Statically Indeterminate Problemsand
Problems Involving Two Materials(Strength of Materials)
Dave Morgan<[email protected]>
Statically Indeterminate Problems and Problems Involving Two Materials – p. 1/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
We have used the laws of statics to analyse
problems such as the one illustrated:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
To find the internal forces in the segment
BC:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
M M
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
To find the internal forces in the segment
BC:Insert a section M-M through
segment BC
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
M M
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
To find the internal forces in the segment
BC:Insert a section M-M through
segment BCConsider only the segment from C to
M-M
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
M M
30 kN
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
To find the internal forces in the segment
BC:Insert a section M-M through
segment BCConsider only the segment from C to
M-MΣFy = 0, so there is an internal force
of 30 kN at M
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
M M
30 kN
We have used the laws of statics to analyse
problems such as the one illustrated:
ΣFy = 0, so there is a reaction force of
20 kN at A
To find the internal forces in the segment
BC:Insert a section M-M through
segment BCConsider only the segment from C to
M-MΣFy = 0, so there is an internal force
of 30 kN at MTBC = 30 kN (tension)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 2/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kNSimilarly, we can find the internal force
within segment AB:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
N N
Similarly, we can find the internal force
within segment AB:
Insert a section N-N through segment
AB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
N N
Similarly, we can find the internal force
within segment AB:
Insert a section N-N through segment
AB
Consider only the segment from A to
N-N
Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
N N
20 kN
Similarly, we can find the internal force
within segment AB:
Insert a section N-N through segment
AB
Consider only the segment from A to
N-N
ΣFy = 0, so there is an internal force
of 20 kN at N-N
Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate Problems
A
B
C
30 kN
10 kN
20 kN
N N
20 kN
Similarly, we can find the internal force
within segment AB:
Insert a section N-N through segment
AB
Consider only the segment from A to
N-N
ΣFy = 0, so there is an internal force
of 20 kN at N-N
TAB = 20 kN (tension)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 3/30
Statically Indeterminate Problems
Structures where forces can be determined using thestatic equilibrium equations alone (ΣFx = 0,ΣFy = 0 andΣMA = 0) are calledstaticallydeterminate structures. The previous example is astatically determinate structure.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate Problems
Structures where forces can be determined using thestatic equilibrium equations alone (ΣFx = 0,ΣFy = 0 andΣMA = 0) are calledstaticallydeterminate structures. The previous example is astatically determinate structure.
Structures where the forces cannot be determined inthis way are calledstatically indeterminatestructures.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate Problems
Structures where forces can be determined using thestatic equilibrium equations alone (ΣFx = 0,ΣFy = 0 andΣMA = 0) are calledstaticallydeterminate structures. The previous example is astatically determinate structure.
Structures where the forces cannot be determined inthis way are calledstatically indeterminatestructures.
Statically indeterminate structures are often analysedusing the conditions of axial deformation given by
δ =PL
AEStatically Indeterminate Problems and Problems Involving Two Materials – p. 4/30
Statically Indeterminate Problems
Example: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls
A C B
500 mm 400 mm
12 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate Problems
Example: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls
A C B
500 mm 400 mm
12 kN
Solution: First, draw a free body diagram:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate Problems
Example: Consider a bar AB supported at bothends by fixed supports, with an axial force of 12 kNapplied at C as illustrated. Find the reactions at thewalls
A C B
500 mm 400 mm
12 kN
Solution: First, draw a free body diagram:
A C B
500 mm 400 mm
12 kNRA RB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 5/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed soδAB = 0.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed soδAB = 0.
⇒ δAC + δCB = 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed soδAB = 0.
⇒ δAC + δCB = 0
⇒−RA×500
AE+ RB×400
AE= 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed soδAB = 0.
⇒ δAC + δCB = 0
⇒−RA×500
AE+ RB×400
AE= 0
⇒ 400RB = 500RA
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kNRA RB
ΣFx = RA + RB − 12 = 0
⇒ RA + RB = 12
Two unknowns and a single equation; the problem isstatically indeterminate.The supports at A and B are fixed soδAB = 0.
