstract The demand of long span bridge is increasing with infrastructure magnification. To achieve maximum central span in bridges is a motivating rational challenge. The bridge with more central span can be achieved using high strength materials and innovative forms of the bridges. The cable-stayed bridge has better structural stiffness and suspension bridge has ability to offer longer span thus combination of above two structural systems could achieve very long span cable-stayed suspension hybrid bridge. To distinguish behaviour and check the feasibility of this innovative form of hybrid bridge, 1400m central span and 312m side span cable-stayed suspension hybrid bridge is considered for analysis. The suspension portion length in central span is also playing important role in behaviour of the entire bridge. Bridge behaviour is presented for variable length of suspension portion in form of suspension portion to main span ratio. The main cable sag in central span is playing important role on behavior of the entire bridge. It directly influences the inclination angles of the main cables, the height of pylon and thus forces in pylon. The axial force in main cable is directly depending on the sag of main cable. The effects of main cable sag is studied by considering dimensionless parameter as sag to main span ratio as 1/9, 1/10 and 1/11. Paper also discusses results of nonlinear static analysis and modal analysis carried out using SAP2000 v14.0.0. The time period of bridge is used to present the behavior of bridge. Key Words: Cable supported long span bridge; cable stayed suspension hybrid bridge; cable sag to main span ratio; dynamic analysis
THEORETICAL BEHAVIOR OF COMPOSITE CONSTRUCTION PRECAST REACTIVE POWDER RC GIR...IAEME Publication
This study displays numerically (or theoretically) investigation by using the finite element models for experimental work of composite behavior for hybrid reinforced concrete slab on girder from locale material in Iraq, ordinary concrete in slab and reactive powder concrete in girder, RPC, with steel fibers of different types (straight, hook, and mix between its), tested as simply supported span subjected under two point loading. Which ANSYS version 15.0 is utilized. By studying the compatibility between the experimental results and the theoretical results. As well as, parametric study of many others variables are investigated by using ANSYS (version 15.0), such as: changing the compressive strength of the slab, changing the main reinforcement of the girder, and changing thickness of resin bond layer between girder and slab.
Study of Eccentrically Braced Outrigger Frame under Seismic ExitationIJTET Journal
Outrigger braced structures has efficient structural form consist of a central core, comprising braced frames with
horizontal cantilever ”outrigger” trusses or girders connecting the core to the outer column. When the structure is loaded
horizontally, vertical plane rotation of the core is restrained by the outriggers through tension in windward column and
compression in leeward column. The effective structural depth of the building is greatly increased, thus augmenting the lateral
stiffness of the building and reducing the lateral deflections and moments in core. In effect, the outriggers join the columns to the
core to make the structure behave as a partly composite cantilever. By providing eccentrically braced system in outrigger frame by
varying the size of links and analyzing it. Push over analysis is carried out by varying the link size using computer programs, Sap
2007 to understand their seismic performance. The ductile behavior of eccentrically braced frame is highly desirable for structures
subjected to strong ground motion. Maximum stiffness, strength, ductility and energy dissipation capacity are provided by
eccentrically braced frame. Studies were conducted on the use of outrigger frame for the high steel building subjected to
earthquake load. Braces are designed not to buckle, regardless of the severity of lateral loading on the frame. Thus eccentrically
braced frame ensures safety against collapse.
Effect of modulus of masonry on initial lateral stiffness of infilled frames ...eSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Effect of configuration on lateral displacement and cost of the structure for...eSAT Journals
Abstract
The choice of a cost effective lateral-force-resisting system for high-rise structures is challenging. There is no streamlined methodology to quantitatively compare the cost-effectiveness of each system beyond the more qualitative perception based evaluation of advantages or disadvantages. Developers currently base their decisions on architectural layout and structural integrity. Cost considerations are often primarily based on experience.
This decision making process has three primary shortfalls.
1) It may not incorporate factors which greatly affect the economy of a particular framing system.
2) It may not allow engineers to carryout designs at the least cost.
3) Comparison of framing systems may not address the specific building types.
This investigation proposes a prototype cost-effective model for selecting either a skeleton framing system or skeleton frame with bracing system for steel structural frames. A model for selecting cost-effective skeleton framing system or skeleton frame with bracing system will be a valuable tool for all decision makers. Engineers, in particular, will be able to select optimal steel framing faster, thus reducing design time and iterations. Furthermore, selection of economic framing system will also result in direct cost savings for steel structural frames.
The study involves the design and cost estimation of steel frames representing skeleton framing system and skeleton frame with bracingsystem. The cost effectiveness of the framing systems are compared based on lateral displacement requirements and cost.The preferred framing system should meet lateral displacement requirements and is lower in cost. The results of this pilot study showed that the Skelton framing system with bracing is the cost-effective choice for 30storeys steel space frames at wind speeds of 55m/sec, 50m/sec and 47m/sec.
Keywords: Bracings, SFS (Skeleton framing system), SFWB (Skeleton frame with bracing system) etc…
Similar to Practical analysis procedures of steel portal frames having different connections rigidities using modified stiffness matrix and end fixity factor concept (20)
HEAP SORT ILLUSTRATED WITH HEAPIFY, BUILD HEAP FOR DYNAMIC ARRAYS.
Heap sort is a comparison-based sorting technique based on Binary Heap data structure. It is similar to the selection sort where we first find the minimum element and place the minimum element at the beginning. Repeat the same process for the remaining elements.
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.
Hybrid optimization of pumped hydro system and solar- Engr. Abdul-Azeez.pdffxintegritypublishin
Advancements in technology unveil a myriad of electrical and electronic breakthroughs geared towards efficiently harnessing limited resources to meet human energy demands. The optimization of hybrid solar PV panels and pumped hydro energy supply systems plays a pivotal role in utilizing natural resources effectively. This initiative not only benefits humanity but also fosters environmental sustainability. The study investigated the design optimization of these hybrid systems, focusing on understanding solar radiation patterns, identifying geographical influences on solar radiation, formulating a mathematical model for system optimization, and determining the optimal configuration of PV panels and pumped hydro storage. Through a comparative analysis approach and eight weeks of data collection, the study addressed key research questions related to solar radiation patterns and optimal system design. The findings highlighted regions with heightened solar radiation levels, showcasing substantial potential for power generation and emphasizing the system's efficiency. Optimizing system design significantly boosted power generation, promoted renewable energy utilization, and enhanced energy storage capacity. The study underscored the benefits of optimizing hybrid solar PV panels and pumped hydro energy supply systems for sustainable energy usage. Optimizing the design of solar PV panels and pumped hydro energy supply systems as examined across diverse climatic conditions in a developing country, not only enhances power generation but also improves the integration of renewable energy sources and boosts energy storage capacities, particularly beneficial for less economically prosperous regions. Additionally, the study provides valuable insights for advancing energy research in economically viable areas. Recommendations included conducting site-specific assessments, utilizing advanced modeling tools, implementing regular maintenance protocols, and enhancing communication among system components.
