” By setting θ=1radin Eq. Consistent Deformation Method and related problems, Slope Deflection Method. Draw bending moment diagram and sketch qualitative elastic curve. Correct Answer View All Answers Show/hide. Method of Joint and Method of Section 33 Problem 1. State the difference between strain energy method and unit load method in the determination of deflection of structures. An excellent extension of the slope- deflection method was presented by Gasper Kani of Germany(4). 1 Introduction; 9. Slope And Deflection Of Beams Problems. Class 23 Tuesday 04-12-11: Real Video. From this equation, any deflection of interest can be found. Wood Page 19 of 26 Additional Slope-Deflection Insights: 1) From the slope deflection method, a few aspects often reappear in other topics a. Piles in trailing rows of pile groups have significantly less lateral load resistance than piles in the lead row. VISCOSITY 1. 5) and y' is the slope of the respective surface profiles; P represents. those not built-in) are zero. the convergence for nonlinear problems. How To Use Modified Slope Deflection Equations Example Problem. Objective • To obtain slope and deflection of beam and frame structures using slope-deflection method 3 4. J ° I 0, D, I I I Figure 3 - Ul11moWII. Classical structural analysis, on the other hand, has been around for a long time and is meant to be performed by hand. When solving deflection equations, if lengths are taken in feet, what is the conversion to find deflection in inches? A) 1728 in3 / 1 ft3 26. Slope deflection method examples - Purdue Engineering Slope-Deflection Method Examples Determine the moments at the ends of each member of the frame. Solution Manual Fundamentals of Structural Analysis (3rd Edition) by Kenneth M. Statically Indeterminate Transversely Loaded Beams LECTURE 18. limit the maximum deflection of a beam to about 1/360 th of its spans. E and I are constant. – From the slope dfldeflection equations it is evident that. Problems 632 16 Slope-Deflection Method 635 16. UNIT I : STRAIN ENERGY METHOD UNIT II : SLOPE DEFLECTION METHOD => Solved Problems: Slope Deflection Method- Structural Analysis UNIT III : MOMENT DISTRIBUTION METHOD => Structural Analysis - Moment Distribution Method with Solved Problems UNIT IV : FLEXIBLITY METHOD => Structural Analysis: Flexibility Method => Solved Problems: Structural Analysis- Flexibility Method. Virtual work method- Dummy unit load method 7. Slope And Deflection Of Beams Problems. Slope-Deflection Method Examples. Answer: (Only the end-moments) M AB = 66. Acces PDF Solution Structural Analysis Kassimali Flexibility Method - Problem 1 - Structural Analysis 2 Problem 1 on Flexibility Method Video Lecture from Chapter Analysis of Indeterminate Slope deflection method for indeterminate structure. b) m - r + 2j. The solution of this equation is complicated because the bending. Matrix Method of Joint 56 Problem 3. Macaulay's Method - Introduction Definition Macaulay' s method (The double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. Consistent Deformation Method and related problems, Slope Deflection Method. However, the bending moment at the fixed end is 4000 in-lb and is thus the maximum moment. 17 ENES 220 ©Assakkaf. 6 Practice Problems; Chapter 9: The Slope Deflection Method. Shear and moment diagrams in beams and frames. In the early stage, approximate modelling establishes whether for the deflection, 6, and end slope, 8, of beams, for various common modes of loading are shown on the facing page. Methods of Analysis Force method of analysis The deflection or slope at any point on a structure as a result of a number of forces, including the reactions, is equal to the algebraic sum of the deflections or slopes at this particular point as a result of these loads acting individually 3. structural analysis problems and solutions. The class is an introduction to the basic tools of structural analysis and design. Manderla (7), E. In slope-deflection method, displacements (like slopes and displacements) are treated as unknowns and hence the method is a „displacement method‟. 2 3 / 2 2 2 dx dy 1 dx d 1 + = ρ. methods (c) differential equation approaches and continuum methods. Sedangkan metode slope deflection yang menggunakan rotasi batang sebagai variabel dikategorikan sebagai metode fleksibilitas (flexibility method). The script files for these problems are formulated in terms of beam and loading parameters to. MECHANICS OF MATERIALS Edition Beer • Johnston • DeWolf 9 - 18 Sample Problem 9. The force- deflection curves, F-δ, obtained in the tested series (with specimens of distinct age) are depicted in Fig. This is done by fixing the joints. 11/5/2014 2 Pendahuluan (lanjutanMetode "slope deflection", seperti kedua metode yang lain bisa digunakan untuk analisis struktur balok statis tak tentu dan portal dengan konsep sebagai berikut : 1. Influence line diagrams. 108 infinity; almost all piles are in this category. The systems offer solutions to regulatory concerns including fire and wind resistance, code approvals, stable R-values, and environmental safety. Download Solution Manual Fundamentals of Structural Analysis. A sketch is illustrated here:. A picture is taken of a man performing a pole vault, and the minimum radius of curvature of the pole is estimated by measurement to be 4. The solution to the first problem leads to a modified structural analysis method which considers plastic behavior and insta­ bility effectso The second problem indicated the need for a large deflection theory for column deflection c. B Johnson, C. Methods for Solving To solve an indeterminate structure it is necessary to satisfy equilibrium, compatibility and force-displacement requirements of the structure. 5 Problems for Solution 145 Reference Further Reading 146 146 6 The Matrix Stiffness Method-Part 2: Coordinate Transformation 147 6. Moment Distribution Applied to Continuous Plates. Slope Deflection Equation For Continuous Beam In Hindi By Parag Pal. Problem 654 For the beam in Fig. The figure below shows the shear forces (kips) in the interior columns of a two-storied frame. Homework 1-Solution Problem 1 (1) Use the slope deflection method to find the bending moment diagram for the continuous beam above (work with the minimum number of unknowns) (2) Find the vertical displacement at mid-spanBC using the moment diagram of (1) and the principal of virtual forces. BEAMS: STATICALLY INDETERMINATE (9. Page 4 of 4 Primer Questions Peter von Buelow Deflection 25. Thus slope-deflection method is a displacement method. Three classical methods are studied in this subject, namely: Moment-area (Mohr's theorems); Slope-Deflection Method; and, Moment Distribution Method. Continuous Beams 2. The figure below shows the applied loads (F 1, F 2. Slope/Deflection Method Before I get into the weeds about what the slope deflection method is, I need to explain what a fixed end moment is. Download link is provided