CAP_12_DEFLEX%C3%83O_EM_VIGAS_E_EIXOS

Prévia do material em texto

Problem 12-01
An L2 steel strap having a thickness of 3 mm and a width of 50 mm is bent into a circular arc of radius
15 m. Determine the maximum bending stress in the strap.
Given: b 50mm:= t 3mm:= ρ 15m:=
E 200GPa:=
Solution:
Section Property :
c
t
2
:= I
b t3⋅
12
:=
Moment - Curvature Relationship :
1
ρ
M
E I⋅
= M
E I⋅
ρ
:=
Bendiug Stress :
σ
M c⋅
I
= σ
E I⋅
ρ
c
I
⎛⎜
⎝
⎞
⎠
⋅= σ
E c⋅
ρ
:=
σ 20 MPa= Ans
Problem 12-02
A picture is taken of a man performing a pole vault, and the minimum radius of curvature of the pole is
estimated by measurement to be 4.5 m. If the pole is 40 mm in diameter and it is made of a glass-
reinforced plastic for which Eg = 131 GPa, determine the maximum bending stress in the pole.
Given: do 40mm:= ρ 4.5m:=
E 131GPa:=
Solution:
Section Property :
c
do
2
:= I
π do
4⋅
64
:=
Moment - Curvature Relationship :
1
ρ
M
E I⋅
= M
E I⋅
ρ
:=
Bendiug Stress :
σ
M c⋅
I
= σ
E I⋅
ρ
c
I
⎛⎜
⎝
⎞
⎠
⋅= σ
E c⋅
ρ
:=
σ 582.2 MPa= Ans
Problem 12-03
Determine the equation of the elastic curve for the beam using the x coordinate that is valid for 
 Specify the slope at A and the beam's maximum deflection. EI is constant.2/0 Lx <≤
Problem 12-04
Determine the equations of the elastic curve for the beam using the x1 and x2 coordinates. Specify the
beam's maximum deflection. EI is constant.
Problem 12-05
Determine the equations of the elastic curve using the x1 and x2 coordinates. EI is constant.
Problem 12-06
Determine the equations of the elastic curve for the beam using the x1 and x3 coordinates. Specify the
beam's maximum deflection. EI is constant.
Problem 12-07
Determine the equations of the elastic curve for the shaft using the x1 and x2 coordinates. Specify the
slope at A and the displacement at the center of the shaft. EI is constant.
Problem 12-08
Determine the equations of the elastic curve for the shaft using the x1 and x3 coordinates. Specify the
slope at A and the deflection at the center of the shaft. EI is constant.
Problem 12-09
The beam is made of two rods and is subjected to the concentrated load P. Determine the maximum
deflection of the beam if the moments of inertia of the rods are IAB and IBC, and the modulus of
elasticity is E.
Problem 12-10
The beam is made of two rods and is subjected to the concentrated load P. Determine the slope at C.
The moments of inertia of the rods are IAB and IBC, and the modulus of elasticity is E.
Problem 12-11
The bar is supported by a roller constraint at B, which allows vertical displacement but resists axial
load and moment. If the bar is subjected to the loading shown, determine the slope at A and the
deflection at C. EI is constant.
Problem 12-12
Determine the deflection at B of the bar in Prob. 12-11.
Problem 12-13
The fence board weaves between the three smooth fixed posts. If the posts remain along the same line,
determine the maximum bending stress in the board. The board has a width of 150 mm. and a
thickness of 12 mm. E = 12 GPa. Assume the displacement of each end of the board relative to its
center is 75 mm.
Given: b 150mm:= t 12mm:= L 2.4m:=
E 12GPa:= δ 75mm:=
Solution:
Support Reactions : By symmetry, RA=RC=R
+ ΣFy=0; 2R P− 0= R 0.5P=
Moment Function : 
M x( ) R x⋅= M x( ) 0.5 P⋅ x⋅=
Mmax R 0.5L( )⋅= Mmax 0.25P L⋅=
Section Property :
c
t
2
:= I
b t3⋅
12
:=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )= E I⋅
d2v
dx2
⋅
P x⋅
2
=
E I⋅
dv
dx
⋅
P x2⋅
4
C1+= (1)
E I⋅ v⋅
P x3⋅
12
C1 x⋅+ C2+= (2)
Boundary Conditions : Due to symmetry, dv/dx=0 at x=L/2. Also, v=0 at x=0.
From Eq. (1): 0
P
4
L
2
⎛⎜
⎝
⎞
⎠
2
⋅ C1+= C1
P L2⋅
16
−=
From Eq. (2): 0 0 0+ C2+= C2 0:=
The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2),
E I⋅ v⋅
P x3⋅
12
P L2⋅
16
x⋅−= v
P x⋅
48E I⋅
4x2 3L2−( )⋅= (3)
Require at x=L/2, v=-δ. From Eq. (3),
δ−
P
48E I⋅
L
2
⎛⎜
⎝
⎞
⎠
⋅ 4
L
2
⎛⎜
⎝
⎞
⎠
2
3L2−
⎡
⎢
⎣
⎤
⎥
⎦
⋅= δ−
P− L3⋅
48 E⋅ I⋅
= P
48 E⋅ I⋅
L3
δ⋅:=
Bending Stress: Mmax 0.25P L⋅:=
cmax
t
2
:= σmax
Mmax cmax⋅
I
:= σmax 11.25 MPa= Ans
Problem 12-14
Determine the equation of the elastic curve for the beam using the x coordinate. Specify the slope at A
and the maximum deflection. EI is constant.
Problem 12-15
Determine the deflection at the center of the beam and the slope at B. EI is constant.
Problem 12-16
A torque wrench is used to tighten the nut on a bolt. If the dial indicates that a torque of 90 N·m is
applied when the bolt is fully tightened, determine the force P acting at the handle and the distance s the
needle moves along the scale. Assume only the portion AB of the beam distorts. The cross section is
square having dimensions of 12 mm by 12 mm. E = 200 GPa.
Given: b 12mm:= h 12mm:= L 0.45m:=
E 200GPa:= δ 75mm:= R 0.3m:=
Tz 90N m⋅:=
Solution:
Equations of Equilibrium : 
+
 
ΣFy=0; Ay P− 0= (1)
ΣΜB=0; Tz P L⋅− 0= (2)
Solving Eqs. (1) and (2): P
Tz
L
:= Ay P:= Ay 200 N=
P 200 N= Ans
Moment Function : M x( ) Ay x⋅ Tz−=
Section Property : I
b h3⋅
12
:=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ Ay x⋅ Tz−=
E I⋅
dv
dx
⋅
Ay x
2⋅
2
Tz x⋅− C1+= (1)
E I⋅ v⋅
Ay x
3⋅
6
Tz x
2⋅
2
− C1 x⋅+ C2+= (2)
Boundary Conditions : Due to symmetry, dv/dx=0 at x=0, and v=0 at x=0.
From Eq. (1): 0 0 0− C1+= C1 0:=
From Eq. (2): 0 0 0− 0+ C2+= C2 0:=
The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2),
E I⋅ v⋅
Ay x
3⋅
6
Tz x
2⋅
2
−= v
x2
6E I⋅
Ay x 3Tz−( )⋅= (3)
At x=R, v=-s. From Eq. (3),
s
R2−
6E I⋅
Ay R 3Tz−( )⋅:= s 9.11 mm= Ans
Problem 12-17
The shaft is supported at A by a journal bearing that exerts only vertical reactions on the shaft and at B
by a thrust bearing that exerts horizontal and vertical reactions on the shaft. Draw the bending-moment
diagram for the shaft and then, from this diagram, sketch the deflection or elastic curve for the shaft's
centerline. Determine the equations of the elastic curve using the coordinates x1 and x2 . EI is constant.
Given: h 60mm:= L1 150mm:=
P 5kN:= L2 400mm:=
Solution: L L1 L2+:=
Support Reactions:
+
 
ΣFy=0; A B+ 0= (1)
ΣΜB=0; P h⋅ A L2⋅+ 0= (2)
Solving Eqs. (1) and (2): A
P− h⋅
L2
:= A 0.75− kN=
B A−:= B 0.75 kN=
Moment Function : M1 x1( ) P h⋅:=
M2 x2( ) B x2⋅:=
Section Property : EI kN m2⋅:=
Slope and Elastic Curve :
EI
d2 v1⋅
dx1
2
⋅ M1 x1( )= EI
d2 v2⋅
dx2
2
⋅ M2 x2( )=
EI
d2v1
dx1
2
⋅ P h⋅= EI
d2v2
dx2
2
⋅ B x2⋅=
EI
dv1
dx1
⋅ P h⋅( ) x1⋅ C1+= (1) EI
dv2
dx2
⋅
B
2
x2
2⋅ C3+= (3)
EI v1⋅
P h⋅ x1
2⋅
2
C1 x1⋅+ C2+= (2) EI v2⋅
B x2
3⋅
6
C3 x2⋅+ C4+= (4)
Boundary Conditions :
v1=0 at x1=0.15m, From Eq. (2): 0
P h⋅ 0.15m( )2⋅
2
C1 0.15m( )⋅+ C2+= (5)
v2=0 at x2=0, From Eq. (4): 0 0 0+ C4+= C4 0:= Ans
v2=0 at x2=0.4m, From Eq. (4): 0
B 0.4m( )3⋅
6
C3 0.4m( )⋅+ C4+=
C3
B
6
− 0.4m( )2⋅:= C3 0.02− kN m
2⋅= Ans
Continuity Condition:
dv1/dx1= - dv2/dx2 at A (x1=0.15m and x2=0.4m)
From Eqs. (1) and (3), P h⋅ 0.15m( )⋅ C1+
B
2
0.4m( )2⋅ C3+=
C1
B
2
0.4m( )2⋅ C3+ P h⋅ 0.15m( )⋅−:= C1 0.005− kN m
2⋅= Ans
From Eq. (5): C2
P h⋅ 0.15m( )2⋅
2
− C1 0.15m( )⋅−:= C2 0.00263− kN m
3⋅= Ans
The Elastic Curve : Substitute the values of C1 and C2 into Eq. (2), and C3 and C4 into Eq. (4),
v1
1
EI
P h⋅
2
x1
2⋅ C1 x1⋅+ C2+
⎛⎜
⎝
⎞
⎠
= Ans
v2
1
EI
B
6
x2
3⋅ C3 x2⋅+ C4+
⎛⎜
⎝
⎞
⎠
= Ans
BMD :
x'1 0 0.01 L1⋅, L1..:= x'2 L1 1.01 L1⋅, L..:=
M'1 x'1( )
P h⋅
kN m⋅
:= M'2 x'2( ) P h⋅ A x'2 L1−( )⋅+⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 0.2 0.4
0
0.2
0.4
Distane (m)
M
om
en
t (
kN
-m
)
M'1 x'1( )
M'2 x'2( )
x'1 x'2,
Problem 12-18
Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope and
deflection at C. EI is constant.
Problem 12-19
Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope at
A. EI is constant.
Problem 12-20
Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope and
deflection at B. EI is constant.Problem 12-21
Determine the equations of the elastic curve using the coordinates x1 and x3 , and specify the slope and
deflection at B. EI is constant.
Problem 12-22
Determine the maximum slope and maximum deflection of the simply-supported beam which is
subjected to the couple moment M0 . EI is constant.
Problem 12-23
The two wooden meter sticks are separated at their centers by a smooth rigid cylinder having a
diameter of 50 mm. Determine the force F that must be applied at each end in order to just make their
ends touch. Each stick has a width of 20 mm and a thickness of 5 mm. Ew = 11 GPa.
Given: do 50mm:= L 0.5m:=
b 20mm:= t 5mm:= E 11GPa:=
Solution:
Section Property: I
b t3⋅
12
:=
Moment Function : M x( ) F− x⋅=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ F− x⋅=
E I⋅
dv
dx
⋅
F x2⋅
2
− C1+= (1)
E I⋅ v⋅
F x3⋅
6
− C1 x⋅+ C2+= (2)
Boundary Conditions :
dv/dx=0 at x=L. From Eq. (1): 0
F L2⋅
2
− C1+= C1
F L2⋅
2
:=
v=0 at x=L. From Eq. (2): 0
F L3⋅
6
− C1 L⋅+ C2+=
C2
F L3⋅
6
C1 L⋅−= C2
F L3⋅
3
−:=
Require : v 0.5− do:= at x=0.
From Eq. (2): 0.5− do E⋅ I⋅ 0− 0
F L3⋅
3
−+=
F
1.5do E⋅ I⋅
L3
:=
F 1.375 N= Ans
Problem 12-24
The pipe can be assumed roller supported at its ends and by a rigid saddle C at its center. The saddle
rests on a cable that is connected to the supports. Determine the force that should be developed in the
cable if the saddle keeps the pipe from sagging or deflecting at its center. The pipe and fluid within it
have a combined weight of 2 kN/m. EI is constant.
Given:
e 0.3m:= L 3.75m:= w
2kN
m
:=
Solution:
Moment Function : M x( ) P x⋅
1
2
w⋅ x2⋅−=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ P x⋅
w
2
x2⋅−=
E I⋅
dv
dx
⋅
P x2⋅
2
w
6
x3⋅− C1+= (1)
E I⋅ v⋅
P x3⋅
6
w
24
x4⋅− C1 x⋅+ C2+= (2)
Boundary Conditions : v=0 at x=0 and at x=L.
From Eq. (2): 0 0 0− 0+ C2+= C2 0:=
0
P L3⋅
6
w
24
L4⋅− C1 L⋅+= (3)
Also, dv/dx=0 at x=L.
From Eq. (1): 0
P L2⋅
2
w
6
L3⋅− C1+= (4)
Solving Eqs. (3) and (4) for P,
P L2⋅
6
w
24
L3⋅− C1+
P L2⋅
2
w
6
L3⋅− C1+=
P L2⋅
3
w
8
L3⋅= P
3w L⋅
8
:= P 2.813 kN=
Equations of Equilibrium : 
+ ΣFy=0; 2P F+ w 2L( )⋅− 0=
F 2w L⋅ 2P−:= F 9375 N=
+At C : ΣFy=0; 2Tcable
e
e2 L2+
⋅ F− 0= Tcable
e2 L2+
2e
F⋅:=
Tcable 58.78 kN= Ans
Problem 12-25
Determine the equations of the elastic curve using the coordinates x1 and x2 , and specify the slope at C
and displacement at B. EI is constant.
Problem 12-26
Determine the equations of the elastic curve using the coordinates x1 and x3 , and specify the slope at B
and deflection at C. EI is constant.
Problem 12-27
Determine the elastic curve for the simply supported beam using the x coordinate . Also,
determine the slope at A and the maximum deflection of the beam. EI is constant. 
2/0 Lx ≤≤
Problem 12-28
Determine the elastic curve for the cantilevered beam using the x coordinate. Also determine the
maximum slope and maximum deflection. EI is constant.
Problem 12-29
The beam is made of a material having a specific weight γ. Determine the displacement and slope at its
end A due to its weight. The modulus of elasticity for the material is E. 
Problem 12-30
The beam is made of a material having a specific weight γ. Determine the displacement and slope at its
end A due to its weight. The modulus of elasticity for the material is E. 
Problem 12-31
The leaf spring assembly is designed so that it is subjected to the same maximum stress throughout its
length. If the plates of each leaf have a thickness t and can slide freely between each other, show that
the spring must be in the form of a circular arc in order that the entire spring becomes flat when a
large enough load P is applied. What is the maximum normal stress in the spring? Consider the spring
to be made by cutting the n strips from the diamond-shaped plate of thickness t and width b. The
modulus of elasticity for the material is E. Hint: Show that the radius of curvature of the spring is
constant.
Problem 12-32
The beam has a constant width b and is tapered as shown. If it supports a load P at its end, determine
the deflection at B.The load P is applied a short distance s from the tapered end B, where s << L. EI is
constant.
Problem 12-33
A thin flexible 6-m-long rod having a weight of 10 N/m rests on the smooth surface. If a force of 15 N
is applied at its end to lift it, determine the suspended length x and the maximum moment developed in
the rod.
Given: P 15N:= w
10N
m
:=
Lmax 6m:=
Solution:
Since the horizontal section has no curvature,
the moment in the rod is zero. Hence, R acts
at the end of the suspended portion and this
portion acts like a simply supported beam.
Thus,
Equations of Equilibrium : 
+ ΣFy=0; P R+ w x⋅− 0= (1)
 
