AR361 - Structural Systems In Architecture III
Lecture 7: Deflections of Beams
& Frames
Res.Asst. Erkan DURMAZGEZER
Department of Civil Engineering
Izmir, TURKEY
erkan.durmazgezer@gediz.edu.tr
DECEMBER 1, 2015
OBJECTIVES
Introduce double integration method and moment - area method to compute deflections & slopes of elastic
beams
Learn the double integration method based on the differential equation of the elastic curve, which relates the
curvature to M / EI along the members’s longitudinal axis.
Learn the moment area method (M / EI) to compute the slope and deflection at any point along the member axis.
INTRODUCTION
When a structure is loaded, its stressed elements deform. In a truss, bars in tension elongate and bars in compression shorten.
As these deformations occur, structure changes shape and points on the structure displace.
In this chapter, we will consider several methods of computing deflections and slopes at points along the axis of beams.
The Elastic Curve
Before the slope or the displacement at a point on a beam (or shaft) is determined, it is often helpful to sketch the deflected
shape of the beam when it is loaded, in order to “visualize” any computed results and thereby partially check these results. The
deflection curve of the longitudinal axis that passes through the centroid of each cross-sectional area of a beam is called the
elastic curve.
INTRODUCTION
If the elastic curve for a beam seems difficult to establish, it is suggested that the moment diagram for the beam be drawn first.
A positive internal moment tends to bend the beam concave upward,
A negative moment tends to bend the beam concave downward.
At a point of zero moment, the curve changes from concave up to concave down is called inflection point.
MOMENT – CURVATURE RELATIONSHIP
The following analysis, here and in the next section, will require the use of three coordinates. The x axis extends
positive to the right, along the initially straight longitudinal axis of the beam. The ʋ axis extends positive upward
from the x axis. It measures the displacement of the elastic curve.
DOUBLE INTEGRATION METHOD
Once moment is expressed as a function of x, then successive integrations of elastic curve equation will yield the
beam’s slope θ = dʋ / dx,
and the equation of the elastic curve ʋ = ʃʃ (M/EI) dx
For each intergration, it is necessary to introduce a constant of integration, and then solve for the constants to obtain a
unique solution for a particular problem.
(Recall if the loading on a beam is discontinuous, it consists of a series of several distributed and concentrated loads. Then
several functions must be written for the internal moment, each valid within the region between the discontinuities.)
Sign Convention: Positive deflection ʋ is upward, the positive slope angle θ will be
measured clockwise from the x (positive to the right) axis.
Since the slope is very small θ = tan θ = dʋ / dx
DOUBLE INTEGRATION METHOD
Boundary Conditions: The constants of integration are determined by evaluating the functions for slope or displacement at a
particular point on the beam where the value of the functions known. These values are called boundary conditions. For
example, if the beam is supported by a roller or pin, then it is required that displacement is zero at these points. Also at a fixed
support the slope and displacement are both zero.
If a single x coordinate cannot be used to express the equation for the beam’s slope or the elastic curve, then continuity
conditions must be used to evaluate some of the integration constants. Consider the beam in Figure. Here the and coordinates
x1 and x2 are valid only within the regions AB and BC, respectively. Once the functions for the slope and deflection are
obtained, they must give the same values for the slope and deflection at point B, x1 = x2 = ʋ so that the elastic curve is
physically continuous. Expressed mathematically, this requires θ1(a) = θ2(a) and ʋ1(a) = ʋ2(a).
These equations can be used to determine two constants of integration.
MOMENT – AREA THEOREMS
Moment – Area theorems provide a semigraphical technique for determining the slope of the elastic curve and its deflection
due to bending.
The moment-area method is based on two theorems, one used to determine the slope and the other to determine the displacement
at a point on the elastic curve.
Theorem 1
Consider the simply supported beam with its associated elastic curve, shown in Figure. A differential segment dx of the beam is
isolated. Here the beam’s internal moment M deforms the element such that the tangents to the elastic curve at each side of the
element intersect at an angle dθ.
MOMENT – AREA THEOREMS
Moment – Area theorems provide a semigraphical technique for determining the slope of the elastic curve and its deflection
due to bending.
The moment-area method is based on two theorems, one used to determine the slope and the other to determine the displacement
at a point on the elastic curve.
Theorem 1
Integrating from a point on the elastic curve to point B, we have
This equation forms the basis for the first moment area theorem.
The angle between the tangents at any two points on the elastic curve equals the area under the M/EI diagram between
these two points.
The notation θB/A is referred to as the angle of the tangent at B measured with respect to the tangent at A.
MOMENT – AREA THEOREMS
Moment – Area theorems provide a semigraphical technique for determining the slope of the elastic curve and its deflection
due to bending.
The moment-area method is based on two theorems, one used to determine the slope and the other to determine the displacement
at a point on the elastic curve.
Theorem 2
The second moment – area theorem is based on the relative deviation of tangents to the elastic curve. The vertical deviation of the
tangent at A with respect to the tangent at B can be found by integration:
Here
is the distance from the vertical axis through A to the centroid of the area between A and B.
MOMENT – AREA THEOREMS
Moment – Area theorems provide a semigraphical technique for determining the slope of the elastic curve and its deflection
due to bending.
The moment-area method is based on two theorems, one used to determine the slope and the other to determine the displacement
at a point on the elastic curve.
Theorem 2
The vertical distance between the tangent at a point (A) on the elastic curve and the tangent extended from another point
(B) equals the moment of the area under the M/EI diagram between these two points (A and B). This moment is calculated
about the point A (the point on the elastic curve) where the vertical deviation tA/B is to be determined.
MOMENT – AREA THEOREMS
It is important to realize that the moment-area theorems can only be used to determine the angles or deviations between two
tangents on the beam’s elastic curve. In general, they do not give a direct solution for the slope or displacement at a point on the
beam. These unknowns must first be related to the angles or vertical deviations of tangents at points on the elastic curve. Usually
the tangents at the supports are drawn in this regard since these points do not undergo displacement and/or have zero slope.
Specific cases for establishing these geometric relationships are given in the example problems.