and students can download the Anna University CE6501 Structural Analysis I (SA-I) Syllabus Question bank Lecture Notes Syllabus Part A 2 marks with answers Part B 16 marks Question Bank with answer, All the materials are listed below for the students to make use of it and score good (maximum) marks with our study. Geometri (compatibility) : titik-titik pertemuan antara balok dan kolom pada suatu portal dianggap kaku, sehingga sudut-sudut. In other words, the "classical" methods of structural analysis, herein, are analytical methods rather than computational methods. This tutorial goes through a slope deflection method example problem that is 4 degrees statically indeterminate and 1 degree kinematically indeterminate. J ° I 0, D, I I I Figure 3 - Ul11moWII. Determine the reactions and draw the shear and bending moment diagrams for the beams shown in figure by using the slope-deflection method. Aerospace Mechanics of Materials (AE1108-II) –Example Problem 17 Example 1a Solution AB q L EI 5) Calculate slopes and deflections 2 2246 24 qz vzLzL EI 2233 6 qz zLzL EI 4 BL(z ) 8 qL vv EI 3 BL(z ) 6 qL EI Determine deflection and slope at B:. 2 Statical indeterminacy of a ring (a) (b) (c) X X X u z u z u x u x u y u y y y x z x z RINGS The simplest approach is to insert constraints in a structure until it becomes a series of completely stiff rings. 1 Basic Concepts of the Displacement Method 29 2. 28) and (16. Nevertheless, one can find extremal values of slopes and deflections using this method, and usually these. strain curve at any selected strain Secant modulus (E. 1 The Plane Truss Element 148 6. The problem is a cantilive beam of length L with a triangular uniform load or W. The application of both met hods requires solving a system of linear equations. Slope deflection method. Before Macaulay’s paper of 1919, the equation for the deflection of beams could not be found in closed form. Download Structural Analysis by Aslam Kassimali Structural analysis is a very historic branch of civil engineering, there are many old and empirical methods of structural analysis along with some analytical ones as well, and some modern methods with the name of fine element analysis etc. slope-deflection method, the moments at the ends of the members are expressed in terms of displacements and end rotations of these ends. Matrix Methods of Analysis. From this equation, any deflection of interest can be found. These double integration method tutorials also show up in the mechanics of materials playlist in the "beam deflection" section. Therefore, the slope and deflection of a beam due to several loads is equal to the sum of those due to the individual loads. Slope on real beam = Shear on conjugate beam Deflection on real beam = Moment on conjugate beam Properties of Conjugate Beam Engr. Useful solutions for standard problems Preface elastic limit, the deflection δ, and the angle of rotation, θ , can be calculated using elastic beam theory (see Further reading in Section 16). Video Solutions - Developed by Professor Jim Hanson, Rose-Hulman Institute of Technology, these are complete, step-by-step solution walkthroughs of representative homework problems from each chapter. The problem is a cantilive beam of length L with a triangular uniform load or W. Problem on stiffness method of analysis for truss. And the maximum slope, or rotation of the beam, by theta b. slope‐deflection eqq,uation, we obtain (1) 4 θ = L EI M “The bending stiffness, , of a member is defined as the moment th t t b li d t d f th b t it K that mus eapplied a an en o ememer o cause a un rotation of that end. Structural Analysis by R. Piles in trailing rows of pile groups have significantly less lateral load resistance than piles in the lead row. Chapter 16 Introduction to general stiffness method Chapter 17 Matrix analysis of trusses by the direct stiffness method Chapter 18 Matrix analysis of beams and frames by the direct stiffness method. Look at the load-deflection graph of some relevant nodes. Slope-Deflection Method Examples. Note: This beam was previously solved as the first slope-deflection example. The one of the method for finding the deflection of the beam is the direct integration method, i. ASSUMPTIONS IN THE SLOPE DEFLECTION METHOD: This method is based on the following simplified assumptions: All the joints of the frame are rigid, Distortion, due to axial and shear stresses, being very small, are neglected. The load on the conjugate beam is the M/EI diagram of the loads on the actual beam. The increasing acceptance of concrete as a major construction material presents the form builder a new range of problems in the development of appropriate sheathing materials and maintenance of rigid tolerances. The method assumes that all deformations are produced by moment. In strain energy method, an imaginary load P is applied at the point where the deflection is. So that would be integrated, and we find out that this is the equation for the deflection. Indeterminate Truss Problems - Method of Consistent Deformations 82 7. Virtual work method- Dummy unit load method 7. For most cases, the global formula will yield good approximate solutions for loads applied past the first segment. Solutions to Chapter 2 Problems. 3 (p193): Energy and Deflection Equations. Useful solutions for standard problems Preface elastic limit, the deflection δ, and the angle of rotation, θ , can be calculated using elastic beam theory (see Further reading in Section 16). The conjugate beam method, developed by Heinrich Muller-Breslau in 1865, is one of the methods used to determine the slope and deflection of a beam. Sketch the deflected shape of the plane frame. of Civil and Environmental Engineering Seoul National University. Support B settles downward 0. Concluding Remarks 90. Müller-Breslau in 1865. Using Calculus to find expressions for the deflection of loaded beams (See Deflection of Beams Part. Methods for finding the deflection: The deflection of the loaded beam can be obtained various methods. The total slope at point C due to bending moment and axial force work is zero. 28) and (16. 1 SLOPE DEFLECTION METHOD - AN OVERVIEW. pdf] - Read File Online - Report Abuse. In most cases, you will be dealing with round numbers, which greatly simplify your calculations. These methods involve several ‘rigid’ steps resulted in unique solution. But we want to find as our worksheet solution the deflection of the beam as a function of x. com - id: 216dbf-NjQ4O. Some of these methods include: Slope-Deflection Method, Moment Distribution Method, and Matrix Stiffness Method. Pendahuluan (lanjutan) Metode slope deflection, seperti kedua metode yang lain bisa digunakan untuk analisis struktur balok statis tak tentu dan portal dengan konsep sebagai berikut : 1. Slope Deflection method and difficult design problems. Use the Portal Method to calculate the corresponding (i) applied loads P 1 and P 2, (ii) column bending moments, (iii) beam axial forces. An important characteristic of the slope-deflection method is that it does not become increasingly complicated to apply as the number of unknowns in the problem increases. 