ΣΜ0=0; P x⋅
w
2
x2⋅− 0= (2)
From Eqs. (2) :
x
2P
w
:= x 3 m= Ans
Maximum moment occurs at center.
Mmax P
x
2
⎛⎜
⎝
⎞
⎠
⋅
w
2
x
2
⎛⎜
⎝
⎞
⎠
2
⋅−:=
Mmax 11.25 N m⋅= Ans
Problem 12-34
The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. The
bearings at A and B exert only vertical reactions on the shaft. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a m:= P kN:= EI kN m2⋅:= EIo 1:=
Solution:
Support Reactions :
+
 
ΣFy=0; A B+ P− 2P− 0= (1)
ΣΜB=0; A 2a( )⋅ P a⋅− 2P a⋅+ 0= (2)
Solving Eqs. (1) and (2): A
P−
2
:= B
7P
2
:=
Moment Function : 
M x( ) A Ψ x 0−( )⋅ P Ψ x a−( )⋅− B Ψ x 2a−( )⋅+=
Slope and Elastic Curve :
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ A Ψ x 0−( )⋅ P Ψ x a−( )⋅− B Ψ x 2a−( )⋅+=
EI
dv
dx
⋅
A
2
Ψ x 0−( )2⋅
P
2
Ψ x a−( )2⋅−
B
2
Ψ x 2a−( )2⋅+ C1+=
EI v⋅
A
6
Ψ x 0−( )3⋅
P
6
Ψ x a−( )3⋅−
B
6
Ψ x 2a−( )3⋅+ C1 x⋅+ C2+=
EI v⋅
A
6
x3⋅
P
6
Ψ x a−( )3⋅−
B
6
Ψ x 2a−( )3⋅+ C1 x⋅+ C2+= (1)
Boundary Conditions : v=0 at x=0
From Eq. (1): 0 0 0− 0+ 0+ C2+= C2 0:=
Also v=0 at x=2a.
From Eq. (1): 0
A
6
2a( )3⋅
P
6
Ψ 2a a−( )3⋅− 0+ C1 2a( )⋅+=
C1
A
12a
− 2a( )3⋅
P
12a
a3⋅+:= C1
5
12
P a2⋅=
Equation of Elastic Curve :
v
1
EI
A
6
x3⋅
P
6
Ψ x a−( )3⋅−
B
6
Ψ x 2a−( )3⋅+ C1 x⋅+
⎛⎜
⎝
⎞
⎠
m=
v x( )
1
EIo
P
12
− x3⋅
P
6
Ψ x a−( )3⋅−
7P
12
Ψ x 2a−( )3⋅+
5
12
P a2⋅ x⋅+⎛⎜
⎝
⎞
⎠
m:= Ans
Problem 12-35
Determine the equation of the elastic curve. Specify the slopes at A and B. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given:
a m:= w
kN
m
:=
Solution: L 2a:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
4
w a( )⋅:= A w a⋅ B−:=
B 0.25 w a⋅= A 0.75 w a⋅=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
EI
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+= (3)
EI v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4):
0 0− 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w 2a( )4⋅
1
24
⎛⎜
⎝
⎞
⎠
w⋅ a4⋅+
A
6
2a( )3⋅+ C1 2a( )⋅+=
C1
1
3
w a3⋅
1
48
w a3⋅−
2A
3
a2⋅−:= C1
3
16
− w a3⋅=
Equations of Elastic Curve and Slope :
v x( )
1
EIo
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
3w a⋅
24
x3⋅+
3
16
w a3⋅ x⋅−⎛⎜
⎝
⎞
⎠
:= Ans
θ x( )
1
EIo
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+
⎛⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=0 into Eq.(3). θ 0( )
3
16
−
w a3⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
Slope at B: Substitute x=L into Eq.(3). θ L( )
7
48
w a3⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
Problem 12-36
The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 1.5m:= P 20kN:= w 6
kN
m
:=
b 3m:=
Solution: L 2a b+:=
Support Reactions :
+
 
ΣFy=0; A B+ P− w a⋅− 0= (1)
ΣΜA=0; w a( )− 0.5a( )⋅ P a b+( )⋅+ B b⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
b
w a( )− 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:=
B 27.75 kN= A 1.25 kN=
Moment Function: 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
E I⋅
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
Ψ x a−( )2⋅+
B
2
Ψ x a− b−( )2⋅+ C1+= (3)
E I⋅ v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
Ψ x a−( )3⋅+
B
6
Ψ x a− b−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=a and x=a+b. From Eq. (4):
0
1
24
− w a4⋅ 0+ 0+ 0+ C1 a⋅+ C2+= (5)
0
1
24
− w a b+( )4⋅
1
24
w b4⋅+
A
6
b3⋅+ 0+ C1 a b+( )⋅+ C2+= (6)
(6)-(5): C1
1
b
1
24
w a b+( )4⋅
1
24
w b4⋅−
A
6
b3⋅−
1
24
w⋅ a4⋅−⎡⎢
⎣
⎤⎥
⎦
:= C1 25.125 kN m
2⋅=
From Eq. (5): C2
1
24
w⋅ a4⋅ C1 a⋅−:= C2 36.422− kN m
3⋅=
Equation of Elastic Curve : ao
w
24
:= a1
A
6
:= a2
B
6
:= a3 C1:= a4 C2:=
ao 0.25
kN
m
= a1 0.2083 kN= a2 4.625 kN= a3 25.13 kN m
2⋅= a4 36.42− kN m
3⋅= Ans
v
1
E I⋅
ao− x
4 ao Ψ x a−( )
4
⋅+ a1 Ψ x a−( )
3
⋅+ a2 Ψ x a− b−( )
3
⋅+ a3 x⋅+ a4+
⎛
⎝
⎞
⎠= Ans
Problem 12-37
The shaft supports the two pulley loads shown. Determine the equation of the elastic curve. The
bearings at A and B exert only vertical reactions on the shaft. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 0.5m:= P1 200N:= P2 300N:=
Solution:
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; A 2a( )⋅ P1 a⋅− P2 a⋅+ 0= (2)
Solving Eqs. (1) and (2): A
P1 P2−
2
:= B
P1 3P2+
2
:=
A 50− N= B 550 N=
Moment Function : 
M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− B Ψ x 2a−( )⋅+=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− B Ψ x 2a−( )⋅+=
E I⋅
dv
dx
⋅
A
2
Ψ x 0−( )2⋅
P1
2
Ψ x a−( )2⋅−
B
2
Ψ x 2a−( )2⋅+ C1+=
E I⋅ v⋅
A
6
Ψ x 0−( )3⋅
P1
6
Ψ x a−( )3⋅−
B
6
Ψ x 2a−( )3⋅+ C1 x⋅+ C2+= (1)
Boundary Conditions : v=0 at x=0
From Eq. (1): 0 0 0− 0+ 0+ C2+= C2 0:=
Also v=0 at x=2a.
From Eq. (1): 0
A
6
Ψ 2a 0−( )3⋅
P1
6
Ψ 2a a−( )3⋅− 0+ C1 2a( )⋅+=
C1
A
12a
− 2a( )3⋅
P1
12a
a3⋅+:= C1 12.50 N m
2⋅=
Equation of Elastic Curve : ao
A
6
:= a1
P1
6
−:= a2
B
6
:= a3 C1:=
ao 8.33− N= a1 33.33− N= a2 91.67 N= a3 12.50 N m
2⋅= Ans
v
1
E I⋅
ao Ψ x 0−( )
3
⋅ a1 Ψ x a−( )
3
⋅+ a2 Ψ x 2a−( )
3
⋅+ a3 x⋅+
⎛
⎝
⎞
⎠= Ans
Problem 12-38
The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 4m:= P 50kN:= w 3
kN
m
:=
b 3m:=
Solution: L a 2 b⋅+:=
Support Reactions :
+
 