Overview of the fundamental roles in Hydropower generation and the components involved in wider Electrical Engineering.
This paper presents the design and construction of hydroelectric dams from the hydrologist’s survey of the valley before construction, all aspects and involved disciplines, fluid dynamics, structural engineering, generation and mains frequency regulation to the very transmission of power through the network in the United Kingdom.
Author: Robbie Edward Sayers
Collaborators and co editors: Charlie Sims and Connor Healey.
(C) 2024 Robbie E. Sayers
Industrial Training at Shahjalal Fertilizer Company Limited (SFCL)MdTanvirMahtab2
This presentation is about the working procedure of Shahjalal Fertilizer Company Limited (SFCL). A Govt. owned Company of Bangladesh Chemical Industries Corporation under Ministry of Industries.
NUMERICAL SIMULATIONS OF HEAT AND MASS TRANSFER IN CONDENSING HEAT EXCHANGERS...ssuser7dcef0
Power plants release a large amount of water vapor into the
atmosphere through the stack. The flue gas can be a potential
source for obtaining much needed cooling water for a power
plant. If a power plant could recover and reuse a portion of this
moisture, it could reduce its total cooling water intake
requirement. One of the most practical way to recover water
from flue gas is to use a condensing heat exchanger. The power
plant could also recover latent heat due to condensation as well
as sensible heat due to lowering the flue gas exit temperature.
Additionally, harmful acids released from the stack can be
reduced in a condensing heat exchanger by acid condensation. reduced in a condensing heat exchanger by acid condensation.
Condensation of vapors in flue gas is a complicated
phenomenon since heat and mass transfer of water vapor and
various acids simultaneously occur in the presence of noncondensable
gases such as nitrogen and oxygen. Design of a
condenser depends on the knowledge and understanding of the
heat and mass transfer processes. A computer program for
numerical simulations of water (H2O) and sulfuric acid (H2SO4)
condensation in a flue gas condensing heat exchanger was
developed using MATLAB. Governing equations based on
mass and energy balances for the system were derived to
predict variables such as flue gas exit temperature, cooling
water outlet temperature, mole fraction and condensation rates
of water and sulfuric acid vapors. The equations were solved
using an iterative solution technique with calculations of heat
and mass transfer coefficients and physical properties.
About
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
Technical Specifications
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
Key Features
Indigenized remote control interface card suitable for MAFI system CCR equipment. Compatible for IDM8000 CCR. Backplane mounted serial and TCP/Ethernet communication module for CCR remote access. IDM 8000 CCR remote control on serial and TCP protocol.
• Remote control: Parallel or serial interface
• Compatible with MAFI CCR system
• Copatiable with IDM8000 CCR
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
Application
• Remote control: Parallel or serial interface.
• Compatible with MAFI CCR system.
• Compatible with IDM8000 CCR.
• Compatible with Backplane mount serial communication.
• Compatible with commercial and Defence aviation CCR system.
• Remote control system for accessing CCR and allied system over serial or TCP.
• Indigenized local Support/presence in India.
• Easy in configuration using DIP switches.
CW RADAR, FMCW RADAR, FMCW ALTIMETER, AND THEIR PARAMETERSveerababupersonal22
It consists of cw radar and fmcw radar ,range measurement,if amplifier and fmcw altimeterThe CW radar operates using continuous wave transmission, while the FMCW radar employs frequency-modulated continuous wave technology. Range measurement is a crucial aspect of radar systems, providing information about the distance to a target. The IF amplifier plays a key role in signal processing, amplifying intermediate frequency signals for further analysis. The FMCW altimeter utilizes frequency-modulated continuous wave technology to accurately measure altitude above a reference point.
Hierarchical Digital Twin of a Naval Power SystemKerry Sado
A hierarchical digital twin of a Naval DC power system has been developed and experimentally verified. Similar to other state-of-the-art digital twins, this technology creates a digital replica of the physical system executed in real-time or faster, which can modify hardware controls. However, its advantage stems from distributing computational efforts by utilizing a hierarchical structure composed of lower-level digital twin blocks and a higher-level system digital twin. Each digital twin block is associated with a physical subsystem of the hardware and communicates with a singular system digital twin, which creates a system-level response. By extracting information from each level of the hierarchy, power system controls of the hardware were reconfigured autonomously. This hierarchical digital twin development offers several advantages over other digital twins, particularly in the field of naval power systems. The hierarchical structure allows for greater computational efficiency and scalability while the ability to autonomously reconfigure hardware controls offers increased flexibility and responsiveness. The hierarchical decomposition and models utilized were well aligned with the physical twin, as indicated by the maximum deviations between the developed digital twin hierarchy and the hardware.
Practical analysis procedures of steel portal frames having different connections rigidities using modified stiffness matrix and end fixity factor concept
1. IOP Conference Series: Materials Science and Engineering
PAPER • OPEN ACCESS
Practical analysis procedures of steel portal frames having different
connections rigidities using modified stiffness matrix and end-fixity factor
concept
To cite this article: Ali Msabawy and Fouad Mohammad 2019 IOP Conf. Ser.: Mater. Sci. Eng. 518 022037
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2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
1
Practical analysis procedures of steel portal frames having
different connections rigidities using modified stiffness matrix
and end-fixity factor concept
Ali Msabawy1,2, *
and Fouad Mohammad1
1
School of Architecture, Design and the Built Environment, Nottingham Trent University,
Nottingham, NG1 5JS, United Kingdom
2
Engineering Works Division, Directorate of Civil Status, Passports and Residence, Ministry of
Interior, Babylon, Iraq
Email: alimsabawy@gmail.com.