5 Force Method for Support Settlements, Temperature Changes and Fabrication Errors; 8. analysis of indeterminate structures by force method 5. releases* equal to the degree of static indeterminacy are applied to the structure. Determine the dead loads acting on beam BF and girder AD. Sometimes, the other method or methods may be not as accurate or as useful, but they can provide. Write the fixed end moments for a beam carrying. 44 KNM = +88. 3 (2) Take partial derivative to obtain deflection Castiglino’s Theorem: ∆=∂U ∂Q Table 5. Class Note for Structural Analysis 2 Fall Semester, 2013 Hae Sung Lee, Professor Dept. Laws of Reciprocity 89 8. Answer: (Only the end-moments) M AB = 66. = lim 1 L +o L L I y! dx; 01 i = 1,2 (1. Manderla (7), E. For the fixed end moments use the relevant formulas in the uploaded pdf file (see screenshots) Show transcribed image text. Preface This is a set of lecture notes on finite elements for the solution of partial differential equations. ASSUMPTIONS IN THE SLOPE DEFLECTION METHOD: This method is based on the following simplified assumptions: All the joints of the frame are rigid, Distortion, due to axial and shear stresses, being very small, are neglected. • Force method and slope-deflection method can be used, with hand calculation, for solving the indeterminate structures when the degree of static or kinematical indeterminacy is small. Determine the dead loads acting on beam BF and girder AD. This is done by fixing the joints. Laterally loaded pile groups have a group efficiency less than 1. Macaulay's or graphical methods can be used to determine zero beam slope if you want to be real accurate, but the maths can get long winded. Slope Deflection Method Study Notes For Civil Engineering. The solutions to these initial. A sketch is illustrated here:. The frame is made from pipe that is rigid connected. of Civil Engg. is referred to as the. 6 Example 4. 5 Analysis of Frames with Sidesway 681 Summary 702 Problems 702 17 Moment-Distribution Method 707 17. • Problems FEM analysis scheme Step 1: Divide the problem domain into non overlapping regions (“elements”) connected to each other through special points (“nodes”) Step 2: Describe the behavior of each element Step 3: Describe the behavior of the entire body by putting together the behavior of each of the elements (this is a process. In other words, the "classical" methods of structural analysis, herein, are analytical methods rather than computational methods. Structures examined in this course will be modeled as planar trusses, beams and/or frame structures. Moment distribution has an arithmetical solution in which a number of successive corrections are applied to an initial set of assumed moments. When solving deflection equations, if lengths are taken in feet, what is the conversion to find deflection in inches? A) 1728 in3 / 1 ft3 26. check that the slope of the beam is zero for x = L/2 and that the elastic curve has a minimum at the midpoint C of the beam (Fig. e the angle between members at the joints are considered not-to change in value as loads are applied, as shown in. lor}3 by slope deflection aO h^r lrn Solution: ln this Problem FEMs: /oow 0e=0, 0e+0, 0c*0, 6=15mm Wab' l(a D c rAB- --f , Wa'b Fro = +11-= ,Bc_ --'q=-41. Slope-deflection method can be applied for simul­ taneous loading of many or all members. Kani's method is able to analyze the simple frame with or without joint sway. Problem 2: The beam shown below is fixed at A and supported by a roller at B and a pin at C Complete the following using the unmodified moment distribution method (whiclh means that you must carry the moments to and from C and then equilibrate them each time). 2 End moments for prismatic members 89 4. Example Problem A w x y #$ Modulus of Elasticity = E Moment of Inertia = I B Find the equation of the elastic curve for the simply supported beam subjected to the uniformly distributed load using the double integration method. E and I are constant. Assume A and C are pinned and B and D are fixed connected. It may be summarized as follows: 1. Support Reactions and Elastic Curve. Introduction to beam deflection and the elastic curve equation 11. Lecture 5: Solution Method for Beam De ections The particular solution of this problem, satisfying the governing equation is w p = q 0x5 60EIl (5. 3 method of consistent deformation • 5. Laws of Reciprocity 89 8. The trainee is presented with slides showing aspects of the analysis, design and erection and asked to consider key aspects with a list of questions. Macaulay's method (The double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. •In the slope-deflection method the individual equations are relatively easy to construct regardless of the number of unknowns Derivation of Slope-Deflection Eqs. Acces PDF Fundamentals Of Structural Analysis Solution Manual 4th Leet Macaulay's Method Problem 1 - Structural Analysis 1 Macaulay's Method Problem 1 Video Lecture from Deflection of Beams Using (in Hindi) Slope deflection method for indeterminate structure. Shear and moment diagrams in beams and frames. but the beam has a non-zero slope. Objectives Slope-deflection method consists of series of simultaneous equations acting at the end of each member considering it to be fixed-end and taking each moment in terms of slope and deflection. 2 introduction • 5. 9 Fixed-end moments 104 Problems 106 5 The Moment-Distribution Method 113. Deflection (in) 50 100 th (in) 150 1 kip Medium Sand Micropile: 7-inch casing t= 0. In gases t hese forces are very weak and cannot hol d the m ass to gether. Slope and Deflection of Beams - Mechanical Engineering (MCQ) questions and answers Home >> Category >> Mechanical Engineering (MCQ) questions and answers >> Slope and Deflection of Beams 1) A simply supported beam carries uniformly distributed load of 20 kN/m over the length of 5 m. This Case Study describes the design of a complex arch at Cribbs Causeway, on the outskirts of Bristol. 1 Basic Concepts of the Displacement Method 29 2. of Civil and Environmental Engineering Seoul National University. 