ASSIGNMENT 2 - DCC 20053

DCC 20053-MECHANICS OF CIVIL
ENGINEERING STRUCTURES

GROUP 10

GROUP MEMBER’S NAME & MATRIX NO: 1)NOR ATIKAH BINTI RAMAINA
(04DKA20F2018)

2)NURUL HUSNA BINTI RAZALI
(04DKA20F2031)

LECTURER’S NAME: PUAN NORHAIZAH BINTI AMBIAH
CLASS: DKA2A

TOPIC 6 : SLOPE AND DEFLECTION OF
BEAM DUE TO SYMMETRICAL BENDING

CONTENTS

❖ INTRODUCTION
❖DEFINITION
❖EXAMPLE QUESTION
❖CONCLUSION
❖REFERENCES

INTRODUCTION

• The slope deflection method is a structural analysis method for beams
and frames introduced in 1915 by George A.Maney. The slope
deflection method was widely used for more than a decade until the
moment distribution method was developed.

• The amount of deflection depends on:-
- the magnitude of the load and
- the stiffness of the member

• The deflection diagram of the longitudinal axis that passes through the
centroid of each cross- sectional area of the beam is called the Elastic
Curve.

• There are many methods to find out the slope and deflection at a
section in a loaded beam.

• The following two methods that used in this chapter is:

- Macaulay Method

- Moment Area Method

DEFINITION SLOPE AND DEFLECTION OF BEAM

• The slope of beam is the angle between deflected beam to the actual
beam at the same point.

• Deflection of beam is defined as the vertical displacement of a point
on a loaded beam.

MACAULAY METHOD

GUIDELINE TO USE MACAULAY METHOD:

1)Determine the reaction of forces moment.
2)Derive moment equation,M = EI 2

2 .

3)Derive slope equation,EI .
4)Derive deflection equation,EIY.

5)Determine the boundary condition to obtain the value of constant C₁

and C₂.

6)Substituting the constant of C₁ and C₂ in the slope equation EI


and deflection equation,EIY.

7)Determine the slope and deflection Y at any specific points given.


8)Applying the flexural rigidity,EI given in the question. If the value is not
given,state in the answer in EI form.

EXAMPLE QUESTION OF MACAULAY METHOD

1)A horizontal beam which is simply supported beam at its end,A and B, have
a uniform cross section and is 5m long. Concentrated load of 20 kN act 1m
from A. Determine the slope and deflection of beam at point C.

20 kN

HA 2m 2m B

VA
A 1m

VA : Σ MB =0 Mx
(VA × 5) – (20 × 4) = 0
VA = 16 kN B
2m
20 kN
HA

VA A 2m
1m x (m)

Moment equation: Mx = 16 (x) – 20 (x – 1)
Macaulay equation: EI 2 = 16 [ x ] – 20 [x – 1]

2

Macaulay equation: EI 2 = 16[ ]1 − 20 [x – 1]

2 1

Slope equation : EI = 16[ ]2 − 20[ − 1]2 + C₁ (equation 1)


22

Deflection equation: EIY = 16 [ ]3 - 20[ − 1]3 + C₁x + C₂ ( equation 2 )
66

When x = 0 m and y = 0, replace at deflection equation(2)
EIY = 16[ ]3 − 20 [ − 1]3 + C₁x + C₂

66
EI (0) = 16[0]3−20[0 − 1]3 + C₁ [0] + C₂

66
C₂ = 0
When x = 5 m and y = 0,replace at deflection equation(2)
EI [0] = 16[5]3 − 20[5 − 1]3 + C₁[5] + 0

66
C₁ = 24

When C₁ = - 24,replace at slope equation (1)

EI = 16[ ]2 −20[ − 1]2 −24(equation 3)


22

When C₁ = - 24 and C₂ = 0, replace at deflection equation (2)

EIY = 16[ ]3− 20[ − 1]3− 24[x] (equation 4)

66

When x = 3m at point C,replace at slope equation (3)

EI = 16[ ]2 − 20[ − 1]2− 24

22

EI = 16[3]2 − 20 3 − 1 2 − 24

22

= 8 radian


EI

When x = 3m at point C, replace at deflection equation (4)
EIY = 16[ ]3− 20[ − 1]3− 24[x] (equation 4)

66
EIY = 16[3]3− 20[3 − 1]3− 24[3]

66
Y = - 26.67 m

EI

EXAMPLE QUESTION OF AREA MOMENT METHOD

2)Get the slope and deflection at point of beam at the distance 8m from
point A, loaded as in figure 4. E = 5 × 106N/ 2,I = 9× 106 4.

5 kN

15kNm 2kN/m

AB

2m 2m 2m 2m 2m


Σ = 0 +

-5(2) + 15 + 15 − 2(2) 2 + 4 =0
2

= 1 kN





A X (m) Ax/EI
1/2 × 10 × 30 = 150 1/3 × 10 500/EI
DIAGRAM
1 -1/2 × 2× 15 = - 15 1/3 × 2 -10/EI
2
3 -1/3 × 6× 36= -72 1/4× 6 -108/EI
4 1/3× 4× 16 = 21.333 1/4 × 4
5 10×8 = 80 1/2 × 8 21.333/EI

320/E1

=723.333
EI

A/EI X(m) Ax/EI
1/2 × 8 × 16 = 64 1/3 × 8 170.667/EI
DIAGRAM
1 -1/3 × 4 × 16 = 21.333 1/4 × 4 -21.333/EI
2 1/3 × 2 × 4 = 2.667 1/4 × 2
3 10 × 6 = 60 1/2 × 6 1.333/EI
4 180/EI

=105.333 = 330.667
EI EI

A FB



F’
B’

FF’ =
8 10
FF’ = × 8 = 723.33× 8

10 EI × 10
FF’ = 578.664

EI
Diketahui = 330.667 dan = 1 −

EI

Maka pesongan pada titik C,
= 578.664 – 330.667

EI

= 247.997 m

EI

MOMENT AREA METHOD

• This is a method of determining the change in slope or the deflection
between two points on a beam. It is expressed as two theorems.

THEOREM I
• Change of slope,( )
• = area of bending moment diagram between of two points ÷ EI
• =

EI
• Unit : Radian

THEOREM II

• Change of deflection vertical,( )

• Y = area bending moment diagram which is between two points ×
distance of centroid ÷ EI

• Y =


• Unit :m

MOMENT AREA METHOD

GUIDELINE OF MOMENT AREA METHOD:

1)Get the reaction force and moment at support.
2)Sketch the bending moment diagram (BMD) which is started from left
to right beam.
3)Get the moment for each load from bending moment diagram(BMD).
4)Sketch diagram distance of deflection (Y) and slope ( ).
5)Get the value of tangent line (T) at right beam.
6)Get the value of tangent line at point which has load.(refer to
question).

7)Get the value of deflection (Y) which used a triangle diagram formula
for simply supported beam.

8)Get the value of slope ( ).

CONCLUSION

• Based of our knowledge,we will be able to learn about Macaulay
Method and Area Moment Method.

• We also be able to solve the questions by using those methods in
correct guidelines.

REFERENCES

❑https://pkb.cidos.edu.my/
❑https://youtu.be/cSzzTbA267I