ΣFy=0; A B+ P− w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ P a b+( )⋅+ B L⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
L
w a( ) 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:=
B 37.4 kN= A 24.6 kN=Moment Function : 
Ans
M x( ) A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−=
Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−=
EI
dv
dx
⋅
A
2
x2⋅
w
6
x3⋅−
w
6
Ψ x a−( )3⋅+
P
2
Ψ x a− b−( )2⋅− C1+= (3)
EI v⋅
A
6
x3⋅
w
24
x4⋅−
w
24
Ψ x a−( )4⋅+
P
6
Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (4)
Boundary Conditions :
v=0 at x=0. From Eq. (4): 0 0 0− 0+ 0+ 0+ C2+= C2 0:=
v=0 at x=L. From Eq. (4): 0
A
6
L3⋅
w
24
L4⋅−
w
24
L a−( )4⋅+
P
6
L a− b−( )3⋅− C1 L⋅+=
C1
A L2⋅
6
−
w
24
L3⋅+
w
24L
L a−( )4⋅−
P
6L
L a− b−( )3⋅+:=
C1 278.70− kN m
2⋅=
Equation of Elastic Curve : ao
A
6
:= a1
w
24
:= a2
P
6
:= a3 C1:= a3 C2:=
ao 4.10 kN= a1 0.125
kN
m
= a2 8.33 kN= a3 0.00 N m
2⋅= Ans
v
1
EI
ao x
3⋅ a1 x
4⋅− a1 Ψ x a−( )
4
⋅+ a2 Ψ x a− b−( )
3
⋅− a3 x⋅+ a4+
⎛
⎝
⎞
⎠=
M x( ) A Ψ x 0−( )⋅ 0.5w Ψ x 0−( )2⋅− 0.5−( )w Ψ x a−( )2⋅− P Ψ x a− b−( )⋅−=
Problem 12-39
The beam is subjected to the load shown. Determine the displacement at x = 7 m and the slope at A.
EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 4m:= P 50kN:= w 3
kN
m
:=
b 3m:=
Solution: L a 2 b⋅+:=
Support Reactions :
+
 
ΣFy=0; A B+ P− w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ P a b+( )⋅+ B L⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
L
w a( ) 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:=
B 37.4 kN= A 24.6 kN=
Moment Function : 
Ans
M x( ) A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−=
Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ A x⋅ 0.5w x2⋅− 0.5w Ψ x a−( )2⋅+ P Ψ x a− b−( )⋅−=
EI
dv
dx
⋅
A
2
x2⋅
w
6
x3⋅−
w
6
Ψ x a−( )3⋅+
P
2
Ψ x a− b−( )2⋅− C1+= (3)
EI v⋅
A
6
x3⋅
w
24
x4⋅−
w
24
Ψ x a−( )4⋅+
P
6
Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (4)
Boundary Conditions :
v=0 at x=0. From Eq. (4): 0 0 0− 0+ 0+ 0+ C2+= C2 0:=
v=0 at x=L. From Eq. (4): 0
A
6
L3⋅
w
24
L4⋅−
w
24
L a−( )4⋅+
P
6
L a− b−( )3⋅− C1 L⋅+=
C1
A L2⋅
6
−
w
24
L3⋅+
w
24L
L a−( )4⋅−
P
6L
L a− b−( )3⋅+:=
C1 278.70− kN m
2⋅=
Equations of Elastic Curve and Slope :
v x( )
1
EI
A
6
x3⋅
w
24
x4⋅−
w
24
Ψ x a−( )4⋅+
P
6
Ψ x a− b−( )3⋅− C1 x⋅+ C2+
⎛⎜
⎝
⎞
⎠
:=
Displacement at x=7m.: Substitute x=7m into Eq.(4). v 7m( ) 834.60−
m
EIo
= Ans
θ x( )
1
EI
A
2
x2⋅
w
6
x3⋅−
w
6
Ψ x a−( )3⋅+
P
2
Ψ x a− b−( )2⋅− C1+
⎛⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=0 into Eq.(3). θ 0( ) 278.70−
1
EIo
=
M x( ) A Ψ x 0−( )⋅ 0.5w Ψ x 0−( )2⋅− 0.5−( )w Ψ x a−( )2⋅− P Ψ x a− b−( )⋅−=
Problem 12-40
The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 2.4m:= P1 10kN:=
MB 6kN m⋅:= P2 20kN:=
Solution:
+
Support Reactions :
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; A 3a( )⋅ P1 2a( )⋅− P2 a⋅− MB+ 0= (2)
Solving Eqs. (1) and (2): A
2P1 P2+
3
MB
3a
−:= B P1 P2+ A−:=
A 12.5 kN= B 17.5 kN=
Moment Function : 
M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x 2a−( )⋅−=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x 2a−( )⋅−=
E I⋅
dv
dx
⋅
A
2
Ψ x 0−( )2⋅
P1
2
Ψ x a−( )2⋅−
P2
2
Ψ x 2a−( )2⋅− C1+=
E I⋅ v⋅
A
6
Ψ x 0−( )3⋅
P1
6
Ψ x a−( )3⋅−
P2
6
Ψ x 2a−( )3⋅− C1 x⋅+ C2+= (1)
Boundary Conditions : v=0 at x=0
From Eq. (1): 0 0 0− 0+ C2+= C2 0:=
Also v=0 at x=3a.
From Eq. (1): 0
A
6
Ψ 3a 0−( )3⋅
P1
6
Ψ 3a a−( )3⋅−
P2
6
Ψ 3a 2a−( )3⋅− C1 3a( )⋅+=
C1
A
18a
− 3a( )3⋅
P1
18a
2a( )3⋅+
P2
18a
a3⋅+:= C1 76.00− kN m
2⋅=
Equation of Elastic Curve : ao
A
6
:= a1
P1
6
−:= a2
P2
6
−:= a3 C1:=
ao 2.08 kN= a1 1.67− kN= a2 3.33− kN= a3 76.00− kN m
2⋅= Ans
v
1
E I⋅
ao Ψ x 0−( )
3
⋅ a1 Ψ x a−( )
3
⋅+ a2 Ψ x 2a−( )
3
⋅+ a3 x⋅+
⎛
⎝
⎞
⎠= Ans
w 6
kN
m
:=
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 1.5m:= b 3m:=
P 20kN:=
Problem 12-41
The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.
Solution: L 2a b+:=
Support Reactions :
+
 
ΣFy=0; A B+ P− w a⋅− 0= (1)
ΣΜA=0; w a( )− 0.5a( )⋅ P a b+( )⋅+ B b⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
b
w a( )− 0.5a( )⋅ P a b+( )⋅+[ ]⋅:= A P w a⋅+ B−:=
B 27.75 kN= A 1.25 kN=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A Ψ x a−( )⋅+ B Ψ x a− b−( )⋅+=
E I⋅
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
Ψ x a−( )2⋅+
B
2
Ψ x a− b−( )2⋅+ C1+= (3)
E I⋅ v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
Ψ x a−( )3⋅+
B
6
Ψ x a− b−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=a and x=a+b. From Eq. (4):
0
1
24
− w a4⋅ 0+ 0+ 0+ C1 a⋅+ C2+= (5)
0
1
24
− w a b+( )4⋅
1
24
w b4⋅+
A
6
b3⋅+ 0+ C1 a b+( )⋅+ C2+= (6)
(6)-(5): C1
1
b
1
24
w a b+( )4⋅
1
24
w b4⋅−
A
6
b3⋅−
1
24
w⋅ a4⋅−⎡⎢
⎣
⎤⎥
⎦
:= C1 25.125 kN m
2⋅=
From Eq. (5): C2
1
24
w⋅ a4⋅ C1 a⋅−:= C2 36.422− kN m
3⋅=
Equation of Elastic Curve : ao
w
24
:= a1
A
6
:= a2
B
6
:= a3 C1:= a4 C2:=
ao 0.25
kN
m
= a1 0.2083 kN= a2 4.625 kN= a3 25.13 kN m
2⋅= a4 36.42− kN m
3⋅= Ans
v
1
E I⋅
ao− x
4 ao Ψ x a−( )
4
⋅+ a1 Ψ x a−( )
3
⋅+ a2 Ψ x a− b−( )
3
⋅+ a3 x⋅+ a4+
⎛
⎝
⎞
⎠= Ans
w3
kN
m
:=
Problem 12-42
The beam is subjected to the load shown. Determine the equation of the slope and elastic curve. EI is
constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 5m:= b 3m:=
Mo 15kN m⋅:=
Solution: L a b+:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− Mo+ 0= (2)
Solving Eqs. (1) and (2): B
1
a
w a( ) 0.5a( )⋅ Mo+⎡⎣ ⎤⎦⋅:= A w a⋅ B−:=
B 10.5 kN= A 4.5 kN=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
Slope and Elastic Curve :
E I⋅
d2 v⋅
dx2
⋅ M x( )=
E I⋅
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
E I⋅
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+
B
2
Ψ x a−( )2⋅+ C1+= (3)
E I⋅ v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+
B
6
Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4):
0 0− 0+ 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w a4⋅ 0+
A
6
a3⋅+ 0+ C1 a( )⋅+= C1
1
24
w a3⋅
A
6
a2⋅−:= C1 3.125− kN m
2⋅=
Equation of Elastic Curve and slope : ao
w
24
:= a1
A
6
:= a2
B
6
:= a3 C1:=
ao 0.125
kN
m
= a1 0.75 kN= a2 1.75 kN= a3 3.125− kN m
2⋅= Ans
v
1
E I⋅
ao− x
4 ao Ψ x a−( )
4
⋅+ a1 x
3⋅+ a2 Ψ x a−( )
3
⋅+ a3 x⋅+
⎛
⎝
⎞
⎠= Ans
θ
1
E I⋅
4− ao⋅ x
3 4 ao⋅ Ψ x a−( )
3
⋅+ 3a1 x
2⋅+ 3a2 Ψ x a−( )
2
⋅+ a3+
⎛
⎝
⎞
⎠= Ans
Problem 12-43
Determine the equation of the elastic curve. Specify the slope at A and the displacement at C. EI is
constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a m:= w
kN
m
:=
Solution: L 2a:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
4
w a( )⋅:= A w a⋅ B−:=
B 0.25 w a⋅= A 0.75 w a⋅=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
EI
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+= (3)
EI v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4):
0 0− 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w 2a( )4⋅
1
24
⎛⎜
⎝
⎞
⎠
w⋅ a4⋅+
A
6
2a( )3⋅+ C1 2a( )⋅+=
C1
1
3
w a3⋅
1
48
w a3⋅−
2A
3
a2⋅−:= C1
3
16
− w a3⋅=
Equations of Elastic Curve and Slope :
v x( )
1
EIo
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
3w a⋅
24
x3⋅+
3
16
w a3⋅ x⋅−⎛⎜
⎝
⎞
⎠
:=
Displacement at C: Substitute x=a into Eq.(4). v a( )
5
48
−
w a4⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
θ x( )
1
EIo
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+
⎛⎜
⎝
⎞
⎠
m:=
Slope at A: Substitute x=0 into Eq.(3). θ 0( )
3
16
− m
w a3⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
Problem 12-44
Determine the equation of the elastic curve. Specify the slope at A and B. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given:
a m:= w
kN
m
:=
Solution: L 2a:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B 2 a⋅( )⋅− 0= (2)
Solving Eqs. (1) and (2): B
1
4
w a( )⋅:= A w a⋅ B−:=
B 0.25 w a⋅= A 0.75 w a⋅=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
Slope and Elastic Curve : EI kN m2⋅:= EIo 1:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+=
EI
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+= (3)
EI v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=2a. From Eq. (4):
0 0− 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w 2a( )4⋅
1
24
⎛⎜
⎝
⎞
⎠
w⋅ a4⋅+
A
6
2a( )3⋅+ C1 2a( )⋅+=
C1
1
3
w a3⋅
1
48
w a3⋅−
2A
3
a2⋅−:= C1
3
16
− w a3⋅=
Equations of Elastic Curve and Slope :
v x( )
1
EIo
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
3w a⋅
24
x3⋅+
3
16
w a3⋅ x⋅−⎛⎜
⎝
⎞
⎠
:= Ans
θ x( )
1
EIo
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+ C1+
⎛⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=0 into Eq.(3). θ 0( )
3
16
−
w a3⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
Slope at B: Substitute x=L into Eq.(3). θ L( )
7
48
w a3⋅
EIo
⎛
⎜
⎝
⎞
⎠
= Ans
Problem 12-45
The beam is subjected to the load shown. Determine the equation of the elastic curve. EI is constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 1.5m:= P 20kN:=
Solution: L 4a:=
Support Reactions :
+
 