9 3 The Slope Deflection Equations Learnaboutstructures Com. This minimization gives equations to calculate the terms of the necessary matrices to solve the problem. example 8 determine the moment at each joint of the gable frame. Sedangkan metode slope deflection yang menggunakan rotasi batang sebagai variabel dikategorikan sebagai metode fleksibilitas (flexibility method). Problems on Lateral Load Analysis by Portal Method 1. These methods involve several ‘rigid’ steps resulted in unique solution. Worked examples multiple random solution (a) we assign a joint probability mass function for x and y as shown in the table below. The global formula simplifies the solution method for stepped beams (eliminating the calculations of slope and deflection of a stepped beam at any location on the beam. The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. At the wall (x=0) the moment felt is the maximum moment or PL, but at the end of the beam, the moment is zero because moments at the locations do not contribute to the overall moments. The method proposed by Kimball and Tsai 57 [4] does not need to locate the inflection point. In the next module, we'll redo the problem using singularity functions to solve for the beam deflection. Simplified Analysis of Continuous Beams Abdulamir Atalla Almayah Ph. Other Practical Design Methods. Truss Analysis: Force Method, Part II 675. COURSE OUTLINE - FALL SEMESTER, 2013 ENCE 353: Introduction to Structural Analysis 9/4/13 8:25 AM problem solutions. Chapter 2 : Part 2 – Unit Load Method • Aims – Determine the slope and deflection by using Unit Load Method • Expected Outcomes : – Able to analyze determinate beam – deflection and slope by Unit Load Method • References – Mechanics of Materials, R. 50 For the beam and loading shown, determine (a) the reaction at the roller support, (b) the deflection at point C. Video Solutions – Developed by Professor Jim Hanson, Rose-Hulman Institute of Technology, these are complete, step-by-step solution walkthroughs of representative homework problems from each chapter. Influence line diagrams. The Arch at Cribbs Causeway - 2 Hours. t ) - The slope of the stress vs. 66 Truss Analysis: Force Method, Part II 67 5. 7 For the beam and loading shown, determine (a) the equation of the elastic curve for portion AB of the beam, (b) the deflection at midspan, (c) the slope at B. 5 Problems for Solution 145 Reference Further Reading 146 146 6 The Matrix Stiffness Method-Part 2: Coordinate Transformation 147 6. Slope-Deflection Method Examples. Virtual work method- Dummy unit load method 7. Use of Macaulay's technique is very convenient for cases of discontinuous and/or discrete loading. Online Study Material, Lecturing Notes, Assignment, Reference, Wiki and important questions and answers => Solved Problems: Slope Deflection Method- Structural Analysis MOMENT DISTRIBUTION METHOD CE6501 Structural Analysis I - Notes - Download Pdf. Download Structural Analysis by Aslam Kassimali Structural analysis is a very historic branch of civil engineering, there are many old and empirical methods of structural analysis along with some analytical ones as well, and some modern methods with the name of fine element analysis etc. of Stiffness or Displacement Method, i. And if you don't get a perfect score, this course is perfect for refreshing your knowledge on these fundamental methods. Influence Lines for Statically Indeterminate Structures Influence Lines for Beams and Trusses / Qualitative Influence Lines by Muller-Breslau's Principle / Summary / Problems 16. Different equations for bending moment were used at. Hibbeler, 7th Edition, Prentice Hall. It was later independently arrived at by Timoshenko (1910), von Mises (1914), and Southwell (1914) in a slightly modified form. Acces PDF Solution Structural Analysis Kassimali Flexibility Method - Problem 1 - Structural Analysis 2 Problem 1 on Flexibility Method Video Lecture from Chapter Analysis of Indeterminate Slope deflection method for indeterminate structure. Structural Analysis by R. 1 Analyse the rigid frame as shown in Fig. Solution: 1) Determine the distribution factors at joints B and C. For the beams shown below determine the reaction forces and draw the shear and moment diagrams using the slope-deflection method. Practice in applying these methods will develop a deeper understanding of the basic engineering sciences of statics and mechanics of materials. of Civil and Environmental Engineering Seoul National University. Use slope deflection equations to find the resultant end moments for the continuous beam shown in figure 7-4(a). The method proposed by Kimball and Tsai 57 [4] does not need to locate the inflection point. However, in the4 slope- deflection method, the slope or rotations are taken as unknowns, and due to this the problem involves three unknown Fixed end moments (b) Slope deflection equation. Truss Deflection 67 Problem 5. Shear and moment diagrams in beams and frames. method was applied by H. In frames axial deformations are much smaller than the bending deformations and are neglected in the analysis. Use the Portal Method to calculate the corresponding (i) applied loads P 1 and P 2, (ii) column bending moments, (iii) beam axial forces. Suggested Readings Reference 1 Chapter 16 2 3. Here is the elastic curve or the deflection curve and the maximum values of deflection and slope, or a concentrated load P where the solutions are as given here. However, in the4 slope- deflection method, the slope or rotations are taken as unknowns, and due to this the problem involves three unknown rotations q A , q B and q C. 2 End moments for prismatic members 89 4. Methods for finding the deflection: The deflection of the loaded beam can be obtained various methods. When solving deflection equations, if lengths are taken in feet, what is the conversion to find deflection in inches? A) 1728 in3 / 1 ft3 26. The class is an introduction to the basic tools of structural analysis and design. t ) - The slope of the stress vs. The slope-deflection method and method of consistent deformation. Slope-Deflection Method: Frames With SideFrames With Side----sssswayway. – But there are some general rules and ideas that are common to the solution of most types of beam problems. The method assumes that all deformations are produced by moment. Slope Deflection Method Notes prepared by: R. Structural Analysis Objective Type Questions and Answers - Set 01 Slope deflection method (B) Column analogy method (C) Moment distribution method (D) None of the above. Contrarily in the moment distribution method, as a first step the slopes at the ends are made zero. 1 Using the slope-deflection method, compute the end moment of members of the beams shown in Figure P11. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. 8 Comparison between slope-deflection and force methods 103 4. example 8 determine the moment at each joint of the gable frame. Take EI as constant for the beam. Find deflection of a simply supported beam with distributed load. The force- deflection curves, F-δ, obtained in the tested series (with specimens of distinct age) are depicted in Fig. To date, quite a number of reports are available on the use of scaffolding practice in many different areas of learning,. The approach taken is mathematical in nature with a strong focus on the. The solution of slope-deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. The slope-deflection method and method of consistent deformation. Number of unknown internal forces in each member of a rigid jointed plane frame is. Lecture 5: Solution Method for Beam De ections The particular solution of this problem, satisfying the governing equation is w p = q 0x5 60EIl (5. Studying this diagram carefully, we note: 1. The conjugate beam method, developed by Heinrich Muller-Breslau in 1865, is one of the methods used to determine the slope and deflection of a beam. Thus, 2 2 dx 1 d y ≈ ρ ⇒ 2 2 dx d y. Laws of Reciprocity 898. Slope And Deflection Of Beams Problems. How to solve statically indeterminate structures using the Slope-Deflection Method, which is a stiffness method. Download Book Fundamentals of Structural Analysis. 