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University of Engineering & Technology
Peshawar, Pakistan
CE301: Structure Analysis II
Module 06:
Analysis of S.I Beams Using stiffness method
By:
Prof. Dr. Bashir Alam
Civil Engineering Department
UET , Peshawar
Topics to be Covered
• Introduction
• Prerequisites for using stiffness method
• Step wise procedure of stiffness method for beam analysis
• Analysis of beam Example 1
• Example 2
• Example 3
• Assignment 03 (b)
Stiffness Method for Beams Analysis
 Introduction:
Beams 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 & bending moment at any section of the
continuous beam.
Stiffness Method for Beams 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
 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 Beams 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 Beams Analysis
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳
Step # 05: Compute the member end actions .
𝑨𝑫 = 𝑨𝑫𝑳 + 𝑺 • 𝑫
Stiffness Method for Beams Analysis
Problem 01: Analyze the given beam using stiffness method.
A
B
C
30k 20k
12ft 12ft
6ft
Take EI = constant
K.I = 2 degree ( neglecting the axial effects )
So two redundant joint displacements should be chosen.
Stiffness Method for Beams Analysis
𝐷 2 ∗ 1 =
𝐷1
𝐷2
=
?
?
𝐴𝐷 2 ∗ 1 =
𝐴𝐷1
𝐴𝐷2
=
0
0
A B C
30k 20k
12ft 12ft
D1 D2
1 2
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
Rotation at B & C is taken as redundant joint displacement.
Stiffness Method for Beams Analysis
• Restrain all the degrees of freedom to get the restrained structure.
Basic kinematic determinate structure ( BKDS) or
restrained structure
A B C
Stiffness Method for Beams 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)
A B
P1
L
ADL1
C
1
2
P2
L
B
ADL2
P1
2
P1
2
P2
2
P2
2
P1L
8
P1L
8
P2L
8
P2L
8
A B C
1
2
30k 20k
12ft 12ft
B
ADL2
15k 15k 10k 10k
45ʹ k 45ʹ k
ADL1
30ʹ k 30ʹ k
Stiffness Method for Beams Analysis
𝐴𝐷𝐿2 = 30ˊ 𝑘
𝐴𝐷𝐿1 = 45 − 30 = 15ˊ 𝑘
𝐴𝐷𝐿 = 𝐴𝐷𝐿1
𝐴𝐷𝐿2
= 15
30
ADL1 = moment in the restrained structure under the actual loads
corresponding to redundant joint displacement 1.
ADL2 = moment in the restrained structure under the actual loads
corresponding to redundant joint displacement 2.
Note:
For rotation corresponding
action is moment.
Stiffness Method for Beams 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 & prevent it 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 Beams Analysis
𝟐𝑬𝑰𝜽
𝑳
A
B
C
1
2
θ=1
𝟒𝑬𝑰𝜽
𝑳
𝟒𝑬𝑰𝜽
𝑳
𝟐𝑬𝑰𝜽
𝑳
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟔𝑬𝑰𝜽
𝑳𝟐
i. 1st a unit rotation is applied at location 1 & prevent it at 2 (D1=1 &
D2=0)as shown. Compute the values of S11 and S21 .
𝟐𝑬𝑰
𝟏𝟐
A
B
C
1
2
θ=1
𝟒𝑬𝑰
𝟏𝟐
𝟒𝑬𝑰
𝟏𝟐
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft
12ft
L
L
Stiffness Method for Beams Analysis
Step # 03 ( i ): Contd…
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
𝑆11 =
4𝐸𝐼
12
+
4𝐸𝐼
12
𝑆11 =
8EI
12
𝑆21 =
2𝐸𝐼
12
𝑆21 =
2EI
12
Stiffness Method for Beams 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 .
where
𝟒𝑬𝑰𝜽
𝑳
A
B C
1
2
θ=1
𝟐𝑬𝑰𝜽
𝑳
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟔𝑬𝑰𝜽
𝑳𝟐
𝟒𝑬𝑰
𝟏𝟐
A
B C
1
2
θ=1
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
L L
12ft 12ft
Stiffness Method for Beams Analysis
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.
𝑆12 =
2𝐸𝐼
12
𝑆12 =
2EI
12
𝑆21 =
4𝐸𝐼
12
𝑆21 =
4EI
12
Step # 03 : Contd…
Stiffness Method for Beams Analysis
𝑆11 =
8𝐸𝐼
12
𝑆21 =
2𝐸𝐼
12
𝑆12 =
2𝐸𝐼
12
𝑆22 =
4𝐸𝐼
12
𝑆 =
𝑆11 𝑆12
𝑆21 𝑆22
S = 𝐸𝐼
8
12
2
12
2
12
4
12
Stiffness coefficient
matrix
Step # 03 : Contd…
Stiffness Method for Beams 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
𝐴𝐷2
=
𝐴𝐷𝐿1
𝐴𝐷𝐿2
+
𝑆11 𝑆12
𝑆21 𝑆22
𝐷1
𝐷2
𝐴𝐷 2 ∗ 1 = 𝐴𝐷𝐿 2 ∗ 1 + 𝑆 2 ∗ 2 • 𝐷 2 ∗ 1
𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳
Stiffness Method for Beams Analysis
𝐷1
𝐷2
=
𝑆11 𝑆12
𝑆21 𝑆22
−1
𝐴𝐷1 − 𝐴𝐷𝐿1
𝐴𝐷2 − 𝐴𝐷𝐿2
𝐷1
𝐷2
=
12
𝐸𝐼
8 2
2 4
−1
0 − 15
0 − 30
𝐷1
𝐷2
=
0
−90
-ive sign shows that our assumed
redundant joint displacement direction is
wrong
Stiffness Method for Beams Analysis
Step # 05(a): Before computing the member end action we can
compute the support reactions directly using
matrix approach .
A B C
AS3 AS1 AS2
𝐴𝑆 = 𝐴𝑆𝐿 + 𝐴𝑆𝐷 𝐷
Stiffness Method for Beams Analysis
A B C
1
2
30k 20
12ft 12ft
B
15k 15k 10k 10k
45ʹ k 45ʹ k 30ʹ k 30ʹ k
i. Compute ASL values.
𝐴𝑆𝐿1
𝐴𝑆𝐿2
𝐴𝑆𝐿3
=
25
10
15
Stiffness Method for Beams Analysis
A B C
ASL3 ASL1 ASL2
P1
P2
ii. Compute the ASD values.
• 1st apply a unit rotation at redundant location 1 and then at 2 in
the restrained structure as shown below.
A B C
ASD31 ASD11 ASD21
Stiffness Method for Beams Analysis
𝟐𝑬𝑰
𝟏𝟐
A
B
C
1
2
θ=1
𝟒𝑬𝑰
𝟏𝟐
𝟒𝑬𝑰
𝟏𝟐
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft
12ft
A B C
ASD32 ASD12 ASD22
Stiffness Method for Beams Analysis
𝟒𝑬𝑰
𝟏𝟐
A
B C
1
2
θ=1
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft 12ft
𝐴𝑆𝐷 3 ∗ 2 =
𝐴𝑆𝐷11 𝐴𝑆𝐷12
𝐴𝑆𝐷21 𝐴𝑆𝐷22
𝐴𝑆𝐷31 𝐴𝑆𝐷32
= EI
0 −0.041
0.041 0.041
−0.5 0
So the ASD values are
𝐴𝑆1
𝐴𝑆2
𝐴𝑆3
=
𝐴𝑆𝐿1
𝐴𝑆𝐿2
𝐴𝑆𝐿3
+
𝐴𝑆𝐷11 𝐴𝑆𝐷12
𝐴𝑆𝐷21 𝐴𝑆𝐷22
𝐴𝑆𝐷31 𝐴𝑆𝐷32
𝐷1
𝐷2
Stiffness Method for Beams Analysis
So the AS values are
𝐴𝑆1
𝐴𝑆2
𝐴𝑆3
=
25
10
15
+EI
0 −0.041
0.041 0.041
−0.5 0
0
−90
𝐴𝑆1
𝐴𝑆2
𝐴𝑆3
=
28.75
6.246
15
30k 20k
Structure with known support reactions
Stiffness Method for Beams Analysis
A B C
15k 28.75k 6.246k
12ft
12ft
A B C
AM1 AM2 AM3
AM4
AM7
AM6
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.