Abstract. The real behaviour of connections in the steel buildings is often underestimated by
designers at the structural analysis and design stages, despite their influences on the structural
performance, deflection limits and a possibility of achieving a reduction in the material
weights, which can significantly reduce the overall cost and amount of energy embodied. This
paper, therefore, described systematic and simplified procedures to conduct a first-order elastic
structural analysis of the semi-rigid steel portal frames in order to implement a design
optimization using a Generalized Reduced Gradient (GRG) algorithm within Solver Add-in
tool in Microsoft Excel. The written program used the robustness and efficiency of the Finite
Element (FE) method with the versatility of a spreadsheet format in Excel. To simulate the
semi-rigid response of the connections, the mathematical representation through End-Fixity
Factor and the Modified Stiffness Matrix were used to incorporate such behaviour into
structural analysis packages. To validate the written program, a computer-based analysis was
conducted using Prokon® software and comparing analysis results with those obtained from
the Excel spreadsheet. It demonstrates that Excel’s results were perfectly accurate.
Consequently, the procedure of establishing spreadsheets as a finite element analysis software
for a certain form of frames demonstrates its validity.
1. Introduction
It is estimated that around 50% of all constructional steel used in the United Kingdom is in erecting
single-story buildings [1]. Steel Portal frames are the most preferable constructional form in pitched-
roof buildings within this major market sector [2]. This form employed for constructing industrial,
distribution, retail and leisure facilities, owing to their cost efficiency, versatility, sustainable
contributions as well as it is the most efficient form that can accommodate changes within the
structure.
In a steel framing system, the structural elements and connections are modelled considering some
idealizations [3]. Ideally, the eave, apex and base connections in a portal frame are categorized into
two idealized types: the fully flexible, ideally pinned, connections and fully-rigid connections.
According to BS EN 1993-1-8 [4], connections can be classified as nominally pinned when are
capable of transmitting axial and shear forces without creating a considerable bending moment as well
as they are able to accept the resulting rotations under the design loads. While connections can be
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classified as fully-rigid in case they are not free to rotate and can transmit all three types of forces:
axial, shear and bending moments.
In practice, the connections of steel structures often behave in a manner that falls somewhere in
between these categories, owing to friction and material behaviour, in which connections can both
transmit moment and experience some rotation that can contribute substantially to overall frame
displacements [5]. The term semi-rigid connections, or joints, are commonly adopted to denote such
form of steel connections. A connection may be classified as semi-rigid by exhibiting a behaviour
intermediate between that of standard pinned and rigid. It is common for the structural engineer to
idealize the frame and behaviour of connections as to simplify the analysis and design processes.
However, the predicted response of the idealized structure may be quite unrealistic compared to the
response for the actual structure [6].
For instance, the fully rigid connection assumption may lead to underestimate of structure drift
and overestimate of structure strength, while the ideally pinned connection assumption may result in
an overdesign of the rafters and an underdesign of the columns [7]. In addition, ideally pinned
connections must have adequate flexibility to accommodate rotations without developing significant
moments which can result in premature failure of the structure or parts of it. Hence, by treating the
connections as semi-rigid, two major advantages can be obtained. Firstly, a more reliable prediction of
structural behaviour. Secondly, the possibility of achieving greater economy by making use of the
stiffness and strength of connections that would otherwise be considered as pinned, as well as by
avoiding the stiffening often required in rigid connections [5]. For illustration, figure 1 presents the
results of a linear analysis obtained for simple and pitched-roof steel portal frames, subjected to a
uniform distributed load, with rigid and semi-rigid beam-column connections. All the above
emerges the necessity of examining the structural performance for semi-rigid frames, in which
their connections can influence the stability, weight and the overall behaviour of the frame.
Figure 1. Bending moment diagrams of portal frames: A) Rigid connections; B) Semi-rigid
connections
2. Modelling and Analysis Semi-Rigid Joints
Since that the real response of steel structures is influenced by the mechanical characteristics of their
connections, namely strength, stiffness and rotational capacity. The stiffness of the connections is a
key element in well-designed steel structure, the fact that the stiffness of connections affects the
deflections of the entire structure, particularly in non-braced frames [8]. The most realistic knowledge
of the joint stiffness behaviour and its properties can basically be observed through the moment-
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rotation curves that are obtained by the experimental investigation [9]. However, the experimental
techniques are usually expensive and time-consuming to be implemented for daily design practice, and
they are normally employed for academic purposes only.
For the theoretical modelling of semi-rigid connection within portal frames, there are two
common approaches to adopt joint stiffness into a structural analysis. The first technique is to
introduce ''additional connection elements'' to simulate the behaviour of such connections directly in
the package programs such as ANSYS and SAP2000. This approach is a time-consuming and required
a complex programming to be incorporated with optimization or parametric studies. The other
difficulty is to achieve a physical sense of the connection member stiffness since it is detached from
the attached end connections. In the literature, the effects of semi-rigidity for connections were
neglected in many cases, owing to the lack of a systematic method of conducting the structural
analysis for such frames. Therefore, the second technique is to model the flexibility of connections, it
employs the stiffness matrix method (displacement method) and its modifications to conduct the
structural analysis. Many researchers and structural engineers successfully adopted the stiffness
method to analyze the two-dimensional (2D) frames due to its efficiency and ease of generating the
required matrixes using computer operations, for instance, [3], [5,6] and [10].
This paper outlines the methodology adopted based on the second approach to implement a
structural analysis for portal frames using the traditional stiffness method and its modifications under
certain loads with different member-end restraint conditions. The conventional form of stiffness
method is as follows:
Q K u= ………………………. (1)
Where: {Q} is the 6×1-member end-force vector in the local coordinate system; [K] is the stiffness
matrix of a member in the local coordinate system; {u} is the 6×1-member displacement vector in the
local coordinate system.