4 Analysis of Frames without Sidesway 673 16. > Category >> Mechanical Engineering (MCQ) questions and answers >> Slope and Deflection of Beams 1) A simply supported beam carries uniformly distributed load of 20 kN/m over the length of 5 m. The solution of slope-deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. Displacement-based methods will also be introduced including slope deflection method and moment distribution. • In this chapter, direct stiffness method (which is also called the displacement method) will be introduced that is a modern method for structural analysis. Pendahuluan (lanjutan) Metode slope deflection, seperti kedua metode yang lain bisa digunakan untuk analisis struktur balok statis tak tentu dan portal dengan konsep sebagai berikut : 1. Leet, Chia-Ming Uang and Annie M. Slope Deflection method and Moment Distribution method. 3-5 A cantilever beam with a uniform load (see figure) has a height h equal to 1/8 of the length L. Solutions to Chapter 2 Problems. In liquids this is strong enough to keep the m ass toget her but not strong enough t o keep i t rigid. PREFACE This report presents a method of frame analysis which is useful in solving a variety of structural problems. The force- deflection curves, F-δ, obtained in the tested series (with specimens of distinct age) are depicted in Fig. – From the slope dfldeflection equations it is evident that. This is done by fixing the joints. The Arch at Cribbs Causeway - 2 Hours. Structural Design and Inspection- Deflection and Slope of Beams By Dr. Slope-deflection method can be applied for simul­ taneous loading of many or all members. In the book, "The Theory and Practice of Modern Framed Structures", written by J. Which of the following methods of structural analysis is a displacement method ? a) moment distribution method b) column analogy method c) three moment. The floor system of a gymnasium consists of a 130 mm thick concrete slab resting on four steel beams (A = 9,100 mm 2) that, in turn, are supported by two steel girders (A = 25,600 mm 2), as shown in Fig. Whatistheequivalentresistanceof. Macaulay's method (The double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. However, in the4 slope- deflection method, the slope or rotations are taken as unknowns, and due to this the problem involves three unknown Fixed end moments (b) Slope deflection equation. deflection v of the beam this method is called method of successive integration Example 9-1 determine the deflection of beam AB supporting a uniform load of intensity q also determine max and A, B flexural rigidity of the beam is EI bending moment in the beam is qLx q x 2 M = CC - CC 2 2. Different equations for bending moment were used at. The equilibrium paths of an initially perfect cylinder and a. Solutions of problems obtained in this course using the FE package can be compared and contrasted against the results from other classical methods discussed in these courses. 6 truss structures • 5. 03 5wL4 3801. The beam is a steel wide-flange section with E 28 106 psi and an allowable bending stress of 17,500 psi in both tension and compression. ANALYSIS OF THE STRUCTURE IN THE INITIAL SITUATION The structure is first analysed in the initial situation, under the accidental load combination. The basic differential equation describing the curvature of the beam at a point Equations for the deflection, δ, and end slope, θ , of beams. 3 Analysis of Continuous Beams 651 16. method are most accurate because of the number of the iterations are more comparatively. = lim 1 L +o L L I y! dx; 01 i = 1,2 (1. Slope deflection equation for continuous beam in HINDI by PARAG PAL Slope deflection method for indeterminate structure Problem 2 On Fixed End Three Moment Theorem. 5 3 Deflection (in) First 5 10 ft critical 50 100 th (in) 150 1 kip First 5-10 ft critical Acceptable deflection under 10 kip (~3/4 inch) 20 kip is close to. The force- deflection curves, F-δ, obtained in the tested series (with specimens of distinct age) are depicted in Fig. For most cases, the global formula will yield good approximate solutions for loads applied past the first segment. The statical indeterminacy of a ring is known and hence. UNIT I : STRAIN ENERGY METHOD UNIT II : SLOPE DEFLECTION METHOD => Solved Problems: Slope Deflection Method- Structural Analysis UNIT III : MOMENT DISTRIBUTION METHOD => Structural Analysis - Moment Distribution Method with Solved Problems UNIT IV : FLEXIBLITY METHOD => Structural Analysis: Flexibility Method => Solved Problems: Structural Analysis- Flexibility Method. the relative qualities of solutions as given by Galerkin's method. This minimization gives equations to calculate the terms of the necessary matrices to solve the problem. To analysis the problem by MDM method is quite preferable because Slope deflection method involve solving a more equation so it is quite tedious to solve the problem by these methods. 2 Statical indeterminacy of a ring (a) (b) (c) X X X u z u z u x u x u y u y y y x z x z RINGS The simplest approach is to insert constraints in a structure until it becomes a series of completely stiff rings. Comprehensive Structural Analysis-I, SLOPE DEFLECTION METHOD 38100 Diagram shown in Fig shows Sketch the bending Solution solve Span BC Span CD Step stiffness. Elastic Deflection Castigliano’s Method (1) Obtain expression for all components of energy Table 5. Therefore, the slope and deflection of a beam due to several loads is equal to the sum of those due to the individual loads. Wink­ ler (8) and O. 2 Slope Œ Deflection Equations Ł Typical Problem 0 0 0 0 A C B P1 P2 L1 L2 w CB 8 0 4 2 1 1 1 1 PL L EI L EI MAB = θA +. Learning Objectives: 1. Whatistheequivalentresistanceof. strain curve at any selected strain Secant modulus (E. · Analysis of statically indeterminate frames (Slope -Deflection Method) The example problems presented in this paper are taken from the text book 1 used for the course. Virtual work method- Dummy unit load method 7. 1 The Plane Truss Element 148 6. Review of what was covered by Dr. Slope-deflection_examples REVISED 2014. Solution In the given problem, joint can also rotate and also translate by an unknown amount. classical methods - slope-deflection method and moment distribution method - before they are given an introduction to the matrix method which is used in computer-aided analysis. 