University of Engineering & Technology
Peshawar, Pakistan
CE301: Structure Analysis II
Module 08:
Analysis of S.I Frames Using stiffness method
By:
Prof. Dr. Bashir Alam
Civil Engineering Department
UET , Peshawar
Topics to be Covered
• Introduction
• Prerequisites for using stiffness method
• Types of frames
• Step wise procedure of stiffness method for frame analysis
• Analysis of Nonsway frames Example 1,2,3
• Assignment 04 (a)
• Analysis of sway frames Example 1,2,3
• Assignment 04 (b)
Stiffness Method for Frames Analysis
 Introduction:
Frames are analyzed with stiffness method due to
•
To solve the problem in matrix notation, which is more systematic
•
To compute reactions at all the supports.
•
To compute internal resisting shear , axial & bending moment at any section
of the frame.
Stiffness Method for Frames Analysis
 Prerequisites for Analysis with stiffness method:
It is necessary that students must have strong background of
the following concepts before starting analysis with stiffness
or any other matrix method.
•
Enough concept of Matrix Algebra
•
Must be able to find the Kinematic Indeterminacy
•
Must know the formulas & concept of fixed end actions
Stiffness Method for Frames Analysis
 Types of Frames :
On the basis of lateral displacement frames are classified in to
following two types
• Frames with Sway
• Frames without Sway
Stiffness Method for Frames Analysis
 Sway:
In statically indeterminate structures the structures the frames
deflect laterally due the presence of lateral load or unsymmetrical
vertical loads or where the frames themselves are unsymmetrical.
• Causes of side sway:
 Unsymmetrical loading (eccentric loading)
 Different end conditions of the columns of the frame
 Non uniform sections of the members
 Horizontal loading on column of the frame
 Settlement of the supports of the frame
Stiffness Method for Frames Analysis
• Frames without side sway:
Stiffness Method for Frames Analysis
• Frames with side sway:
Stiffness Method for Frames Analysis
Part I
Analysis of Frames without side sway
Stiffness Method for Frames Analysis
 Step wise Solution Procedure using Stiffness method method:
The following steps must be followed while solving a structure using
Stiffness method.
• Step # 01: Make the structure kinametically determinate, by
restraining the joints i.e. select the redundant joint displacement.
Stiffness Method for Frames Analysis
Step # 02: Apply the actual external loads on the BKDS (Basic
kinametically determinate structure) and find the
actions at the locations of redundant joints ( compute
fixed end actions) this will generate ADL matrix.
Step # 03: Apply the redundant joint displacement on the BKDS (To
standardize the procedure, only a unit displacement
is applied in the +ve direction) this will generate
stiffness coefficient matrix.
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝑨𝑫
𝑫
𝑨𝑫𝑳
𝑺
𝑺 • 𝑫
𝟏
•
𝑨𝑫
𝑨𝑫𝑳
Step # 05: Compute the member end actions .
Stiffness Method for Frames Analysis
Problem 01: Analyze the given Frame using stiffness method.
B
2k/ft
C
10ft
Take EI = constant
A
12ft
K.I = 2 degree ( neglecting the axial effects )
So two redundant joint displacements should be chosen.
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D1
B
12ft
C
D2
10ft
A
Rotation at B & C is taken as redundant joint displacement.
𝐷
2∗1=
𝐷1
?
=
𝐷2
?
𝐴𝐷
2∗1=
0
𝐴𝐷1
=
0
𝐴𝐷2
Stiffness Method for Frames Analysis
• Restrain all the degrees of freedom to get the restrained structure.
C
B
12ft
10ft
A
Basic kinematic determinate structure ( BKDS) or
restrained structure
Stiffness Method for Frames Analysis
• Step # 02 : Restrained structure acted upon by the actual loads.
compute the values of actions in the restrained structure
corresponding to the redundant locations. This will
generate ADL matrix.( Fixed end actions)
12k
0ʹ k
0k
B
1
ADL1
= 24ʹ k
B
10ft
12k
0k
0ʹ k
12k
2k/ft
C
24ʹ k
12ft
𝐴𝐷𝐿1
A
2
𝐴𝐷𝐿 =
ADL2
12k
24ˊ 𝑘
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿2
=
24ˊ 𝑘
24
24
Stiffness Method for Frames Analysis
Step # 03 : Primary structure acted upon by a unit value of D &
computation of stiffness coefficients “ S” values in the
BKDS corresponding to the redundant joint displacement
locations.
•
1st a unit rotation is applied at location 1 & prevented at 2 as shown.
Compute the values of S11 and S21 .
•
Then apply a unit rotation at the redundant displacement location 2
and prevented at 1 as shown. Compute the values of S12 & S22 .
Stiffness Method for Frames Analysis
i.
1st a unit rotation is applied at location 1 & prevented at 2 (D1=1 &
D2=0)as shown. Compute the values of S11 and S21 .
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟒𝑬𝑰𝜽
𝑳
𝟒𝑬𝑰
𝟏𝟎
1
𝟒𝑬𝑰
𝟏𝟐
B
12ft
θ=1
θ=1
10ft
𝟔𝑬𝑰
𝟏𝟐𝟐
2
𝟐𝑬𝑰
𝟏𝟎
A
𝟔𝑬𝑰
𝟏𝟐𝟐
C
𝟐𝑬𝑰𝜽
𝑳
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
Step # 03 ( i ): Contd…
11
21
11
21
S11 = Action(sum of moments in this case) in the BKDS at redundant displacement
location 1 due to unit rotation at that location.
S21 = Moment in the BDS at redundant displacement location 2 due to a unit
rotation applied at location 1
Stiffness Method for Frames Analysis
ii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 ( D2=1 & D1=0 ) as shown.
Compute the values of S12 & S22 .
𝟐𝑬𝑰𝜽
𝑳
1
B
𝟐𝑬𝑰
𝟏𝟐
2
θ=1
12ft
𝟔𝑬𝑰𝜽
𝑳𝟐
10ft
A
𝟔𝑬𝑰
𝟏𝟐𝟐
C
𝟒𝑬𝑰𝜽
𝑳
𝟎. 𝟎𝟒𝟏𝟕
𝟒𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
Step # 03 : Contd…
12
22
12
22
S12 = Moment in the BKDS at redundant displacement location 1 due to
S22 =
unit rotation applied at redundant displacement location 2.
Moment in the BKDS at redundant displacement location 2 due to a
unit rotation applied at that location.
Stiffness Method for Frames Analysis
Step # 03 : Contd…
𝑆11
0.733𝐸𝐼
𝑆12
0.1667𝐸𝐼
𝑆21
0.1667𝐸𝐼
𝑆22
0.333𝐸𝐼
𝑆 =
S = 𝐸𝐼
𝑆11
𝑆21
𝑆12
𝑆22
0.733 0.1667
0.1667 0.333
Stiffness coefficient
matrix
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷1
𝐴𝐷𝐿1
𝑆11𝐷1
𝑆12𝐷2
𝐴𝐷2
𝐴𝐷𝐿2
𝑆21𝐷1
𝑆22𝐷2
𝐴𝐷1
𝐴𝐷𝐿1
=
𝐴𝐷2
𝐴𝐷𝐿2
𝐴𝐷
2∗1
𝑫
𝐴𝐷𝐿
𝑺
+
𝑆11
𝑆21
𝑆
2∗1
𝟏
•
𝑆12 𝐷1
𝑆22 𝐷2
𝑨𝑫
2∗2
𝑨𝑫𝑳
• 𝐷
2∗1
Stiffness Method for Frames Analysis
𝑆11
𝑆21
𝐷1
𝐷2
𝐷1
𝐷2
𝑆12
𝑆22
𝐴𝐷1
𝐴𝐷2
1 0.733 0.1667
𝐸𝐼 0.1667 0.333
𝐷1
𝐷2
55.40 1
99.741 𝐸𝐼
𝐴𝐷𝐿1
𝐴𝐷𝐿2
0
0
24
24
-ive sign shows that our
assumed redundant joint
displacement direction is
wrong
Stiffness Method for Frames Analysis
Step # 05: Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐴𝐷
Member AB:
AM4
AM6
B
A
AM3
AM5
AM2
AM1
AMn = is the member end action in the indeterminate structure at different location
specified with a number “n”. n shows the number of member end actions.