Step # 05(b): Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Stiffness Method for Beams Analysis
i. Compute AML values.
A B
P1
L
C
2
P2
L
B
AML1 AML2 AML3
AML4
AML7
AML6
AML5
A B C
1
2
30k 20
12ft 12ft
B
15k 15k 10k 10k
45ʹ k 45ʹ k 30ʹ k 30ʹ k
Stiffness Method for Beams Analysis
𝐴𝑀𝐿1
𝐴𝑀𝐿2
𝐴𝑀𝐿3
𝐴𝑀𝐿4
𝐴𝑀𝐿5
𝐴𝑀𝐿6
𝐴𝑀𝐿7
=
15
15
10
10
−45
45
−30
So AML values from the previous slide is.
Stiffness Method for Beams Analysis
θ=1
A
B
C
AMD11
AMD21 AMD31
AMD41
AMD61
AMD51
1
2
AMD71
b) Compute the AMD values.
• 1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
𝟐𝑬𝑰
𝟏𝟐
A
B
C
1
2
θ=1
𝟒𝑬𝑰
𝟏𝟐
𝟒𝑬𝑰
𝟏𝟐
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
Stiffness Method for Beams Analysis
θ=1
A
B
C
AMD12
AMD22 AMD32
AMD42
AMD62
AMD51
1
2
AMD72
Stiffness Method for Beams Analysis
𝟒𝑬𝑰
𝟏𝟐
A
B C
1
2
θ=1
𝟐𝑬𝑰
𝟏𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
𝟔𝑬𝑰
𝟏𝟐𝟐
12ft 12ft
𝐴𝑀𝐷 7 ∗ 2 =
𝐴𝑀𝐷11 𝐴𝑀𝐷12
𝐴𝑀𝐷21 𝐴𝑀𝐷22
𝐴𝑀𝐷31 𝐴𝑀𝐷32
𝐴𝑀𝐷41 𝐴𝑀𝐷42
𝐴𝑀𝐷51 𝐴𝑀𝐷52
𝐴𝑀𝐷61 𝐴𝑀𝐷62
𝐴𝑀𝐷71 𝐴𝑀𝐷72
=
−0.041 0
0.041 0
−0.041 −0.041
0.041 0.041
0.16 0
0.33 0
0.33 0.16
So the AMD values are
Stiffness Method for Beams Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
𝐴𝑀7
=
𝐴𝑀𝐿1
𝐴𝑀𝐿2
𝐴𝑀𝐿3
𝐴𝑀𝐿4
𝐴𝑀𝐿5
𝐴𝑀𝐿6
𝐴𝑀𝐿7
+
𝐴𝑀𝐷11 𝐴𝑀𝐷12
𝐴𝑀𝐷21 𝐴𝑀𝐷22
𝐴𝑀𝐷31 𝐴𝑀𝐷32
𝐴𝑀𝐷41 𝐴𝑀𝐷42
𝐴𝑀𝐷51 𝐴𝑀𝐷52
𝐴𝑀𝐷61 𝐴𝑀𝐷62
𝐴𝑀𝐷71 𝐴𝑀𝐷72
𝐷1
𝐷2
Stiffness Method for Beams Analysis
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
𝐴𝑀7
=
15
15
10
10
−45
45
−30
+
−0.041 0
0.041 0
−0.041 −0.041
0.041 0.041
0.16 0
0.33 0
0.33 0.16
0
−90
=
15 k
15 k
13.75 k
6.247 k
−45 ˊ k
45ˊ k
−45ˊ k
Complete analyzed structure. Shear force and bending moment diagrams are
given on next slide.
A B C
15k 15k 13.75k 6.247k
45ʹ k 45ʹ k 45ʹ k
12ft 12ft
Stiffness Method for Beams Analysis
A B C
15k 15k 13.75k 6.247k
45ʹ k 45ʹ k 44.4ʹ k
12ft 12ft
30k 20k
15
45
13.75
6.247
SFD
15
15
15
6.247
45
45
37.5
BMD
0
0
Now Shear force and Bending moment diagram
Stiffness Method for Beams Analysis
Stiffness Method for Beams Analysis
Problem 02: Analyze the given beam using stiffness method, if
support A rotates by 0.002 rad clockwise, support B settles down
by 0.75 in & support C settles down by 0.5 in.
Take
E = 30000 ksi
I = 800 in4
EI = 166666.6 k-ft2
K.I = 2 degree
So two redundant joint displacements should be chosen.(ignoring the axial effects).
A B C
30k
θ=0.002 rad
Δ1= 0.75ʺ
2k/ft
Δ2= 0.5ʺ
A
B C
30k 2k/ft
25ft 25ft
10ft
6ft
𝐷 2 ∗ 1 =
𝐷1
𝐷2
=
?
?
𝐴𝐷 2 ∗ 1 =
𝐴𝐷1
𝐴𝐷2
=
0
36
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
Rotation at B & C is taken as redundant joint displacement.
Stiffness Method for Beams Analysis
The effect of
overhanging part
is taken as AD2
D2
A
B C
30k 2k/ft
15ft 25ft
10ft
D1 2k/ft
6ft
36ʹ k
12k
• Restrain all the degrees of freedom to get the restrained structure.
Basic kinematic determinate structure ( BKDS) or
restrained structure
A B C
Stiffness Method for Beams Analysis
• Step # 02 : Compute ADL matrix.
i. Due to direct loadings ( ADLʹ)
Stiffness Method for Beams Analysis
A B C
1
2
30k
2k/ft
10ft 25ft
B
19.44k 10.56k
25k 25k
168.4ʹ k 72ʹ k 104.17ʹ k 104.17ʹ k
ADL1ʹ ADL2ʹ
15ft
𝐴𝐷𝐿ˊ =
𝐴𝐷𝐿1ˊ
𝐴𝐷𝐿2ˊ
=
−32.17
104.17
Note:
For Fixed end actions see
formulas tables in module 5.
• Step # 02 : Compute ADL matrix.
ii. Due to indirect loadings i.e due to 0.002 rad C.W rotation at A
( ADLʺ)
Stiffness Method for Beams Analysis
𝐴𝐷𝐿ˊˊ =
𝐴𝐷𝐿1ˊˊ
𝐴𝐷𝐿2ˊˊ
=
26.66
0
A B
C
1
2
θ=0.002 rad
𝟒𝑬𝑰θ
𝟐𝟓
= 53.33ʹ k
3.2k
26.66ʹ k
3.2k
25ft
25ft
• Step # 02 : Compute ADL matrix.
iii. Due to indirect loadings i.e due to settlement at B ( ADLʺʹ)
Stiffness Method for Beams Analysis
𝐴𝐷𝐿ˊˊˊ =
𝐴𝐷𝐿1ˊˊˊ
𝐴𝐷𝐿2ˊˊˊ
=
0
100
Δ = 0.75ʺ = 0.0625 ft
𝑴𝑨 =
𝟔𝑬𝑰Δ𝑨
𝑳𝟐 = 100ʹ k
A B C
25ft
25ft
8k
8k 8k
8k
100ʹ k 100ʹ k
100ʹ k
ADL1ʹʺ
ADL2ʹʹʹ
• Step # 02 : Compute ADL matrix.