4) where tan. These methods involve several 'rigid' steps resulted in unique solution. 9 Slope-deflection method. B Johnson, C. Indeterminate Truss Problems Method of Consistent Deformations 827. Objectives Slope-deflection method consists of series of simultaneous equations acting at the end of each member considering it to be fixed-end and taking each moment in terms of slope and deflection. To analysis the problem by MDM method is quite preferable because Slope deflection method involve solving a more equation so it is quite tedious to solve the problem by these methods. 1 SLOPE DEFLECTION METHOD - AN OVERVIEW. Methods for finding the deflection: The deflection of the loaded beam can be obtained various methods. The contribution to the slope at point C from the axial energy is slope is zero. 9 Slope-deflection method. The RRM is an energy minimization (partial derivatives with respect to the unknown coefficients) of the energy terms involved in the problem. Laws of Reciprocity 89 8. This video is part 1 of 3 for the problem. State the difference between strain energy method and unit load method in the determination of deflection of structures. 45 minus the interest on the loan and are made at the end of each year for 10 years. How To Use Modified Slope Deflection Equations Example Problem. It is important for students to see the entire range of methods, both classical and modern, so as to place the entire sphere of structural analysis in proper perspective. It may be summarized as follows: 1. pdf] - Read File Online - Report Abuse. 1 Analyse the rigid frame as shown in Fig. Flexibility Method - Problem 1 - Structural Analysis 2 Problem 1 on. The basic equations for calculation by the slope­ deflection method are joint equations (following. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. Ritz method (RRM). In this case two spans (AC and CD) of the beam are to be considered. equation is not enough to solve this problem. Which of the following methods of structural analysis is a force method ? a) slope deflection method b) column analogy method c) moment distribution method d) none of the above Ans: b 43. Second Moment Therorem - Deflection is equal to the moment of the moment area about the point you want the deflection. In problems involving both bending and axial deformation, be careful with the units. Solved Problems: Slope Deflection Method- Structural Analysis When. 9 Fixed-end moments 104 Problems 106 5 The Moment-Distribution Method 113. MBC MCB MCD 0 MBA MCD Slope Deflection Method (frame analysis-no sideway) Example 1 Subs all the equations will give us: 0. Matrix Methods of Analysis. Sketch the deflected shape of the plane frame. Beam and Frame Analysis: Force Method, Part I 931. Moment distribution has an arithmetical solution in which a number of successive corrections are applied to an initial set of assumed moments. Influence Lines for Statically Indeterminate Structures Influence Lines for Beams and Trusses / Qualitative Influence Lines by Muller-Breslau's Principle / Summary / Problems 16. Finally, beams and frames having non-prismatic members are considered in Chapter 13. Done Problem set 4 deflection. Practical Analysis of RC Floor Slabs. Structural Analysis by R. Mar 07, 2020 - Chapter 17 : The Slope-Deflection Method : Frames with Sidesway Civil Engineering (CE) Notes | EduRev is made by best teachers of Civil Engineering (CE). find the conditional pdf, slope-deflection method examples. the method using the differential equation which we have derived. Acces PDF Solution Structural Analysis Kassimali Flexibility Method - Problem 1 - Structural Analysis 2 Problem 1 on Flexibility Method Video Lecture from Chapter Analysis of Indeterminate Slope deflection method for indeterminate structure. Thus for a span AB of a beam. pdf] - Read File Online - Report Abuse. The unknowns in this method are degree of freedom i. Figure 7-4(a) Solution: The given beam in figure 7-4(a) is statically indeterminate of degree 1. 44 KNM = +88. Moment Distribution Applied to Continuous Plates. Calculating Static Deflection and Natural Frequency of Stepped Cantilever Beam Using Modified Rayleigh Method 109 Figure 1: The Dividing Scheme of the Stepping Cantilever Beam By calculating the deflection of the beam(y(x)) using the following steps [21, 25, 26, 27]: Dividing the length of the beam into (n) parts (i. Slope/Deflection Method Before I get into the weeds about what the slope deflection method is, I need to explain what a fixed end moment is. The classical solution to the buckling problem of axially loaded cylinders was obtained by Lorenz (1908). So we get minus M End times L over 6 for C1. ∑M cut = 0 = - (wL/2) x + w x (x/2) + M. As the floors are supported by the primary frames, the secondary beams only support their self-weight. The Arch at Cribbs Causeway - 2 Hours. 2-1 The deflection curve for a simple beam AB (see figure) is given by the following equation: Describe the load acting on the beam. Kohutek in class 20 - Slope Deflection Equations. t ) - The slope of the stress vs. 00016 m 8 x 60 x 10 300 x 4 8EI wL y 6 4 Deflection is 0. Theory & Examples * Moment-Curvature Relation (developed earlier): EI 1 M = ρ. 7 Practice Problems. Objectives Slope-deflection method consists of series of simultaneous equations acting at the end of each member considering it to be fixed-end and taking each moment in terms of slope and deflection. This slope can be designed using the friction circle BANK-EXC program written by the author of this course. Yoo, Sung C. s ) - The slope of the line connecting the origin to any point on the stress vs. (3) Slope at the centre of symmetrically loaded and supported beams is zero. 3 Double Integration Method The Double Integration Method, also known as Macaulay's Method is a powerful tool in solving deflection and slope of a beam at any point because we will be. 1 through Figure P11. The slope-deflection method and method of consistent deformation. 2 Statical indeterminacy of a ring (a) (b) (c) X X X u z u z u x u x u y u y y y x z x z RINGS The simplest approach is to insert constraints in a structure until it becomes a series of completely stiff rings. Chapter 2 : Part 2 – Unit Load Method • Aims – Determine the slope and deflection by using Unit Load Method • Expected Outcomes : – Able to analyze determinate beam – deflection and slope by Unit Load Method • References – Mechanics of Materials, R. This method is particularly useful when deflection at a specific point on the beam is required. Hibbeler 8th Edition. Specifically, the slope θ at point A with respect to a tangent line at point B on a beam deflection curve See Beloow). Other Practical Design Methods. 