Stiffness Method for Frames Analysis
Compute AML values.
12k
AML4
AML6
B
0ʹ k
AML5
0k
B
1
10ft
2
3
4
AML3
A
AML1
AML2
5
A
z
i.
0ʹ k
12k
0k
6
Stiffness Method for Frames Analysis
b) Compute the AMD values.
•
1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
𝟔𝑬𝑰
𝟏𝟐𝟐
AMD41
AMD61
B
AMD51
𝟒𝑬𝑰
𝟏𝟎
1
𝟒𝑬𝑰
𝟏𝟐
B
AMD31
AMD11
AMD21
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft
θ=1
θ=1
10ft
A
2
𝟐𝑬𝑰
𝟏𝟎
A
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟐𝑬𝑰
𝟏𝟐
C
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
AMD42
AMD62
B
AMD52
𝟐𝑬𝑰𝜽
𝑳
1
B
𝟐𝑬𝑰
𝟏𝟐
2
θ=1
12ft
𝟔𝑬𝑰𝜽
𝑳𝟐
10ft
AMD32
A
AMD12
AMD22
A
𝟔𝑬𝑰
𝟏𝟐𝟐
C
𝟒𝑬𝑰𝜽
𝑳
𝟎. 𝟎𝟒𝟏𝟕
𝟒𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
So the AMD values are
𝐴𝑀𝐷
6∗2
=
𝐴𝑀𝐷11
𝐴𝑀𝐷21
𝐴𝑀𝐷31
𝐴𝑀𝐷41
𝐴𝑀𝐷51
𝐴𝑀𝐷61
𝐴𝑀𝐷12
𝐴𝑀𝐷22
𝐴𝑀𝐷32
𝐴𝑀𝐷42
𝐴𝑀𝐷52
𝐴𝑀𝐷62
=
0.0417
0.06
0.2
0.0417
0.06
0.4
0.0417
0
0
0.0417
0
0
Stiffness Method for Frames Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐴𝐷
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
12
0
0
12
0
0
0.0417
0.06
0.2
0.0417
0.06
0.4
0.0417
0
0
0.0417
0
0
55.40
99.714
13.8 k
3.3 k
11.1ˊ k
13.8 k
3.3 k
22.2ˊ k
Stiffness Method for Frames Analysis
Final Analyzed Member
13.3k
AM4
AM6
B
A
AM3
AM1
22.2ʹ k
AM5
AM2
B
11.1ʹ k
A
13.8k
3.3k
3.3k
Stiffness Method for Frames Analysis
Now Shear force and bending moment diagrams
13.3k
22.2ʹ k
B
11.1ʹ k
A
13.8k
3.3k
0
0.33
0
0.33
0
3.3k
SFD
11.1
0
BMD
22.22
Stiffness Method for Frames Analysis
Step # 05: Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐴𝐷
Member BC:
AM3
AM1
B
AM2
AM6
C
AM4
AM5
AMn = is the member end action in the indeterminate structure at different location
specified with a number “n”. n shows the number of member end actions.
Stiffness Method for Frames Analysis
i.
Compute AML values.
AML6
AML3
AML1
B
C
AML4
AML5
AML2
1
2
3
4
5
= 24ʹ k
B
2k/ft
C
24ʹ k
12ft
12k
12k
ADL2
6
Stiffness Method for Frames Analysis
b) Compute the AMD values.
•
1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
AMD31
B
𝟔𝑬𝑰
𝟏𝟐𝟐
AMD61
C
AMD41
AMD11
AMD21
𝟒𝑬𝑰
𝟏𝟎
1
𝟒𝑬𝑰
𝟏𝟐
B
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft
θ=1
θ=1
10ft
AMD51
2
𝟐𝑬𝑰
𝟏𝟎
A
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟐𝑬𝑰
𝟏𝟐
C
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
AMD32
B
AMD62
C
AMD42
AMD12
AMD22
1
𝟐𝑬𝑰𝜽
𝑳
B
𝟐𝑬𝑰
𝟏𝟐
2
θ=1
12ft
AMD52
𝟔𝑬𝑰
𝟏𝟐𝟐
10ft
A
𝟔𝑬𝑰
𝟏𝟐𝟐
C
𝟒𝑬𝑰
𝟏𝟐
𝟎. 𝟎𝟒𝟏𝟕
Stiffness Method for Frames Analysis
So the AMD values are
𝐴𝑀𝐷
6∗2
=
𝐴𝑀𝐷11
𝐴𝑀𝐷21
𝐴𝑀𝐷31
𝐴𝑀𝐷41
𝐴𝑀𝐷51
𝐴𝑀𝐷61
𝐴𝑀𝐷12
𝐴𝑀𝐷22
𝐴𝑀𝐷32
𝐴𝑀𝐷42
𝐴𝑀𝐷52
𝐴𝑀𝐷62
=
0.06
0.0417
0.333
0.06
0.0417
0.0167
0
0.0417
0.167
0
0.0417
0.333
Stiffness Method for Frames Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐴𝐷
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
0
12
24
0
12
24
0.06
0.0417
0.333
0.06
0.0417
0.0167
0
0.0417
0.167
0
0.0417
0.333
55.40
99.714
3.3 k
13.8 k
22.2ˊ k
3.3 k
10.2 k
0ˊ k
Stiffness Method for Frames Analysis
Final Analyzed Member
AM6
AM3
AM1
3.3k
B
C
AM2
AM5
22.2ʹk
0
B
2k/ft
C
12ft
13.8k
10.2k
AM4
3.3k
Stiffness Method for Frames Analysis
Now Shear force and bending moment diagrams
22.2ʹk
3.3k
B
2k/ft
C
0
3.3k
12ft
10.2k
13.8k
10.2
2
13.8
0
SFD
25.41
0
22.2
5.1′
10.2
BMD
Stiffness Method for Frames Analysis
Combined shear force & bending moment diagrams:
BMD
SFD
Stiffness Method for Frames Analysis
Problem 02: Analyze the given Frame using stiffness method.
1k/ft
B
C
6ft
5k
12ft
Take EI = constant
6ft
A
20ft
K.I = 2 degree ( neglecting the axial effects )
So two redundant joint displacements should be chosen.
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D1
B
20ft
C
D2
12ft
A
Rotation at B & C is taken as redundant joint displacement.
𝐷
2∗1=
𝐷1
?
=
𝐷2
?
𝐴𝐷
2∗1=
0
𝐴𝐷1
=
0
𝐴𝐷2
Stiffness Method for Frames Analysis
• Restrain all the degrees of freedom to get the restrained structure.
C
B
20ft
12ft
A
Basic kinematic determinate structure ( BKDS) or
restrained structure
Stiffness Method for Frames Analysis
• Step # 02 : Restrained structure acted upon by the actual loads.
Compute ADL matrix.( Fixed end actions)
= 7.5ʹ k
10k
2.5 k
B
5k
= 33.33ʹ k
B
1
ADL1
2.5 k
10k
𝐴𝐷𝐿1
z
7.5ʹk
10k
1k/ft
C
2.5 k
𝐴𝐷𝐿 =
ADL2
10k
25ˊ 𝑘 𝐴𝐷𝐿2
𝐴𝐷𝐿1
𝐴𝐷𝐿2
33.33ʹ k
2.5 k
20ft
12ft
A
2
=
33.33ˊ 𝑘
25
33.33
Stiffness Method for Frames Analysis
Step # 03 : Primary structure acted upon by a unit value of D &
computation of stiffness coefficients “ S” values in the
BKDS corresponding to the redundant joint displacement
locations.