⇒ δAC + δCB = 0
⇒−RA×500
AE+ RB×400
AE= 0
⇒ 400RB = 500RA
Now we have two equations and two unknowns; we cansolve forRA andRB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 6/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
⇒ RA = 12 − RB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
⇒ RA = 12 − RB
⇒ 400RB = 500 (12 − RB)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
⇒ RA = 12 − RB
⇒ 400RB = 500 (12 − RB)
⇒ 900RB = 6000
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
⇒ RA = 12 − RB
⇒ 400RB = 500 (12 − RB)
⇒ 900RB = 6000
⇒ RB = 6.667 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
RA + RB = 12
400RB = 500RA
⇒ RA = 12 − RB
⇒ 400RB = 500 (12 − RB)
⇒ 900RB = 6000
⇒ RB = 6.667 kN
⇒ RA = 5.333 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 7/30
Statically Indeterminate Problems
A C B
500 mm 400 mm
12 kN5.333 kN 6.667 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 8/30
Statically Indeterminate Problems
Exercise: FindRA andRB for the problem illustrated:
A C B
a b
P
Statically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate Problems
Exercise: FindRA andRB for the problem illustrated:
A C B
a b
P
Solution: Draw free body diagram
Statically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate Problems
Exercise: FindRA andRB for the problem illustrated:
A C B
a b
P
Solution: Draw free body diagram
A C B
a b
RA RBP
Statically Indeterminate Problems and Problems Involving Two Materials – p. 9/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
⇒ aRA = bRB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
⇒ aRA = bRB
⇒ a (P − RB) = bRB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
⇒ aRA = bRB
⇒ a (P − RB) = bRB
⇒ aP = (a + b)RB
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
⇒ aRA = bRB
⇒ a (P − RB) = bRB
⇒ aP = (a + b)RB
⇒ RB =(
aa+b
)
P
Statically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Statically Indeterminate Problems
A C B
a b
RA RBP
Solution: RA + RB − P = 0
⇒ RA = P − RB
δAC + δCB = 0
⇒RA×aAE
+ −RB×bAE
= 0
⇒ aRA = bRB
⇒ a (P − RB) = bRB
⇒ aP = (a + b)RB
⇒ RB =(
aa+b
)
P
⇒ RA =(
ba+b
)
PStatically Indeterminate Problems and Problems Involving Two Materials – p. 10/30
Problems Involving Two Materials
Steel-reinforced concrete is used in the construction ofmany structures:
Bridges
Basements
High-Rise Buildings
Stadia, such as the SaddleDome or theSpeed-Skating Oval
Statically Indeterminate Problems and Problems Involving Two Materials – p. 11/30
Problems Involving Two Materials
Concrete has a high load-bearing capacity incompression but is not very strong under a tensileload.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two Materials
Concrete has a high load-bearing capacity incompression but is not very strong under a tensileload.
Steel rod has high load-bearing capacity in tensionbut buckles easily under compression.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two Materials
Concrete has a high load-bearing capacity incompression but is not very strong under a tensileload.
Steel rod has high load-bearing capacity in tensionbut buckles easily under compression.
Combining steel rod and concrete gives a buildingmaterial with both good tensile and compressiveload-bearing qualities.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 12/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
When a column is loaded, it deforms(
δ = PLAE
)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
When a column is loaded, it deforms(
δ = PLAE
)
Under compression,δ is negative and there isnegative axial strain
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
When a column is loaded, it deforms(
δ = PLAE
)
Under compression,δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt = −µǫa)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
When a column is loaded, it deforms(
δ = PLAE
)
Under compression,δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt = −µǫa)
The concrete is under tension laterally
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
Steel in a concrete column also helps the concrete’scompressive strength:
When a column is loaded, it deforms(
δ = PLAE
)
Under compression,δ is negative and there isnegative axial strainConsequently, there is a positive transverse strain(ǫt = −µǫa)
The concrete is under tension laterallyHorizontal steel-reinforcing increases the lateraltensile strength of the column
Statically Indeterminate Problems and Problems Involving Two Materials – p. 13/30
Problems Involving Two Materials
A concrete footing ispoured:
It contains steel rebarthroughout
Steel extrudes fromthe top of the footing
This will be attachedto the steel for thecolumn.