50 For the beam and loading shown, determine (a) the reaction at the roller support, (b) the deflection at point C. 45 minus the interest on the loan and are made at the end of each year for 10 years. Slope-deflection equations for mnd Moments: Modified slope-deflection equation when far end is supported by a roller or pin: Practice Problems. Released structure. Take EI as constant for the beam. The conjugate beam method, developed by Heinrich Muller-Breslau in 1865, is one of the methods used to determine the slope and deflection of a beam. Using moment- displacement relations, moments are then known. Ostenfeld (10). Introduction to beam deflection and the elastic curve equation 11. Answer: (Only the end-moments) M AB = 66. 3 Example 4. The geometric methods previously discussed are very good and quite straight-forward for simple loadings. Williot Mohr diagram method 4. It may be summarized as follows: 1. Slope deflection method examples - Purdue Engineering Slope-Deflection Method Examples Determine the moments at the ends of each member of the frame. SOLUTION i. Chapter 2 Design Loads Chapter 3 Statistics of structures-Reactions Chapter 4 Trusses Chapter 5 Beams and Frames Chapter 6 Cables Chapter 7 Arches. Whatistheequivalentresistanceof. Wink­ ler (8) and O. Step-by-step solution: Chapter: CHB CHC CH2 CH3 CH4 CH5 CH6 CH7 CH8 CH9 CH10 CH12 CH13 CH14 CH15 CH16 CH17 CH18 Problem: 1P 2P 3P 4P 5P 6P 7P 8P 9P 10P 11P 12P 13P 14P 15P 16P 17P 18P 19P 20P 21P 22P 23P 24P. The structure is thus reduced to a determinate structure. A jump in the slope gives an in nite curvature, and thus an in nite bending moments. 2-1 Simple beam. The moment distribution method falls into the category of displacement method of structural analysis. Riprap tends to increase the speed of water flow along an armored. 2-1 The deflection curve for a simple beam AB (see figure) is given by the following equation: Describe the load acting on the beam. Truss Analysis: Force Method, Part II 675. The class is an introduction to the basic tools of structural analysis and design. ASSUMPTIONS IN THE SLOPE DEFLECTION METHOD: This method is based on the following simplified assumptions: All the joints of the frame are rigid, Distortion, due to axial and shear stresses, being very small, are neglected. 28) and (16. Of these methods, the first two are the ones that are commonly used. Solution In the given problem, joint can also rotate and also translate by an unknown amount. Method of elastic weights 6. Methods of Analysis Force method of analysis The deflection or slope at any point on a structure as a result of a number of forces, including the reactions, is equal to the algebraic sum of the deflections or slopes at this particular point as a result of these loads acting individually 3. 2 Slope Œ Deflection Equations Ł Typical Problem 0 0 0 0 A C B P1 P2 L1 L2 w CB 8 0 4 2 1 1 1 1 PL L EI L EI MAB = θA +. Sometimes, the other method or methods may be not as accurate or as useful, but they can provide. These double integration method tutorials also show up in the mechanics of materials playlist in the "beam deflection" section. Finally, beams and frames having non-prismatic members are considered in Chapter 13. 6 The Slope. Lee, in Stability of Structures, 2011. In problems involving both bending and axial deformation, be careful with the units. method was applied by H. Conjugate beam is defined as the imaginary beam with the same dimensions (length) as that of the original beam but load at any point on the conjugate beam is equal to the bending moment at that point divided by EI. The solution is to find the slope and deflection at the end by moment area method. Laterally loaded pile groups have a group efficiency less than 1. Slope/Deflection Method Before I get into the weeds about what the slope deflection method is, I need to explain what a fixed end moment is. Summary and Conclusions. Example 17. Using moment- displacement relations, moments are then known. In this module, we'll use superposition techniques for finding the deflection. example, the problem is complex if the piles are driven into soft clay. Problems 27 CHAPTER 2 - DISPLACEMENT METHOD OF ANALYSIS: SLOPE-DEFLECTION METHOD 29 2. Mar 07, 2020 - Chapter 17 : The Slope-Deflection Method : Frames with Sidesway Civil Engineering (CE) Notes | EduRev is made by best teachers of Civil Engineering (CE). 3 Normal force. d) m + r - 3j. COURSE OUTLINE - FALL SEMESTER, 2013 ENCE 353: Introduction to Structural Analysis 9/4/13 8:25 AM problem solutions. Degrees of freedom In a structure, the numbers of independent joint displacements that the structure can undergoes are known as degrees of freedom. 44 KNM = +88. Slope Deflection Method Study Notes For Civil Engineering. The double integration method is a procedure to establish the equations for slope and deflection at points along the elastic curve of a loaded beam. Please cite this article as: A. 14845934 Structural Analysis Force Method Indeterminate Beams - Free download as Powerpoint Presentation (. Matrix Method of Joint 56 Problem 3. 23 Friday Classes Meet Nov. Slope Deflection method and Moment Distribution method. SLOPE-DEFLECTION METHOD - SUGGESTED PROBLEMS Problem 1: Determine all support reactions and the moment and shear diagrams of the structure by the slope-deflection method. Equivalent Frame Method Applied to Flat Slabs. For all problems use the slope-deflection method. Solution In the given problem, joints Band C rotate and also translate by an amountΔ. Slope/Deflection Method Before I get into the weeds about what the slope deflection method is, I need to explain what a fixed end moment is. From the shear force diagram, we find V = 0 at x =25” and, thus the maximum bending moment is 2250 in-lb at x=25”. The list of steps given below for the solution of deflection problems by the area-moment method may prove helpful. In a warpage analysis, if you have over-constrained the model so that the shrinkage strains are in conflict with constraints, this will often cause the large deflection analysis solution to fail, even if the linear result is reasonable. 28-Mar Midterm Exam: Frames, Moment Area Method, Influence Lines Lecture 10 4-Apr Example Problems: Principle of Virtual Work Lecture 11 11-Apr Slope Deflection Method Lecture 12 18-Apr Moment Distribution Method Lecture 13 25-Apr Example Problems: Slope Deflection Method, Moment Distribution Method Lecture 14. Beams –SFD and BMD: Example (3) Draw the SFD and BMD for the beam Solution: Draw FBD of the beam and Calculate the support reactions Draw the SFD and the BMD starting from any one end ∑M A = 0 R A = 60 N ∑M B = 0 R B = 60 N 60 N 120 Nm V 60 N-60 N M-120 Nm ME101 - Division III Kaustubh Dasgupta 9. 