•
1st a unit rotation is applied at location 1 & prevented at 2 as shown.
Compute the values of S11 and S21 .
•
Then apply a unit rotation at the redundant displacement location 2
and prevented at 1 as shown. Compute the values of S12 & S22 .
Stiffness Method for Frames Analysis
i.
1st a unit rotation is applied at location 1 & prevented at 2 (D1=1 &
D2=0)as shown. Compute the values of S11 and S21 .
𝟔𝑬𝑰
𝟏𝟐𝟐
1
= 0.042EI
𝟒𝑬𝑰
𝟏𝟐
𝟒𝑬𝑰
=
𝟐𝟎
B
20ft
θ=1
𝟎. 𝟑𝟑𝟑𝑬𝑰
𝟎. 𝟏𝑬𝑰
𝟎. 𝟎𝟒𝟐𝑬𝑰
C
𝟔𝑬𝑰𝜽
𝑳𝟐
θ=1
12ft
𝟎. 𝟏𝟔𝟕𝑬𝑰
𝟎. 𝟎𝟒𝟐𝑬𝑰
2
0.2EI
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
𝟔𝑬𝑰
𝟐𝟎𝟐
= 0.015EI
Stiffness Method for Frames Analysis
Step # 03 ( i ): Contd…
11
21
11
21
S11 = Action(sum of moments in this case) in the BKDS at redundant displacement
location 1 due to unit rotation at that location.
S21 = Moment in the BDS at redundant displacement location 2 due to a unit
rotation applied at location 1
Stiffness Method for Frames Analysis
ii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 ( D2=1 & D1=0 ) as shown.
Compute the values of S12 & S22 .
𝟐𝑬𝑰
𝟏𝟐
1
𝟎. 𝟏𝑬𝑰
2
θ=1
B
20ft
𝟎. 𝟎𝟏𝟓𝑬𝑰
12ft
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
C
𝟒𝑬𝑰𝜽
𝑳
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟒𝑬𝑰
𝟐𝟎
𝟔𝑬𝑰
𝟐𝟎𝟐
0.2EI
𝟎. 𝟎𝟏𝟓𝑬𝑰
Stiffness Method for Frames Analysis
Step # 03 : Contd…
12
22
12
22
S12 = Moment in the BKDS at redundant displacement location 1 due to
S22 =
unit rotation applied at redundant displacement location 2.
Moment in the BKDS at redundant displacement location 2 due to a
unit rotation applied at that location.
Stiffness Method for Frames Analysis
Step # 03 : Contd…
𝑆11
0.533𝐸𝐼
𝑆12
0.1𝐸𝐼
𝑆21
0.1𝐸𝐼
𝑆22
0.2𝐸𝐼
𝑆 =
S = 𝐸𝐼
𝑆11
𝑆21
𝑆12
𝑆22
0.533 0.1
0.1
0.2
Stiffness coefficient
matrix
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷1
𝐴𝐷𝐿1
𝑆11𝐷1
𝑆12𝐷2
𝐴𝐷2
𝐴𝐷𝐿2
𝑆21𝐷1
𝑆22𝐷2
𝐴𝐷1
𝐴𝐷𝐿1
=
𝐴𝐷2
𝐴𝐷𝐿2
𝐴𝐷
2∗1
𝑫
𝐴𝐷𝐿
𝑺
+
𝑆11
𝑆21
𝑆
2∗1
𝟏
•
𝑆12 𝐷1
𝑆22 𝐷2
𝑨𝑫
2∗2
𝑨𝑫𝑳
• 𝐷
2∗1
Stiffness Method for Frames Analysis
𝑆11
𝑆21
𝐷1
𝐷2
𝐷1
𝐷2
𝑆12
𝑆22
1 0.533 0.1
0.2
𝐸𝐼 0.1
𝐷1
𝐷2
𝐴𝐷1
𝐴𝐷2
𝐴𝐷𝐿1
𝐴𝐷𝐿2
0
87.93 1
210.61 𝐸𝐼
0
25.83
33.33
-ive sign shows that our
assumed redundant joint
displacement direction is
wrong
Stiffness Method for Frames Analysis
Step # 05: Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Member AB:
AM5
AM6
B
A
AM3
AM4
AM1
AM2
AMn = is the member end action in the indeterminate structure at different location
specified with a number “n”. n shows the number of member end actions.
Stiffness Method for Frames Analysis
i.
Compute AML values.
AML5
AML6
B
= 7.5ʹ k
AML4
10k
2.5 k
B
1
2
5k
3
12ft
4
5
AML3
A
AML2
A
AML1
2.5 k
7.5ʹk
10k
6
Stiffness Method for Frames Analysis
b) Compute the AMD values.
•
1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
AMD51
AMD61
B
AMD41
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟒𝑬𝑰
𝟏𝟐
1
= 0.042EI
𝟒𝑬𝑰
=
𝟐𝟎
B
2
0.2EI
20ft
𝟎. 𝟏𝑬𝑰
C
θ=1
𝟎. 𝟑𝟑𝟑𝑬𝑰
θ=1
12ft
AMD31
A
AMD21
𝟎. 𝟏𝟔𝟕𝑬𝑰
AMD11
𝟎. 𝟎𝟒𝟐𝑬𝑰
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟔𝑬𝑰
𝟐𝟎𝟐
= 0.015EI
Stiffness Method for Frames Analysis
AMD42
AMD62
B
AMD52
𝟐𝑬𝑰
𝟏𝟐
1
𝟎. 𝟏𝑬𝑰
θ=1
B
20ft
𝟎. 𝟎𝟏𝟓𝑬𝑰
12ft
AMD32
A
AMD22
AMD12
2
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
C
𝟒𝑬𝑰𝜽
𝑳
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟒𝑬𝑰
𝟐𝟎
𝟔𝑬𝑰
𝟐𝟎𝟐
0.2EI
𝟎. 𝟎𝟏𝟓𝑬𝑰
Stiffness Method for Frames Analysis
So the AMD values are
𝐴𝑀𝐷
6∗2
=
𝐴𝑀𝐷11
𝐴𝑀𝐷21
𝐴𝑀𝐷31
𝐴𝑀𝐷41
𝐴𝑀𝐷51
𝐴𝑀𝐷61
𝐴𝑀𝐷12
𝐴𝑀𝐷22
𝐴𝑀𝐷32
𝐴𝑀𝐷42
𝐴𝑀𝐷52
𝐴𝑀𝐷62
=
0.42
0.015
0.167
0.042
0.015
0.333
0
0.015
0
0
0.015
0
Stiffness Method for Frames Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐴𝐷
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
2.5
10
7.5
2.5
10
7.5
0.42
0.015
0.167
0.042
0.015
0.333
0
0.015
0
0
0.015
0
87.93
210.61
1.193 k
11.84 k
7.18ˊ k
6.193 k
11.84 k
36.78ˊ k
Stiffness Method for Frames Analysis
Final Analyzed Member
11.84k
AM4
AM6
B
A
AM3
AM1
36.78ʹ k
AM5
AM2
B
7.18ʹ k
A
11.84k
6.193k
1.193k
Stiffness Method for Frames Analysis
Now Shear force and bending moment diagrams
11.84k
36.78ʹ k
5k
0
6.193k
B
7.18ʹ k
6.193
0
12ft
A
11.84k
1.193k
0 1.193
SFD
7.18
0
BMD
36.78
Stiffness Method for Frames Analysis
Step # 05: Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Member BC:
AM3
AM1
B
AM2
AM6
C
AM4
AM5
AMn = is the member end action in the indeterminate structure at different location
specified with a number “n”. n shows the number of member end actions.
Stiffness Method for Frames Analysis
i.
Compute AML values.