ΣFy=0; A B+ 2P− 0= (1)
ΣΜB=0; A 2a( )⋅ P 3a( )⋅− P a⋅+ 0= (2)
Solving Eqs. (1) and (2): A P:= B P:=
Moment Function : 
M x( ) P− Ψ x 0−( )⋅ A Ψ x a−( )⋅+ B Ψ x 3a−( )⋅+=
Slope and Elastic Curve : EI kN m2⋅:=
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ P− Ψ x 0−( )⋅ A Ψ x a−( )⋅+ B Ψ x 3a−( )⋅+=
EI
dv
dx
⋅
P
2
− Ψ x 0−( )2⋅
A
2
Ψ x a−( )2⋅+
B
2
Ψ x 3a−( )2⋅+ C1+= (1)
EI v⋅
P
6
− Ψ x 0−( )3⋅
A
6
Ψ x a−( )3⋅+
B
6
Ψ x 3a−( )3⋅+ C1 x⋅+ C2+=
EI v⋅
P
6
− x3⋅
A
6
Ψ x a−( )3⋅+
B
6
Ψ x 3a−( )3⋅+ C1 x⋅+ C2+= (2)
Boundary Conditions : Due to symmetry, dv/dx=0 at x=2a
From Eq. (1): 0
P
2
− 2a( )2⋅
A
2
Ψ 2a a−( )2⋅+ 0+ C1+= C1 2 P⋅
A
2
−⎛⎜
⎝
⎞
⎠
a2⋅:=
C1 67.50 kN m
2⋅=Also v=0 at x=a.
From Eq. (2): 0
P
6
− a3⋅ 0+ 0+ C1 a( )⋅+ C2+= C2
P
6
a3⋅ C1 a⋅−:=
C2 90− kN m
3⋅=
Equation of Elastic Curve : ao
P
6
:= a1
A
6
:= a2 C1:= a3 C2:=
ao 3333.33 N= a1 3.3333 kN= a2 67.5 kN m
2⋅= a3 90− kN m
3⋅= Ans
v
1
E I⋅
ao− x
3⋅ a1 Ψ x a−( )
3
⋅+ a1 Ψ x 3a−( )
3
⋅+ a2 x⋅+ a3+
⎛
⎝
⎞
⎠= Ans
w 2
kN
m
:=
Problem 12-46
The beam is subjected to the load shown. Determine the equation of the slope and elastic curve. EI is
constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 5m:= b 3m:=
P 8kN:=
Solution: L a b+:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− P− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− P L⋅+ 0= (2)
Solving Eqs. (1) and (2): B
1
a
w a( ) 0.5a( )⋅ P L⋅+[ ]⋅:=
B 17.8 kN=
A w a⋅ P+ B−:= A 0.2 kN=
Moment Function : 
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+=
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
Slope and Elastic Curve :
EI
d2 v⋅
dx2
⋅ M x( )=
EI
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
EI
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+
B
2
Ψ x a−( )2⋅+ C1+= (3)
EI v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+
B
6
Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4):
0 0− 0+ 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w a4⋅ 0+
A
6
a3⋅+ 0+ C1 a( )⋅+= C1
1
24
w a3⋅
A
6
a2⋅−:= C1 9.583 kN m
2⋅=
Equation of Elastic Curve and slope : ao
w
24
:= a1
A
6
:= a2
B
6
:= a3 C1:=
ao 0.0833
kN
m
= a1 0.0333 kN= a2 2.9667 kN= a3 9.583 kN m
2⋅= Ans
v
1
EI
ao− x
4 ao Ψ x a−( )
4
⋅+ a1 x
3⋅+ a2 Ψ x a−( )
3
⋅+ a3 x⋅+
⎛
⎝
⎞
⎠= Ans
θ
1
EI
4− ao⋅ x
3 4 ao⋅ Ψ x a−( )
3
⋅+ 3a1 x
2⋅+ 3a2 Ψ x a−( )
2
⋅+ a3+
⎛
⎝
⎞
⎠= Ans
EI
d2v
dx2
⋅ 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
w 2
kN
m
:=
Problem 12-47
The beam is subjected to the load shown. Determine the slope at A and the displacement at C. EI is
constant.
Use Macaulay Function: Ψ z( ) Φ z( ) z( )⋅:=
Given: a 5m:= b 3m:=
P 8kN:=
Solution: L a b+:=
Support Reactions :
+
 
ΣFy=0; A B+ w a⋅− P− 0= (1)
ΣΜA=0; w a( ) 0.5a( )⋅ B a⋅− P L⋅+ 0= (2)
Solving Eqs. (1) and (2): B
1
a
w a( ) 0.5a( )⋅ P L⋅+[ ]⋅:= A w a⋅ P+ B−:=
B 17.8 kN= A 0.2 kN=Moment Function : 
M x( ) 0.5− w x2⋅ 0.5w Ψ x a−( )2⋅+ A x⋅+ B Ψ x a−( )⋅+=
Slope and Elastic Curve : EI kN m2⋅:=
EIo 1:=EI
d2 v⋅
dx2
⋅ M x( )=
EI
dv
dx
⋅
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+
B
2
Ψ x a−( )2⋅+ C1+= (3)
EI v⋅
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+
B
6
Ψ x a−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=0 and x=a. From Eq. (4):
0 0− 0+ 0+ 0+ 0+ C2+= C2 0:=
0
1
24
− w a4⋅ 0+
A
6
a3⋅+ 0+ C1 a( )⋅+= C1
1
24
w a3⋅
A
6
a2⋅−:= C1 9.583 kN m
2⋅=
Equation of Elastic Curve and slope :
v x( )
1
EI
1
24
− w x4⋅
1
24
w Ψ x a−( )4⋅+
A
6
x3⋅+
B
6
Ψ x a−( )3⋅+ C1 x⋅+
⎛⎜
⎝
⎞
⎠
:=
Displacement at C: Substitute x=L into Eq.(4). v L( ) 160.75−
m
EIo
= AnsAns
θ x( )
1
EI
1
6
− w x3⋅
1
6
w Ψ x a−( )3⋅+
A
2
x2⋅+
B
2
Ψ x a−( )2⋅+ C1+
⎛⎜
⎝
⎞
⎠
:=
Slope atA: Substitute x=0 into Eq.(3). θ 0( ) 9.58
1
EIo
= Ans
M x( ) 0.5− w Ψ x 0−( )2⋅ 0.5−( )w Ψ x a−( )2⋅− A Ψ x 0−( )⋅+ B Ψ x a−( )⋅+=
Problem 12-48
The beam is subjected to the load shown. Determine the equation of the elastic curve.
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: L1 2m:= L2 3m:= wo 150
kN
m
:=
Solution: L L1 L2+:=
Support Reactions :
+ ΣFy=0; A B+ wo L1⋅−
wo
2
L2⋅− 0= (1)
 
ΣΜA=0;
wo
2
− L1
2⋅
wo
2
L2⋅
L2
3
⎛
⎜
⎝
⎞
⎠
⋅+ B L2⋅− 0= (2)
Solving Eqs. (1) and (2): B
3− L1
2 L2
2+
6L2
wo⋅:= A wo L1 0.5L2+( )⋅ B−:=
B 25− kN= A 550 kN=
Moment Function : 
M x( )
wo
2
− Ψ x 0−( )2⋅
wo
6L2
Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+=
Slope and Elastic Curve : EI
d2 v⋅
dx2
⋅ M x( )= EI kN m2⋅:= EIo 1:=
EI
d2v
dx2
⋅
wo
2
− Ψ x 0−( )2⋅
wo
6L2
Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+=
EI
dv
dx
⋅
wo
6
− Ψ x 0−( )3⋅
wo
24 L2⋅
Ψ x L1−( )4⋅+
A
2
Ψ x L1−( )2⋅+ C1+= (3)
EI v⋅
wo
24
− Ψ x 0−( )4⋅
wo
120 L2⋅
Ψ x L1−( )5⋅+
A
6
Ψ x L1−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=L1 Given
(5)From Eq. (4): 0
wo
24
− L1
4⋅ 0+ 0+ C1 L1⋅+ C2+=
Also v=0 at x=L.
From Eq. (4): 0
wo
24
− L4⋅
wo
120 L2⋅
L L1−( )5⋅+
A
6
L L1−( )3⋅+ C1 L⋅+ C2+= (6)
Solving Eqs. (5) and (6): Guess C1 1kN m
2⋅:= C2 1kN m
3⋅:=
C1
C2
⎛⎜
⎜⎝
⎞
⎠
Find C1 C2,( ):= C1 410 kN m2⋅= C2 720− kN m3⋅=
Equation of Elastic Curve : ao
wo
24
−:= a1
wo
120 L2⋅
:= a2
A
6
:= a3 C1:= a4 C2:=
ao 6.25−
1
m
kN= a1 0.42
1
m2
kN= a2 91.67 kN= Ans
a3 410.00 kN m
2⋅= a4 720.00− kN m
3⋅= Ans
v
1
E I⋅
ao Ψ x 0−( )
4
⋅ a1 Ψ x L1−( )5⋅+ a2 Ψ x L1−( )3⋅+ a3 x⋅+ a4+⎛⎝ ⎞⎠= Ans
wo 150
kN
m
:=
A B+ wo L1⋅−
wo
2
L2⋅− 0=
Problem 12-49
Determine the displacement at C and the slope at A of the beam.
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: L1 2m:= L2 3m:=
Solution: L L1 L2+:=
Support Reactions :
+ ΣFy=0; (1)
 
ΣΜA=0;
wo
2
− L1
2⋅
wo
2
L2⋅
L2
3
⎛
⎜
⎝
⎞
⎠
⋅+ B L2⋅− 0= (2)
Solving Eqs. (1) and (2): B
3− L1
2 L2
2+
6L2
wo⋅:= A wo L1 0.5L2+( )⋅ B−:=
B 25− kN= A 550 kN=
Moment Function : 
M x( )
wo
2
− Ψ x 0−( )2⋅
wo
6L2
Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+=
Slope and Elastic Curve : EI
d2 v⋅
dx2
⋅ M x( )= EI kN m2⋅:= EIo 1:=
EI
d2v
dx2
⋅
wo
2
− Ψ x 0−( )2⋅
wo
6L2
Ψ x L1−( )3⋅+ A Ψ x L1−( )⋅+=
EI
dv
dx
⋅
wo
6
− Ψ x 0−( )3⋅
wo
24 L2⋅
Ψ x L1−( )4⋅+
A
2
Ψ x L1−( )2⋅+ C1+= (3)
EI v⋅
wo
24
− Ψ x 0−( )4⋅
wo
120 L2⋅
Ψ x L1−( )5⋅+
A
6
Ψ x L1−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : v=0 at x=L1 Given
(5)From Eq. (4): 0
wo
24
− L1
4⋅ 0+ 0+ C1 L1⋅+ C2+=
Also v=0 at x=L.
From Eq. (4): 0
wo
24
− L4⋅
wo
120 L2⋅
L L1−( )5⋅+
A
6
L L1−( )3⋅+ C1 L⋅+ C2+= (6)
Solving Eqs. (5) and (6): Guess C1 1kN m
2⋅:= C2 1kN m
3⋅:=
C1
C2
⎛⎜
⎜⎝
⎞
⎠
Find C1 C2,( ):= C1 410 kN m2⋅= C2 720− kN m3⋅=
Equation of Elastic Curve and Slope :
v x( )
1
EI
wo
24
− Ψ x 0−( )4⋅
wo
120 L2⋅
Ψ x L1−( )5⋅+
A
6
Ψ x L1−( )3⋅+ C1 x⋅+ C2+
⎛
⎜
⎝
⎞
⎠
:= (7)
Displacement at C: Substitute x=0 into Eq.(7). v 0m( ) 720−
m
EIo
= Ans
θ x( )
1
EI
wo
6
− Ψ x 0−( )3⋅
wo
24 L2⋅
Ψ x L1−( )4⋅+
A
2
Ψ x L1−( )2⋅+ C1+
⎛
⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=L1 into Eq.(3). θ L1( ) 210
1
EIo
= Ans
A B+ w L⋅− 0=
w
kN
m
:=
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: L m:= EIo 1:=
Problem 12-50
Determine the equation of the elastic curve. Specify the slope at A. EI is constant.
Solution:
Support Reactions :
+
 
ΣFy=0; (1)
ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2)
Solving Eqs. (1) and (2): B 0.5− w L:=
A w L⋅ B−:= A 1.5 w L⋅=
Moment Function : 
M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
Slope and Elastic Curve :
Ans
EI kN m2⋅:=
EI
d2v
dx2
⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
EI
dv
dx
⋅
w
6
− Ψ x 0−( )3⋅
w
6
Ψ x L−( )3⋅+
A
2
Ψ x L−( )2⋅+ C1+= (3)
EI v⋅
w
24
− Ψ x 0−( )4⋅
w
24
Ψ x L−( )4⋅+
A
6
Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : Given
v=0 at x=L From Eq. (4): 0
w
24
− L4⋅ 0+ 0+ C1 L⋅+ C2+= (5)
v=0 at x=2L From Eq. (4): 0
w
24
− 2L( )4⋅
w
24
L4⋅+
A
6
L3⋅+ C1 2L( )⋅+ C2+= (6)
Solving Eqs. (5) and (6): Guess C1 1kN m
2⋅:= C2 1kN m
3⋅:=
C1
C2
⎛⎜
⎜⎝
⎞
⎠
Find C1 C2,( ):= C1
1
3
w L3⋅= C2
7
24
− w L4⋅=
Equation of Elastic Curve and Slope :
v x( )
1
EI
w
24
− x4⋅
w
24
Ψ x L−( )4⋅+
3w L⋅
12
Ψ x L−( )3⋅+
1
3
w L3⋅ x⋅+
7
24
w L4⋅−⎛⎜
⎝
⎞
⎠
:= Ans
θ x( )
1
EI
w
6
− Ψ x 0−( )3⋅
w
6
Ψ x L−( )3⋅+
9w L⋅
12
Ψ x L−( )2⋅+ C1+
⎛⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=L into Eq.(3). θ L( )
1
6
w L3⋅
EI
=
EI
d2 v⋅
dx2
⋅ M x( )=
w
kN
m
:=
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: L m:=
Problem 12-51
Determine the equation of the elastic curve. Specify the deflection at C. EI is constant.
Solution:
Support Reactions :
+
 