2 have constant flexural rigidity EI. As shown in Fig. Able to analyse plane frames restrained against sidesway by slope-deflection equations. slope and deflection; Combinely these slopes and deflections are known as displacements. Slope deflection method and Moment distribution method are both stiffness methods. Primary structure must be chosen such that it is geometrically stable and statically determinate. Solution To Problem 410 Shear And Moment Diagrams. Application of Slope-Deflection Equations to Statically Indeterminate BeamsIndeterminate Beams: ::: The procedure is the same whether it is applied to beams or frames. The method is based on the principle of statics. Sketch the loaded beam, the moment or M/(EI) diagrams (either by parts or. Moment-Distribution Method. by Influence Lines for Beams and Trusses / Qualitative Influence Lines by Muller-Breslau's Principle / Summary / Problems 16. Lecture 5: Solution Method for Beam De ections The particular solution of this problem, satisfying the governing equation is w p = q 0x5 60EIl (5. Some of these methods include: Slope-Deflection Method, Moment Distribution Method, and Matrix Stiffness Method. Here is the elastic curve or the deflection curve and the maximum values of deflection and slope, or a concentrated load P where the solutions are as given here. The figure below shows the applied loads (F 1, F 2. To analysis the problem by MDM method is quite preferable because Slope deflection method involve solving a more equation so it is quite tedious to solve the problem by these methods. Truss analysis. For problems below, find the deflection or slope at the assigned locations using (a) moment-area method, (b) conjugate-beam method. This is problem 9-3, page 551, from bok Problem Solvers, strength of materials and mechanics of materials by REA. Solution To Problem 410 Shear And Moment Diagrams. 2-2 q 0 x 360LEI (7L4 10L2x2 + 3x4) 9 Deflections of Beams 707 y x A B L Solution 9. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. Aerospace Mechanics of Materials (AE1108-II) –Example Problem 17 Example 1a Solution AB q L EI 5) Calculate slopes and deflections 2 2246 24 qz vzLzL EI 2233 6 qz zLzL EI 4 BL(z ) 8 qL vv EI 3 BL(z ) 6 qL EI Determine deflection and slope at B:. Influence line diagrams. This figure shows that, just after the deflection δ L, a load decay occurred with an amplitude that also increased with the SFRSCC age, since a higher load had to be. Macaulay's method (The double integration method) is a technique used in structural analysis to determine the deflection of Euler-Bernoulli beams. As shown in Fig. 3 Analysis of Continuous Beams 651 16. The deflection of the beam v(x). In problems involving both bending and axial deformation, be careful with the units. Worked examples multiple random solution (a) we assign a joint probability mass function for x and y as shown in the table below. This is a very powerful and convenient method since solutions for many support and. Degrees of freedom In a structure, the numbers of independent joint displacements that the structure can undergoes are known as degrees of freedom. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. Online Study Material, Lecturing Notes, Assignment, Reference, Wiki and important questions and answers => Solved Problems: Slope Deflection Method- Structural Analysis MOMENT DISTRIBUTION METHOD CE6501 Structural Analysis I - Notes - Download Pdf. 4 Example 4. Solutions of problems obtained in this course using the FE package can be compared and contrasted against the results from other classical methods discussed in these courses. The deposits to the sinking fund are equal to 1,627. Slope deflection equation for continuous beam in HINDI by PARAG PAL Slope deflection method for indeterminate structure Problem 2 On Fixed End Three Moment Theorem. Able to analyse plane frames restrained against sidesway by slope-deflection equations. The solution of slope-deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. 7 For the beam and loading shown, determine the slope and deflection at point B. Slope/Deflection Method Before I get into the weeds about what the slope deflection method is, I need to explain what a fixed end moment is. 3 (p193): Energy and Deflection Equations. State the difference between strain energy method and unit load method in the determination of deflection of structures. Released structure. The slope-deflection method and method of consistent deformation. The figure below shows the applied loads (F 1, F 2. SLOPE-DEFLECTION METHOD - SUGGESTED PROBLEMS Problem 1: Determine all support reactions and the moment and shear diagrams of the structure by the slope-deflection method. 6 Failure of Cylindrical Shells. Released structure. The systems offer solutions to regulatory concerns including fire and wind resistance, code approvals, stable R-values, and environmental safety. These methods involve several ‘rigid’ steps resulted in unique solution. Calculate the slope and deflection at the free end. Wink­ ler (8) and O. The method is shown to be applicable to no-sway, plane-frame structures that may derive part or all of their support from soils. Problems arise because the effects of riprap do not stop at the point of installation. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. Structural Analysis, 3rd Edition / Edition 3. 14) Indeterminate Structures: Influence Lines. For many years now, engineers have been using matrix methods to analyze structures. It may be summarized as follows: 1. , Large deflection of cantilever beams with geometric non-linearity: Analytical and numerical approaches, Int. and related problems, Moment Distribution Method. UNIT I : STRAIN ENERGY METHOD UNIT II : SLOPE DEFLECTION METHOD => Solved Problems: Slope Deflection Method- Structural Analysis UNIT III : MOMENT DISTRIBUTION METHOD => Structural Analysis - Moment Distribution Method with Solved Problems UNIT IV : FLEXIBLITY METHOD => Structural Analysis: Flexibility Method => Solved Problems: Structural Analysis- Flexibility Method. Slope Deflection Method Notes prepared by: R. Shear and moment diagrams in beams and frames. SOLUTION i. Slope-Deflection Method Slope-Deflection Equations / Basic Concept of the Slope-Deflection Method / Analysis of Continuous Beams / Analysis of Frames without Sidesway. Apago PDF Enhancer. 5 Problems for Solution 145 Reference Further Reading 146 146 6 The Matrix Stiffness Method-Part 2: Coordinate Transformation 147 6. Identify all kinematic degrees of freedom for the given problem. Truss Deflection 67. The approach taken is mathematical in nature with a strong focus on the. Method • Aims – Determine the slope and deflection by using Moment Area Method • Expected Outcomes : – Able to analyze determinate beam – deflection and slope by Moment Area Method. Slope Deflection Method Equation Structural Ysis. solutions to industry concerns such as slope-to-drain, moisture resistance, high compressive strength, dimensional stability, and the ability to mechanically fasten the roofing membrane to the insulation. The class is an introduction to the basic tools of structural analysis and design. Develop an organized approach to solving structural analysis and design problems. The solution of slope-deflection equations along with equilibrium equations gives the values of unknown rotations of the joints. So that's our answer.