AML6
AML3
AML1
B
C
AML5
AML2
= 33.33ʹ k
2.5k
AML4
B
1k/ft
C
20ft
10k
10k
33.33ʹ k
2.5k
𝐴𝑀𝐿1
𝐴𝑀𝐿2
𝐴𝑀𝐿3
𝐴𝑀𝐿4
𝐴𝑀𝐿5
𝐴𝑀𝐿6
2.5
10
33.33
2.5
10
33.33
Stiffness Method for Frames Analysis
b) Compute the AMD values.
•
1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
1
0.042EI
AMD31
B
AMD61
𝟎. 𝟑𝟑𝟑𝑬𝑰
AMD21
B
2
0.2EI
20ft
θ=1
C
θ=1
C
AMD41
AMD11
𝟒𝑬𝑰
=
𝟐𝟎
12ft
AMD51
𝟎. 𝟏𝟔𝟕𝑬𝑰
𝟎. 𝟎𝟒𝟐𝑬𝑰
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
0.015EI
𝟎. 𝟏𝑬𝑰
0.042EI
Stiffness Method for Frames Analysis
AMD32
B
AMD62
C
AMD42
AMD12
AMD22
AMD52
𝟎. 𝟏𝑬𝑰
1
B
2
θ=1
20ft
C
𝟎. 𝟎𝟏𝟓𝑬𝑰
12ft
A
𝟎. 𝟎𝟏𝟓𝑬𝑰
𝟎. 𝟎𝟏𝟓𝑬𝑰
0.2EI
Stiffness Method for Frames Analysis
So the AMD values are
𝐴𝑀𝐷
6∗2
=
𝐴𝑀𝐷11
𝐴𝑀𝐷21
𝐴𝑀𝐷31
𝐴𝑀𝐷41
𝐴𝑀𝐷51
𝐴𝑀𝐷61
𝐴𝑀𝐷12
𝐴𝑀𝐷22
𝐴𝑀𝐷32
𝐴𝑀𝐷42
𝐴𝑀𝐷52
𝐴𝑀𝐷62
=
0.042
0.015
0.2
0.042
0.015
0.1
0
0.015
0.1
0
0.015
0.2
Stiffness Method for Frames Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
2.5
10
33.33
2.5
10
33.33
0.042
0.015
0.2
0.042
0.015
0.1
0
0.015
0.1
0
0.015
0.2
87.93
210.61
6.193 k
11.8 k
36.8ˊ k
6.193 k
8.16 k
0ˊ k
Stiffness Method for Frames Analysis
Final Analyzed Member
AM6
AM3
AM1
6.193k
B
C
AM2
AM5
36.81ʹk
0
B
1k/ft
C
20ft
11.8k
8.16k
AM4
6.193k
Stiffness Method for Frames Analysis
Now Shear force and bending moment diagrams
36.81ʹk
B
6.193k
0
1k/ft
C
6.193k
20ft
8.16k
11.8k
11.84
0
8.16
1
8.16′
SFD
8.16
0
36.81
BMD
Stiffness Method for Frames Analysis
Combined shear force & bending moment diagrams:
BMD
SFD
Stiffness Method for Frames Analysis
Problem 03: Analyze the given Frame using stiffness method.
D
C
12ft
Take
EI = constant
2k/ft
B
E
12ft
A
F
24ft
K.I = 2 degree ( neglecting the axial effects )
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D
C
B
D2
D1
E
A
𝐷
Rotation at B & C
is taken as redundant
joint displacement.
F
2∗1=
𝐷1
?
=
𝐷2
?
𝐴𝐷
2∗1=
0
𝐴𝐷1
=
0
𝐴𝐷2
Stiffness Method for Frames Analysis
• Step # 02 : Compute ADL matrix.( Fixed end actions)
C
B
D
1
E
B
2k/ft
= 96ʹ k
B
A
2
E
96ʹ k
24ft
24k
24k
𝐴𝐷𝐿 =
𝐴𝐷𝐿1
𝐴𝐷𝐿2
=
96
96
E
F
Stiffness Method for Frames Analysis
Step # 03 : Primary structure acted upon by a unit value of D &
computation of stiffness coefficients “ S” values in the
BKDS corresponding to the redundant joint displacement
locations.
•
1st a unit rotation is applied at location 1 & prevented at 2 as shown.
Compute the values of S11 and S21 .
•
Then apply a unit rotation at the redundant displacement location 2
and prevented at 1 as shown. Compute the values of S12 & S22 .
Stiffness Method for Frames Analysis
i.
1st D1=1 & D2 = 0 & Compute the values of S11 and S21 .
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
C
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟒𝑬𝑰𝜽
𝑳
𝟒𝑬𝑰
𝟏𝟐
θ=1
B
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟒𝑬𝑰
𝟏𝟐
D
𝟒𝑬𝑰
𝟐𝟒
24ft
𝟐𝑬𝑰
𝟐𝟒
12ft
E
θ=1
θ=1
A
𝟔𝑬𝑰
𝟐𝟒𝟐
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟐𝟒𝟐
12ft
F
Stiffness Method for Frames Analysis
Step # 03 ( i ): Contd…
11
21
11
21
S11 = Action(sum of moments in this case) in the BKDS at redundant displacement
location 1 due to unit rotation at that location.
S21 = Moment in the BDS at redundant displacement location 2 due to a unit
rotation applied at location 1
Stiffness Method for Frames Analysis
ii. Now D2 = 1 & D1 = 0 as shown. Compute the values of S12
& S22 .
𝟐𝑬𝑰
𝟏𝟐
C
12ft
𝟒𝑬𝑰
𝟐𝟒
𝟐𝑬𝑰
𝟐𝟒
B
D
θ=1
θ=1
E
24ft
θ=1
12ft
A
𝟔𝑬𝑰
𝟐𝟒𝟐
𝟒𝑬𝑰
𝟏𝟐
𝟒𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟐𝟒𝟐
𝟐𝑬𝑰
𝟏𝟐
F
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Frames Analysis
Step # 03 : Contd…
12
22
12
22
S12 = Moment in the BKDS at redundant displacement location 1 due to
S22 =
unit rotation applied at redundant displacement location 2.
Moment in the BKDS at redundant displacement location 2 due to a
unit rotation applied at that location.
Stiffness Method for Frames Analysis
Step # 03 : Contd…
𝑆11
0.833𝐸𝐼
𝑆12
0.0833𝐸𝐼
𝑆21
0.0833𝐸𝐼
𝑆22
0.833𝐸𝐼
𝑆 =
𝑆11
𝑆21
𝑆12
𝑆22
0.833 0.0833
S = 𝐸𝐼
0.0833 0.833
Stiffness coefficient
matrix
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷1
𝐴𝐷𝐿1
𝑆11𝐷1
𝑆12𝐷2
𝐴𝐷2
𝐴𝐷𝐿2
𝑆21𝐷1
𝑆22𝐷2
𝐴𝐷1
𝐴𝐷𝐿1
=
𝐴𝐷2
𝐴𝐷𝐿2
𝐴𝐷
2∗1
𝑫
𝐴𝐷𝐿
𝑺
+
𝑆11
𝑆21
𝑆
2∗1
𝟏
•
𝑆12 𝐷1
𝑆22 𝐷2
𝑨𝑫
2∗2
𝑨𝑫𝑳
• 𝐷
2∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷1
𝐷2
𝑆11
𝑆21
𝑆12
𝑆22
𝐴𝐷1
𝐴𝐷2
1 0.833 0.0833
𝐸𝐼 0.0833 0.833
𝐷1
𝐷2
128 1
128 𝐸𝐼
𝐴𝐷𝐿1
𝐴𝐷𝐿2
0
0
96
96
-ive sign shows that our
assumed redundant joint
displacement direction is
wrong
Stiffness Method for Frames Analysis
Step # 05: Compute the member end actions & you will get.
