...9 – Beam Deflection Experiment Report 1) L(mm) L3 680 314432000 700 343000000 800 512000000 900 729000000 1000 1E+09 1100 1.331E+09 1200 1.728E+09 This graph shows a strong linear relationship the deflection and L3. The black line on the graph shows a trend line found from the regression of the data, it lines up very well with the experimental data showing it is fairly accurate. The graph shows y to be proportional to L3 (y ∝ L3), hence y=K L3 where K is a constant found to be 1.08 x10-9. 2) D 1/d3 y (mm) 6 0.00463 1.59 7.92 0.002013 0.7 10 0.001 0.35 This graph shows an extremely strong linear relationship, however this may be due to a lack of data as the required materials for a more concise test were not available. The graph shows deflection to be inversely proportional to d3 (y ∝ 1/d3). Hence y=K/d3 where K is 341.33 as found from the gradient of the graph. 3) y(mm) L b d l^3 d^3 W WL3/4bd3 Beam 1 1.72 680 25 6 314432000 216 27.468 399852.6933 1.88 700 25 6 343000000 216 27.468 436181.6667 2.83 800 25 6 512000000 216 27.468 651093.3333 3.98 900 25 6 729000000 216 27.468 927045 5.44 1000 25 6 1E+09 216 27.468 1271666.667 7.2 1100 25 6 1.331E+09 216 27.468 1692588.333 9.17 1200 25 6 1.728E+09 216 27.468 2197440 Beam 2 1.76 900 25.32 7.92 729000000 496.7931 27.468 397974.5768 Beam 3 0.87 900 25 10 729000000 1000 27.468 200241.72 This graph shows the relationship between Y, the deflection of the...
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...ME Analysis Project #2 Christian Seymour Due Date: December 12, 2013 Abstract: The diameter for my shaft design was hand calculated to be 0.8756” for a factor of safety or 2. This factor of safety was expected to be the outcome in FEA SolidWorks and it was not. The reason for this is simply SolidWorks is not the best tool to use for complex designs and loads as ours was. It worked great for the simple beam deflection in the last project. Many times hand calculations are the long way of doing things. In this case, it took much longer to figure out the correct way of adding fixtures, load, and connections in SolidWorks. Many of the material properties in SolidWorks are also not the same as they are in our book. Simply said, choosing which method to use is the hardest decision to make. Sometimes FEA analysis is better, and in some cases, hand calculations are the easiest. Background: We have been given the task to design a shaft to specific guidelines using inch size pillow blocks. We will perform appropriate calculations using modeling techniques and verify the design in SolidWorks FEA program. The problem is 7-5 in the book and is shown below: The problem statement: A geared industrial roll shown in the figure is driven at 300 rev/ min by force F acting on a 3-in-diamter pitch circle as shown. The roll exerts a normal force of 30 lbf/in of roll length on the material being pulled through. The material passes under the roll. Design a shaft for the situation...
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...Experiment 7: Deflection of beams (Effect of beam length and width) 1. OBJECTIVE The objective of this laboratory experiment is to find the relationship between the deflection (y) at the centre of a simply supported beam and the span, width. 2. MATERIALS - APPARATUS Steel Beams, Deflection measuring device, 500g weight 3. INTRODUCTORY INFORMATION The deflection of a beam, y, will depend on many factors such as: - • The applied load F (F=m•g). • The span L. • The width of the beam b, and its thickness h. Other factors such as position, method of loading, the material of which the beam is made will also influence the deflection. If we wish to find the relationship between y and one of the possible variables it is necessary to keep all the other possible variables constant throughout the experiment. 1. Length calculation In this experiment the same beam is used throughout and the centrally applied point load is kept constant. Thus keeping all possible variables other than the deflection y and the span L constant we may investigate the relationship between y and L. Let y[pic]Ln where n is to be found Then y = k•Ln where k is a constant Taking logarithms: log y = n log L + log k which is in the straight line form (y = mx + C). Thus plotting logy against log L will give a straight-line graph of slope “n” and “k” may be determined. 2. Width calculation In this experiment beams of the same material but of different width are...