In 2D plane, the frame element with elastic restraint has three degrees of freedom at each end,
namely, horizontal displacement, vertical displacements and rotations. The axial force, shear force and
bending moment represented the corresponding forces to these displacements respectively. To
establish a relationship between forces and displacements, the stiffness matrices for the element are
derived by assuming having fully rigid connections. Therefore, figure 2 is shown a semi-rigid member
that described by Xu [10] consisted of a finite-length beam-column member with a zero-length
rotational spring at both ends of the member. The connection flexibilities are modelled by rotational
springs of stiffness R1 and R2 at both ends of the member. The relative stiffness of the beam–column
member and the rotational end-spring connection is measured by an end-fixity factor that developed
by Monforton and Wu [11]. The mathematical expression of the end-fixity factor rj is as follows:
( )
1
1,2
1 3
j
j
r j
EI R L
= =
+
…………… (2)
Where: Rj is end-connection spring stiffness; E is the modulus of elasticity of the prismatic member,
m; I is the second moment of area of the prismatic member, m; L is the length of the prismatic
member, m.
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Figure 2. Model of undeformed semi-rigid member.
The values of the end-fixity factor are ranging from zero up to 1. The connection rotational stiffness of
a pinned type is idealized as zero thus the end-fixity is zero, rj= 0. The connection rotational stiffness
of a rigid type is taken to be infinite and the end-fixity factor has a value of unity, rj= 1. Hence, the
semi-rigid connections are modelled with end-fixity factors between zero and one, 0 ˂ rj ˂ 1. This
mathematical model made the structural analysis of semi-rigid structures a systematic and
straightforward process due to its connectivity to the stiffness matrix method. In addition, it could
easily be incorporated into the analytical model written in Excel spreadsheets that can handle large
matrices. In which various member-end restraint conditions are modelled by setting a suitable
combination such as rigid-pinned, rigid–semi-rigid or pinned–semi-rigid, with their appropriate values
of the end-fixity factors at both ends of the member. The classical stiffness matrices of a rigid
member, equation 1, modified by so-called correction matrix, equation 3, that embeds the end-fixity
factor within it [10-11]. Consequently, the modified stiffness matrix, equation 4, was produced for
member m having semi-rigid end-connections. The implementation of end-fixity factor approach into
structure analysis is straightforward process due to its connectivity to the stiffness matrix method.
To simplify the calculation, this paper adopted the linear representation of the moment-rotation curve
of such connections. This assumption was to avoid iterative the process that is too complex to be
modelled into the written program. This is the simplest connection model to be used in the linear,
vibration and bifurcation analysis where the deflections are relatively small [12]. The adopted
methodology also assumed that all members are straight and prismatic as well as connection
dimensions are assumed to be negligible compared to the lengths of the rafters and columns.
( )
( )
( )
( )
1 22 1 1 2
1 2 1 2
1 21 2
1 2 1 2
2 11 2 1 2
1 2 1 2
2 11 2
1 2 1 2
1 0 0 0 0 0
2 14 2
0 0 0 0
4 4
3 26( )
0 0 0 0
(4 ) 4
0 0 0 1 0 0
2 14 2
0 0 0 0
4 4
3 26( )
0 0 0 0
(4 ) 4
Lr rr r r r
r r r r
r rr r
L r r r rC
Lr rr r r r
r r r r
r rr r
L r r r r
−− +
−
− −
−−
− −=
−− +
− −
−−
− −
….. (3)
SR
K K C= ………………………………………… (4)
2.1. Formulations for End-Reactions of a Semirigid Member
This paper adopted the modified formulations of fixed-end reactions for various loading types making
the analysis of a semi-rigid member by stiffness matrix method more capable of using end-fixity factor
concept. These modified formulations are applied to simulate the behavior of semi-rigid connections
within steel portal frames. This paper refers to [10] and [13] for further details about the end reactions
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of a loaded member, that applied to different restraint conditions at the ends depending on the values
of the end-fixity factor.
3. Finite Element Analysis Procedures
Finite element method (FEM) is a numerical approach for solving engineering and mathematical
problems, it is a process of modelling complex problems by subdividing into an equivalent system of
smaller and simpler parts [14]. The most common engineering problems are solvable by utilizing the
FEM which is an essential tool in structural analysis, fluid flow, mass transport. However, the
common idea in mind of structural engineers is that Finite Element Analysis (FEA) is complex and
normally requires a high-priced commercial software.
Therefore, and for practical needs, the spreadsheets that available in Microsoft Excel were employed
in this paper to establish FEA program. Microsoft Excel spreadsheets are free and most of the
structural engineers are already familiar with Excel. Besides to the affordability and versatility of
Excel spreadsheets, it presents the input information, the output results, the intermediate calculations
and the secondary and supporting formulas on a single spreadsheet. This contrasts with commercial
software which normally does not present intermediate steps, nor the implied formulas. Ease of
modifying the spreadsheets is another feature to accommodate users who want to customize the
spreadsheets for particular requirements. It can readily be connected with different spreadsheets,
sequenced with others, adjustable for use with other configurations of 2D frames.
Hence, this paper developed ''Finite Element Analysis Software '' in a simplified spreadsheet form to
conduct a first-order elastic analysis of steel portal frames with different connections rigidities, by
combining the robustness and efficiency of FE method with the versatility of a spreadsheet format in
Microsoft Excel. The written program was formulated by the authors in Microsoft Excel Spreadsheets,
it is capable of analyzing cold-formed or hot-rolled steel portal frames having different end-
connection, pinned, rigid or semi-rigid joints, between their attached members. Furthermore, this
spreadsheet can implement the structural analysis of portal frames with different cases of applied loads
in the global or local axes systems.
3.1. Analytical Model and Cartesian Coordinate Systems
Based on basic concepts in FEA that presented in Section 3, 2D frame was divided into members and
joints for conducting a structural analysis. The first and possibly the most significant step in such an
analysis is the process of preparing a so-called Analytical Model. It is an idealized representation of a
real frame and its purpose is to facilitate the analysis of complex structures through presenting details
about joints, members, etc. Six key steps of establishing an analytical model can be described. Define
systems of coordinates firstly and for the sake of convention, this paper followed Hibbeler's approach
[15] for specifying coordinate systems of both the structural members individually and the entire
frame.
Hence, two systems of the coordinates were established. The first was the Local Coordinates System
(LGS) that identifies the direction of the internal forces for each member within the frame and their
corresponding displacement as shown in figure 3. The system was denoted by the subscripts x՚, y՚ and
z՚. The beginning i of the member m represents the origin of x՚-axis that coincides with the centroidal
axis (geometric centre axis) of the member in the undeformed case. The local x՚-axis is positive when it
be oriented from the i end toward the j end. The y՚-axis is always perpendicular to x՚-axis.
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Figure 3. The local coordinates system of a member.
