Strength of Materials - Lecture Notes / Mehmet Zor

1

23 Agust 2024

9. Deflection of Beams

(ELASTIC CURVE)

9.3 Moment Area Method

(tvids: 9..3)

Strength of Materials - Lecture Notes / Mehmet Zor

2

23 Agust 2024

y

x

ds

A

B

dθ

9.3 Moment Area Method

r

r

K

N

Elastic curve

Orginal beam axis

This method allows us to directly find the deflection or slope of a certain point and is more practical for singular loads.

tangent-K

tangentt-N

dθ

tangent-B

Static moment (Q) of area 𝐴′ with respect to the vertical axis passing through B

A

B

C

Bending moment diagram

We can perform the following operations according to the figures in left side:

dx

(Equation 9.2.1)

Angle between tangents of sections A and B

of the beam (or difference between slopes):

The length of the vertical line passing through B, between the tangents of A and B:

The area under the moment diagram between two sections of the beam, A and B.

A beam subjected to vertical loads

For the part of length dx :

tangent-A

(9.3.1.a)

(9.3.1.b)

dx

G

P

Figure 9.3.1.a

(b)

Figure 9.3.1.b

Deflection of Beams / Moment Area Method

Strength of Materials - Lecture Notes / Mehmet Zor

3

23 Agust 2024

Example 9.3.1

For the cantilever beam in the figure, find

a-) rotation angle value and b-) collapse values

​​of the free end B, according to the Moment Field Method.

Solution

From equ. (9.3.1.a):

From equ. (9.3.1.b) ..>>

b-)

a-)

Bending Moment Diagram

tangent-B

tangent-A

Figure 9.3.2

Figure 9.3.3

(a)

(b)

Figure 9.3.4

Deflection of Beams / Moment Area Method