Statically Indeterminate Problems and Problems Involving Two Materials – p. 14/30
Problems Involving Two Materials
Steel is tied for the column
Statically Indeterminate Problems and Problems Involving Two Materials – p. 15/30
Problems Involving Two Materials
A frame is built around thesteel and the concretecolumn is poured
Statically Indeterminate Problems and Problems Involving Two Materials – p. 16/30
Problems Involving Two Materials
Statically Indeterminate Problems and Problems Involving Two Materials – p. 17/30
Problems Involving Two Materials
We can useδC = PC ·LC
AC ·ECto calculate the deformation
of concrete under a load
Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two Materials
We can useδC = PC ·LC
AC ·ECto calculate the deformation
of concrete under a load
We can useδS = PS ·LS
AS ·ESto calculate the deformation
of steel under a load
Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two Materials
We can useδC = PC ·LC
AC ·ECto calculate the deformation
of concrete under a load
We can useδS = PS ·LS
AS ·ESto calculate the deformation
of steel under a load
How can we calculate the deformation of asteel-reinforced concrete column?
Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two Materials
We can useδC = PC ·LC
AC ·ECto calculate the deformation
of concrete under a load
We can useδS = PS ·LS
AS ·ESto calculate the deformation
of steel under a load
How can we calculate the deformation of asteel-reinforced concrete column?
EC is not the same asES so we cannot simplyapplyδ = PL
AEfor the whole column
Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two Materials
We can useδC = PC ·LC
AC ·ECto calculate the deformation
of concrete under a load
We can useδS = PS ·LS
AS ·ESto calculate the deformation
of steel under a load
How can we calculate the deformation of asteel-reinforced concrete column?
EC is not the same asES so we cannot simplyapplyδ = PL
AEfor the whole column
We cannot solve this problem directly using theequations of statics, so this is astatically-indeterminate problem
Statically Indeterminate Problems and Problems Involving Two Materials – p. 18/30
Problems Involving Two Materials
Example: A concrete column has a
diameter of300 mm. The column has6 steel
reinforcing rods.
Each rod has a cross-sectional area of
200 mm2. (See plan view)
ES = 210 GPa andEC = 25 GPa
The column is1.15 m long and has a load of
1.37 MN is applied to a rigid steel plate at
the top of the column (the plate distributes
the load evenly over the top of the column).
1.37 MN
1.15 m
A = 200 mm2
D = 300 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 19/30
Problems Involving Two Materials
Example: A concrete column has a
diameter of300 mm. The column has6 steel
reinforcing rods.
Each rod has a cross-sectional area of
200 mm2. (See plan view)
ES = 210 GPa andEC = 25 GPa
The column is1.15 m long and has a load of
1.37 MN is applied to a rigid steel plate at
the top of the column (the plate distributes
the load evenly over the top of the column).
Find the stress in the steel and in the
concrete, and the deformation under the load
1.37 MN
1.15 m
A = 200 mm2
D = 300 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 19/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
ΣFy = PS + PC − 1370 = 0
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
ΣFy = PS + PC − 1370 = 0
PS + PC = 1370 kN
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
ΣFy = PS + PC − 1370 = 0
PS + PC = 1370 kN
We have a single equation with two unknowns, so
the problem is statically indeterminate.
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
ΣFy = PS + PC − 1370 = 0
PS + PC = 1370 kN
We have a single equation with two unknowns, so
the problem is statically indeterminate.
The concrete and the steel rods deform (contract)
by the same amount,δ, so...
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution: Let PS be the total reaction force of the
six steel rods andPC the reaction force of the
concrete.
ΣFy = PS + PC − 1370 = 0
PS + PC = 1370 kN
We have a single equation with two unknowns, so
the problem is statically indeterminate.
The concrete and the steel rods deform (contract)
by the same amount,δ, so...