ΣFy=0; A B+ w L⋅− 0= (1)
ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2)
Solving Eqs. (1) and (2): B 0.5− w L:=
A w L⋅ B−:= A 1.5 w L⋅=
Moment Function : 
M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
Slope and Elastic Curve :
Ans
EI kN m2⋅:= EIo 1:=
EI
d2v
dx2
⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
EI
dv
dx
⋅
w
6
− Ψ x 0−( )3⋅
w
6
Ψ x L−( )3⋅+
A
2
Ψ x L−( )2⋅+ C1+= (3)
EI v⋅
w
24
− Ψ x 0−( )4⋅
w
24
Ψ x L−( )4⋅+
A
6
Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : Given
v=0 at x=L From Eq. (4): 0
w
24
− L4⋅ 0+ 0+ C1 L⋅+ C2+= (5)
v=0 at x=2L From Eq. (4): 0
w
24
− 2L( )4⋅
w
24
L4⋅+
A
6
L3⋅+ C1 2L( )⋅+ C2+= (6)
Solving Eqs. (5) and (6): Guess C1 1kN m
2⋅:= C2 1kN m
3⋅:=
C1
C2
⎛⎜
⎜⎝
⎞
⎠
Find C1 C2,( ):= C1
1
3
w L3⋅= C2
7
24
− w L4⋅=
Equation of Elastic Curve :
v x( )
1
EI
w
24
− x4⋅
w
24
Ψ x L−( )4⋅+
3w L⋅
12
Ψ x L−( )3⋅+
1
3
w L3⋅ x⋅+
7
24
w L4⋅−⎛⎜
⎝
⎞
⎠
:= Ans
v 0( )
7
24
−
w L4⋅
EI
=
EI
d2 v⋅
dx2
⋅ M x( )=
w
kN
m
:=
Problem 12-52
Determine the equation of the elastic curve. Specify the slope at B. EI is constant.
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: L m:=
Solution:
Support Reactions :
+
 
ΣFy=0; A B+ w L⋅− 0= (1)
ΣΜA=0; 0.5L( )− w L⋅ B L⋅− 0= (2)
Solving Eqs. (1) and (2): B 0.5− w L:=
A w L⋅ B−:= A 1.5 w L⋅=
Moment Function : 
M x( ) 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
Slope and Elastic Curve :
Ans
EI kN m2⋅:= EIo 1:=
EI
d2v
dx2
⋅ 0.5w( )− Ψ x 0−( )2⋅ 0.5− w( ) Ψ x L−( )2⋅− A Ψ x L−( )⋅+=
EI
dv
dx
⋅
w
6
− Ψ x 0−( )3⋅
w
6
Ψ x L−( )3⋅+
A
2
Ψ x L−( )2⋅+ C1+= (3)
EI v⋅
w
24
− Ψ x 0−( )4⋅
w
24
Ψ x L−( )4⋅+
A
6
Ψ x L−( )3⋅+ C1 x⋅+ C2+= (4)
Boundary Conditions : Given
v=0 at x=L From Eq. (4): 0
w
24
− L4⋅ 0+ 0+ C1 L⋅+ C2+= (5)
v=0 at x=2L From Eq. (4): 0
w
24
− 2L( )4⋅
w
24
L4⋅+
A
6
L3⋅+ C1 2L( )⋅+ C2+= (6)
Solving Eqs. (5) and (6): Guess C1 1kN m
2⋅:= C2 1kN m
3⋅:=
C1
C2
⎛⎜
⎜⎝
⎞
⎠
Find C1 C2,( ):= C1
1
3
w L3⋅= C2
7
24
− w L4⋅=
Equation of Elastic Curve and Slope :
v x( )
1
EI
w
24
− x4⋅
w
24
Ψ x L−( )4⋅+
3w L⋅
12
Ψ x L−( )3⋅+
1
3
w L3⋅ x⋅+
7
24
w L4⋅−⎛⎜
⎝
⎞
⎠
:= Ans
θ x( )
1
EI
w
6
− Ψ x 0−( )3⋅
w
6
Ψ x L−( )3⋅+
9w L⋅
12
Ψ x L−( )2⋅+ C1+
⎛⎜
⎝
⎞
⎠
:=
Slope at A: Substitute x=2L into Eq.(3). θ 2L( )
1
12
−
w L3⋅
EI
=
EI
d2 v⋅
dx2
⋅ M x( )=
Problem 12-53
The shaft is made of steel and has a diameter of 15 mm. Determine its maximum deflection. The
bearings at A and B exert only vertical reactions on the shaft. Est = 200 GPa.
Use Macaulay Function: Ψ x( ) Φ x( ) x⋅:=
Given: a 200mm:= P1 250N:=
b 300mm:= P2 80N:=
do 15mm:= E 200GPa:=
Solution: L 2a b+:=
Section Property: I
πdo
4
64
:=
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; A L⋅ P1 a b+( )⋅− P2 b⋅− 0= (2)
Solving Eqs. (1) and (2): A
P1 a b+( )⋅ P2 a⋅+
L
:= A 201.429 N=
B P1 P2+ A−:= B 128.571 N=
Moment Function : 
M x( ) A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x a− b−( )⋅−=
Slope and Elastic Curve :
E I⋅
d2v
dx2
⋅ A Ψ x 0−( )⋅ P1 Ψ x a−( )⋅− P2 Ψ x a− b−( )⋅−=
E I⋅
dv
dx
⋅
A
2
Ψ x 0−( )2⋅
P1
2
Ψ x a−( )2⋅−
P2
2
Ψ x a− b−( )2⋅− C1+= (1)
E I⋅ v⋅
A
6
Ψ x 0−( )3⋅
P1
6
Ψ x a−( )3⋅−
P2
6
Ψ x a− b−( )3⋅− C1 x⋅+ C2+= (2)
Boundary Conditions :
v=0 at x=0 From Eq. (2): 0 0 0− 0+ C2+= C2 0:=
v=0 at x=L From Eq. (2): 0
A6
L3⋅
P1
6
L a−( )3⋅−
P2
6
a3⋅− C1 L( )⋅+=
C1
A
6
− L2⋅
P1
6L
L a−( )3⋅+
P2
6L
a3⋅+:= C1 8.857− N m
2⋅=
Equation of Elastic Curve :
v x( )
1
E I⋅
A
6
x3⋅
P1
6
Ψ x a−( )3⋅−
P2
6
Ψ x a− b−( )3⋅− C1 x⋅+
⎛
⎜
⎝
⎞
⎠
:= (3)
Maximum Deflection: Assume vmax occurs at a < x < a+b. Given
E I⋅
d2 v⋅
dx2
⋅ M x( )=
dv/dx=0 at x', From Eq. (1): 0
A
2
x'2⋅
P1
2
x' a−( )2⋅− 0− C1+= (4)
Solving Eq.(3): Guess x' 300 mm⋅:= x' Find x'( ):=
x' 330.05 mm=
For vmax, substitute x' into Eq.(3). v x'( ) 3.64− mm= Ans
Note: The negative sign indicates downward displacement.
Problem 12-54
Determine the slope and deflection at C. EI is constant.
Given: L1 8m:= L2 4m:= P 75kN:=
Solution:
Support Reactions : L L1 L2+:=
+ ΣFy=0; A B+ P− 0= (1)
 
ΣΜB=0; A L1⋅ P L2⋅+ 0= (2)
Solving Eqs. (1) and (2): A P−
L2
L1
⋅:= B P
L1 L2+
L1
⋅:=
A 37.5− kN= B 112.5 kN=
M/EI Diagram:
Set EI kN m2⋅:= EIo 1:=
x1 0 0.01 L1⋅, L1..:= x2 L1 1.01 L1⋅, L..:=
M'1 x1( ) A x1⋅( )
1
EI
⋅:= M'2 x2( ) A x2⋅ B x2 L1−( )⋅+⎡⎣ ⎤⎦
1
EI
⋅:=
0 5 10
200
0
Distance (m)
M
/E
I (
1/
m
)
M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Slopes :
tB.A
1
2
M'1 L1( ) L1⋅
L1
3
⎛
⎜
⎝
⎞
⎠
⋅:=
θA
tB.A
L1
:= θA 400
1
EIo
=
θC.A
1
2
M'1 L1( ) L1⋅
1
2
M'1 L1( ) L2⋅+:=
θC θC.A θA−:= θC 1400
1
EIo
= Ans
Deflections :
tC.A
1
2
M'1 L1( ) L1⋅
L1
3
L2+
⎛
⎜
⎝
⎞
⎠
⋅
1
2
M'1 L1( ) L2⋅
2L2
3
⎛
⎜
⎝
⎞
⎠
⋅+:=
∆C tC.A
L
L1
tB.A⋅−:= ∆C 4800
m
EIo
= Ans
Problem 12-55
Determine the slope and deflection at B. EI is constant.
Problem 12-56
If the bearings exert only vertical reactions on the shaft, determine the slope at the bearings and the
maximum deflection of the shaft. EI is constant.
Problem 12-57
Determine the slope at B and the deflection at C. EI is constant.
Problem 12-58
Determine the slope at C and the deflection at B. EI is constant.
Problem 12-59
The 60-kg gymnast stands on the center of the simply supported balance beam. If the beam is made of
wood and has the cross section shown, determine the maximum deflection.The supports at A and B
are assumed to be rigid. Ew = 12 GPa.
Given: mo 60 kg⋅:= L 2.7m:=
bt 125mm:= bb 75mm:=
h 150mm:= E 12GPa:=
Solution: W mo g⋅:=
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R W− 0=
2.7 m 5.4 m
m
794.34
R 0.5W:= R 294.20 N=
Section Property : ∆b bt bb−:=
yc
Σ yi
⎯
Ai⋅( )⋅
Σ Ai( )⋅
= yc
bb h⋅( ) 0.5h( )⋅ 0.5∆b h⋅
h
3
⎛⎜
⎝
⎞
⎠
⋅+
bb h⋅ 0.5∆b h⋅+
:= yc 68.75 mm=
I
1
12
bb⋅ h
3⋅ bb h⋅( ) 0.5h yc−( )2⋅+
1
36
∆b⋅ h3⋅+ 0.5 ∆b⋅ h⋅( ) h
3
yc−
⎛⎜
⎝
⎞
⎠
2
⋅+:=
I 27539062.50 mm4=
M/EI Diagram:
x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:=
M'1 x1( ) R x1⋅( )
1
E I⋅
⋅⎡⎢
⎣
⎤⎥
⎦
:= M'2 x2( ) R x2⋅ W x2 L−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅⎡⎢
⎣
⎤⎥
⎦
:=
0 2 4
0
0.002
Distance (m)
M
/E
I M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Deflections :
tA.C
1
2
M'1 L( ) L⋅
2L
3
⎛⎜
⎝
⎞
⎠
⋅:=
∆C tA.C:=
∆C 5.84 mm=
Due to symmetry, the slope at
midspan (point C) is zero. Hence,
∆max ∆C:=
∆max 5.84 mm= Ans
Problem 12-60
The shaft is supported by a journal bearing at A, which exerts only vertical reactions on the shaft, and
by a thrust bearing at B, which exerts both horizontal and vertical reactions on the shaft. Determine the
slope of the shaft at the bearings. EI is constant.
Given: L 0.3m:= a 0.1m:= P 400N:=
Solution:
Support Reactions : L 2L:=
Due to anti-symmetry, 
 