𝟏𝟐𝒌
𝟏𝟐𝒌
D
C
𝟓. 𝟑𝟑𝒌
𝟐𝟏. 𝟑𝟑ˊ𝒌
𝟓. 𝟑𝟑𝒌
𝟐𝟏. 𝟑𝟑ˊ𝒌
12ft
𝟖𝟓. 𝟑𝟑ˊ𝒌
𝟒𝟐. 𝟔𝟕ˊ𝒌
𝟖𝟓. 𝟑𝟑ˊ𝒌
B
𝟒𝟐. 𝟔𝟕ˊ𝒌
𝟓. 𝟑𝟑𝒌
24ft
𝟒𝟐. 𝟔𝟕ˊ𝒌
E
𝟒𝟐. 𝟔𝟕ˊ𝒌
12ft
𝟐𝟏. 𝟑𝟑ˊ𝒌
𝟐𝟏. 𝟑𝟑ˊ𝒌
A
𝟏𝟐𝒌
F
𝟏𝟐𝒌
𝟓. 𝟑𝟑𝒌
Stiffness Method for Frames Analysis
Step # 06: Draw Shear force & bending moment diagram.
Stiffness Method for Frames Analysis
Problem 04: Analyze the given Frame using stiffness method.
60k
A
5ft
3k/ft
B
10ft
C
10ft
Take EI = constant
D
15ft
20ft
K.I = 2 degree ( neglecting the axial effects )
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D1
A
D2
C
B
D
𝐷
2∗1=
𝐷1
?
=
𝐷2
?
𝐴𝐷
2∗1=
0
𝐴𝐷1
=
0
𝐴𝐷2
Stiffness Method for Frames Analysis
• Restrain all the degrees of freedom to get the restrained structure.
A
B
C
D
Basic kinematic determinate structure ( BKDS) or
restrained structure
Stiffness Method for Frames Analysis
• Step # 02 : Compute ADL matrix.( Fixed end actions)
133.33ʹ k
B
5ft
44.4k
60k
C
66.67ʹ k
= 100ʹ k
B
1
ADL1
10ft
2
3k/ft
C
100ʹ k
20ft
15.56k
0ʹ k
0k
B
30k
30k
ADL2
10ft
A
0k
0ʹ k
𝐴𝐷𝐿 =
𝐴𝐷𝐿1
𝐴𝐷𝐿2
=
33.33
100
Stiffness Method for Frames Analysis
Step # 03 : Primary structure acted upon by a unit value of D &
computation of stiffness coefficients “ S” values in the
BKDS corresponding to the redundant joint displacement
locations.
•
1st a unit rotation is applied at location 1 & prevented at 2 as shown.
Compute the values of S11 and S21 .
•
Then apply a unit rotation at the redundant displacement location 2
and prevented at 1 as shown. Compute the values of S12 & S22 .
Stiffness Method for Frames Analysis
i.
D1 = 1 & D2 = 0) as shown. Compute the values of S11 and S21 .
1
𝟎. 𝟐𝟔𝟕𝑬𝑰
𝟎. 𝟏𝟑𝟑𝟓𝑬𝑰
B
θ=1
A
𝟎. 𝟐𝑬𝑰
𝟎. 𝟏𝑬𝑰
20ft
C
θ=1
15ft
𝟎. 𝟒𝑬𝑰
θ=1
10ft
𝟎. 𝟐𝑬𝑰
D
𝑆11
𝑆11
0.2
0.4
0.867𝐸𝐼
2
0.267 𝐸𝐼
21
Stiffness Method for Frames Analysis
ii. Now D2 = 1 & D1 = 0 & Compute the values of S12 & S22 .
𝟎. 𝟐𝑬𝑰 2
1
B
A
𝑆12
0.1𝐸𝐼
𝑆22
0.2𝐸𝐼
𝑆 =
𝟎. 𝟏𝑬𝑰
𝟎. 𝟎𝟏𝟓𝑬𝑰
θ=1
C
𝟎. 𝟎𝟏𝟓𝑬𝑰
D
𝑆11
𝑆21
𝑆12
𝑆22
S = 𝐸𝐼
0.867 0.1
0.1
0.2
Stiffness coefficient
matrix
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷1
𝐴𝐷𝐿1
𝑆11𝐷1
𝑆12𝐷2
𝐴𝐷2
𝐴𝐷𝐿2
𝑆21𝐷1
𝑆22𝐷2
𝐴𝐷1
𝐴𝐷𝐿1
=
𝐴𝐷2
𝐴𝐷𝐿2
𝐴𝐷
2∗1
𝑫
𝐴𝐷𝐿
𝑺
+
𝑆11
𝑆21
𝑆
2∗1
𝟏
•
𝑆12 𝐷1
𝑆22 𝐷2
𝑨𝑫
2∗2
𝑨𝑫𝑳
• 𝐷
2∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷1
𝐷2
𝑆11
𝑆21
𝑆12
𝑆22
𝐴𝐷1
𝐴𝐷2
1 0.867 0.1
0.2
𝐸𝐼 0.1
𝐷1
𝐷2
102.6 1
551 𝐸𝐼
𝐴𝐷𝐿1
𝐴𝐷𝐿2
0
0
33.3
100
-ive sign shows that our
assumed redundant joint
displacement direction is
wrong
Step # 05: Compute the member end actions & then draw shear force
and bending moment diagram.
Stiffness Method for Frames Analysis
Assignment 02(contd).
Q4. Find the member end actions and the draw shear force and
bending moment diagram for the frames in problem 3 & 4
Q5,6. Analyze the frames given below using stiffness method.
Take EI = Constant
D
12ft
24k
A
2k/ft
6ft
12ft
B
Q5
12ft
C
Stiffness Method for Frames Analysis
Assignment 02(contd).
C
20k
12ft
2k/ft
B
15ft
15ft
12ft
A
Q6
D
Stiffness Method for Frames Analysis
Part II
Analysis of Frames with side sway
Stiffness Method for Frames Analysis
Problem 05: Analyze the given Frame using stiffness method.
10k
2k/ft
B
C
2I
12ft
I
I
Take EI = constant
D
A
40ft
4ft
K.I = 3 degree ( neglecting the axial effects )
Stiffness Method for Frames Analysis
C
B
2I
I
I
A
D
Deflected shape
Stiffness Method for Frames Analysis
C
2k/ft
B
40ʹk
2I
10k
12ft
I
I
D
A
40ft
The effect of overhanging portion can be taken as moment
and force at point C as shown above.
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D1
D3
D2
B
C
𝐷1
?
𝐷 3 ∗ 1 = 𝐷2 = ?
𝐷3
?
2I
I
A
I
0
𝐴𝐷1
𝐴𝐷 2 ∗ 1 = 𝐴𝐷2 = 40ˊ𝑘
0
𝐴𝐷3
D
Rotation at B , C and Sway is taken as redundant joint displacement.
Stiffness Method for Frames Analysis
• Step # 02 : Restrained structure acted upon by the actual loads.
compute the values of actions in the restrained structure
corresponding to the redundant locations. This will
generate ADL matrix.( Fixed end actions)
40k
0ʹ k
0k
= 266.67ʹ k
2k/ft
B
C
266.67ʹ k
40ft
B
40k
12ft
A
0ʹ k
0k
B
40k
12ft
A
0k
0ʹ k
40k
40k
𝐴𝐷𝐿 =
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿3
=
266.67
266.67
0
0k
0ʹ k
40k
Stiffness Method for Frames Analysis
Step # 03 : Primary structure acted upon by a unit value of D &
computation of stiffness coefficients “ S” values in the
BKDS corresponding to the redundant joint displacement
locations.
•
1st a unit rotation is applied at location 1 & prevented at 2 & 3 as
shown. Compute the values of S11 , S21 & S31.
•
Then a unit rotation is applied at redundant displacement location 2
and prevented at 1 & 3 as shown. Compute the values of S12 , S22 &
S32 .
• Then a unit is applied at 3 and prevented at 1 & 2 as shown.
Compute the values of S12 , S22 & S32 .
Stiffness Method for Frames Analysis
i.
1st a unit rotation is applied at location 1 & prevented at 2 & 3.