The second system was the Global Coordinates System (GCS) for plane frames and was denoted by
the symbols x, y and z. Generally, it is proper to locate the origin of this system at the lower left joint
of the proposed structure as demonstrated in figure 4. The x-axis oriented in the horizontal direction
where the right side is considered the positive direction while the y-axis oriented in the vertical
direction where upward direction is considered the positive direction. All the external loads, support
reactions and joint displacements of the frame were specified using this system. In terms of the third
axis (z or z՚), it is perpendicular into the plane x-y or x՚- y՚. The rotation about the mentioned axes is
considered positive if it acts counterclockwise when seen in the positive direction of the axis.
Figure 4. The analytical model of a portal frame showing global, local coordinate systems and code
numbers at the nodes
Secondly, the frame must be sorted and then divided it into members and joints in accordance with
FE concept, all the members must be straight and prismatic. The joints were identified alphabetically
as A, B, C, etc. The structural members were divided numerically as 1, 2, 3, etc. The members’
numbers were enclosed within circles to distinguish them from others. Thirdly, identify the beginning
and the end of each member, in which the codes that used were i and j for the beginning and the end of
a member respectively, in both the local and global systems. The fourth step was to establish a table of
locations of all joints and connectivity points of the members. This table is beneficial to draw a portal
frame using Excel. The table defined the locations of each joint as coordinates of x and y. For
example, the coordinates of the joints A and B were (0,0) and (0, y) respectively where the origin of
the global coordinate system was located at A for both x and y coordinates. In relation to the members,
their connectivity points were defined according to the joints at their beginning and end. For instance,
the member 1 and member 2 connectivity points were (A, B) and (B, C) respectively.The next step
was to define the degrees of freedom of the entire structure (DOFS) which are basically the unknown
displacements of its all joints. In 2D rigid frames, unsupported (unrestrained) joints can move in any
direction in the x-y plane in which that will produce two linear displacements, horizontal translation
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and vertical translation along the x-axis and y-axis respectively as well as it can rotate about the z-axis.
These displacements were necessary to identify the deformed position of the frame element, it called
the unknown displacements. Regarding joints attached to support, it can be identified according to
their type. A joint attached to a fixed support, for example, can neither translate nor rotate;
consequently, it does not have any unrestrained DOF. These degrees of freedom are named ''known
degrees of freedom'' considering values of their displacements are equal to zero. Thus, the number of
degrees of freedom of the entire structure (NDOFS) can be calculated using the equation as follows:
( )NDOFS 3*NJ NR= − ………………… (5)
Where: NR is number of joint displacements restrained by supports (equal to number of support
reactions or number of DOFs have displacement equals zero); NJ is number of joints in the structure.
Lastly, figure 4 is shown the structure coordinates (coding components) that were specified
numerically on the analytical model at each node (joint or support) of the frame. To implement this
step, designating numbers to the arrows that were drawn beside the joints in the positive directions of
the joint displacements (unconstrained and/or restrained coordinates). The code numbers at the nodes
of the frame were designated with numbers assigned first to the unconstrained (unknown) degrees of
freedom followed by the remaining joints that have the highest unknown of displacements.
3.2. Displacement and Force Transformation Matrices
Unlike beams, the 2D frames normally included elements that can be moved or rotated in various
directions within the plane of a structure. Therefore, transformation matrices that were given in [15-
16] were utilized in this paper to be able to transform the internal member loads Qf and displacements
from local to global coordinates systems. The first transformation matrix was called the displacement
transformation matrix [T] that can be expressed as follows:
0 0 0 0
0 0 0 0
0 0 1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0 1
x y
y x
x y
y x
T
−
=
−
………………….. (6)
Where:
( ) ( )2 2
j i
x
x x xj xi
L
xj xi yj yi
− −
= =
− + −
………………….. (7)
( ) ( )2 2
y
yj yi yj yi
L
xj xi yj yi
− −
= =
− + −
…………………… (8)
Where: x and y are global coordinates for the beginning point i and the end point j of the member m in
figure 3.
Therefore, the transformation relationship of an element can be expressed as follows:
u T v= …………………………………….……… (9)
Where: {v} is the member end displacements vector in the global coordinate system.
While the second matrix was the force transformation matrix [T]T
that is the transpose of the
transformation matrix [T] that can be expressed as follows:
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0 0 0 0
0 0 0 0
0 0 1 0 0 0
0 0 0 0
0 0 0 0
0 0 0 0 0 1
x y
y x
T
x y
y x
T
−
= −
……………………… (10)
Therefore, the transformation relationship of a frame element as follows:
F Q
T
f fT= ………………………………………….. (11)
Where: {Ff} is member global fixed-end force vector; {Qf} is member local fixed-end force vector.
These matrices were adopted to transform the stiffness matrix of an element in the equation 4 from
local to global coordinate systems. Hence, the final expression of the modified stiffness matrix of
elements with semi-rigid joints in the global system SR
g
K were as follows:
T SRSR
gK T K T= ……. …………………… (12)
It is worth noting that it is essential for designers to follow the same order that presented in equation
12 when multiplying the matrices because that any different order will result in incorrect solutions.
3.3. Member Loads and Member Local Fixed-End Force
Figure 5 shows that the structural elements of planar frames might be subjected to loads oriented in
various directions within plane of the structure. Hence and before moving to the computation of the
fixed-end forces Qf, the loads W acting on the structural members in inclined directions must be
resolved into their rectangular components in the directions of the local axes of member m as follows:
sinWx W = …………………………………………. (13)
cosWy W = ..………………………………………… (14)
Figure 5: The relationship between member loads and member local fixed-end forces
This procedure was essential to analyze pitched-roof portal frames that normally subjected to different
types of loads and due to their inclined members. In addition, the fixed-end forces equations that
presented in [13] are only to process the loads that are perpendicular to the longitudinal axis of the
structural elements. The directions for the member local fixed-end forces Qf, axial and shear forces,
were counted as positive when moving in the positive directions of the member’s local system.
Simultaneously, the local fixed-end moments were counted as positive when rotating
counterclockwise. Conversely, the member loads, Wx՚or Wy՚, are often defined to be positive in the
directions opposite to Qf directions. In other words, the member axial and perpendicular loads are
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counted as positive if it was in the negative directions of the member’s local x՚-axis and y՚-axis,
respectively, and the external couples considered as positive when rotating clockwise.