B A−=
ΣΜB=0; A 2L( )⋅ P 2a( )⋅− 0=
A
a
L
⎛⎜
⎝
⎞
⎠
P:= B A−:=
A 66.67 N= B 66.67− N=
M/EI Diagram:
Set EI kN m2⋅:= EIo 1:=
x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:=
M'1 x1( ) A x1⋅( )
1
EI
⋅:= M'2 x2( ) A x2⋅ P 2a( )⋅−⎡⎣ ⎤⎦
1
EI
⋅:=
0 0.5 1
0
Distance (m)
M
/E
I (
1/
m
)
M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Slopes :
tB.A
1−
2
M'1 L( ) L⋅
2L
3
⎛⎜
⎝
⎞
⎠
⋅
1
2
M'1 L( ) L⋅
L
3
L+⎛⎜
⎝
⎞
⎠
⋅+:=
θA
tB.A
L
:= θA 0.008
1
EIo
= Ans
Due to anti-symmetry (or in a similar manner),
θB θA:= θB 0.008
1
EIo
= Ans
Problem 12-61
The beam is subjected to the loading shown. Determine the slope at A and the displacement at C.
Assume the support at A is a pin and B is a roller. EI is constant.
Problem 12-62
The rod is constructed from two shafts for which the moment of inetia of AB is I and BC is 2I.
Determine the maximum slope and deflection of the rod due to the loading. The modulus of elasticity is
E.
Problem 12-63
Determine the deflection and slope at C. EI is constant.
Problem 12-64
If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at A. EI is
constant.
Problem 12-65
If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at C. EI is
constant.
Problem 12-66
Determine the deflection at C and the slope of the beam at A, B, and C. EI is constant.
Given: L1 6m:= L2 3m:= Mo 8kN m⋅:=
Solution: L L1 L2+:=
Support Reactions :
+
 
ΣFy=0; A B+ 0= (1)
ΣΜB=0; A L1⋅ Mo+ 0= (2)
Solving Eqs. (1) and (2): A
Mo
L1
−:= B A−:=
A 1.333− kN= B 1.333 kN=
M/EI Diagram:
Set EI kN m2⋅:= EIo 1:=
x1 0 0.01 L1⋅, L1..:= x2 L1 1.01 L1⋅, L..:=
M'1 x1( ) A x1⋅( )
1
EI
⋅:= M'2 x2( ) A x2⋅ B x2 L1−( )⋅+⎡⎣ ⎤⎦
1
EI
⋅:=
0 5
10
5
0
Distance (m)
M
/E
I (
1/
m
)
M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Deflections :
tB.A
1
2
M'1 L1( ) L1⋅
L1
3
⎛
⎜
⎝
⎞
⎠
⋅:=
tC.A
1
2
M'1 L1( ) L1⋅
L1
3
L2+
⎛
⎜
⎝
⎞
⎠
⋅ M'1 L1( ) L2⋅
L2
2
⎛
⎜
⎝
⎞
⎠
⋅+:=
∆C tC.A
L
L1
tB.A⋅−:=
∆C 84
m
EIo
= Ans
Slopes :
θA
tB.A
L1
:= θA 8
1
EIo
= Ans
θB.A
1
2
M'1 L1( ) L1⋅:=
θB θB.A θA−:= θB 16
1
EIo
= Ans
θC.A
1
2
M'1 L1( ) L1⋅ M'1 L1( ) L2⋅+:=
θC θC.A θA−:= θC 40
1
EIo
= Ans
Problem 12-67
The bar is supported by the roller constraint at C, which allows vertical displacement but resists axial
load and moment. If the bar is subjected to the loading shown, determine the slope and displacement at
A. EI is constant.
Problem 12-68
The acrobat has a weight of 750 N (~75 kg), and suspends himself uniformly from the center of the
high bar. Determine the maximum bending stress in the pipe (bar) and its maximum deflection. The
pipe is made of L2 steel and has an outer diameter of 25 mm and a wall thickness of 3 mm.
Given: a 0.9m:= do 25mm:= W 750N:=
b 0.45m:= t 3mm:= E 200GPa:=
Solution: L 2a b+:=
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R W− 0=
R 0.5W:= R 375.00 N=
Section Property : ro 0.5do:= ri ro t−:=
I
π
4
ro
4 ri
4−⎛⎝
⎞
⎠⋅:=
Maximum Bendiug Stress :
Mmax R a⋅:= σmax
Mmax ro⋅
I
:= σmax 330.17 MPa= Ans
< σY = 703 MPa (O.K. !)
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) R x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) R x2⋅
W
2
x2 a−( )⋅−⎡⎢⎣
⎤⎥
⎦
1
E I⋅
⋅:=
M'3 x3( ) R x3⋅ W x3
L
2
−⎛⎜
⎝
⎞
⎠
⋅−⎡⎢
⎣
⎤⎥
⎦
1
E I⋅
⋅:=
0 1 2
0
0.1
0.2
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
Moment-Area Theorems :
Deflections :
tA.C
1
2
M'1 a( ) a⋅
2a
3
⎛⎜
⎝
⎞
⎠
⋅ M'1 a( )
b
2
⎛⎜
⎝
⎞
⎠
⋅
b
4
a+⎛⎜
⎝
⎞
⎠
⋅+:=
∆C tA.C:=
∆C 65.74 mm=
Due to symmetry, the slope at midspan (point C) is zero. Hence,
∆max ∆C:= ∆max 65.74 mm= Ans
Problem 12-69
Determine the value of a so that the displacement at C is equal to zero. EI is constant.
Problem 12-70
The beam is made of a ceramic material. In order to obtain its modulus of elasticity, it is subjected to
the elastic loading shown. If the moment of inertia is I and the beam has a measured maximum
deflection ∆, determine E. The supports at A and D exert only vertical reactions on the beam.
Problem 12-71
Determine the maximum deflection of the shaft. EI is constant. The bearings exert only vertical
reactions on the shaft.
Problem 12-72
The beam is subjected to the load P as shown. Determine the magnitude of force F that must be
applied at the end of the overhang C so that the deflection at C is zero. EI is constant.
Problem 12-73
At what distance a should the bearing supports at A and B be placed so that the deflection at the center
of the shaft is equal to the deflection at its ends? The bearings exertonly vertical reactions on the shaft.
EI is constant.
0 1 2
0
0.01
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
Given: a 0.5m:= b 0.8m:= c 1.2m:= do 50mm:=
P1 600N:= P2 1200N:= E 200GPa:=
Solution: L a b+ c+:=
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; A L⋅ P1 b c+( )⋅− P2 c⋅− 0= (2)
Solving Eqs. (1) and (2): A
P1 b c+( )⋅ P2 c⋅+
L
:= B P1 P2+ A−:=
A 1.056 kN= B 0.744 kN=
Section Property : I
π do
4⋅
64
:=
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) A x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) A x2⋅ P1 x2 a−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
M'3 x3( ) A x3⋅ P1 x3 a−( )⋅− P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
Problem 12-74
Determine the slope of the 50-mm-diameter A-36 steel shaft at the bearings at A and B. The bearings
exert only vertical reactions on the shaft.
Moment-Area Theorems :
Slopes :
tB.A
1
2
M'3 L c−( ) c⋅
2c
3
⋅
1
2
M'2 a b+( ) M'1 a( )−( ) b⋅
b
3
c+⎛⎜
⎝
⎞
⎠
⋅+
...
M'1 a( ) b⋅
b
2
c+⎛⎜
⎝
⎞
⎠
⋅
1
2
M'1 a( ) a⋅ L
2a
3
−⎛⎜
⎝
⎞
⎠
⋅++
...
:=
θA
tB.A
L
:= θA 0.01046 rad=
Ans
θB
tA.B
L
:=
θB 0.00968 rad= Ans
tA.B
1
2
M'3 L c−( ) c⋅ L
2c
3
−⎛⎜
⎝
⎞
⎠
⋅
1
2
M'2 a b+( ) M'1 a( )−( ) b⋅
2 b⋅
3
a+⎛⎜
⎝
⎞
⎠
⋅+
...
M'1 a( ) b( )⋅
b
2
a+⎛⎜
⎝
⎞
⎠
⋅
1
2
M'1 a( ) a⋅
2a
3
⎛⎜
⎝
⎞
⎠
⋅++
...
:=
0 1 2
0
0.01
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
Given: a 0.5m:= b 0.8m:= c 1.2m:= do 50mm:=
P1 600N:= P2 1200N:= E 200GPa:=
Solution: L a b+ c+:=
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; A L⋅ P1 b c+( )⋅− P2 c⋅− 0= (2)
Solving Eqs. (1) and (2): A
P1 b c+( )⋅ P2 c⋅+
L
:= B P1 P2+ A−:=
A 1.056 kN= B 0.744 kN=
Section Property : I
π do
4⋅
64
:=
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) A x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) A x2⋅ P1 x2 a−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
M'3 x3( ) A x3⋅ P1 x3 a−( )⋅− P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
Problem 12-75
Determine the maximum deflection of the 50-mm diameter A-36 steel shaft. It is supported by bearings
at its ends A and B which only exert vertical reactions on the shaft.
Moment-Area Theorems :
Deflections :
tB.A
1
2
M'3 L c−( ) c⋅
2c
3
⋅
1
2
M'2 a b+( ) M'1 a( )−( ) b⋅
b
3
c+⎛⎜
⎝
⎞
⎠
⋅+
...
M'1 a( ) b⋅
b
2
c+⎛⎜
⎝
⎞
⎠
⋅
1
2
M'1 a( ) a⋅ L
2a
3
−⎛⎜
⎝
⎞
⎠
⋅++
...
:=
θA
tB.A
L
:= θA 0.01046 rad= Ans
Maximum Deflection: Assume vmax occurs at point E ( a < x' < a+b ).
Slopes :
θE.A
1
2
M'1 a( ) a⋅ M'1 a( ) x'⋅+
1
2
M'2 a b+( ) M'1 a( )−( )
x'
b
⋅ x'⋅+=
θE.A
1
2
A a⋅
E I⋅
a⋅
A a⋅
E I⋅
x'⋅+
1
2E I⋅
A a b+( )⋅ P1 b⋅− A a⋅−⎡⎣ ⎤⎦
x'
b
⋅ x'⋅+=
θE.A
1
2E I⋅
A a2⋅ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+⎡⎣ ⎤⎦=
θE θA− θE.A+=
θE θA−
1
2E I⋅
A a2⋅ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+⎡⎣ ⎤⎦+=
At point E, where θE=0. Given
0 θA− 2E I⋅( )⋅ A a
2⋅+ 2A a⋅ x'⋅+ A P1−( ) x'2⋅+= (3)
Solving Eq.(3): Guess x' 0.7 m⋅:= x' Find x'( ):=
x' 733.27 mm= < 0.8m (O.K. !)
Deflection:
tA.E
1
2
M'1 a( ) a⋅
2a
3
⎛⎜
⎝
⎞
⎠
⋅ M'1 a( ) x'⋅
x'
2
a+⎛⎜
⎝
⎞
⎠
⋅+
1
2
M'2 a b+( ) M'1 a( )−( )
x'
b
⋅ x'⋅
2 x'⋅
3
a+⎛⎜
⎝
⎞
⎠
⋅+⎡⎢
⎣
⎤⎥
⎦
:=
∆max tA.E:=
∆max 8.161 mm= Ans
0 0.5 1
0.1
0.05
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
a 0.2m:= b 0.3m:= c 0.5m:= do 20mm:=
Problem 12-76
Determine the slope of the 20-mm-diameter A-36 steel shaft at the bearings at A and B. The bearings
exert only vertical forces on the shaft.
P1 800N:= P2 350N:= E 200GPa:=
Solution: L a b+ c+:=
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; P1− L⋅ A b c+( )⋅+ P2 c⋅− 0= (2)
Solving Eqs. (1) and (2): A
P1 L⋅ P2 c⋅+
b c+
:= B P1 P2+ A−:=
A 1218.75 N= B 68.75− N=
Section Property : I
π do
4⋅
64
:=
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) P1− x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) P1− x2⋅ A x2 a−( )⋅+⎡⎣ ⎤⎦
1
E I⋅
⋅:=
M'3 x3( ) P1− x3⋅ A x3 a−( )⋅+ P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
Given:
Moment-Area Theorems :
Slopes :
tB.A
1
2
M'3 a b+( ) c⋅
2c
3
⋅ M'3 a b+( ) b⋅
b
2
c+⎛⎜
⎝
⎞
⎠
⋅+
1
2
M'2 a( ) M'3 a b+( )−( ) b⋅
2 b⋅
3
c+⎛⎜
⎝
⎞
⎠
⋅+
...:=
θA
tB.A
b c+
:= θA 0.01811 rad= Ans
tA.B
1
2
M'3 a b+( ) c⋅
c
3
b+⎛⎜
⎝
⎞
⎠
⋅ M'3 a b+( ) b⋅
b
2
⎛⎜
⎝
⎞
⎠
⋅+
1
2
M'2 a( ) M'3 a b+( )−( ) b⋅
b
3
⎛⎜
⎝
⎞
⎠
⋅+
...:=
θB
tA.B
b c+
:= θB 0.00592 rad= Ans
0 0.5 1
0.1
0.05
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
Given: a 0.2m:= b 0.3m:= c 0.5m:= do 20mm:=
P1 800N:= P2 350N:= E 200GPa:=
Solution: L a b+ c+:=
Support Reactions :
+
 