iv. Due to indirect loadings i.e due to settlement at B ( ADLʺʹʹ)
Stiffness Method for Beams Analysis
𝐴𝐷𝐿ˊˊˊˊ =
𝐴𝐷𝐿1ˊˊˊˊ
𝐴𝐷𝐿2ˊˊˊˊ
=
−66.67
−66.67
Δ = 0.5ʺ = 0.041 ft
A
B
C
Δ = 0.5in = 0.041ft
𝑴𝑨 =
𝟔𝑬𝑰Δ𝑨
𝑳𝟐
= 𝟔𝟔. 𝟔𝟕ˊ𝒌
5.33k
66.67ʹ k
5.33k
25ft
25ft
𝐴𝐷𝐿1 = 𝐴𝐷𝐿1ˊ + 𝐴𝐷𝐿1ˊˊ + 𝐴𝐷𝐿1ˊˊˊ + 𝐴𝐷𝐿1ˊˊˊˊ
𝐴𝐷𝐿 = 𝐴𝐷𝐿1
𝐴𝐷𝐿2
= −72.17
137.50
Stiffness Method for Beams Analysis
𝐴𝐷𝐿2 = 𝐴𝐷𝐿2ˊ + 𝐴𝐷𝐿2ˊˊ + 𝐴𝐷𝐿2ˊˊˊ + 𝐴𝐷𝐿2ˊˊˊˊ
𝐴𝐷𝐿1 = −32.17 + 26.67 + 0 − 67.67 = −72.17
𝐴𝐷𝐿2 = 104.17 + 0 + 100 − 67.67 = 137.50
Step # 03 : Computation of stiffness coefficients matrix
• 1st a unit rotation is applied at location 1 & prevent it at 2
( D1= 1 & D2=0) as shown. Compute the values of S11 and S21 .
Stiffness Method for Beams Analysis
A
B
C
1
2
θ=1
26666.67ˊk
26666.67ˊk
13333.33ˊk
13333.33ˊk
1600k
1600k
1600k
1600k
𝑆11 = 53333.33 𝑆21 = 13333.33
ii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 (D1 = 0 & D2 = 1) as shown.
Compute the values of S12 & S22 .
Stiffness Method for Beams Analysis
A
B C
1
2
θ=1
25ft 25ft
26666.67ˊk
13333.33ˊk
1600k
1600k
𝑆12 = 13333.33 𝑆22 = 26666.62
S =
53333.33 13333.33
13333.33 26666.62
𝑆 =
𝑆11 𝑆12
𝑆21 𝑆22
⇒
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
𝐴𝐷1
𝐴𝐷2
=
𝐴𝐷𝐿1
𝐴𝐷𝐿2
+
𝑆11 𝑆12
𝑆21 𝑆22
𝐷1
𝐷2
Stiffness Method for Beams Analysis
𝐷1
𝐷2
=
𝑆11 𝑆12
𝑆21 𝑆22
−1
𝐴𝐷1 − 𝐴𝐷𝐿1
𝐴𝐷2 − 𝐴𝐷𝐿2
𝐷1
𝐷2
=
53333.33 13333.33
13333.33 26666.62
−1 0 − (−72.17)
36 − (137.50)
𝐷1
𝐷2
=
0.0026
0.0051
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.
Step # 05(b): Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Stiffness Method for Beams Analysis
A B C
AM1 AM2 AM3
AM4
AM7
AM6
AM5 AM8
i. Compute AML values.
Stiffness Method for Beams Analysis
𝐴𝑀𝐿1
𝐴𝑀𝐿2
𝐴𝑀𝐿3
𝐴𝑀𝐿4
𝐴𝑀𝐿5
𝐴𝑀𝐿6
𝐴𝑀𝐿7
𝐴𝑀𝐿8
=
24.24
5.76
22.34
27.64
−154.67
−1.33
−70.81
137.50
So AML values from the previous
slides is.
A B C
AML1 AML2 AML3
AML4
AML7
AML6
AML5 AML8
25ft 25ft
b) Compute the AMD values.
• 1st apply a unit rotation at redundant location 1 and then at 2 as
shown below.
Stiffness Method for Beams Analysis
θ=1
A
B
C
AMD11
AMD21 AMD31
AMD41
AMD61
AMD51
1
2
AMD71 AMD81
A
B
C
1
2
θ=1
26666.67ˊk
26666.67ˊk
13333.33ˊk
13333.33ˊk
1600k
1600k
1600k
1600k
Stiffness Method for Beams Analysis
θ=1
A
B
C
AMD12
AMD22 AMD32
AMD42
AMD62
AMD51
1
2
AMD72
AMD82
A
B C
1
2
θ=1
25ft 25ft
26666.67ˊk
13333.33ˊk
1600k
1600k
𝐴𝑀𝐷 7 ∗ 2 =
𝐴𝑀𝐷11 𝐴𝑀𝐷12
𝐴𝑀𝐷21 𝐴𝑀𝐷22
𝐴𝑀𝐷31 𝐴𝑀𝐷32
𝐴𝑀𝐷41 𝐴𝑀𝐷42
𝐴𝑀𝐷51 𝐴𝑀𝐷52
𝐴𝑀𝐷61 𝐴𝑀𝐷62
𝐴𝑀𝐷71 𝐴𝑀𝐷72
𝐴𝑀𝐷81 𝐴𝑀𝐷82
=
−1600 0
1600 0
−1600 −1600
1600 1600
13333.33 0
26666.67 0
26666.67 13333.33
13333.33 26666.67
So the AMD values are
Stiffness Method for Beams Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Stiffness Method for Beams Analysis
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
𝐴𝑀7
𝐴𝑀8
=
24.24
5.76
22.34
27.64
−154.67
−1.33
−70.81
137.50
+
−1600 0
1600 0
−1600 −1600
1600 1600
13333.33 0
26666.67 0
26666.67 13333.33
13333.33 26666.67
0.0026
0.0051
=
20.03 k
9.93 k
26.32 k
23.68 k
−119.55ˊk
68.91ˊ k
−68.91ˊ k
36ˊ k
Stiffness Method for Beams Analysis
Class activity: Draw Shear force and bending moment diagrams.
Complete analyzed structure.
30k
2k/ft
A B C
20.03k 9.93k 26.32k 23.68k
119.55ʹ k 68.91ʹ k 68.91ʹ k
25ft 25ft
36ʹ k
6ft
12k From over
hanging portion
Stiffness Method for Beams Analysis
Problem 03: Analyze the given beam using stiffness method.
Take
EI = constant
K.I = 3 degree
So three redundant joint displacements should be chosen.(ignoring the axial effects).
A
B
1.5k/ft
10ft 20ft
• As the I.H is not at the middle of the beam so θ1 ≠ θ2 so θ1 , θ2 and Δ
need to be restrained.
A
B
Δ
θ1 θ2
𝐷 3 ∗ 1 =
𝐷1
𝐷2
𝐷3
=
?
?
?
𝐴𝐷 3 ∗ 1 =
𝐴𝐷1
𝐴𝐷2
𝐴𝐷3
=
0
0
0
• Step # 01: Selection of redundant Joint displacements and assign
coordinates at those locations. Also compute AD values.
Two Rotations & one translation at I.H is taken as redundant joint
displacement as shown.
Stiffness Method for Beams Analysis
A
B
1.5k/ft
10ft 20ft
D1 D2
D3
• Restrain all the degrees of freedom to get the restrained structure.
Basic kinematic determinate structure ( BKDS) or
restrained structure
A B
Stiffness Method for Beams Analysis
• Step # 02 : Compute ADL matrix.
Stiffness Method for Beams Analysis
Note:
For Fixed end actions see
formulas tables in module 5.