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...Answer all questions. 1. When the deflections of a statically determinant beam are calculated using singularity functions, it is necessary to know the values of 2 boundary conditions. For each of the beams below, what are the boundary conditions? 2. Calculate the reactions and draw the SFD and BMD for the cantilever beams below. Using singularity functions, calculate the deflection at 2 metres, the deflection at the tip, and sketch the deflected shape. The cross section of the beam is 300 mm deep by 200 mm wide, and it is made of concrete with a Young’s modulus of 30,000 MPa. 4. Using singularity functions, derive (in terms of P) the equation for the upwards deflection at midspan of the beam below. EI = 10 x 106 Nm2 5. Determine the location and value of the maximum deflection for the beam below. How far from the centre is the point of maximum deflection (expressed as a percentage of the span length)? EI = 10 x 106 Nm2 6. Calculate the reactions and draw the shear force and bending moment diagrams for the beam below. EI = 10 x 106 Nm2 (Note that this has four reactions, so the 3 equations of equilibrium do not give sufficient information to solve the problem – it is statically indeterminate. The answer requires you to use the results from questions 4 and 5). 7. For the beam below, use singularity functions to determine an equation for the deflected shape, expressed in terms of x (measured from the left) and EI. If the beam is made of timber (E = 10,000 MPa)...
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...emphasizes on Beams for the construction of a residential apartment, which is closely interrelated with columns and slabs. Controls: 1) Configurations of beam span. (Width, length, shape, bracing) 2) Material particularities (steel, bar spacing, concrete, admixtures) 3) Known/ Estimated Loads on beam Pattern of the geometry could be modified at any stage to optimize the objective. Along with the choice of materials; the quantities can make substantial difference in efficiency. States/Outputs: 1) Vertical Deflections 2) Internal Stress/Strains 3) Durability 4) Fire-safety-durations (Fibre-Reinforced Polymer) State variables are correlated to Control; control variables are primarily chosen to satisfy the conditions of States. Minimum cover, trial depth etc. are some early estimates which are altered overtime to meet certain standards. Model: 1) Key modelling features include 2 prime equations: i) Deflection equation- δmax = 5*ω*l4/384*E*I [Where δmax- maximum sustainable deflection, ω- uniform load, l- length of the beam, E- Young’s modulus, I- second moment of inertia] ii) Strength equation- R = (*Rn [Where R- load (dead or live; force, moment or stress),Rn- nominal strength (design strength), (- reduction factor] Deflection equations are derived according to the nature of load and type of beams. In this case a uniform load applied to fixed beam was assumed...
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...LINE FOR DEFLECTION 2 1.1. INTORDUCTION 2 1.2. OBJECTIVES 2 1.3. EQUIPMENT NEEDED 2 1.4. PROCEDURE 3 1.5. RESULTS 4 1.6. ANALYSIS OF THE RESULTS 4 1.7. DISCUSSION 6 1.8. CONCLUSION 7 1.9. REFERENCES 7 LIST OF FIGURES and CHARTS Figure 1: Force diagram for influence line of deflection Figure 2: Set-up for influence line of deflection Chart 1: Deflection curves (0 N, 5 N, 10 N) Chart 2: Deflection curves (15 N, 20 N, 25 N) EXPERIMENT 1: INFLUENCE LINE FOR DEFLECTION 1.1. INTORDUCTION Structural members subjected to a system of stationary loads are easily to analysis and determine completely the maximum design loads, deflection and rotation at supports of the member. However, in some members, such as bridge girders and crane gantry, which are subjected to moving loads, it is not easy to analyse since the maximum design loads and deflections depends on the position of the applied load(s) with time. An analytical procedure which deals specifically with the determination of the location of the moving load that will produce the design loads of the highest magnitude in the members was reached by using influence lines. Influence line can be defined as a graphical representation of the variation of internal member force or deflection in a fixed member section due to a unit load moving along or traversing a member of a given structural. It may also be defined as a response function of support reaction, axial force, shear force, bending moment or deflection. A deflection...
