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Statically Indeterminate Frame ExampleStevenVukazich
SanJoseStateUniversity
Steps in Solving an Indeterminate Structure using the Force Method
Determine degree of Indeterminacy Let n =degree of indeterminacy
(i.e. the structure is indeterminate to the nth degree)
Define Primary Structure and the n Redundants
Define the Primary Problem
Solve for the nRelevant
Deflections in Primary Problem
Define the nRedundant Problems
Solve for the nRelevant Deflections in each Redundant
Problem
Write the nCompatibility Equations at
Relevant Points
Solve the nCompatibility
Equations to find the n Redundants
Use the Equations of Equilibrium to
solve for the remaining unknowns
Construct Internal Force
Diagrams (if necessary)
10 ft
30 kB10 k
5 ft 5 ft
A
C
D
Example Problem
For the indeterminate frame subjected to the point loads shown, find the support reactions and draw the bending moment diagram for the frame. EIis the same for both the horizontal and vertical members.
EI
EI
FBD of the Frame
10 ft
30 kB10 k
5 ft 5 ft
C
D
EI
EI
Ax
Ay
MA
Dy
X = 4
3n = 3(1) = 3
Frame is stable
Statically Indeterminate to the 1st degree
Define Primary Structure and Redundant• Remove all applied loads from the actual structure;• Remove support reactions or internal forces to define a primary structure;• Removed reactions or internal forces are called redundants;• Same number of redundants as degree of indeterminacy• Primary structure must be stable and statically determinate;• Primary structure is not unique – there are several choices.
Primary Structure Redundant
B
A
CMA
D
B
A
CD
Dy
Define and Solve the Primary Problem• Apply all loads on actual structure to the primary structure;• Define a reference coordinate system;• Calculate relevant deflections at points where redundants were
removed.
10 ft
30 kB10 k
5 ft 5 ft
A
C
D
EI
EI Δ"
Need to find Δ"Use the Principle of Virtual Work
Solve the Primary Problem
10 ft
30 kB10 k
5 ft 5 ft
A
C
D
EI
EI Δ"
Real System
Need to construct the Mpdiagram
Solve the Primary Problem
10 ft
B
5 ft 5 ft
A
C
D
EI
EI
Virtual System to measure Δ"
Need to construct the MQdiagram
1
FBD of the Primary Problem
#𝑀%
�
�
= 0+
#𝐹+
�
�
= 0+
#𝐹,
�
�
= 0+
MA = 250 k-ft
Ax = – 10 k
Ay = 30 k
10 ft
30 kB10 k
5 ft 5 ft
C
D
EI
EI
C
Ax
Ay
MA
B
A
D
C
150 k-ft
0
250 k-ft
0
MP Diagram for the Primary Problem
Moment diagram is drawn on the compression side of the member
FBD of the Virtual System
#𝑀%
�
�
= 0+
#𝐹+
�
�
= 0+
#𝐹,
�
�
= 0+
MA = 10 ft
Ax = 0
Ay = –1
10 ft
1
B10 k
5 ft 5 ft
C
D
EI
EI
C
Ax
Ay
MA
B
A
D
C
10 ft
10 ft
0
MQ Diagram for the Primary Problem
Moment diagram is drawn on the compression side of the member
10 ft5 ft
1 . ∆"= 1𝐸𝐼3 𝑀4𝑀5
6
7𝑑𝑥
Solve the Primary Problem
BA C10 ft 5 ftB
10 ft 10 ft5 ft
150 k-ft250 k-ft
150 k-ft
0
−16
5ft + 2 10ft 150k−ft 5ft
−12𝑀A 𝑀B + 𝑀C 𝐿
−12
10ft 250k−ft + 150k−ft 10ft
−16𝑀A + 2𝑀E 𝑀B𝐿
M1
M3M4
M1M2
∆"= −23,125k−ft3
𝐸𝐼
M3
L L
–20,000 k-ft3 –3,125 k-ft3
Define and Solve the Redundant Problem
• Write the redundant deflection in terms of the flexibility coefficient and the redundant for each redundant problem.