PS · LS
AS · ES
= δ =PC · LC
AC · EC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 20/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
⇒PS×1150
1200×(200×103)= PC×1150
“
π×3002
4−1200
”
×(25×103)
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
⇒PS×1150
1200×(200×103)= PC×1150
“
π×3002
4−1200
”
×(25×103)
⇒PS
1200×200 = PC
69486×25PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
⇒PS×1150
1200×(200×103)= PC×1150
“
π×3002
4−1200
”
×(25×103)
⇒PS
1200×200 = PC
69486×25
⇒ PS = 1200×20069486×25 · PC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
⇒PS×1150
1200×(200×103)= PC×1150
“
π×3002
4−1200
”
×(25×103)
⇒PS
1200×200 = PC
69486×25
⇒ PS = 1200×20069486×25 · PC
⇒ PS = 0.13816PC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:PS ·LS
AS ·ES= PC ·LC
AC ·EC
⇒PS×1150
(6×200)×ES= PC×1150
“
π×3002
4−(6×200)
”
×EC
⇒PS×1150
1200×(200×103)= PC×1150
“
π×3002
4−1200
”
×(25×103)
⇒PS
1200×200 = PC
69486×25
⇒ PS = 1200×20069486×25 · PC
⇒ PS = 0.13816PC
We now have two equations for the two unknowns,
PS andPC .
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 21/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
⇒ 0.13816PC + PC = 1370
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
⇒ 0.13816PC + PC = 1370
⇒ PC = 13701+0.13816
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
⇒ 0.13816PC + PC = 1370
⇒ PC = 13701+0.13816
⇒ PC = 1203.7 kN
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
⇒ 0.13816PC + PC = 1370
⇒ PC = 13701+0.13816
⇒ PC = 1203.7 kN
⇒ PS = 166.3 kN PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution:
PS + PC = 1370
PS = 0.13816PC
⇒ 0.13816PC + PC = 1370
⇒ PC = 13701+0.13816
⇒ PC = 1203.7 kN
⇒ PS = 166.3 kN
We can now calculate the stress in the steel and in
the concrete
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 22/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
Find the stress in the steel:
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
Find the stress in the steel:
PS = 152 kN PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
Find the stress in the steel:
PS = 152 kN
⇒ σS = 152(6×200)
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
Find the stress in the steel:
PS = 152 kN
⇒ σS = 152(6×200)
⇒ σS = 0.1267 kN
mm2
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the stress in the concrete:
PC = 1098 kN
⇒ σC = PC
A
⇒ σC = 1098π×3002
4−(6×200)
⇒ σC = 0.0580 kN
mm2
⇒ σC = 58.0 MPa
Find the stress in the steel:
PS = 152 kN
⇒ σS = 152(6×200)
⇒ σS = 0.1267 kN
mm2
⇒ σS = 126.7 MPa
PC
PS
6
PS
6
1370 kN
Statically Indeterminate Problems and Problems Involving Two Materials – p. 23/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
⇒ δC = 0.00221 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
⇒ δC = 0.00221 mm
Find the deformation in the steel (if we’ve done our calculations correctly,
thenδS = δC):
δS = PS ·LS
AS ·ES
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
⇒ δC = 0.00221 mm
Find the deformation in the steel (if we’ve done our calculations correctly,
thenδS = δC):
δS = PS ·LS
AS ·ES
⇒ δS =152×(3.5×103)1200×(200×103)
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
⇒ δC = 0.00221 mm
Find the deformation in the steel (if we’ve done our calculations correctly,
thenδS = δC):
δS = PS ·LS
AS ·ES
⇒ δS =152×(3.5×103)1200×(200×103)
⇒ δS = 0.00222 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Solution: Find the deformation in the concrete:
δC = PC ·LC
AC ·EC
⇒ δC =1098×(3.5×103)
(π×3002
4−1200)×(25×103)
⇒ δC = 0.00221 mm
Find the deformation in the steel (if we’ve done our calculations correctly,
thenδS = δC):
δS = PS ·LS
AS ·ES
⇒ δS =152×(3.5×103)1200×(200×103)
⇒ δS = 0.00222 mm
The small difference in deformation is due to rounding errors
Statically Indeterminate Problems and Problems Involving Two Materials – p. 24/30
Problems Involving Two Materials
Exercise: A hollow square steel
structural section has outside dimensions of
115 mm× 115 mm and inside dimensions of
105 mm× 105 mm. It is filled with concrete,
as shown in plan view (upper right). The
section is3.5 m and supports a compressive
load of250 kN.