A 1
1.5k/ft
10ft 20ft
B
7.5k 7.5k 15k 15k
12.5ʹ k 12.5ʹ k 50ʹ k 50ʹ k
2
1.5k/ft
𝐴𝐷𝐿 =
𝐴𝐷𝐿1
𝐴𝐷𝐿2
𝐴𝐷𝐿3
=
12.5
−50
−22.5
Step # 03 : Computation of stiffness coefficients matrix
• 1st a unit rotation is applied at location 1 & prevent it at 2 & 3.
( D1= 1 , D2=0 & D3=0) as shown.
Compute the values of S11 , S21 & S31.
Stiffness Method for Beams Analysis
𝑆11 = 0.4𝐸𝐼 𝑆21 = 0 𝑆31 = −0.06𝐸𝐼
θ=1
A B
1 2
0.06EI
0.4EI
0.06EI
3
0.2EI
10ft 20ft
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
Stiffness Method for Beams Analysis
𝑆12 = 0 𝑆22 = 0.2𝐸𝐼 𝑆32 = 0.015𝐸𝐼
A
B
1 2
10ft
20ft
0.1𝐸𝐼
0.2𝐸𝐼
0.015𝐸𝐼
θ=1
0.015𝐸𝐼
3
ii. Now a unit rotation is applied at the redundant displacement
location 2 and prevented at 1 & 3 (D3 = 1 & D1 = D2 = 0) as shown.
Compute the values of S13 , S23 & S33
Stiffness Method for Beams Analysis
𝑆13 = −0.06𝐸𝐼 𝑆23 = 0.015𝐸𝐼 𝑆32 = 0.0135𝐸𝐼
A B
20ft
10ft
0.015EI
Δ
0.015EI
0.06EI
0.06EI
0.0015EI
0.0015EI
0.012EI
0.012EI
1 2
3
𝑆11 = 0.4𝐸𝐼 𝑆21 = 0 𝑆31 = −0.06𝐸𝐼
𝑆12 = 0 𝑆22 = 0.2𝐸𝐼 𝑆32 = 0.015𝐸𝐼
𝑆13 = −0.06𝐸𝐼 𝑆23 = 0.015𝐸𝐼 𝑆32 = 0.0135𝐸𝐼
Stiffness Method for Beams Analysis
So Stiffness coefficient matrix will be
S =
0.4 0 −0.06
0 0.2 0.015
−0.06 0.015 0.0135
Step # 04: Apply equilibrium condition at the location of the
redundant joint displacement to write equilibrium
equations and solve for unknown joint displacement.
Stiffness Method for Beams Analysis
𝐷1
𝐷2
𝐷3
=
1
𝐸𝐼
0.4 0 −0.06
0 0.2 0.015
−0.06 0.015 0.0135
−1 0 − 12.5
0 − −50
0 − (−22.5)
𝐷1
𝐷2
𝐷3
=
718.75
−125
5000
1
𝐸𝐼
𝐴𝐷 3 ∗ 1 = 𝐴𝐷𝐿 3 ∗ 1 + 𝑆 3 ∗ 3 • 𝐷 3 ∗ 1
𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳
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.
Step # 05(b): Compute the member end actions. As we know that
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Stiffness Method for Beams Analysis
A B
AM1
AM2
AM5
AM4
AM3 AM6
10ft 20ft
i. Compute AML values.
Stiffness Method for Beams Analysis
𝐴𝑀𝐿1
𝐴𝑀𝐿2
𝐴𝑀𝐿3
𝐴𝑀𝐿4
𝐴𝑀𝐿5
𝐴𝑀𝐿6
=
7.5
15
−12.5
12.5
−50
50
A 1
1.5k/ft
10ft 20ft
B
7.5k 7.5k 15k 15k
12.5ʹ k 12.5ʹ k 50ʹ k 50ʹ k
2
1.5k/ft
A B
AML1
AML2
AML5
AML4
AML3 AML6
b) Compute the AMD values.
• 1st apply a unit rotation at redundant location 1 then at 2 then a
unit translation at 3 as shown below.
Stiffness Method for Beams Analysis
A B
1 2
0.06EI
0.4EI
0.06EI
3
0.2EI
A B
AMD11
AMD21
AMD51
AMD41
AMD31 AMD61
Stiffness Method for Beams Analysis
A
B
1 2
10ft
20ft
0.1𝐸𝐼
0.2𝐸𝐼
0.015𝐸𝐼
θ=1
0.015𝐸𝐼
3
A B
AMD12
AMD22
AMD52
AMD42
AMD32 AMD62
10ft 20ft
Stiffness Method for Beams Analysis
A B
20ft
10ft
0.015EI
Δ
0.015EI
0.06EI
0.06EI
0.0015EI
0.0015EI
0.012EI
0.012EI
1 2
3
A B
AMD13
AMD23
AMD53
AMD43
AMD33 AMD63
𝐴𝑀𝐷 =
𝐴𝑀𝐷11 𝐴𝑀𝐷12 𝐴𝑀𝐷13
𝐴𝑀𝐷21 𝐴𝑀𝐷22 𝐴𝑀𝐷23
𝐴𝑀𝐷31 𝐴𝑀𝐷32 𝐴𝑀𝐷33
𝐴𝑀𝐷41 𝐴𝑀𝐷42 𝐴𝑀𝐷43
𝐴𝑀𝐷51 𝐴𝑀𝐷52 𝐴𝑀𝐷53
𝐴𝑀𝐷61 𝐴𝑀𝐷62 𝐴𝑀𝐷63
=
−0.06 0 0.012
0 0.015 0.0015
0.2 0 −0.06
0.4 1600 −0.06
0 0.2 0.015
0 0.1 0.015
So the AMD values are
Stiffness Method for Beams Analysis
Now member end actions will be computed as given below
𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷
Stiffness Method for Beams Analysis
𝐴𝑀1
𝐴𝑀2
𝐴𝑀3
𝐴𝑀4
𝐴𝑀5
𝐴𝑀6
=
7.5
15
−12.5
12.5
−50
50
+
−0.06 0 0.012
0 0.015 0.0015
0.2 0 −0.06
0.4 1600 −0.06
0 0.2 0.015
0 0.1 0.015
718.75
−125
5000
=
24. 38k
20.63 k
−168.75ˊk
0
0
112.5ˊ k
Complete analyzed structure.
A B
24.38k 20.63k
0
0
168.75ˊk 112.5ˊ k
10ft 20ft
1.5k/ft
Now Shear force and Bending moment diagram
Stiffness Method for Beams Analysis
24.38
168.75
13.75
SFD
15
20.63
45
45
29.37
BMD
0
0
A B
24.38k 20.63k
0
0
168.75ˊk 112.5ˊ k
10ft 20ft
1.5k/ft
20.63
1.5
= 13.75ʹ
112.5
Assignment # 03(b) : Analyze the given beams using stiffness method.
Stiffness Method for Beams Analysis
A
B C
30k
2k/ft
10ft
10ft
20ft
D1 Take
EI = constant
Take
EI = constant
A
B
1.5k/ft
15ft 15ft
Take
E = 30000 ksi
I = 800 in4
EI = 166666.6 k-ft2
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

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Module 6.pdf

  • 1. University of Engineering & Technology Peshawar, Pakistan CE301: Structure Analysis II Module 06: Analysis of S.I Beams Using stiffness method By: Prof. Dr. Bashir Alam Civil Engineering Department UET , Peshawar
  • 2. Topics to be Covered • Introduction • Prerequisites for using stiffness method • Step wise procedure of stiffness method for beam analysis • Analysis of beam Example 1 • Example 2 • Example 3 • Assignment 03 (b)
  • 3. Stiffness Method for Beams Analysis  Introduction: Beams 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 & bending moment at any section of the continuous beam.