(D1=1 & D2 = D3 = 0)as shown. Compute the values of S11 , S21 &
S31 .
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
B
𝟎. 𝟐𝑬𝑰
𝟎. 𝟏𝑬𝑰
40ft
C
θ=1
𝟎. 𝟑𝟑𝑬𝑰
θ=1
𝟎. 𝟎𝟎𝟕𝟓𝑬𝑰
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
A
𝟎. 𝟎𝟏𝟎𝟕𝟓𝑬𝑰
12ft
D
𝟎. 𝟏𝟔𝟓𝑬𝑰
Stiffness Method for Frames Analysis
Step # 03 ( i ): Contd…
11
21
31
S11 = Action(sum of moments in this case) in the BKDS at redundant displacement
location 1 due to unit rotation at that location.
S21 = Moment in the BDS at redundant displacement location 2 due to a unit
rotation applied at location 1
S31 = Moment in the BDS at redundant displacement location 3 due to a unit
rotation applied at location 1
Stiffness Method for Frames Analysis
ii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 & 3( D2=1 & D1 = D3 = 0 ) as shown.
Compute the values of S12 , S22 & S32 .
𝟎. 𝟐𝑬𝑰
𝟎. 𝟏𝑬𝑰
θ=1
B
C
40ft
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
𝟎. 𝟑𝟑𝑬𝑰
𝟎. 𝟎𝟎𝟕𝟓𝑬𝑰
θ=1
𝟎. 𝟎𝟎𝟕𝟓𝑬𝑰
12ft
A
D
𝟎. 𝟎𝟏𝟔𝟕𝑬𝑰
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
Stiffness Method for Frames Analysis
Step # 03 : Contd…
12
22
32
S12 = Moment in the BKDS at redundant displacement location 1 due to
unit rotation applied at redundant displacement location 2.
S22 = Moment in the BKDS at redundant displacement location 2 due to a
unit rotation applied at that location.
S23 = Moment in the BKDS at redundant displacement location 3 due to a
unit rotation applied at redundant location 2.
Stiffness Method for Frames Analysis
iii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 ( D3=1 & D1 = D2 = 0 ) as shown.
Compute the values of S13 , S23 & S33 .
D3 =1
D3 =1
B
C
2I
𝟎. 𝟎𝟎𝟔𝟗𝑬𝑰
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
𝟎. 𝟎𝟎𝟔𝟗𝑬𝑰
I
I
12ft
𝟎. 𝟎𝟎𝟔𝟗𝑬𝑰
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
A
𝟎. 𝟎𝟒𝟏𝟕𝑬𝑰
D
𝟎. 𝟎𝟎𝟔𝟗𝑬𝑰
Stiffness Method for Frames Analysis
Step # 03 : Contd…
13
23
33
S13 = Moment in the BKDS at redundant displacement location 1 due to
unit rotation applied at redundant displacement location 2.
S23 = Moment in the BKDS at redundant displacement location 2 due to a
unit rotation applied at that location.
S33 = Moment in the BKDS at redundant displacement location 3 due to a
unit rotation applied at redundant location 2.
Stiffness Method for Frames Analysis
Step # 03 : Contd…
11
21
31
12
22
32
𝑆 =
33
23
13
𝑆11
𝑆21
𝑆31
𝑆12
𝑆22
𝑆32
𝑆13
𝑆23 = 𝐸𝐼
𝑆33
0.533
0.1
0.0417
Stiffness coefficient matrix
0.1
0.533
0.0417
0.0417
0.0417
0.0138
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷
3∗1
𝑫
𝐴𝐷𝐿
𝑺
𝑆
3∗1
𝟏
•
𝑨𝑫
3∗3
𝑨𝑫𝑳
• 𝐷
3∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷3
𝐷1
𝐷2
𝐷3
1
𝐸𝐼
𝑆11
𝑆21
𝑆31
0.533
0.1
0.0417
𝐷1
𝐷2
𝐷3
𝑆12
𝑆22
𝑆32
𝑆13
𝑆23
𝑆33
0.1
0.533
0.0417
1
𝐸𝐼
𝐴𝐷1
𝐴𝐷2
𝐴𝐷2
0.0417
0.0417
0.0138
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿2
0
40
266
266.67
0 0
621.90
571.45
313.41
Step # 05: Compute the member end actions & then draw shear force
and bending moment diagram. CLASS ACTIVITY
Stiffness Method for Frames Analysis
Problem 06: Analyze the given Frame using stiffness method if the
yielding of support D to the right downwards in t-m units are &
respectively.
11.12t
C
B
4I
I
10m
4I
5m
D
A
10m
K.I = 3 degree ( neglecting the axial effects )
Take EI = con
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D2
D1
D3
B
C
4I
𝐷1
?
𝐷 3 ∗ 1 = 𝐷2 = ?
𝐷3
?
I
4I
A
11.12𝑡
𝐴𝐷1
𝐴𝐷 3 ∗ 1 = 𝐴𝐷2 =
0
0
𝐴𝐷3
D
Rotation at B , C and Sway is taken as redundant joint displacement.
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
As there is no external load acting on the frame that will
cause fixed end actions so the ADL matrix will consist
of restraining forces due to in direct loadings only.
1. Induce actions due to horizontal displacement of 20/EI (towards
right) at support D.
2. Induce actions due to vertical displacement of 50/EI (downward) at
support D.
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
1. Induce actions due to horizontal displacement of 20/EI (towards
right) at support D and get the values of ADLʹ.
B
4I, 10m
C
4.8 t-m
1.92 t
I 5m
10m 4I
D
= 1.92 t
∗
A
𝐴𝐷𝐿1ˊ
1.92𝑡
= 4.8 t-m
Δ=
𝐴𝐷𝐿2ˊ
0
𝐴𝐷𝐿3ˊ
4.8𝑡
𝑚
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
1. Induce actions due to vertical displacement of 50/EI (downward) at
support D and get the values of ADLʹʹ.
B
4I, 10m
C
∗
= 12 t-m
12 t-m
2.4 t
10m 4I
= 2.4 t
I,5m
D
Δ=
A
𝐴𝐷𝐿1ˊˊ
0𝑡
𝐴𝐷𝐿2ˊˊ
12𝑡
𝑚
𝐴𝐷𝐿3ˊˊ
12𝑡
𝑚
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
𝐴𝐷𝐿1ˊ
𝐴𝐷𝐿1ˊˊ
1.92𝑡
0𝑡
𝐴𝐷𝐿2ˊ
𝐴𝐷𝐿2ˊˊ
0
12𝑡
𝑚
𝐴𝐷𝐿3ˊ
𝐴𝐷𝐿3ˊˊ
𝐴𝐷𝐿1
𝐴𝐷𝐿1ˊ
𝐴𝐷𝐿1ˊˊ
1.92𝑡
𝐴𝐷𝐿2
𝐴𝐷𝐿2ˊ
𝐴𝐷𝐿2ˊˊ
12𝑡
𝑚
𝐴𝐷𝐿3
𝐴𝐷𝐿3ˊ
𝐴𝐷𝐿3ˊˊ
7.2𝑡
𝑚
𝐴𝐷𝐿 =
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿3
=
1.92
12
7.2
4.8𝑡
12𝑡
𝑚
𝑚
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
i. When D1 =1 & D2 = D3 = 0 as shown.
D1 =1
D1 =1
B
𝟎. 𝟎𝟒𝟖𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
C
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟎𝟗𝟔𝑬𝑰
5m,I
D
4I,10m
𝟎. 𝟎𝟗𝟔𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟎𝟒𝟖𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
A
𝑆11
0.48
0.096 𝐸𝐼
0.144𝐸𝐼
𝑆21
0.24𝐸𝐼
𝑆31
0.24𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
ii. When D2 =1 & D1 = D3 = 0 as shown.