• Calculate the flexibility coefficient associated with the relevant deflections for each redundant problem;
∆""= 𝐷,𝛿""
Redundant Problem
10 ft
B
5 ft 5 ft
A
C D
EI
EIΔ""
Dy
Define and Solve the Redundant Problem
• Write the redundant deflection in terms of the flexibility coefficient and the redundant for each redundant problem.
• Calculate the flexibility coefficient associated with the relevant deflections for each redundant problem;
∆""= 𝐷,𝛿""
Flexibility Coefficient
10 ft
B
5 ft 5 ft
A
C D
EI
EI𝛿""
1
Need to find 𝛿""Use the Principle of Virtual Work
Solve the Flexibility Coefficient Problem
Real System
Need to construct the Mpdiagram
10 ft
B
5 ft 5 ft
A
C D
EI
EI𝛿""
1
Solve the Flexibility Coefficient Problem
Virtual System to measure 𝛿""
Need to construct the MQdiagram
10 ft
B
5 ft 5 ft
A
C D
EI
EI𝛿""
1 Note that for the flexibility coefficient problem the real and virtual systems are identical
FBD of the Flexibility Coefficient Problem
#𝑀%
�
�
= 0+
#𝐹+
�
�
= 0+
#𝐹,
�
�
= 0+
MA = 10 ft
Ax = 0
Ay = –1
10 ft
1
B10 k
5 ft 5 ft
C
D
EI
EI
C
Ax
Ay
MA
B
A
D
C
10 ft
10 ft
0
MP Diagram for the Flexibility Coefficient Problem
Moment diagram is drawn on the compression side of the member
10 ft
B
A
D
C
10 ft
10 ft
0
MQ Diagram for the Flexibility Coefficient Problem
Moment diagram is drawn on the compression side of the member
10 ft
1 . δ"" = 1𝐸𝐼3 𝑀4𝑀5
6
7𝑑𝑥
Solve the Flexibility Coefficient Problem
BA D10 ft 10 ftB
10 ft 10 ft
0
M1
M3
M1
𝛿"" =1333ft3
𝐸𝐼
M3
L L
1000 ft3 333.333 ft3
10 ft10 ft
13𝑀A𝑀B𝐿𝑀A𝑀B𝐿
10ft 10ft 10ft 13
10ft 10ft 10ft
∆""= 𝐷,1333ft3
𝐸𝐼
∆""= 𝐷,𝛿""
Compatibility Equation at Point D
∆" + ∆""= 0
Compatibility at Point D
Compatibility Equation in terms of Redundant and Flexibility Coefficient
Δ" + 𝐷,𝛿"" = 0
−23,125k−ft3
𝐸𝐼+ 𝐷,
1333ft3
𝐸𝐼= 0
𝐷, =23,125k−ft3
𝐸𝐼𝐸𝐼
1333ft3𝐷, = 17.34k
Solve for Dy
FBD of the Frame
10 ft
30 kB10 k
5 ft 5 ft
C
D
EI
EI
Ax
Ay
MA
17.34 k
𝐷, = 17.34k
#𝑀%
�
�
= 0+
#𝐹+
�
�
= 0+
#𝐹,
�
�
= 0+
MA = 76.6 k-ft
Ax = – 10 k
Ay = 12.66 k
B
A
D
C
76.6 k-ft
23.4 k-ft0
Moment Diagram for the Frame
Moment diagram is drawn on the compression side of the member
86.7 k-ft
B
A
DC
0
250 k-ft
0
Moment Diagrams for the Primary and Redundant Problems
150 k-ft
B DC
173.4 k-ft
173.4 k-ft
86.7 k-ft
Moment diagram is drawn on the compression side of the member
0
A
++ +
For horizontal member BDE
Choose sign convention for internal forces for bothhorizontal and vertical members
For vertical member ABC++ +
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