Deflections - جامعة نزوى€¦ · Conjugate-Beam Method The conjugate-beam method was...

Deflections

Dr. Nasrellah H A 1

Deflection Diagrams and the Elastic Curve Deflections of structures can occur from various sources, such as loads, temperature, fabrication errors, or settlement. In design, deflections must be limited in order to provide integrity and stability . Furthermore, a structure must not vibrate or deflect severely in order to “appear” safe for its occupants. More important, though, deflections at specified points in a structure must be determined if one is to analyze statically indeterminate structures.

Dr. Nasrellah H A 2

Restrictions to slope or displacement that often occur at a support or a connection:

Dr. Nasrellah H A 3

Before the slope or displacement of a point on a beam or frame is determined, it is often helpful to sketch the deflected shape of the structure when it is loaded in order to partially check the results. This deflection diagram represents the elastic curve

Dr. Nasrellah H A 4

Dr. Nasrellah H A 5

Dr. Nasrellah H A 6

Elastic-Beam Theory (Double Integration Method): In this section we will develop two important differential equations that relate the internal moment in a beam to the displacement and slope of its elastic curve.

Dr. Nasrellah H A 7

Dr. Nasrellah H A 8

Dr. Nasrellah H A 9

Dr. Nasrellah H A 10

The cantilevered beam shown in Fig. below is subjected to a couple moment at its end. Determine the equation of the elastic curve. EI is constant.

Example 1:

Solution:


Example 2:

For the beam shown in Figures below. Determine the equation of the elastic curve. EI is constant. And find the maximum deflection.


Example 3:


Example 4:


Example 5: The beam in Fig. below is subjected to a load P at its end. Determine the displacement at C. EI is constant.



Moment-Area Theorems

The initial ideas for the two moment-area

Theorems were developed by Otto Mohr and later

stated formally by Charles E. Greene in 1873.

These theorems provide a semigraphical technique

for determining the slope of the elastic curve and

its deflection due to bending.


From the elastic theory : Since

and

or


Example 6:

For the beam shown in Figures below. find the maximum deflection. EI is constant.


Example 7:


Solution:


Example 7:


Solution:


Example 8:


Solution:


Example 9:


Solution:


Conjugate-Beam Method

The conjugate-beam method was developed by

H. Müller-Breslau in 1865. Essentially, it

requires the same amount of computation as the

moment-area theorems to determine a beam’s

slope or deflection; however, this method relies

only on the principles of statics, and hence

its application will be more familiar.


we can write these equations as follows:


The conjugate beam is “loaded” with the M/EI diagram derived from the load w on the real beam. From the above comparisons, we can state two theorems related to the conjugate beam, namely,


Theorem 1: The slope at a point in the real beam is

numerically equal to the shear at the corresponding

point in the conjugate beam.

Theorem 2: The displacement of a point in the real

beam is numerically equal to the moment at the

corresponding point in the conjugate beam.


Conjugate-Beam Supports:-

When drawing the conjugate beam it is important

that the shear and moment developed at the supports

of the conjugate beam account for the corresponding

slope and displacement of the real beam at its

supports,

Deflections - جامعة نزوى€¦ · Conjugate-Beam Method The conjugate-beam method was...

Documents

Transcript of Deflections - جامعة نزوى€¦ · Conjugate-Beam Method The conjugate-beam method was...