𝟎. 𝟐𝟒𝑬𝑰
𝟏. 𝟔𝑬𝑰
B
𝟎. 𝟖𝑬𝑰
10m,4I
C
θ=1
𝟏. 𝟔𝑬𝑰
5m,I
θ=1
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
D
10m,4I
A
𝟎. 𝟐𝟒𝑬𝑰
𝑆12
0.24𝐸𝐼
𝟎. 𝟖𝑬𝑰
𝑆22
1.6𝐸𝐼
1.6𝐸𝐼
3.2𝐸𝐼
𝑆32
0.8𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iii. When D3 =1 & D1 = D2 = 0 as shown.
𝟏. 𝟔𝑬𝑰
𝟎. 𝟖𝑬𝑰
B
θ=1
C
10m,4I
𝟎. 𝟖𝑬𝑰
θ=1
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
10m,4I
𝟎. 𝟐𝟒𝑬𝑰
5m,I
D
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟒𝑬𝑰
A
𝑆13
0.24𝐸𝐼
𝑆23
0.8𝐸𝐼
𝑆32
1.6𝐸𝐼
0.8𝐸𝐼
2.4𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Contd…
11
22
12
23
13
𝑆 =
31
21
𝑆11
𝑆21
𝑆31
𝑆12
𝑆22
𝑆32
𝑆13
0.144
𝑆23 = 𝐸𝐼
0.24
𝑆33
0.24
32
33
0.24
3.20
0.80
0.24
0.80
2.40
Stiffness
coefficient
matrix
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷
3∗1
𝑫
𝐴𝐷𝐿
𝑺
𝑆
3∗1
𝟏
•
𝑨𝑫
3∗3
𝑨𝑫𝑳
• 𝐷
3∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷3
𝐷1
𝐷2
𝐷3
1
𝐸𝐼
𝑆11
𝑆21
𝑆31
0.144
0.24
0.24
𝑆12
𝑆22
𝑆32
𝑆13
𝑆23
𝑆33
0.24
3.20
0.80
𝐷1
𝐷2
𝐷3
𝐴𝐷1
𝐴𝐷2
𝐴𝐷2
0.24
0.80
2.40
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿2
11.12
0
0
1.92
12
7.2
1 128.369
10.276
𝐸𝐼
12.408
Step # 05: Compute the member end actions & then draw shear force
and bending moment diagram. HOME WORK
Stiffness Method for Frames Analysis
Problem 07: Analyze the given Frame using stiffness method.
9t
4.5t/m
B
10t
E
C
2I
2m,I
2I
4t
5m
6t/m
6t/m
3m
F
2I
2m,I
D
A
2m
4m
6m
6m
Take EI = constant
K.I = 4 degree ( neglecting the axial effects )
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
D2
D1
B
C
6m ,2I
D3
2m,I
E
6m ,2I
2I
D3
3m
F
5m 2I
2m,I
D
A
𝐷1
?
𝐷2
?
𝐷 4∗1=
=
𝐷3
?
𝐷4
?
10𝑡
𝐴𝐷1
0
𝐴𝐷2
𝐴𝐷 4 ∗ 1 =
=
0
𝐴𝐷3
0
𝐴𝐷4
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
12𝑡𝑚
4.5t/m
B
4m
12.5tm
15t
C
4m
A
12.5tm
15t
2tm
8tm
2m
2t
9t
E
2m
3m
4t
D
E
4.5tm
2tm
C
5m
9t
C
2m
B
6t/m
4𝑡𝑚
8tm
2m
F
2t
4.5tm
6t/m
9t
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
𝐴𝐷𝐿1
15
𝐴𝐷𝐿2
12.5
𝐴𝐷𝐿3
8
2
𝐴𝐷𝐿4
8
4.5
𝐴𝐷𝐿 =
2
9
8.0𝑡
12
4
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿3
𝐴𝐷𝐿4
0.5𝑡
6.0𝑡
3.5𝑡
=
𝑚
𝑚
𝑚
266.67
266.67
0
3.5
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
i. When D1 =1 & D2 = D3 = D4 = 0 as shown.
D1 =1
𝟎. 𝟏𝟗𝟐𝑬𝑰
B
𝟎. 𝟒𝟖𝑬𝑰
D1 =1
6m ,2I
C
6m ,2I
E
D1 =1
𝟏. 𝟑𝟑𝟑𝑬𝑰
𝟎. 𝟑𝟕𝟓𝑬𝑰
𝟎. 𝟖𝟖𝑬𝑰
𝟎. 𝟏𝟖𝟕𝟓𝑬𝑰
4m,I
3m, 2I
F
2I,5m
D
𝟎. 𝟎𝟗𝟔𝑬𝑰
𝟎. 𝟐𝟒𝑬𝑰
𝟎. 𝟒𝟖𝑬𝑰
A
𝑆11
𝑆21
0.48𝐸𝐼
0.192
𝑆31
0.1875
0.88 𝐸𝐼
0.375𝐸𝐼
𝑆41
1.2683𝐸𝐼
1.333𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
ii. When D2 =1 & D1 = D3 = D4 = 0 as shown.
𝟎. 𝟔𝟔𝟔𝟕𝑬𝑰
𝟎. 𝟒𝟖𝑬𝑰
𝟏. 𝟔𝑬𝑰
E
C
6m ,2I
θ=1
B
6m ,2I
𝟏. 𝟑𝟑𝑬𝑰
3m ,2I
4m ,I
θ=1
F
5m ,2I
D
A
𝟎. 𝟒𝟖𝑬𝑰
𝑆22
𝟎. 𝟖𝑬𝑰
1.6𝐸𝐼
1.33𝐸𝐼
𝑆12
2.933𝐸𝐼
𝑆32
0.6667𝐸𝐼
0.48𝐸𝐼
𝑆42
0
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iii. When D3 =1 & D1 = D2 = D4 = 0 as shown.
𝟏. 𝟑𝟑𝑬𝑰
𝟎. 𝟔𝟔𝟔𝟕𝑬𝑰
C
θ=1
6m ,2I
B
𝟎. 𝟑𝟕𝟓𝑬𝑰
𝟏. 𝟑𝟑𝑬𝑰
θ=1
𝟎. 𝟔𝟔𝟔𝟕𝑬𝑰
6m ,2I
θ=1
E
𝟏. 𝟎𝑬𝑰
4m ,I
3m ,2I
F
5m ,2I
D
𝟎. 𝟑𝟕𝟓𝑬𝑰
𝟎. 𝟓𝑬𝑰
A
𝑆13
𝑆32
1.33𝐸𝐼
1.33𝐸𝐼
1𝐸𝐼
0.375𝐸𝐼
𝑆23
0.6667𝐸𝐼
3.6667𝐸𝐼
𝑆43
0.6667𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iv. When D4 =1 & D1 = D2 = D3 = 0 as shown.