  • 4. Stiffness Method for Beams 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
  • 5.  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 Beams Analysis
  • 6. 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 Beams Analysis
  • 7. Step # 04: Apply equilibrium condition at the location of the redundant joint displacement to write equilibrium equations and solve for unknown joint displacement. 𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳 Step # 05: Compute the member end actions . 𝑨𝑫 = 𝑨𝑫𝑳 + 𝑺 • 𝑫 Stiffness Method for Beams Analysis
  • 8. Problem 01: Analyze the given beam using stiffness method. A B C 30k 20k 12ft 12ft 6ft Take EI = constant K.I = 2 degree ( neglecting the axial effects ) So two redundant joint displacements should be chosen. Stiffness Method for Beams Analysis
  • 9. 𝐷 2 ∗ 1 = 𝐷1 𝐷2 = ? ? 𝐴𝐷 2 ∗ 1 = 𝐴𝐷1 𝐴𝐷2 = 0 0 A B C 30k 20k 12ft 12ft D1 D2 1 2 • Step # 01: Selection of redundant Joint displacements and assign coordinates at those locations. Also compute AD values. Rotation at B & C is taken as redundant joint displacement. Stiffness Method for Beams Analysis
  • 10. • Restrain all the degrees of freedom to get the restrained structure. Basic kinematic determinate structure ( BKDS) or restrained structure A B C Stiffness Method for Beams Analysis
  • 11. • 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) A B P1 L ADL1 C 1 2 P2 L B ADL2 P1 2 P1 2 P2 2 P2 2 P1L 8 P1L 8 P2L 8 P2L 8 A B C 1 2 30k 20k 12ft 12ft B ADL2 15k 15k 10k 10k 45ʹ k 45ʹ k ADL1 30ʹ k 30ʹ k Stiffness Method for Beams Analysis
  • 12. 𝐴𝐷𝐿2 = 30ˊ 𝑘 𝐴𝐷𝐿1 = 45 − 30 = 15ˊ 𝑘 𝐴𝐷𝐿 = 𝐴𝐷𝐿1 𝐴𝐷𝐿2 = 15 30 ADL1 = moment in the restrained structure under the actual loads corresponding to redundant joint displacement 1. ADL2 = moment in the restrained structure under the actual loads corresponding to redundant joint displacement 2. Note: For rotation corresponding action is moment. Stiffness Method for Beams Analysis
  • 13. 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 & prevent it 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 Beams Analysis
  • 14. 𝟐𝑬𝑰𝜽 𝑳 A B C 1 2 θ=1 𝟒𝑬𝑰𝜽 𝑳 𝟒𝑬𝑰𝜽 𝑳 𝟐𝑬𝑰𝜽 𝑳 𝟔𝑬𝑰𝜽 𝑳𝟐 𝟔𝑬𝑰𝜽 𝑳𝟐 𝟔𝑬𝑰𝜽 𝑳𝟐 𝟔𝑬𝑰𝜽 𝑳𝟐 i. 1st a unit rotation is applied at location 1 & prevent it at 2 (D1=1 & D2=0)as shown. Compute the values of S11 and S21 . 𝟐𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟒𝑬𝑰 𝟏𝟐 𝟒𝑬𝑰 𝟏𝟐 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 12ft 12ft L L Stiffness Method for Beams Analysis
  • 15. Step # 03 ( i ): Contd… 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 𝑆11 = 4𝐸𝐼 12 + 4𝐸𝐼 12 𝑆11 = 8EI 12 𝑆21 = 2𝐸𝐼 12 𝑆21 = 2EI 12 Stiffness Method for Beams Analysis
  • 16. 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 . where 𝟒𝑬𝑰𝜽 𝑳 A B C 1 2 θ=1 𝟐𝑬𝑰𝜽 𝑳 𝟔𝑬𝑰𝜽 𝑳𝟐 𝟔𝑬𝑰𝜽 𝑳𝟐 𝟒𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 L L 12ft 12ft Stiffness Method for Beams Analysis
  • 17. 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. 𝑆12 = 2𝐸𝐼 12 𝑆12 = 2EI 12 𝑆21 = 4𝐸𝐼 12 𝑆21 = 4EI 12 Step # 03 : Contd… Stiffness Method for Beams Analysis
  • 18. 𝑆11 = 8𝐸𝐼 12 𝑆21 = 2𝐸𝐼 12 𝑆12 = 2𝐸𝐼 12 𝑆22 = 4𝐸𝐼 12 𝑆 = 𝑆11 𝑆12 𝑆21 𝑆22 S = 𝐸𝐼 8 12 2 12 2 12 4 12 Stiffness coefficient matrix Step # 03 : Contd… Stiffness Method for Beams Analysis
  • 19. 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 𝐴𝐷2 = 𝐴𝐷𝐿1 𝐴𝐷𝐿2 + 𝑆11 𝑆12 𝑆21 𝑆22 𝐷1 𝐷2 𝐴𝐷 2 ∗ 1 = 𝐴𝐷𝐿 2 ∗ 1 + 𝑆 2 ∗ 2 • 𝐷 2 ∗ 1 𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳 Stiffness Method for Beams Analysis
  • 20. 𝐷1 𝐷2 = 𝑆11 𝑆12 𝑆21 𝑆22 −1 𝐴𝐷1 − 𝐴𝐷𝐿1 𝐴𝐷2 − 𝐴𝐷𝐿2 𝐷1 𝐷2 = 12 𝐸𝐼 8 2 2 4 −1 0 − 15 0 − 30 𝐷1 𝐷2 = 0 −90 -ive sign shows that our assumed redundant joint displacement direction is wrong Stiffness Method for Beams Analysis
  • 21. Step # 05(a): Before computing the member end action we can compute the support reactions directly using matrix approach . A B C AS3 AS1 AS2 𝐴𝑆 = 𝐴𝑆𝐿 + 𝐴𝑆𝐷 𝐷 Stiffness Method for Beams Analysis
  • 22. A B C 1 2 30k 20 12ft 12ft B 15k 15k 10k 10k 45ʹ k 45ʹ k 30ʹ k 30ʹ k i. Compute ASL values. 𝐴𝑆𝐿1 𝐴𝑆𝐿2 𝐴𝑆𝐿3 = 25 10 15 Stiffness Method for Beams Analysis A B C ASL3 ASL1 ASL2 P1 P2
