Change in slope and tangential deviation between points A and B

Moment-Area Method

Beam and moment curve

M/EI curve between points A and B

Moment Area Theorems
•The change in slope between any two points on a smooth continuous elastic curve is equal to the area under the M/EI curve between these points •The tangential deviation at a point B on a smooth continuous elastic curve from the tangent line drawn to the elastic curve at the second point A is equal to the moment about B of the area under the M/EI curve between these two points.

Moment-Area Method

Horizontal, therefore the vertical distance between tangent line and elastic curve are displacements

Cantilever, point of tangency at fixed support

Moment-Area Method

Symmetric members with symmetric loading, point of tangency at intersection of axis of symmetry and elastic curve

Moment-Area Method

Point of tangency at left end of member AB
t BA L tan θ A = θ A in radians tan θ A =

θA=

t BA L

Caution 2.The theorem is applicable for continuous elastic curve 3.Presence of hinge breaks continuity of elastic curve 4.If hinge is present on beam or a frame – then work on either side of hinge (left or right side)

P 9.6 (3rd Edition)

Derive the equations for slope and deflection for the beam shown. Determine the slope at each support and value of deflection at mid span. Hint: Take advantage of symmetry; slope is zero at midspan.

P 9.13

Compute the slope at support A and the deflection at point B. Treat the rocker at D as a roller. Express the answers in terms of EI.

P 9.15 Determine the slop and deflection of point C...

...
FACULTY OF ENGINBERING AND BUILT ENVIRONMENT
BEng (Hons) Civil Engineering
Structure I
Deflection
Contents:
Introduction 3
Objectives 3
Apparatus 4
Procedure 4
Results 4
Discussion 7
Conclusion 7
References 8
Introduction:
The deflections of a beam are an engineering concern as they can create
an unstable structure if they are large. People don’t want to work in a
building in which the floor beams deflect an excessive amount, even
though it may be in no danger of failing. Consequently, limits are often
placed upon the allowable deflections of a beam, as well as upon the
stresses.
When loads are applied to a beam their originally straight axes become
curved. Displacements from the initial axes are called bending or flexural
deflections. The amount of flexural deflection in a beam is related to the
beams area moment of inertia (I), the single applied concentrated load (P),
length of the beam (L), the modulus of elasticity (E), and the position of the
applied load on the beam. The amount of deflection due to a single
concentrated load P, is given by:
Objective:
Is to find the relationship between the deflection at the center of a simply supported beam and the span, width.
Apparatus:
Frame with Movable Knife Edge Supports.
Steel rectangular beam.
Weights....

...The Report of Deflections of Beams and Cantilevers
Summary:
There are four parts in this big experiment, including deflection of a cantilever, deflection of a simply supported beam, the shape of a deflected beam, and circular bending. In these four parts, a same set of laboratory instrument and apparatus is used, concluding a bracket, a moveable digital dial test indicator, U-section channel, moveable knife-edge, and three material beams: brass, aluminum, and steel. The experiment methods, and fixed point to the beam are the differences between these four small experiments. The aim of this experiment is to improve the ability to use the precision engineering components like moveable digital dial test indicator, also understand the formula: Deflection= WL＾3/3EI.
To explain this formula: W is load, its unit is N, L is distance from support to position of loading (m), E is Young’s modulus for cantilever material, and its unit is Nm＾-2, I is the second moment of area of the cantilever, its unit is m＾4. In addition, the experiment safety is very important.
Objective:
(1) Operation techniques. In this experiment, measuring data is very important, because of comparing the actual deflection to theoretical deflection. Every step of this experiment should be precise. To obtain the correct data, you must be sure that the all components are secure and fastenings are sufficiently tight. Also...

...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.
3.1 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 yLn 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.
3.2 Width calculation
In this...

...on measuring and comparing results on deflection on a beam.
Intro: This assignment consists of predictions to theories on measuring and comparing results on deflection on a beam.
Beam Defection Experiment
1) This graph and its table below showed the resultant forces which were achieved when the test on the relationship between deflection (Y) and the spacing achieved (L3) using a load of my choice which was 2.5kg (constant). The scientific instruments used in the lab for this experiment were a digital gauge to measure the final beam deflection and also a hanger to freelance the weight. Beam depth (d) of 0.0063 m. A prediction was made that this beam would indeed prove to be one with a high deflection point due to its depth. Gradient is identical to deflection.
This graph and its table below showed the resultant forces which were achieved when the test on the relationship between deflection (Y) and the spacing achieved (L3) using a load of my choice which was 2.5kg (constant). The scientific instruments used in the lab for this experiment were a digital gauge to measure the final beam deflection and also a hanger to freelance the weight. Beam depth (d) of 0.0063 m. A prediction was made that this beam would indeed prove to be one with a high deflection point due to its depth. Gradient is identical to deflection.
The slope...