4. Structural Analysis Using Microsoft Excel Spreadsheet
The key steps that required for conducting a first-order elastic structural analysis of portal frames by
utilizing Microsoft Excel Platform to establish Finite Element Program were described in this section.
Step 1 was to define and draw the geometric shape of the proposed portal frame. Define node, joint or
support, locations as a point in the xy-plane as explained in Section 3.1. Defining locations of these
nodes was to facilitate the diagrammatic representation of the proposed portal frame by employing
Scatter charts that available in Microsoft Excel.
Step 2 was to define the names of structural members and calculate their lengths. By employing
Pythagoras' theorem and subtract the numerical values of the coordinates, x and y, of a member and
the attached member at points of the start and end of these members. This procedure was to modify
the arrangement of portal frames automatically. The cells in figure 6 show the formulas for computing
the length of each member. In the formula SQRT ((D10-D9) ^2 + (E10-E9) ^2), for example, the
SQRT refers to the square root function in Excel while the D10 and D9 are the cells that return the x-
coordinates of the number 2 and 1 respectively. While the E10 and E9 are the cells that return the y-
coordinates of the member 2 and 1 respectively. The Excel will calculate members’ lengths
automatically for any new arrangement just by changing the value of the cells.
Figure 6: Examples of formulas for computing the length of members in Excel spreadsheet
Step 3 was to establish a table for defining the parameters of each member of the portal frame. Setting
up such table was to facilitate the structural analysis process throughout its stages. Table of the
parameters must be inputted manually by the structural engineer. It was entered either as a numerical
value such as the loads on joints, cross-sectional area, modulus of elasticity and the second moment of
the area or formulated such as λx, λy and α. λx and λy outcomes were compared with the built-in
trigonometric formulae in Excel COS(RADIANS(number)) and SIN(RADIANS(number)) where the
“number” was the angle value that calculated using inverse cosine or inverse sine of the angle. To
obtain angles in degree system, these relationships must be preceded by function ''Degrees'' to convert
the result from radians system that used in Excel as DEGREES(ACOS((D10-D9)/C15)).
Step 4 was to formulate stiffness matrices [K], of each member of the frame in LCS. [K] of the
prismatic member, m is a relationship between four parameters, E, A, I and L. Each one of these
parameters was entered in a table of the parameters as in step 3. It is important to notice that [K] must
be 6 rows x 6 columns as well as it must be symmetric about the main diagonal where the upper left is
symmetric with the lower right. Besides, all the elements of the diagonal of stiffness matrix must be
positive to generate so-called the positive definite. Otherwise, it is a sign that the structural engineer
might be made mathematical errors somewhere in the formulations of such matrices.
Step 5 was to formulate correction matrix [C] of each member. [C] was formed as a relationship
between end-fixity factors at both ends of the member and its length. The structural engineers will be
able to increase or decrease values for end-fixity factors and length of the member just changing the
cells related to these parameters in step 3, and then the correction matrix will be recomputed
accordingly.
Member Name Member Length(m) Joint No. Joint No.
Member 1 =SQRT((D10-D9)^2 + (E10-E9)^2) A B
Member 2 =SQRT(SUM((E11-E10)^2, D11^2)) B C
Member 3 =SQRT(SUM((E11-E10)^2, D11^2)) C D
Member 4 =E12 D E
ϴ(degree) (Roof pitch) =DEGREES(ACOS((D11)/C16)) =DEGREES(ASIN((E11-E10)/C16))
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Step 6 was to compute the modified stiffness matrix [K]SR
for each member. [K]SR
was calculated by
multiplying the direct stiffness matrix [K] and correction matrix [C] of a member in their local system.
Excel's function ‘'MMULT(array1, array2) was utilized to implement this multiplication. For the sake
of validity of the formulations within this stage, the modified stiffness matrix of a member was
compared with the generalized form of the modified stiffness matrix that presented in [5].
Step 7 was to generate the transformation matrices [T] and [T]T
for each member by utilizing equations
6 and 10 respectively. [T] was implemented just by selecting the cells that contain λx and λy within the
table of the parameters for the concerned member. [T]T
was formulated either with the same steps for
[T] or by using the commands MINVERSE(array) or TRANSPOSE(array) where the ''array'' here
represents the matrix. Step 8 was to compute the global stiffness matrix SR
g
K for each semi-rigid
member using equation 12. One of limitations in Excel is multiplying a triple product of matrices as
SR
g
K . Consequently, this multiplication must be done in two stages. Firstly, [K]SR
was multiplied by
[T] to produce an intermediate matrix. Secondly, Then, [T]T
was multiplied by the intermediate matrix.
For the sake of validity of the formulations within this stage, SR
g
K was compared with the generalized
form of the global stiffness matrix that presented in [15]. By assigning the end-fixity factors for both
ends of the member in spreadsheet equals to 1 because the mentioned matrix is specified for rigid
member. Step 9 was to compute the member local fixed-end force vector {Qf} and the member global
fixed-end force vector {Ff}. {Qf} was calculated as explained in Section 2.1. {Ff} was calculated using
equation 11. It is essential to write the coding for each component of the vector as set up in the
analytical model because it will be required to be assembled later., see Figure 7.
Figure 7: Example of {Qf} and {Ff} in Excel spreadsheet.
Step 10 was to compute and define the joint load vector {P} and the structure fixed-joint force vector
{Pf}. These vectors are dealing with forces acting on the structural members and joints in GCS. {P}
represents all the loads that applied to the joints directly, the number of its components equals
NDOFS. {Pf} calculated directly by algebraically adding {Ff} of each of degrees of freedom having a
similar coding within the frame in the same location and direction. The number of its components
equals the total number of all degrees of freedom of the frame.
Step 11 was to assembly of the structure stiffness matrix [S]. It was formulated directly by
algebraically adding the pertinent components of SR
gK for each member within the frame according to
their coding numbers. Alternatively, it can be formulating this matrix in spreadsheet by employing
three sophisticated functions in Excel which are ''IFERROR(value, value_if_error)'', VLOOKUP
(value, table, col_index, [range_lookup]) and MATCH(lookup_value, lookup_array,[match_type]).
The square form is a distinguishing characteristic for this matrix where the number of its rows and the
number of its columns equal to the total number of all degrees of freedom, known and unknown.