  • 23. ii. Compute the ASD values. • 1st apply a unit rotation at redundant location 1 and then at 2 in the restrained structure as shown below. A B C ASD31 ASD11 ASD21 Stiffness Method for Beams Analysis 𝟐𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟒𝑬𝑰 𝟏𝟐 𝟒𝑬𝑰 𝟏𝟐 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 12ft 12ft
  • 24. A B C ASD32 ASD12 ASD22 Stiffness Method for Beams Analysis 𝟒𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 12ft 12ft
  • 25. 𝐴𝑆𝐷 3 ∗ 2 = 𝐴𝑆𝐷11 𝐴𝑆𝐷12 𝐴𝑆𝐷21 𝐴𝑆𝐷22 𝐴𝑆𝐷31 𝐴𝑆𝐷32 = EI 0 −0.041 0.041 0.041 −0.5 0 So the ASD values are 𝐴𝑆1 𝐴𝑆2 𝐴𝑆3 = 𝐴𝑆𝐿1 𝐴𝑆𝐿2 𝐴𝑆𝐿3 + 𝐴𝑆𝐷11 𝐴𝑆𝐷12 𝐴𝑆𝐷21 𝐴𝑆𝐷22 𝐴𝑆𝐷31 𝐴𝑆𝐷32 𝐷1 𝐷2 Stiffness Method for Beams Analysis
  • 26. So the AS values are 𝐴𝑆1 𝐴𝑆2 𝐴𝑆3 = 25 10 15 +EI 0 −0.041 0.041 0.041 −0.5 0 0 −90 𝐴𝑆1 𝐴𝑆2 𝐴𝑆3 = 28.75 6.246 15 30k 20k Structure with known support reactions Stiffness Method for Beams Analysis A B C 15k 28.75k 6.246k 12ft 12ft
  • 27. A B C AM1 AM2 AM3 AM4 AM7 AM6 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. Step # 05(b): Compute the member end actions. As we know that 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 Stiffness Method for Beams Analysis
  • 28. i. Compute AML values. A B P1 L C 2 P2 L B AML1 AML2 AML3 AML4 AML7 AML6 AML5 A B C 1 2 30k 20 12ft 12ft B 15k 15k 10k 10k 45ʹ k 45ʹ k 30ʹ k 30ʹ k Stiffness Method for Beams Analysis
  • 30. θ=1 A B C AMD11 AMD21 AMD31 AMD41 AMD61 AMD51 1 2 AMD71 b) Compute the AMD values. • 1st apply a unit rotation at redundant location 1 and then at 2 as shown below. 𝟐𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟒𝑬𝑰 𝟏𝟐 𝟒𝑬𝑰 𝟏𝟐 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 Stiffness Method for Beams Analysis
  • 31. θ=1 A B C AMD12 AMD22 AMD32 AMD42 AMD62 AMD51 1 2 AMD72 Stiffness Method for Beams Analysis 𝟒𝑬𝑰 𝟏𝟐 A B C 1 2 θ=1 𝟐𝑬𝑰 𝟏𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 𝟔𝑬𝑰 𝟏𝟐𝟐 12ft 12ft
  • 32. 𝐴𝑀𝐷 7 ∗ 2 = 𝐴𝑀𝐷11 𝐴𝑀𝐷12 𝐴𝑀𝐷21 𝐴𝑀𝐷22 𝐴𝑀𝐷31 𝐴𝑀𝐷32 𝐴𝑀𝐷41 𝐴𝑀𝐷42 𝐴𝑀𝐷51 𝐴𝑀𝐷52 𝐴𝑀𝐷61 𝐴𝑀𝐷62 𝐴𝑀𝐷71 𝐴𝑀𝐷72 = −0.041 0 0.041 0 −0.041 −0.041 0.041 0.041 0.16 0 0.33 0 0.33 0.16 So the AMD values are Stiffness Method for Beams Analysis
  • 33. Now member end actions will be computed as given below 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 𝐴𝑀1 𝐴𝑀2 𝐴𝑀3 𝐴𝑀4 𝐴𝑀5 𝐴𝑀6 𝐴𝑀7 = 𝐴𝑀𝐿1 𝐴𝑀𝐿2 𝐴𝑀𝐿3 𝐴𝑀𝐿4 𝐴𝑀𝐿5 𝐴𝑀𝐿6 𝐴𝑀𝐿7 + 𝐴𝑀𝐷11 𝐴𝑀𝐷12 𝐴𝑀𝐷21 𝐴𝑀𝐷22 𝐴𝑀𝐷31 𝐴𝑀𝐷32 𝐴𝑀𝐷41 𝐴𝑀𝐷42 𝐴𝑀𝐷51 𝐴𝑀𝐷52 𝐴𝑀𝐷61 𝐴𝑀𝐷62 𝐴𝑀𝐷71 𝐴𝑀𝐷72 𝐷1 𝐷2 Stiffness Method for Beams Analysis
  • 34. 𝐴𝑀1 𝐴𝑀2 𝐴𝑀3 𝐴𝑀4 𝐴𝑀5 𝐴𝑀6 𝐴𝑀7 = 15 15 10 10 −45 45 −30 + −0.041 0 0.041 0 −0.041 −0.041 0.041 0.041 0.16 0 0.33 0 0.33 0.16 0 −90 = 15 k 15 k 13.75 k 6.247 k −45 ˊ k 45ˊ k −45ˊ k Complete analyzed structure. Shear force and bending moment diagrams are given on next slide. A B C 15k 15k 13.75k 6.247k 45ʹ k 45ʹ k 45ʹ k 12ft 12ft Stiffness Method for Beams Analysis
  • 35. A B C 15k 15k 13.75k 6.247k 45ʹ k 45ʹ k 44.4ʹ k 12ft 12ft 30k 20k 15 45 13.75 6.247 SFD 15 15 15 6.247 45 45 37.5 BMD 0 0 Now Shear force and Bending moment diagram Stiffness Method for Beams Analysis
  • 36. Stiffness Method for Beams Analysis Problem 02: Analyze the given beam using stiffness method, if support A rotates by 0.002 rad clockwise, support B settles down by 0.75 in & support C settles down by 0.5 in. Take E = 30000 ksi I = 800 in4 EI = 166666.6 k-ft2 K.I = 2 degree So two redundant joint displacements should be chosen.(ignoring the axial effects). A B C 30k θ=0.002 rad Δ1= 0.75ʺ 2k/ft Δ2= 0.5ʺ A B C 30k 2k/ft 25ft 25ft 10ft 6ft
  • 37. 𝐷 2 ∗ 1 = 𝐷1 𝐷2 = ? ? 𝐴𝐷 2 ∗ 1 = 𝐴𝐷1 𝐴𝐷2 = 0 36 • Step # 01: Selection of redundant Joint displacements and assign coordinates at those locations. Also compute AD values. Rotation at B & C is taken as redundant joint displacement. Stiffness Method for Beams Analysis The effect of overhanging part is taken as AD2 D2 A B C 30k 2k/ft 15ft 25ft 10ft D1 2k/ft 6ft 36ʹ k 12k
  • 38. • Restrain all the degrees of freedom to get the restrained structure. Basic kinematic determinate structure ( BKDS) or restrained structure A B C Stiffness Method for Beams Analysis
  • 39. • Step # 02 : Compute ADL matrix. i. Due to direct loadings ( ADLʹ) Stiffness Method for Beams Analysis A B C 1 2 30k 2k/ft 10ft 25ft B 19.44k 10.56k 25k 25k 168.4ʹ k 72ʹ k 104.17ʹ k 104.17ʹ k ADL1ʹ ADL2ʹ 15ft 𝐴𝐷𝐿ˊ = 𝐴𝐷𝐿1ˊ 𝐴𝐷𝐿2ˊ = −32.17 104.17 Note: For Fixed end actions see formulas tables in module 5.