ΣFy=0; A B+ P1− P2− 0= (1)
ΣΜB=0; P1− L⋅ A b c+( )⋅+ P2 c⋅− 0= (2)
Solving Eqs. (1) and (2): A
P1 L⋅ P2 c⋅+
b c+
:= B P1 P2+ A−:=
A 1218.75 N= B 68.75− N=
Section Property : I
π do
4⋅
64
:=
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) P1− x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) P1− x2⋅ A x2 a−( )⋅+⎡⎣ ⎤⎦
1
E I⋅
⋅:=
M'3 x3( ) P1− x3⋅ A x3 a−( )⋅+ P2 x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
Problem 12-77
Determine the displacement of the 20-mm-diameter A-36 steel shaft at D. The bearings at A and B
exert only vertical reactions on the shaft.
Moment-Area Theorems :
Deflection :
tD.B
1
2
M'3 a b+( ) c⋅ L
2c
3
−⎛⎜
⎝
⎞
⎠
⋅ M'3 a b+( ) b⋅
b
2
a+⎛⎜
⎝
⎞
⎠
⋅+
1
2
M'2 a( ) M'3 a b+( )−( ) b⋅
b
3
a+⎛⎜
⎝
⎞
⎠
⋅
1
2
M'1 a( ) a⋅
2a
3
⋅++
...:=
tA.B
1
2
M'3 a b+( ) c⋅
c
3
b+⎛⎜
⎝
⎞
⎠
⋅ M'3 a b+( ) b⋅
b
2
⎛⎜
⎝
⎞
⎠
⋅+
1
2
M'2 a( ) M'3 a b+( )−( ) b⋅
b
3
⎛⎜
⎝
⎞
⎠
⋅+
...:=
∆D tD.B
L
b c+
tA.B−:=
∆D 4.98 mm= Ans
Problem 12-78
The beam is subjected to the loading shown. Determine the slope at B and deflection at C. EI is
constant.
Problem 12-79
Determine the slope at B and the displacement at C. The bearings at A and B exert only vertical
reactions on the shaft. EI is constant.
Problem 12-80
Determine the displacement at D and the slope at C. The bearings at A and B exert only vertical
reactions on the shaft. EI is constant.
Fh 1kN:=
∆max ∆C:=
Given: a 50mm:= do 32mm:= Fv 4.5kN:=
Support Reactions :
b 650mm:= h 450mm:= E 200GPa:=
Solution: L a b+:=
∆max 1.78 mm=
+
 
ΣFy=0; Fv A− B+ 0= (1)
ΣΜA=0; Fv a⋅ Fh h⋅+ B b⋅− 0= (2)
Solving Eqs. (1) and (2): B
Fv a⋅ Fh h⋅+
b
:= A Fv B+:=
Ans
Section Property : I
π
64
do
4⋅:=
M/EI Diagram:
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:=
M'1 x1( ) Fh h⋅ Fv x1⋅+( )
1
E I⋅
⋅:=
0 0.2 0.4 0.6
0
0.05
Distance (m)
M
/E
I (
1/
m
)
M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Deflections :
tA.B
1
2
M'1 a( ) b⋅
b
3
⎛⎜
⎝
⎞
⎠
⋅:=
θB
tA.B
b
= θB
M'1 a( ) b⋅
6
:=
The maximum displacement occurs at point C,
where θC = 0. Let x' = L - x
θC.B
1
2
M'1 a( )
x'
b
⎛⎜
⎝
⎞
⎠
⋅⎡⎢
⎣
⎤⎥
⎦
⋅ x'⋅=
θC θB θC.B+=
0
M'1 a( ) b⋅
6
−
1
2
M'1 a( )⋅
x'2
b
⋅+=
x' b
1
3
⋅:= x' 375.28 mm= < b = 650 mm (O.K. !)
∆C tB.C= tB.C θB.C
2x'
3
⎛⎜
⎝
⎞
⎠
⋅= θB.C θC.B=
∆C
1
2
M'1 a( )
x'
b
⎛⎜
⎝
⎞
⎠
⋅⎡⎢
⎣
⎤⎥
⎦
⋅ x'⋅
2x'
3
⎛⎜
⎝
⎞
⎠
⋅:=
Problem 12-81
The two force components act on the tire of the automobile as shown. The tire is fixed to the axle,
which is supported by bearings at A and B. Determine the maximum deflection of the axle. Assume
that the bearings resist only vertical loads. The thrust on the axle is resisted at C. The axle has a
diameter of 32 mm and is made of A-36 steel. Neglect the effect of axial load on deflection.
M'2 x2( ) Fh h⋅ Fv x2⋅+ A x2 a−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
EI kN m2⋅:=
Given: a 0.3m:= P 20kN:=
b 0.9m:=
Solution: L 2a b+:=
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R 2P− 0=
R P:=
Maximum Bendiug Stress : Mmax R a⋅:=
M/EI Diagram:
Set
Ans
EIo 1:=
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
M'1 x1( ) R x1⋅( )
1
EI
⋅:= M'2 x2( ) R x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦
1
EI
⋅:=
M'3 x3( ) R x3⋅ P x3 a−( )⋅− P x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
EI
⋅:=
0 0.5 1 1.5
0
5
10
Distance (m)
M
/E
I (
1/
m
) M'1 x1( )
M'2 x2( )
M'3 x3( )
x1 x2, x3,
Moment-Area Theorems :
Deflections :
tA.C
1
2
M'1 a() a⋅
2a
3
⎛⎜
⎝
⎞
⎠
⋅ M'1 a( )
b
2
⎛⎜
⎝
⎞
⎠
⋅
b
4
a+⎛⎜
⎝
⎞
⎠
⋅+:=
∆C tA.C:=
∆C 1.5975
m
EIo
=
Due to symmetry, the slope at midspan (point C) is zero. Hence,
∆max ∆C:= ∆max 1.5975
m
EIo
=
Problem 12-82
Determine the displacement at D and the slope at C. The bearings at A and B exert only vertical
reactions on the shaft. EI is constant.
Problem 12-83
Beams made of fiber-reinforced plastic may one day replace many of those made of A-36 steel since
they are one-fourth the weight of steel and are corrosion resistant. Using the table in Appendix B, with
σallow = 160 MPa and τallow = 84 MPa, select the lightest-weight steel wide-flange beam that will safely
support the 25-kN load, then compute its maximum deflection. What would be the maximum deflection
of this beam if it were made of a fiber-reinforced plastic with Ep = 126 GPa and had the same moment
of inertia as the steel beam?
Given: σallow 160MPa:= P 25kN:= L 3m:=
τallow 84MPa:= E 126GPa:=
Solution:
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R P− 0= R 0.5P:= R 12.50 kN=
Maximum Moment and Shear: 
Vmax R:= Vmax 12.5 kN=
Mmax R L⋅:= Mmax 37.5 kN m⋅=
Bending Stress:
Sreq'd
Mmax
σallow
:= Sreq'd 234375.00 mm
3=
Select W 310x21 : Sx 244 10
3( )⋅ mm3:= d 303mm:= tw 5.08mm:= I Sx 0.5d( )⋅:=
Shear Stress : Provide a shear stress check.
τmax
Vmax
tw d⋅
:= τmax 8.12 MPa= < τallow =84 MPa (O.K.!)
Hence, Use W 310x21 Ans
M/EI Diagram:
x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:=
M'1 x1( ) R x1⋅( )
1
E I⋅
⋅⎡⎢
⎣
⎤⎥
⎦
:= M'2 x2( ) R x2⋅ P x2 L−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅⎡⎢
⎣
⎤⎥
⎦
:=
0 2 4 6
0
0.005
Distance (m)
M
/E
I M'1 x1( )
M'2 x2( )
x1 x2,
Moment-Area Theorems :
Deflections :
tA.C
1
2
M'1 L( ) L⋅
2L
3
⎛⎜
⎝
⎞
⎠
⋅:=
∆C tA.C:=
∆C 24.15 mm=
Due to symmetry, the slope at
midspan (point C) is zero. Hence,
∆max ∆C:=
∆max 24.15 mm= Ans
Problem 12-84
The simply supported shaft has a moment of inertia of 2I for region BC and a moment of inertia I for
regions AB and CD. Determine the maximum deflection of the shaft due to the load P. The modulus of
elasticity is E.
Problem 12-85
The A-36 steel shaft is used to support a rotor that exerts a uniform load of 5 kN/m within the region
CD of the shaft. Determine the slope of the shaft at the bearings A and B. The bearings exert only
vertical reactions on the shaft.
Problem 12-86
The beam is subjected to the loading shown. Determine the slope at B and deflection at C. EI is
constant.
Problem 12-87
Determine the slope of the shaft at A and the displacement at D. The bearings at A and B exert only
vertical reactions on the shaft. EI is constant.
Problem 12-88
Determine the slope at B and the displacement at C. The member is an A-36 steel structural tee for
which I = 30(106) mm4.
Given: P 25kN:= L 1m:= w 25
kN
m
:=
E 200GPa:= I 30 106( )mm4:=
Solution:
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R' P− 0= 2R'' w 2L( )⋅− 0=
R' 0.5P:= R'' w L⋅:=
R' 12.50 kN= R'' 25.00 kN=
M/EI Diagram:
x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:=
For point load P:
M'1 x1( ) R' x1⋅( )
1
E I⋅
⋅:= M'2 x2( ) R' x2⋅ P x2 L−( )⋅−⎡⎣ ⎤⎦
1
E I⋅
⋅:=
For uniform load w :
M''1 x1( ) R'' x1⋅ 0.5w x12⋅−⎛⎝ ⎞⎠
1
E I⋅
⋅:= M''2 x2( ) R'' x2⋅ 0.5 w⋅ x22⋅−⎛⎝ ⎞⎠
1
E I⋅
⋅:=
0 1 2
0
0.002
Distance (m)
M
/E
I M'1 x1( )
M'2 x2( )
x1 x2,
0 1 2
0
0.002
Distance (m)
M
/E
I M''1 x1( )
M''2 x2( )
x1 x2,
Moment-Area Theorems :
Slope :
Due to symmetry, the slope at midspan (point C) is zero. Hence, θB = θB/C
θB.C
1
2
M'1 L( )⋅ L⋅
2
3
M''1 L( )⋅ L⋅+:= θB θB.C:= θB 0.00243 rad= Ans
Deflection :
tA.C
1
2
M'1 L( )⋅ L⋅
2L
3
⎛⎜
⎝
⎞
⎠
⋅
2
3
M''1 L( )⋅ L⋅
5L
8
⎛⎜
⎝
⎞
⎠
⋅+⎡⎢
⎣
⎤⎥
⎦
:=
∆C tA.C:=
∆max ∆C:=
∆max 1.56 mm= Ans
Problem 12-89
The W200 X 71 cantilevered beam is made of A-36 steel and is subjected to the loading shown.
Determine the displacement at its end A.
Given: P 6kN:= Mo 3kN m⋅:=
E 200GPa:= a 2.4m:=
Solution: L 2a:=
Use W 200x71 : I 76.6 106( )⋅ mm4:=
Elsastic Curve : The elastic curves for the concentrated load and
couple moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
∆A1
P L3⋅
3E I⋅
:= ∆C2
Mo a
2⋅
2E I⋅
:=
θC2
Mo a⋅
E I⋅
:=
∆A2 ∆C2 θC2( ) a⋅+:=
Hence, the displacement at A is
∆A ∆A1 ∆A2+:=
∆A 16.13 mm= Ans
Problem 12-90
The W200 X 71 cantilevered beam is made of A-36 steel and is subjected to the loading shown.
Determine the displacement at C and the slope at A.
Given: P 6kN:= Mo 3kN m⋅:=
E 200GPa:= a 2.4m:=
Solution: L 2a:=
Use W 200x71 : I 76.6 106( )⋅ mm4:=
Elsastic Curve : The elastic curves for the concentrated load and
couple moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
∆C1
P a2⋅
6E I⋅
3L a−( )⋅:= ∆C2
Mo a
2⋅
2E I⋅
:=
θA1
P L2⋅
2E I⋅
:= θC2
Mo a⋅
E I⋅
:=
θA2 θC2:=
Hence, the slope at A is
θA θA1 θA2+:=
θA 0.00498 rad= Ans
Hence, the displacement at C is
∆C ∆C1 ∆C2+:=
∆C 5.08 mm= Ans
Problem 12-91
The W360 X 64 simply supported beam is made of A-36 steel and is subjected to the loading shown.
Determine the deflection at its center C.
Given: a 3m:= Mo 60kN m⋅:=
E 200GPa:= w 30
kN
m
:=
Solution: L 2a:=
Use W 360x64: I 179 106( )⋅ mm4:=
Elsastic Curve : The elastic curves for the uniform load and couple
moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required displacements are,
∆C1
5w L4⋅
768E I⋅
:= ∆C2
Mo
6 E⋅ I⋅
a
L
⎛⎜
⎝
⎞
⎠
⋅ a2 3 L⋅ a⋅− 2 L2⋅+( )⋅:=
Hence, the displacement at C is
∆C ∆C1 ∆C2+:=
∆C 10.84 mm= Ans
Problem 12-92
The W360 X 64 simply supported beam is made of A-36 steel and is subjected to the loading shown.
Determine the slope at A and B.
Given: a 3m:= Mo 60kN m⋅:=
E 200GPa:= w 30
kN
m
:=
Solution: L 2a:=
Use W 360x64: I 179 106( )⋅ mm4:=
Elsastic Curve : The elastic curves for the uniform load and couple
moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required slopes are,
θA1
7w L3⋅
384E I⋅
:= θA2
Mo L⋅
6 E⋅ I⋅
:=
Hence, the slope at A is
θA θA1 θA2+:=
θA 0.00498 rad=
θA 0.285 deg= Ans
θB1
3w L3⋅
128E I⋅
:= θB2
Mo L⋅
3 E⋅ I⋅
:=
Hence, the slope at B is
θB θB1 θB2+:=
θB 0.00759 rad=
θB 0.435 deg= Ans
Problem 12-93
Determine the moment M0 in terms of the load P and dimension a so that the deflection at the center of
the beam is zero. EI is constant.
Problem 12-94
The beam supports the loading shown. Code restrictions, due to a plaster ceiling, require the maximum
deflection not to exceed 1/360 of the span length. Select the lightest-weight A-36 steel wide-flange
beam from Appendix B that will satisfy this requirement and safely support the load. The allowable
bending stress is σallow = 168 MPa and the allowable shear stress is τallow = 100 MPa. Assume A is a
roller and B is a pin.
Given: σallow 168MPa:= a 3.6m:=
τallow 100MPa:= w 60
kN
m
:=
E 200GPa:=
Solution: L 2a:=
+
 