=(D159*D152)/2
=(((H155+H158)/D152))+((D160*D152)/2)
=((D160*(D152^2))/12)*(((3*D155*((2-D156))/(4-(D155*D156)))))
=(D159*D152)/2
=-(((H155+H158)/D152))+((D160*D152)/2)
=-((D160*(D152^2))/12)*(((3*D156*((2-D155))/(4-(D155*D156)))))
Coding in GCS
=(H153*D168)-(H154*D169) 1
=(H153*D169)+(H154*D168) 2
=H155 3
=(H156*D168)-(H157*D169) 4
=(H156*D169)+(H157*D168) 5
=H158 6
The Member Global Fixed End Force Vector
Member Local Fixed-End Force Vector
{Qf}
{Ff}
12. 2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
11
Moreover, these matrices for linear elastic structures are similar to local stiffness matrices where it
must be symmetric about the main diagonal. Figure 8 is shown a part of [S] in Excel spreadsheet.
At this point, the analysis results were determined. Step 12 considered the first in obtaining analysis
results. Compute the displacement vector that represents the unknown displacements in GCS of the
frame. The values of these displacements were computed using the equations below:
P - Pf S D= ……………………………………… (15)
It also can be expressed as:
11 12 u
u 21 22 k
DQ
Q D
k K K
K K
=
………………………………….. (16)
Partition this matrix and substituting {Dk} to zero will result in:
11 uQ Dk K= ………………………………………. (17)
u 21 uQ DK= …………………………………………. (18)
Where: {Du} is the unknown displacements vector that corresponds to the force vector in GCS {Qk};
{Dk} is the known displacements vector that corresponds to the support reaction vector in GCS {Qu};
[Kij] is the four components of [S], as seen in figure 8.
Figure 8: Example of partial structure stiffness matrix in Excel spreadsheet.
Thus, {Du} was determined from the following equation;
1
u 11D QkK
−
= ……………………………….… (19)
Where: [K11]-1
is the inverse of the part of [S] that is pertinent to the force vector {Qk}.
Step 13 was to compute {Qu} in GCS, that was calculated directly using equation 18 and then to
calculate the member end-forces vector {Q} in LGS for each member. By using the equations below:
Q Q
SR
fK u= + ………………………………….… (20)
u T v= ……………………………………………… (21)
Q Q
SR
fK T v= + ………………………………… (22)
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
=I433 1 =(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C456,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C456,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C456,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C456,$C$217:$I$=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C456,$C$=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C456,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C456,$C$129:$I$134,M=(IFERROR(VLOOKUP($C456,$C$129:
=I434 2 =(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C457,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C457,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C457,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C457,$C$217:$I$=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C457,$C$=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C457,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C457,$C$129:$I$134,M=(IFERROR(VLOOKUP($C457,$C$129:
=I435 3 =(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C458,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C458,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C458,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C458,$C$217:$I$=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C458,$C$=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C458,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C458,$C$129:$I$134,M=(IFERROR(VLOOKUP($C458,$C$129:
=I436 4 =(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C459,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C459,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C459,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C459,$C$217:$I$=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C459,$C$=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C459,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C459,$C$129:$I$134,M=(IFERROR(VLOOKUP($C459,$C$129:
=I437 5 =(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C460,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C460,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C460,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C460,$C$217:$I$=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C460,$C$=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C460,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C460,$C$129:$I$134,M=(IFERROR(VLOOKUP($C460,$C$129:
=I438 6 =(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C461,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C461,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C461,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C461,$C$217:$I$=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C461,$C$=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C461,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C461,$C$129:$I$134,M=(IFERROR(VLOOKUP($C461,$C$129:
=I439 7 =(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C462,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C462,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C462,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C462,$C$217:$I$=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C462,$C$=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C462,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C462,$C$129:$I$134,M=(IFERROR(VLOOKUP($C462,$C$129:
=I440 8 =(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C463,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C463,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C463,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C463,$C$217:$I$=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C463,$C$=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C463,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C463,$C$129:$I$134,M=(IFERROR(VLOOKUP($C463,$C$129:
=I441 9 =(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C464,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C464,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C464,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C464,$C$217:$I$=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C464,$C$=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C464,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C464,$C$129:$I$134,M=(IFERROR(VLOOKUP($C464,$C$129:
R10 10 =(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C465,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C465,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C465,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C465,$C$217:$I$=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C465,$C$=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C465,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C465,$C$129:$I$134,M=(IFERROR(VLOOKUP($C465,$C$129:
R11 11 =(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C466,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C466,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C466,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C466,$C$217:$I$=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C466,$C$=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C466,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C466,$C$129:$I$134,M=(IFERROR(VLOOKUP($C466,$C$129:
R12 12 =(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C467,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C467,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C467,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C467,$C$217:$I$=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C467,$C$=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C467,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C467,$C$129:$I$134,M=(IFERROR(VLOOKUP($C467,$C$129:
R13 13 =(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C468,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C468,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C468,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C468,$C$217:$I$=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C468,$C$=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C468,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C468,$C$129:$I$134,M=(IFERROR(VLOOKUP($C468,$C$129:
R14 14 =(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C469,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C469,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C469,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C469,$C$217:$I$=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C469,$C$=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C469,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C469,$C$129:$I$134,M=(IFERROR(VLOOKUP($C469,$C$129:
R15 15 =(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(D$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C470,$C$217:$I$222,MATCH(D$455,$C$216:=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(E$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C470,$C$217:$I$222,MATCH(E$455=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(F$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C470,$C$217:$I$222,MAT=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(G$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C470,$C$217:$I$=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(H$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C470,$C$=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(I$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLOOKUP($C=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(J$455,$C$128:$I$128,0),FALSE),0))+(IFERROR(VLO=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(K$455,$C$128:$I$128,0),FALSE),0))+(IFER=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(L$455,$C$128:$I$128,0),FALSE),0)=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(M$455,$C$128:$I$128,0),F=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(N$455,$C$128:$I$1=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(O$455,$C$1=(IFERROR(VLOOKUP($C470,$C$129:$I$134,MATCH(P$45=(IFERROR(VLOOKUP($C470,$C$129:$I$134,M=(IFERROR(VLOOKUP($C470,$C$129:
11K 12K
21K 22K
kQ
uQ
13. 2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
12
Equation 22 consists of two components in which the first component was a triple product. As
explained previously, one of the limitations in Excel is multiplying a triple product of matrices. Thus,
solving such an equation was done in four stages. Firstly, extracting the member end displacements
from {Du}. Secondly, [T] was multiplied by the vector {v}. Thirdly, the resultant vector that is
member displacement vector {u} multiplied by the modified stiffness matrix [K]SR
. Finally, the
resultant vector from the previous action was algebraically added to {Qf} and the result will be the
member end-force vector {Q} in LCS. In this stage, the formulation of establishing an FE program has
ended and the designer is capable of modifying any parameter within the spreadsheet. FE program in
this stage has the ability to repeat all steps automatically. Equilibrium check must be satisfied for each
element in the frame to ensure that the calculations of the structural analysis were conducted correctly.