  • 40. • Step # 02 : Compute ADL matrix. ii. Due to indirect loadings i.e due to 0.002 rad C.W rotation at A ( ADLʺ) Stiffness Method for Beams Analysis 𝐴𝐷𝐿ˊˊ = 𝐴𝐷𝐿1ˊˊ 𝐴𝐷𝐿2ˊˊ = 26.66 0 A B C 1 2 θ=0.002 rad 𝟒𝑬𝑰θ 𝟐𝟓 = 53.33ʹ k 3.2k 26.66ʹ k 3.2k 25ft 25ft
  • 41. • Step # 02 : Compute ADL matrix. iii. Due to indirect loadings i.e due to settlement at B ( ADLʺʹ) Stiffness Method for Beams Analysis 𝐴𝐷𝐿ˊˊˊ = 𝐴𝐷𝐿1ˊˊˊ 𝐴𝐷𝐿2ˊˊˊ = 0 100 Δ = 0.75ʺ = 0.0625 ft 𝑴𝑨 = 𝟔𝑬𝑰Δ𝑨 𝑳𝟐 = 100ʹ k A B C 25ft 25ft 8k 8k 8k 8k 100ʹ k 100ʹ k 100ʹ k ADL1ʹʺ ADL2ʹʹʹ
  • 42. • Step # 02 : Compute ADL matrix. iv. Due to indirect loadings i.e due to settlement at B ( ADLʺʹʹ) Stiffness Method for Beams Analysis 𝐴𝐷𝐿ˊˊˊˊ = 𝐴𝐷𝐿1ˊˊˊˊ 𝐴𝐷𝐿2ˊˊˊˊ = −66.67 −66.67 Δ = 0.5ʺ = 0.041 ft A B C Δ = 0.5in = 0.041ft 𝑴𝑨 = 𝟔𝑬𝑰Δ𝑨 𝑳𝟐 = 𝟔𝟔. 𝟔𝟕ˊ𝒌 5.33k 66.67ʹ k 5.33k 25ft 25ft
  • 43. 𝐴𝐷𝐿1 = 𝐴𝐷𝐿1ˊ + 𝐴𝐷𝐿1ˊˊ + 𝐴𝐷𝐿1ˊˊˊ + 𝐴𝐷𝐿1ˊˊˊˊ 𝐴𝐷𝐿 = 𝐴𝐷𝐿1 𝐴𝐷𝐿2 = −72.17 137.50 Stiffness Method for Beams Analysis 𝐴𝐷𝐿2 = 𝐴𝐷𝐿2ˊ + 𝐴𝐷𝐿2ˊˊ + 𝐴𝐷𝐿2ˊˊˊ + 𝐴𝐷𝐿2ˊˊˊˊ 𝐴𝐷𝐿1 = −32.17 + 26.67 + 0 − 67.67 = −72.17 𝐴𝐷𝐿2 = 104.17 + 0 + 100 − 67.67 = 137.50
  • 44. Step # 03 : Computation of stiffness coefficients matrix • 1st a unit rotation is applied at location 1 & prevent it at 2 ( D1= 1 & D2=0) as shown. Compute the values of S11 and S21 . Stiffness Method for Beams Analysis A B C 1 2 θ=1 26666.67ˊk 26666.67ˊk 13333.33ˊk 13333.33ˊk 1600k 1600k 1600k 1600k 𝑆11 = 53333.33 𝑆21 = 13333.33
  • 45. ii. Now a unit rotation is applied at the redundant displacement location 2 and prevented at 1 (D1 = 0 & D2 = 1) as shown. Compute the values of S12 & S22 . Stiffness Method for Beams Analysis A B C 1 2 θ=1 25ft 25ft 26666.67ˊk 13333.33ˊk 1600k 1600k 𝑆12 = 13333.33 𝑆22 = 26666.62 S = 53333.33 13333.33 13333.33 26666.62 𝑆 = 𝑆11 𝑆12 𝑆21 𝑆22 ⇒
  • 46. Step # 04: Apply equilibrium condition at the location of the redundant joint displacement to write equilibrium equations and solve for unknown joint displacement. 𝐴𝐷1 𝐴𝐷2 = 𝐴𝐷𝐿1 𝐴𝐷𝐿2 + 𝑆11 𝑆12 𝑆21 𝑆22 𝐷1 𝐷2 Stiffness Method for Beams Analysis 𝐷1 𝐷2 = 𝑆11 𝑆12 𝑆21 𝑆22 −1 𝐴𝐷1 − 𝐴𝐷𝐿1 𝐴𝐷2 − 𝐴𝐷𝐿2 𝐷1 𝐷2 = 53333.33 13333.33 13333.33 26666.62 −1 0 − (−72.17) 36 − (137.50) 𝐷1 𝐷2 = 0.0026 0.0051
  • 47. 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. Step # 05(b): Compute the member end actions. As we know that 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 Stiffness Method for Beams Analysis A B C AM1 AM2 AM3 AM4 AM7 AM6 AM5 AM8
  • 48. i. Compute AML values. Stiffness Method for Beams Analysis 𝐴𝑀𝐿1 𝐴𝑀𝐿2 𝐴𝑀𝐿3 𝐴𝑀𝐿4 𝐴𝑀𝐿5 𝐴𝑀𝐿6 𝐴𝑀𝐿7 𝐴𝑀𝐿8 = 24.24 5.76 22.34 27.64 −154.67 −1.33 −70.81 137.50 So AML values from the previous slides is. A B C AML1 AML2 AML3 AML4 AML7 AML6 AML5 AML8 25ft 25ft
  • 49. b) Compute the AMD values. • 1st apply a unit rotation at redundant location 1 and then at 2 as shown below. Stiffness Method for Beams Analysis θ=1 A B C AMD11 AMD21 AMD31 AMD41 AMD61 AMD51 1 2 AMD71 AMD81 A B C 1 2 θ=1 26666.67ˊk 26666.67ˊk 13333.33ˊk 13333.33ˊk 1600k 1600k 1600k 1600k
  • 50. Stiffness Method for Beams Analysis θ=1 A B C AMD12 AMD22 AMD32 AMD42 AMD62 AMD51 1 2 AMD72 AMD82 A B C 1 2 θ=1 25ft 25ft 26666.67ˊk 13333.33ˊk 1600k 1600k
  • 51. 𝐴𝑀𝐷 7 ∗ 2 = 𝐴𝑀𝐷11 𝐴𝑀𝐷12 𝐴𝑀𝐷21 𝐴𝑀𝐷22 𝐴𝑀𝐷31 𝐴𝑀𝐷32 𝐴𝑀𝐷41 𝐴𝑀𝐷42 𝐴𝑀𝐷51 𝐴𝑀𝐷52 𝐴𝑀𝐷61 𝐴𝑀𝐷62 𝐴𝑀𝐷71 𝐴𝑀𝐷72 𝐴𝑀𝐷81 𝐴𝑀𝐷82 = −1600 0 1600 0 −1600 −1600 1600 1600 13333.33 0 26666.67 0 26666.67 13333.33 13333.33 26666.67 So the AMD values are Stiffness Method for Beams Analysis
  • 52. Now member end actions will be computed as given below 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 Stiffness Method for Beams Analysis 𝐴𝑀1 𝐴𝑀2 𝐴𝑀3 𝐴𝑀4 𝐴𝑀5 𝐴𝑀6 𝐴𝑀7 𝐴𝑀8 = 24.24 5.76 22.34 27.64 −154.67 −1.33 −70.81 137.50 + −1600 0 1600 0 −1600 −1600 1600 1600 13333.33 0 26666.67 0 26666.67 13333.33 13333.33 26666.67 0.0026 0.0051 = 20.03 k 9.93 k 26.32 k 23.68 k −119.55ˊk 68.91ˊ k −68.91ˊ k 36ˊ k
  • 53. Stiffness Method for Beams Analysis Class activity: Draw Shear force and bending moment diagrams. Complete analyzed structure. 30k 2k/ft A B C 20.03k 9.93k 26.32k 23.68k 119.55ʹ k 68.91ʹ k 68.91ʹ k 25ft 25ft 36ʹ k 6ft 12k From over hanging portion
  • 54. Stiffness Method for Beams Analysis Problem 03: Analyze the given beam using stiffness method. Take EI = constant K.I = 3 degree So three redundant joint displacements should be chosen.(ignoring the axial effects). A B 1.5k/ft 10ft 20ft • As the I.H is not at the middle of the beam so θ1 ≠ θ2 so θ1 , θ2 and Δ need to be restrained. A B Δ θ1 θ2