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...width, height and deflection of a beam, placed on two bear affected by a concentrated load at the center. 2. To ascertain the coefficient of elasticity for steel, brass, aluminum and wood. Theory The stress-strain behavior of brittle materials (e.g. ceramic, low toughness composite material) is not usually ascertained by tensile tests as outline in Exp. 1. A more suitable transverse bending test is most frequently employed, in which a rod specimen either a circular or rectangular cross section is bent until fracture using a three- or four-point loading technique. The assessments are conducted according to ASTM Standard C 1161, “Standard Test Method for Flexural Strength of Advanced Ceramics at Ambient Temperature.” In this module, the apparatus has been design to enable students to carry out experiments on simply supported and cantilever beams in order to investigate:(a) the relationship between the deflections and the applied loads (b) the effect of variations in 1ength and cross sectional i.e. deflection per unit load. Simply supported beam with central point load For this arrangement, it can be shown that the deflection under the load i.e. maximum deflection Wl 3 ∆= 48 EI 15 where I = bd 3 12 ∆ l3 = W 4Ebd 3 ∴beam compliance Cantilever beam with end point load For this arrangement, it can be shown that the central deflection relative to the supports, i.e. maximum deflection between the supports:Wl 3 ∆= 3EI bd 3 where I = 12 ∆ 4l 3 ∴beam compliance = W Ebd...
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...1 Chapter 4 Shear Forces and Bending Moments 4.1 Introduction Consider a beam subjected to transverse loads as shown in figure, the deflections occur in the plane same as the loading plane, is called the plane of bending. In this chapter we discuss shear forces and bending moments in beams related to the loads. 4.2 Types of Beams, Loads, and Reactions Type of beams a. simply supported beam (simple beam) b. cantilever beam (fixed end beam) c. beam with an overhang 2 Type of loads a. concentrated load (single force) b. distributed load (measured by their intensity) : uniformly distributed load (uniform load) linearly varying load c. couple Reactions consider the loaded beam in figure equation of equilibrium in horizontal direction Fx = 0 HA - P1 cos = 0 HA = P1 cos MB = 0 - RA L + (P1 sin ) (L - a) + P2 (L - b) + q c2 / 2 = 0 (P1 sin ) (L - a) P2 (L - b) q c2 RA = CCCCCCC + CCCC + CC L L 2 L (P1 sin ) a P2 b q c2 RB = CCCCC + CC + CC L L 2 L for the cantilever beam Fx = 0 HA = 5 P3 / 13 12 P3 (q1 + q2) b Fy = 0 RA = CC + CCCCC 13 2 3 12 P3 q1b q1 b MA = 0 MA = CC + CC (L – 2b/3) + CC (L – b/3) 13 2 2 for the overhanging beam MB = 0 - RA L + P4 (L– a) + M1 = 0 MA = 0 - P4 a + RB L + M1 = 0 P4 (L– a) + M1 P4 a - M1 RA = CCCCCC RB = CCCC L L 4.3 Shear Forces and Bending Moments Consider a cantilever beam with a concentrated load P applied at the end A, at the cross section mn, the shear force and bending moment are found ...