By applying the equilibrium conditions to the free body of each member after calculating{Ff}, each
joint after calculating {P} and then for whole structure after calculating its support reactions {Qu}.
Figure 9 includes a flow-chart summarizing the procedure of establishing the FE program in this paper
that was developed by the authors.
14. 2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
13
Figure 9: Computational Flow-chart of establishing the FE program in this paper.
15. 2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
14
5. Validity of Structural Analysis Procedure
A single-span symmetrical pitched roof steel portal frame with rigid joints was modelled and analyzed
using the analysis procedures in Section 4 to prove the validity of suggested procedure. The frame was
presented and analyzed in Kassimali's book [16, P.332].
This paper also employed Prokon® software to conduct computer-based analysis against the hand
calculations that presented in the mentioned book. The same values of the unfactored forces and
material properties that include cross-sectional area, modulus of elasticity and second moment of area
for columns and rafters were used. This paper used the analytical model that established in Section
3.1. The obtained joint displacements and support reactions using Excel spreadsheet, Kassimali
example [16] and Prokon® Software are tabulated as follows:
Table 1: The joint displacements using three different method
Joint Displacements
Coding of unknown displacements FE Program Kassimali [16] Prokon®
d1 (in) 3.447 3.447 3.446
d2 (in) -0.009 -0.009 -0.009
d3 (rad) -0.019 -0.019 -0.019
d4 (in) 3.952 3.952 3.952
d5 (in) -1.315 -1.315 -1.315
d6 (rad) 0.007 0.007 0.007
d7 (in) 4.424 4.424 4.424
d8 (in) -0.021 -0.021 -0.021
d9 (rad) -0.009 -0.009 -0.009
d12 (in) -0.023 -0.023 -0.023
Table 2: Support reactions using three different method
Support Reactions
Coding of support reaction FE Program Kassimali [16] Prokon®
R10 (kip) -33.502 -33.501 -33.502
R11 (kip) 76.195 76.195 76.196
R12 (k-in) 0.000 0.000 0.000
R13 (kip) -67.355 -67.356 -67.356
R14 (kip) 33.015 33.014 33.016
R15 (k-in) 13788.655 13789 13788.65
By comparing the analysis results that obtained through the three methods, it showed that Excel’s
results were perfectly accurate. Consequently, the proposed procedure of establishing spreadsheets as
a Finite Element Analysis Software for a certain form of frames demonstrates its validity and
efficiency.
16. 2nd International Conference on Sustainable Engineering Techniques (ICSET 2019)
IOP Conf. Series: Materials Science and Engineering 518 (2019) 022037
IOP Publishing
doi:10.1088/1757-899X/518/2/022037
15
6. Conclusions
A first-order elastic structural analysis was performed in this paper. The effects of semi-rigidity for
steel connections were taken into consideration, by incorporating the well-known direct stiffness
matrix with the mathematical method employing the end-fixity factor. A computer program based on a
finite element method was developed and formulated by the authors utilizing Microsoft Excel
spreadsheet. The developed FEA spreadsheet was capable of analysing frames having different end-
connection, such as pinned, rigid joints or semi-rigid joints, between their attached members. The FE
program was verified through two different methods and the results showed that the Excel
spreadsheets as a Finite Element Software are perfectly accurate and capable of conducting the
structural analysis of two-dimensional (2D) frames with different end-rigidity factors.
7. References
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to Eurocode 3 (P397) (Steel Construction Institute)
[2] Salter P, Malik A S and King C M 2004 Design of Single-Span Steel Portal Frames to BS 5950-1 :
2000 (Berkshire: Steel Construction Institute)
[3] Hayalioglu M S and Degertekin S O 2005 Minimum Cost Design of Steel Frames with Semi-Rigid
Connections and Column Bases via Genetic Optimization Computers & Structures 83 1849–63
[4] European Standard 2005 Eurocode 3: Design of steel structures - Part 1-8: Design of joints
(Brussels)
[5] Simões L M C 1996 Optimization of Frames with Semi-Rigid Connections Computers &
Structures 60 531–9
[6] Xu L and Grierson D E 1993 Computer‐Automated Design of Semirigid Steel Frameworks Journal
of Structural Engineering 119 1740–60
[7] Lui E M and Chen W F 1986 Analysis and Behaviour of Flexibly-Jointed Frames Engineering
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[8] Ivanyi M and Baniotopoulos C C 2000 Semi-Rigid Joints in Structural Steelwork vol 134 (Vienna:
Springer Vienna)
[9] Faella C, Piluso V and Rizzano G 2000 Structural steel semirigid connections: theory, design, and
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[10] Xu L 2005 Semirigid Frame Structures Handbook of Structural Engineering ed W-F Chen and E
M Lui (CRC Press)
[11] Monforton G R and Wu T S 1963 Matrix Analysis of Semi-Rigid Connected Frames, Journal of
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[12] Chan S L and Chui P P T 2000 Non-linear static analysis allowing for plastic hinges and semi-
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(Oxford) pp 123–94
[13] Msabawy A 2017 Minimum Weight of Cold-Formed Steel Portal Frames with Semi-Rigid Joints
Using GRG Algorithm (Nottingham Trent University)
[14] Logan D L 2017 A First Course in the Finite Element Method (Cengage Learning US)
[15] Hibbeler R C 2018 Structural Analysis (Pearson)
[16] Kassimali A 2011 Matrix Analysis of Structures (Stamford, CT: Cengage Learning Custom
Publishing)