𝟏. 𝟑𝟑𝑬𝑰
𝟎. 𝟔𝟔𝟔𝟕𝑬𝑰
6m ,2I
B
C
E
θ=1
6m ,2I
𝟏. 𝟑𝟑𝑬𝑰
𝟐. 𝟔𝑬𝑰
4m ,I
θ=1
3m ,2I
F
5m ,2I
𝟏. 𝟑𝟑𝑬𝑰
𝟏. 𝟑𝑬𝑰
D
A
𝑆14
𝑆34
0.6667𝐸𝐼
1.333𝐸𝐼
𝑆43
1.333𝐸𝐼
𝑆24
2.666𝐸𝐼
0𝐸𝐼
4.0𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Contd…
𝑆21
0.48𝐸𝐼
𝑆11
1.2683𝐸𝐼
𝑆12
0.48𝐸𝐼
𝑆22
2.933𝐸𝐼
𝑆13
0.375𝐸𝐼
𝑆23
0.6667𝐸𝐼
𝑆14
1.333𝐸𝐼
𝑆24
𝑆 =
𝑆11
𝑆21
𝑆31
𝑆41
𝑆12
𝑆22
𝑆32
𝑆42
𝑆13
𝑆23
𝑆33
𝑆43
0
𝑆31
𝑆32
𝑆41
0.375𝐸𝐼
0.6667𝐸𝐼
1.333𝐸𝐼
𝑆42
0
𝑆33
3.6667𝐸𝐼
𝑆43
0.6667𝐸𝐼
𝑆34
0.6667𝐸𝐼
𝑆44
4.00𝐸𝐼
𝑆14
1.2683
𝑆24
0.48
= 𝐸𝐼
𝑆34
0.375
𝑆44
1.333
0.48
2.933
0.6667
0
0.375
0.6667
3.6667
0.6667
1.333
0
0.6667
4.00
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷
3∗1
𝑫
𝐴𝐷𝐿
𝑺
𝑆
3∗1
𝟏
•
𝑨𝑫
3∗3
𝑨𝑫𝑳
• 𝐷
3∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷3
𝐷4
𝐷1
𝐷2
𝐷3
𝐷4
1
𝐸𝐼
𝑆11
𝑆21
𝑆31
𝑆41
1.2683
0.48
0.375
1.333
𝑆12
𝑆22
𝑆32
𝑆42
0.48
2.933
0.6667
0
𝐷1
𝐷2
𝐷3
𝐷4
𝑆13
𝑆23
𝑆33
𝑆43
0.375
0.6667
3.6667
0.6667
1
𝐸𝐼
𝐴𝐷1
𝐴𝐷2
𝐴𝐷3
𝐴𝐷4
𝑆14
𝑆24
𝑆34
𝑆44
1.333
0
0.6667
4.00
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿3
𝐴𝐷𝐿4
10
0
0
0
8.0
0.5
6.0
3.5
22.40
3.786
1.271
6.802
Step # 05: Compute the member end actions & then draw shear force
and bending moment diagram. Class WORK
Stiffness Method for Frames Analysis
Problem 08: Analyze the given Frame using stiffness method.
K.I = 6 degree
Take EI = constant
Stiffness Method for Frames Analysis
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
𝐴𝐷1
2𝑡
𝐴𝐷2
4𝑡
𝐴𝐷3
0
𝐴𝐷 6 ∗ 1 =
=
𝐴𝐷4
0
𝐴𝐷5
0
0
𝐴𝐷6
𝐷1
𝐷2
𝐷3
𝐷 6∗1=
= ?
𝐷4
𝐷5
𝐷6
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
Stiffness Method for Frames Analysis
• Step # 02 : Computation of ADL matrix.( Fixed end actions).
1
2
=
3
4
5
6
=
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
i. When D1 =1 & D2 = D3 = D4 = D5 = D6 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
i. When D1 =1 & D2 = D3 = D4 = D5 = D6 = 0 as shown.
𝑆11
𝑆21
0.444
0.444 𝐸𝐼
0.888𝐸𝐼
𝑆31
0.666𝐸𝐼
𝑆41
0.666𝐸𝐼
𝑆51
0.666𝐸𝐼
𝑆61
0.666𝐸𝐼
0.888𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
ii. When D2 =1 & D1 = D3 = D4 = D5 = D6 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
ii. When D2 =1 & D1 = D3 = D4 = D5 = D6 = 0 as shown.
𝑆12
𝑆22
0.888𝐸𝐼
1.387𝐸𝐼
𝑆32
0.666𝐸𝐼
𝑆42
0.666𝐸𝐼
𝑆52
0 𝐸𝐼
𝑆62
0.5𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iii. When D3 =1 & D1 = D2 = D4 = D5 = D6 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iii. When D3 =1 & D1 = D2 = D4 = D5 = D6 = 0 as shown.
𝑆13
0.666𝐸𝐼
𝑆23
0.666𝐸𝐼
𝑆33
2.666𝐸𝐼
𝑆43
0.666𝐸𝐼
𝑆53
0.666 𝐸𝐼
𝑆63
0 𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iv. When D4 =1 & D1 = D2 = D3 = D5 = D6 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
iv. When D4 =1 & D1 = D2 = D3 = D5 = D6 = 0 as shown.
𝑆14
0.666𝐸𝐼
𝑆24
0.666𝐸𝐼
𝑆34
0.666𝐸𝐼
𝑆44
2.666𝐸𝐼
𝑆54
0𝐸𝐼
𝑆64
0.666 𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
v. When D5 =1 & D1 = D2 = D3 = D4 = D6 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
v. When D5 =1 & D1 = D2 = D3 = D4 = D6 = 0 as shown.
𝑆15
0.666𝐸𝐼
𝑆25
0𝐸𝐼
𝑆35
0.666𝐸𝐼
𝑆45
0𝐸𝐼
𝑆55
4.0𝐸𝐼
𝑆65
0.666 𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
vi. When D6 =1 & D1 = D2 = D3 = D4 = D5 = 0 as shown.
Stiffness Method for Frames Analysis
Step # 03 : Computation of stiffness coefficient matrix
vi. When D6 =1 & D1 = D2 = D3 = D4 = D5 = 0 as shown.
𝑆16
0.666𝐸𝐼
𝑆26
0.5𝐸𝐼
𝑆36
0𝐸𝐼
𝑆46
0.666𝐸𝐼
𝑆56
0.666𝐸𝐼
𝑆66
3.332 𝐸𝐼
Stiffness Method for Frames Analysis
Step # 03 : Contd…
𝑆 =
= 𝐸𝐼
0.888
0.888
0.666
0.666
0.666
0.666
𝑆11
𝑆21
𝑆31
𝑆41
𝑆51
𝑆61
𝑆12
𝑆22
𝑆32
𝑆42
𝑆52
𝑆62
0.888
1.387
0.666
0.666
0
0.5
𝑆13
𝑆23
𝑆33
𝑆43
𝑆53
𝑆63
𝑆14
𝑆24
𝑆34
𝑆44
𝑆54
𝑆64
0.666
0.666
2.666
0.666
0.666
0
𝑆15
𝑆25
𝑆35
𝑆45
𝑆55
𝑆65
𝑆15
𝑆26
𝑆35
𝑆46
𝑆56
𝑆66
0.666
0.666
0.666
2.666
0
0.666
0.666
0
0.666
0
4.0
0.666
0.666
0.5
0
0.666
0.666
3.332
Stiffness Method for Frames Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷
3∗1
𝑫
𝐴𝐷𝐿
𝑺
𝑆
3∗1
𝟏
•
𝑨𝑫
3∗3
𝑨𝑫𝑳
• 𝐷
3∗1
Stiffness Method for Frames Analysis
𝐷1
𝐷2
𝐷3
𝐷4
𝐷5
𝐷6
1
𝐸𝐼
0.888
0.888
0.666
0.666
0.666
0.666
0.888
1.387
0.666
0.666
0
0.5
𝐷1
𝐷2
𝐷3
𝐷4
𝐷5
𝐷6
0.666
0.666
2.666
0.666
0.666
0
1
𝐸𝐼
0.666
0.666
0.666
2.666
0
0.666
0.666
0
0.666
0
4.0
0.666
0.666
0.5
0
0.666
0.666
3.332
2𝑡 0
4𝑡 0
0 9
0 9
0 18
0 18
6.6186
2.8619
3.3155
1.9350
5.9258
5.3151
Step # 05: Compute the member end actions & then draw shear force
and bending moment diagram.
Stiffness Method for Frames Analysis
Assignment 02(contd).
Q7. Find the member end actions and the draw shear force and
bending moment diagram for the frames in problem 5,6,7 & 8.
Q8. Develop stiffness matrix for the frame shown in fig. on next
slide.
Stiffness Method for Frames Analysis
Assignment 02(contd).
Q8.
6m
6m
References
• Structural Analysis by R. C. Hibbeler
•
Matrix structural analysis by William Mc Guire
•
Matrix analysis of frame structures by William Weaver
• Online Civil Engineering blogs