ES = 200 GPa andEC = 20 GPa.
FindσS , σC andδ
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 25/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
AC = 105 × 105
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
AC = 105 × 105
= 11025 mm2
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
AC = 105 × 105
= 11025 mm2
Let PS be the reaction force of the steel and
PC the reaction force of the concrete.
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
AC = 105 × 105
= 11025 mm2
Let PS be the reaction force of the steel and
PC the reaction force of the concrete. Then,
ΣFy = PS + PC − 250 = 0
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: Find the areas of the steel and
of the concrete:
AS = (2 × 115 × 5) + (2 × 105 × 5)
= 2200 mm2
AC = 105 × 105
= 11025 mm2
Let PS be the reaction force of the steel and
PC the reaction force of the concrete. Then,
ΣFy = PS + PC − 250 = 0
PS + PC = 250 kN
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 26/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
⇒ PC = 11025×202200×200 · PS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
⇒ PC = 11025×202200×200 · PS
⇒ PC = 0.5011PS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
⇒ PC = 11025×202200×200 · PS
⇒ PC = 0.5011PS
ΣFy = 0 so
PC + PS − 250 = 0
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
⇒ PC = 11025×202200×200 · PS
⇒ PC = 0.5011PS
ΣFy = 0 so
PC + PS − 250 = 0
⇒ PC = 250 − PS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution: The steel casing and the
concrete both deform by the same amount
PC ·LC
AC ·EC= PS ·LS
AS ·ES
⇒(PC×103)×175
11025×(200×103)=
(PS×103)×175
2200×(200×103)
⇒PC
11025×20 = PS
2200×200
⇒ PC = 11025×202200×200 · PS
⇒ PC = 0.5011PS
ΣFy = 0 so
PC + PS − 250 = 0
⇒ PC = 250 − PS
Now we have two equations for the two
unknowns,PC andPS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 27/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
⇒ PC = 83.5 kN
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
⇒ PC = 83.5 kN
Now, find the stress in the steel:
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
⇒ PC = 83.5 kN
Now, find the stress in the steel:
σS = PS
AS
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
⇒ PC = 83.5 kN
Now, find the stress in the steel:
σS = PS
AS
⇒ σS = 166.5×103
2200
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution:
PC = 0.5011PS
PC = 250 − PS
⇒ 250 − PS = 0.5011PS
⇒ PS = 2501+0.5011
⇒ PS = 166.5 kN
⇒ PC = 83.5 kN
Now, find the stress in the steel:
σS = PS
AS
⇒ σS = 166.5×103
2200
⇒ σS = 75.7 MPa
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 28/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
⇒ σC = 7.57 MPa
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
⇒ σC = 7.57 MPa
Now, find the deformation:
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
⇒ σC = 7.57 MPa
Now, find the deformation:
δ = PC ·LC
AC ·EC
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
⇒ σC = 7.57 MPa
Now, find the deformation:
δ = PC ·LC
AC ·EC
⇒ δ =(83.5×103)×175
11025×(20×103)
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Problems Involving Two Materials
Solution: Now, find the stress in the
concrete:
σC = PC
AC
⇒ σC = 83.5×103
11025
⇒ σC = 7.57 MPa
Now, find the deformation:
δ = PC ·LC
AC ·EC
⇒ δ =(83.5×103)×175
11025×(20×103)
⇒ δ = 0.0663 mm
105 mm
115 mm
250 kN
175 mm
Statically Indeterminate Problems and Problems Involving Two Materials – p. 29/30
Statically Indeterminate Problems and Problems Involving Two Materials
Created by Dave Morgan using LATEX 2εandProsper on January 26, 2006
Statically Indeterminate Problems and Problems Involving Two Materials – p. 30/30
Top Related