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...mraChapter 9 9.1 Introduction Deflections of Beams in this chapter, we describe methods for determining the equation of the deflection curve of beams and finding deflection and slope at specific points along the axis of the beam 9.2 Differential Equations of the Deflection Curve consider a cantilever beam with a concentrated load acting upward at the free end the deflection in the y v is the displacement direction of the axis the angle of rotation (also called slope) is the angle between the x axis and the tangent to the deflection curve point m1 is located at distance x point m2 is located at distance x + dx slope at slope at m1 m2 is is +d denote O' the center of curvature and the radius of curvature, then d = ds is and the curvature 1 = 1 C = d C ds the sign convention is pictured in figure slope of the deflection curve dv C dx for = tan or ds j dx d C dx d C = dx = cos j 1 and d 2v CC dx2 = dv tan-1 C dx tan dv C dx j , then small = 1 C = 1 C = = if the materials of the beam is linear elastic = 1 C = M C EI [chapter 5] then the differential equation of the deflection curve is obtained d C dx d2v = CC dx2 = M C EI and v dV CC dx d 4v CC dx4 = -q q -C EI it can be integrated to find ∵ dM CC dx d 3v CC dx3 = V then V = C EI = 2 sign conventions for M, V and q are shown the above equations can be written in a simple form EIv" = M EIv"' = V EIv"" = -q this equations are valid only when...
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...Laboratory 2: Build a Beam for 3-Point Bending Abstract: This laboratory report contains the design process of a simple foam I-beam. An analysis of the beam is conducted after the application of a 3-point bending from an ATS machine. This beam experienced a linear stiffness of 262 lbf./in. at an ultimate strength of 53.9 lbf. and deflection of 0.351 in. to which it immediately failed. Objective: This laboratory required the design and construction of a beam that spans 30 inches in order to determine the maximum strength and carry load at mid-span. The beam was created exclusively using a 10 by 40 inches piece of foam board of thickness 3/16 inches, 4 hot glue sticks, and Exacto knives. Design Rationale: From the materials mentioned, an I-beam was constructed. The cuts necessary to produce the I-beam are shown in Figured 1. 32 in. 1 in. 5 in. 5 in. Figure 1: Foam Beam Cut Schematic. When deciding how to cut foam board, the thought behind it was to target the areas of the beam that would experience the most applied load force. The final design consisted of five 1-inch units hot glued together at three equally spaced sections between two 32-inch long boards. The top and bottom sections 1 were supposed to keep the board from failing at its ends and splitting apart and the middle section was to keep the board from splitting in the middle and dipping inward. The final design is shown in Figure 2. Top View Side View Front View Figure 2: Finished I-Beam Views The focuses...
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...Module 10 Compression Members Version 2 CE IIT, Kharagpur Lesson 27 Slender Columns Version 2 CE IIT, Kharagpur Instructional Objectives: At the end of this lesson, the student should be able to: • • • • • • • • define a slender column, give three reasons for its increasing importance and popularity, explain the behaviour of slender columns loaded concentrically, explain the behaviour of braced and unbraced single column or a part of rigid frame, bent in single or double curvatures, roles and importance of additional moments due to P- Δ effect and moments due to minimum eccentricities in slender columns, identify a column if sway or nonsway type, understand the additional moment method for the design of slender columns, apply the equations or use the appropriate tables or charts of SP-16 for the complete design of slender columns as recommended by IS 456. 11.27.1 Introduction Slender and short are the two types of columns classified on the basis of slenderness ratios as mentioned in sec.10.21.5 of Lesson 21. Columns having both lex/D and ley/b less than twelve are designated as short and otherwise, they are slender, where lex and ley are the effective lengths with respect to major and minor axes, respectively; and D and b are the depth and width of rectangular columns, respectively. Short columns are frequently used in concrete structures, the design of such columns has been explained in Lessons 22 to 26, loaded concentrically or eccentrically about one or both...