  • 55. 𝐷 3 ∗ 1 = 𝐷1 𝐷2 𝐷3 = ? ? ? 𝐴𝐷 3 ∗ 1 = 𝐴𝐷1 𝐴𝐷2 𝐴𝐷3 = 0 0 0 • Step # 01: Selection of redundant Joint displacements and assign coordinates at those locations. Also compute AD values. Two Rotations & one translation at I.H is taken as redundant joint displacement as shown. Stiffness Method for Beams Analysis A B 1.5k/ft 10ft 20ft D1 D2 D3
  • 56. • Restrain all the degrees of freedom to get the restrained structure. Basic kinematic determinate structure ( BKDS) or restrained structure A B Stiffness Method for Beams Analysis
  • 57. • Step # 02 : Compute ADL matrix. Stiffness Method for Beams Analysis Note: For Fixed end actions see formulas tables in module 5. A 1 1.5k/ft 10ft 20ft B 7.5k 7.5k 15k 15k 12.5ʹ k 12.5ʹ k 50ʹ k 50ʹ k 2 1.5k/ft 𝐴𝐷𝐿 = 𝐴𝐷𝐿1 𝐴𝐷𝐿2 𝐴𝐷𝐿3 = 12.5 −50 −22.5
  • 58. Step # 03 : Computation of stiffness coefficients matrix • 1st a unit rotation is applied at location 1 & prevent it at 2 & 3. ( D1= 1 , D2=0 & D3=0) as shown. Compute the values of S11 , S21 & S31. Stiffness Method for Beams Analysis 𝑆11 = 0.4𝐸𝐼 𝑆21 = 0 𝑆31 = −0.06𝐸𝐼 θ=1 A B 1 2 0.06EI 0.4EI 0.06EI 3 0.2EI 10ft 20ft
  • 59. 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 Stiffness Method for Beams Analysis 𝑆12 = 0 𝑆22 = 0.2𝐸𝐼 𝑆32 = 0.015𝐸𝐼 A B 1 2 10ft 20ft 0.1𝐸𝐼 0.2𝐸𝐼 0.015𝐸𝐼 θ=1 0.015𝐸𝐼 3
  • 60. ii. Now a unit rotation is applied at the redundant displacement location 2 and prevented at 1 & 3 (D3 = 1 & D1 = D2 = 0) as shown. Compute the values of S13 , S23 & S33 Stiffness Method for Beams Analysis 𝑆13 = −0.06𝐸𝐼 𝑆23 = 0.015𝐸𝐼 𝑆32 = 0.0135𝐸𝐼 A B 20ft 10ft 0.015EI Δ 0.015EI 0.06EI 0.06EI 0.0015EI 0.0015EI 0.012EI 0.012EI 1 2 3
  • 61. 𝑆11 = 0.4𝐸𝐼 𝑆21 = 0 𝑆31 = −0.06𝐸𝐼 𝑆12 = 0 𝑆22 = 0.2𝐸𝐼 𝑆32 = 0.015𝐸𝐼 𝑆13 = −0.06𝐸𝐼 𝑆23 = 0.015𝐸𝐼 𝑆32 = 0.0135𝐸𝐼 Stiffness Method for Beams Analysis So Stiffness coefficient matrix will be S = 0.4 0 −0.06 0 0.2 0.015 −0.06 0.015 0.0135
  • 62. Step # 04: Apply equilibrium condition at the location of the redundant joint displacement to write equilibrium equations and solve for unknown joint displacement. Stiffness Method for Beams Analysis 𝐷1 𝐷2 𝐷3 = 1 𝐸𝐼 0.4 0 −0.06 0 0.2 0.015 −0.06 0.015 0.0135 −1 0 − 12.5 0 − −50 0 − (−22.5) 𝐷1 𝐷2 𝐷3 = 718.75 −125 5000 1 𝐸𝐼 𝐴𝐷 3 ∗ 1 = 𝐴𝐷𝐿 3 ∗ 1 + 𝑆 3 ∗ 3 • 𝐷 3 ∗ 1 𝑫 = 𝑺 −𝟏 • 𝑨𝑫 − 𝑨𝑫𝑳
  • 63. 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. Step # 05(b): Compute the member end actions. As we know that 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 Stiffness Method for Beams Analysis A B AM1 AM2 AM5 AM4 AM3 AM6 10ft 20ft
  • 64. i. Compute AML values. Stiffness Method for Beams Analysis 𝐴𝑀𝐿1 𝐴𝑀𝐿2 𝐴𝑀𝐿3 𝐴𝑀𝐿4 𝐴𝑀𝐿5 𝐴𝑀𝐿6 = 7.5 15 −12.5 12.5 −50 50 A 1 1.5k/ft 10ft 20ft B 7.5k 7.5k 15k 15k 12.5ʹ k 12.5ʹ k 50ʹ k 50ʹ k 2 1.5k/ft A B AML1 AML2 AML5 AML4 AML3 AML6
  • 65. b) Compute the AMD values. • 1st apply a unit rotation at redundant location 1 then at 2 then a unit translation at 3 as shown below. Stiffness Method for Beams Analysis A B 1 2 0.06EI 0.4EI 0.06EI 3 0.2EI A B AMD11 AMD21 AMD51 AMD41 AMD31 AMD61
  • 66. Stiffness Method for Beams Analysis A B 1 2 10ft 20ft 0.1𝐸𝐼 0.2𝐸𝐼 0.015𝐸𝐼 θ=1 0.015𝐸𝐼 3 A B AMD12 AMD22 AMD52 AMD42 AMD32 AMD62 10ft 20ft
  • 67. Stiffness Method for Beams Analysis A B 20ft 10ft 0.015EI Δ 0.015EI 0.06EI 0.06EI 0.0015EI 0.0015EI 0.012EI 0.012EI 1 2 3 A B AMD13 AMD23 AMD53 AMD43 AMD33 AMD63
  • 68. 𝐴𝑀𝐷 = 𝐴𝑀𝐷11 𝐴𝑀𝐷12 𝐴𝑀𝐷13 𝐴𝑀𝐷21 𝐴𝑀𝐷22 𝐴𝑀𝐷23 𝐴𝑀𝐷31 𝐴𝑀𝐷32 𝐴𝑀𝐷33 𝐴𝑀𝐷41 𝐴𝑀𝐷42 𝐴𝑀𝐷43 𝐴𝑀𝐷51 𝐴𝑀𝐷52 𝐴𝑀𝐷53 𝐴𝑀𝐷61 𝐴𝑀𝐷62 𝐴𝑀𝐷63 = −0.06 0 0.012 0 0.015 0.0015 0.2 0 −0.06 0.4 1600 −0.06 0 0.2 0.015 0 0.1 0.015 So the AMD values are Stiffness Method for Beams Analysis
  • 69. Now member end actions will be computed as given below 𝐴𝑀 = 𝐴𝑀𝐿 + 𝐴𝑀𝐷 𝐷 Stiffness Method for Beams Analysis 𝐴𝑀1 𝐴𝑀2 𝐴𝑀3 𝐴𝑀4 𝐴𝑀5 𝐴𝑀6 = 7.5 15 −12.5 12.5 −50 50 + −0.06 0 0.012 0 0.015 0.0015 0.2 0 −0.06 0.4 1600 −0.06 0 0.2 0.015 0 0.1 0.015 718.75 −125 5000 = 24. 38k 20.63 k −168.75ˊk 0 0 112.5ˊ k Complete analyzed structure. A B 24.38k 20.63k 0 0 168.75ˊk 112.5ˊ k 10ft 20ft 1.5k/ft
  • 70. Now Shear force and Bending moment diagram Stiffness Method for Beams Analysis 24.38 168.75 13.75 SFD 15 20.63 45 45 29.37 BMD 0 0 A B 24.38k 20.63k 0 0 168.75ˊk 112.5ˊ k 10ft 20ft 1.5k/ft 20.63 1.5 = 13.75ʹ 112.5
  • 71. Assignment # 03(b) : Analyze the given beams using stiffness method. Stiffness Method for Beams Analysis A B C 30k 2k/ft 10ft 10ft 20ft D1 Take EI = constant Take EI = constant A B 1.5k/ft 15ft 15ft Take E = 30000 ksi I = 800 in4 EI = 166666.6 k-ft2
  • 72. 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