...1.0 BACKGROUND OF STUDY
The deflections of a beam are an engineering concern as they can create an unstable structure if they are large. People don’t want to work in a building in which the floor beams deflect an excessive amount, even though it may be in no danger of failing. Consequently, limits are often placed upon the allowable deflections of a beam, as well as upon the stresses.
When loads are applied to a beam their originally straight axes become curved. Displacements from the initial axes are called bending or flexural deflections. The amount of flexural deflection in a beam is related to the beams area moment of inertia I, the single applied concentrated load P, length of the beam l, the modulus of elasticity E, and the position of the applied load on the beam. The amount of deflection due to a single concentrated load P, is given by δ=PL3kEI whereby k is a constant based on the position of the load, and on the end conditions of the beam.
The bending stress at any location of a beam section is determined by the flexure formula, σ=MyI whereby M is the moment at the section, y is the distance from the neutral axis to the point of interest and I is the moment of inertia.
2.0 OBJECTIVES
2.1 EXPERIMENT 1
To investigate, for a simply supported beam carrying a central point load,
a) The relationship between the deflection and the applied loads
b) The effect of...

...Slope deflection method
From Wikipedia, the free encyclopedia
The slope deflection method is a structural analysis method for beams and frames introduced in 1914 by George A. Maney.[1] The slope deflection method was widely used for more than a decade until the moment distribution method was developed.
Contents
[hide]
1 Introduction
2 Slope deflection equations
2.1 Derivation of slope deflection equations
3 Equilibrium conditions
3.1 Joint equilibrium
3.2 Shear equilibrium
4 Example
4.1 Degrees of freedom
4.2 Fixed end moments
4.3 Slope deflection equations
4.4 Joint equilibrium equations
4.5 Rotation angles
4.6 Member end moments
5 See also
6 Notes
7 References
Introduction[edit]
By forming slope deflection equations and applying joint and shear equilibrium conditions, the rotation angles (or the slope angles) are calculated. Substituting them back into the slope deflection equations, member end moments are readily determined.
Slope deflection equations[edit]
The slope deflection equations can also be written using the stiffness factor and the chord rotation :
Derivation of slope deflection equations[edit]
When a simple beam of length and flexural rigidity is loaded at each end with clockwise moments and , member end rotations occur in the same direction. These rotation angles can be calculated using the unit dummy...

...applying different masses at a specified length across the beam and then measuring deflection. The measuring device was set a specified distance from the clamped end. The following procedure was employed for both the steel and aluminum beam. The second part of the experiment required placing a single known mass at various lengths across the supported beam and then measuring the resulting deflection. This method was only completed for the steel beam. Thedeflections from both parts of the experiment were then averaged independently to ascertain final conclusions. The first part of the experiment resulted in a much greater deflection for the aluminum beam, with its greatest deflection spanning to an average of 2.8 mm. Moreover, the deflection for the steel beam was much less, concluding that steel has a larger structural stiffness. In fact, the structural stiffness that was found for steel was 3992 N/m, compared to aluminum, which was 1645 N/m. In addition, the theoretical values of structural stiffness for steel and aluminum were calculated to be 1767.9 N/m and 5160.7 N/m, respectively. There was a large error between the theoretical and experimental values for steel, close to 29%. This could have been due to human error, or a defective beam. The second part of the experiment resulted in validating the fact that the values of deflection are proportional to length cubed. It was...

...Laboratory Three: Parallam Beam Deflection
Lab Group - 1st Mondays, Late: Jesse Bertrand, Ryan Carmichael, Anne Krikorian, Noah Marks, Ann Murray Report by Ryan Carmichael and Anne Krikorian
E6 Laboratory Report – Submitted 12 May 2008 Department of Engineering, Swarthmore College
Abstract:
In this laboratory, we determined six different values for the Elastic Flexural Modulus of a 4-by10 (100” x 3.50” x 9.46”) Parallam wood-composite test beam. To accomplish this, we loaded the beam at 1/3 span with 1200 psi in five load increments in both the upright (9.46 inch side vertical) and flat (9.46 inch side horizontal) orientations. We then used three different leastsquare methods (utilizing Matlab and Kaleidagraph) on the data for each orientation to fit the data, resulting in the following:
E: Upright Orientation Units Method One Method Two Method Three
E: Flat Orientation 10 ksi 103 ksi
3
0.981 ± 0.100 1.253 ± 0.198 1.065 ± 0.247
1.880 ± 0.046 2.080 ± 0.083 1.881 ± 0.106
1
Purpose:
The purpose of this lab is to determine the flexural elastic modulus of a Parallam woodcomposite beam by examining its behavior when simply supported and under flexural stress, and to analyze deflection data using different least-squares methods to fit theoretical deflection curves.
Theory:
In theory, a beam’s deflection can be mapped by the governing equation of beam flexure: EI d2y/dx2 = M(x), where E is the...