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...Mathematics – I & ENG Mathematics II: 2D & 3D Coordinate Geometry; Differential Calculus; Infinite Series; Matrices,Ordinary differential equations of first and second order; Laplace Transforms; Vector Calculus • Elements of Mechanical Engineering & Elements of Civil Engineering Principle of virtual work,Rectilinear & curvilinear translation; Rotation of a rigid body about a fixed axis; Plane motion of a rigid body,Classification of force systems; Principle of physical independence of forces, Principle of superposition of forces, Principle of transmissibility of forces; Equivalent force – couple system; Resolution of forces, composition of forces; Types of supports, statically determinate beams, Numerical problems on support reactions for statically determinate beams and analysis of simple trusses ,Friction. • Engineering Physics Interference, diffraction and polarization of light; Nuclear fission, fusion, particle accelerators; Wave Particle Duality • Engineering Chemistry Physical Chemistry: Atoms, molecules and solids; phase equilibria; Galvanic & Fuel cells • Organic Chemistry: Types of reactions and reaction mechanisms; Concept of armaticity Computer Concepts & C Programming Introduction to digital computers; problem solving using computers; Programming in Fortran 77: Constants, variables,expressions, statements, control statements, arrays, functions, concept of files and file operations. • Computer Aided Engineering...
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...slabs stating the limits of ly /lx ratios for one and two-way slabs, explain the share of loads by the supporting beams of one- and two-way slabs when subjected to uniformly distributed vertical loads, explain the roles of the total depth in resisting the bending moments, shear force and in controlling the deflection, state the variation of design shear strength of concrete in slabs of different depths with identical percentage of steel reinforcement, assume the depth of slab required for the control of deflection for different support conditions, determine the positive and negative bending moments and shear force, determine the amount of reinforcing bars along the longer span, state the maximum diameter of a bar that can be used in a particular slab of given depth, decide the maximum spacing of reinforcing bars along two directions of one-way slab, design one-way slab applying the design principles and following the stipulated guidelines of IS 456, draw the detailing of reinforcing bars of one-way slabs after the design. Version 2 CE IIT, Kharagpur 8.18.1 Introduction Version 2 CE IIT, Kharagpur Slabs, used in floors and roofs of buildings mostly integrated with the supporting beams, carry the distributed loads primarily by bending. It has been mentioned in sec. 5.10.1 of Lesson 10 that a part of the integrated slab is considered as flange of T- or L-beams because of monolithic construction. However, the remaining part of the slab needs design considerations. These...
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...Results and Description For the first part of the experiment, the results were recorded as a table of deflection measurements from each beam resulting from various loads. This table can be seen in Table 1, in Tables and Figures. The data from this table was then plotted using MATLAB. After plotting, trendlines were drawn to model the data. The graph of the trendlines for each beam, as well as their respective equations can be seen in Figure 1, in Figures and Tables. The best representation of the beam’s strength is its Young’s Modulus, or Modulus of Elasticity, which models an object’s resistance to strain under stress. The higher the value of a beam’s Modulus of Elasticity, the stronger it will be. Using the data described above, the Modulus of Elasticity of each beam can be calculated using Equation 1. Table 2, in Tables and Attachments, shows the Modulus of Elasticity of each of the four beams analyzed. For the second part of the experiment, the results of the Finite Element Analysis yielded displacement values in the same way as in the material analysis in part 1 of the experiment. From these displacement values, the same calculations can be made, and non-experimental values for the Modulus of Elasticity of each material result...
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...Structural Analysis III Chapter 3 – Characteristics of Structures Chapter 3 - Characteristics of Structures 3.1 Introduction ......................................................................................................... 2 3.1.1 Background .................................................................................................... 2 3.2 Basic Statical Determinacy ................................................................................. 5 3.2.1 Introduction.................................................................................................... 5 3.2.2 Plane Beams and Frames ............................................................................... 6 3.2.3 Plane Trusses ............................................................................................... 15 3.3 Stability ............................................................................................................... 20 3.3.1 Introduction.................................................................................................. 20 3.3.2 Exceptions to Basic Rule ............................................................................. 21 3.3.3 Examples...................................................................................................... 23 3.4 Further Statical Determinacy .......................................................................... 25 3.4.1 Internal and External Determinacy ...........................................
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