ΣFy=0; A B+ w a⋅− 0= (1)
ΣΜB=0; A L⋅ 0.5−⋅ w a
2⋅ 0= (2)
Solving Eqs. (1) and (2): A
w a2
2L
:= B w a⋅ A−:=
A 54 kN= B 162 kN=
Maximum Moment and Shear: 
Vmax max A B,( ):= Vmax 162 kN=
Mmax occurs at x where V(x) = 0:
V x( ) A w Φ x a−( )⋅−=
0 A w Φ x a−( )⋅−= x
A
w
a+:=
Mmax A x⋅ 0.5w x a−( )
2⋅−:= Mmax 218.7 kN m⋅=
Strength criterion:
Sreq'd
Mmax
σallow
:= Sreq'd 1301785.71 mm
3=
Select W 410x74 : Sx 1330 10
3( )mm3:= d 413mm:= tw 9.65mm:= I Sx 0.5d( )⋅:=
Shear Stress : Provide a shear stress check.
τmax
Vmax
tw d⋅
:= τmax 40.65 MPa= < τallow =100 MPa (O.K.!)
Deflection criterion:
∆allow
L
360
:= ∆allow 20 mm=
Using the table in Appendix C, ∆max 0.006563
w L4⋅
E I⋅
:=
∆max 19.27 mm= < ∆allow =20mm (O.K.!)
Hence, Use W 410x74 Ans
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:=
V1 x1( ) A( )
1
kN
:= V2 x2( ) A w x2 a−( )⋅−⎡⎣ ⎤⎦
1
kN
⋅:=
M1 x1( ) A x1⋅( )
1
kN m⋅
:= M2 x2( ) A x2⋅ 0.5 w⋅ x2 a−( )2⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 5
200
0
Distance(m)
Sh
ea
r (
kN
)
V1 x1( )
V2 x2( )
x1 x2,
0 5
0
200
Distance (m)
M
 (k
N
-m
)
M1 x1( )
M2 x2( )
x1 x2,
Problem 12-95
The simply supported beam carries a uniform load of 29 kN/m. Code restrictions, due to a plaster
ceiling, require the maximum deflection not to exceed 1/360 of the span length. Select the lightest-
weight A-36 steel wide-flange beam from Appendix B that will satisfy this requirement and safely
support the load. The allowable bending stress is σallow = 168 MPa and the allowable shear stress is
τallow = 100 MPa. Assume A is a pin and B a roller support.
Given: σallow 168MPa:= a 1.2m:= b 2.4m:=
τallow 100MPa:= P 40kN:=
E 200GPa:= w 29
kN
m
:=
Solution: L 2a b+:=
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R 2P− w L⋅− 0=
R 0.5 2P w L⋅+( ):= R 109.60 kN=
Maximum Moment and Shear: 
Vmax R:= Vmax 109.6 kN=
Mmax occurs at midspan.
Mmax R 0.5L( )⋅ P 0.5L a−( )⋅− 0.5w 0.5L( )
2⋅−:=
Mmax 131.52 kN m⋅=
Strength criterion:
Sreq'd
Mmax
σallow
:= Sreq'd 782857.14 mm
3=
Select W 360x51 : Sx 794 10
3( )mm3:= d 355mm:= tw 7.24mm:= I Sx 0.5d( )⋅:=
Shear Stress : Provide a shear stress check.
τmax
Vmax
tw d⋅
:= τmax 42.64 MPa= < τallow =100 MPa (O.K.!)
Deflection criterion:
∆allow
L
360
:= ∆allow 13.33 mm=
Using the table in Appendix C, 
∆max
5w L4⋅
384E I⋅
2
P a⋅ b⋅
6E I⋅ L⋅
⎛⎜
⎝
⎞
⎠
⋅ L2 a2− 0.5L( )2−⎡⎣ ⎤⎦⋅+:=
∆max 11.61 mm= < ∆allow =20mm (O.K.!)
Hence, Use W 360x51 Ans
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, a b+( )..:= x3 a b+( ) 1.01 a b+( )⋅, L..:=
V1 x1( ) R w x1⋅−( )
1
kN
:= V2 x2( ) R w x2⋅− P−( )
1
kN
⋅:= V3 x3( ) R w x3⋅− 2P−( )
1
kN
⋅:=
M1 x1( ) R x1⋅ 0.5 w⋅ x12−⎛⎝ ⎞⎠
1
kN m⋅
:= M2 x2( ) R x2⋅ 0.5 w⋅ x22− P x2 a−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M3 x3( ) R x3⋅ 0.5 w⋅ x32− P x3 a−( )⋅− P x3 a− b−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 2 4
100
0
100
Distance (m)
Sh
ea
r (
kN
) V1 x1( )
V2 x2( )
V3 x3( )
x1 x2, x3,
0 2 4
0
100
Distance (m)
M
 (k
N
-m
) M1 x1( )
M2 x2( )
M3 x3( )
x1 x2, x3,
Problem 12-96
The W250 X 45 cantilevered beam is made of A-36 steel and is subjected to unsymmetrical bending
caused by the applied moment. Determine the deflection of the centroid at its end A due to the loading.
Hint: Resolve the moment into components and use superposition.
Given: L 4.5m:= Mo 2.5kN m⋅:=
E 200GPa:= θ 30deg:=
Solution:
Use W 250x45: Ix 71.1 10
6( )⋅ mm4:=
Iy 7.03 10
6( )⋅ mm4:=
Displacements :
Using the table in Appendix C, the required displacements are,
∆x_max
Mo sin θ( )⋅ L
2⋅
2E Iy⋅
:= ∆y_max
Mo cos θ( )⋅ L
2⋅
2E Ix⋅
:=
Hence, the displacement at A is
∆A ∆x_max ∆y_max+:=
∆A 10.54 mm= Ans
Problem 12-97
The assembly consists of a cantilevered beam CB and a simply supported beam AB. If each beam is
made of A-36 steel and has a moment of inertia about its principal axis of Ix = 46(106) mm4, determine
the displacement at the center D of beam BA.
Given: P 75kN:= I 46 106( )⋅ mm4:=
E 200GPa:= LDA 2.4m:=
LBC 4.8m:= LBA 4.8m:=
Solution:
Consider beam AB :
Support Reactions : By symmetry, A=B=R
+ ΣFy=0; 2R P− 0= R 0.5P:=
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
Consider beam BC : ∆B
R LBC
3⋅
3E I⋅
:=
Consider beam AB : ∆D1
P LBA
3⋅
48E I⋅
:= ∆D2
LDA
LBA
∆B⋅:=
Hence, the displacement at D is
∆D ∆D1 ∆D2+:=
∆D 93.91 mm= Ans
Problem 12-98
The rod is pinned at its end A and attached to a torsional spring having a stiffness k, which measures
the torque per radian of rotation of the spring. If a force P is always applied perpendicular to the end of
the rod, determine the displacement of the force. EI is constant.
Problem 12-99
The relay switch consists of a thin metal strip or armature AB that is made of red brass C83400 and is
attracted to the solenoid S by a magnetic field. Determine the smallest force F required to attract the
armature at C in order that contact is made at the free end B. Also, what should the distance a be for
this to occur? The armature is fixed at A and has a moment of inertia of I = 0.18(10-12) m4.
Given: ∆B 2mm:= L 50mm:=
E 101GPa:= I 0.18 10 12−( )m4:=
Solution: LAC L:= LCB L:=
Elsastic Curve : The elastic curves for the concentrated load and
couple moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
θC
F LAC
2⋅
2E I⋅
=
∆C
F LAC
3⋅
3E I⋅
= (1)
Hence, the displacement at B is
∆B ∆C θC LCB⋅+=
∆B
F LAC
3⋅
3E I⋅
F LAC
2⋅
2E I⋅
LCB⋅+=
F
6∆B E⋅ I⋅
2 LAC
3⋅ 3 LCB⋅ LAC
2⋅+
:= F 0.349 N= Ans
From Eq.(1), ∆C
F LAC
3⋅
3E I⋅
:= ∆C 0.800 mm= Ans
Problem 12-100
Determine the vertical deflection and slope at the end A of the bracket. Assume that the bracket is fixed
supported at its base, and neglect the axial deformation of segment AB. EI is constant.
Given: h 75mm:= L 100mm:=
P 400N:= w 4
kN
m
:=
Solution:
Set EI 1kN m2:=
Elsastic Curve :
The elastic curves for the concentrated load,
uniform distributed load and couple moment
are drawn separately as shown. 
MB 0.5w L
2⋅:=
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
θA1
w L3⋅
6EI
:=
θB2
MB h⋅
EI
:= θA2 θB2:=
θB3
P h2⋅
2EI
:= θA3 θB3:=
∆A1
w L4⋅
8EI
:=
∆A2 θB2 L( )⋅:=
∆A3 θB3 L( )⋅:=
Hence, the slope at A is
θA θA1 θA2+ θB3+:=
θA 0.00329 rad= θA
0.00329
EI
= Ans
The displacement at A is
∆A ∆A1 ∆A2+ ∆A3+:=
∆A 0.313 mm= ∆A
0.313
EI
= Ans
Problem 12-101
The W610 X 155 A-36 steel beam is used to support the uniform distributed load and a concentrated
force which is applied at its end. If the force acts at an angle with the vertical as shown, determine the
horizontal and vertical displacement at point A.
Given: L 3m:= P 25kN:= E 200GPa:=
cx
3
5
:= cy
4
5
:= w 30
kN
m
:=
Solution:
Use W 610x155: Ix 1290 10
6( )⋅ mm4:=
Iy 108 10
6( )⋅ mm4:=
Elsastic Curve :
The elastic curves for the concentrated load and
uniform load are drawn separately as shown. 
Py P cy⋅:= Py 20 kN=
Px P cx⋅:= Px 15 kN=
Method of Superposition :
Using the table in Appendix C, the required displacement are,
∆A1
w L4⋅
8E Ix⋅
:=
∆A2
Py L
3⋅
3E Ix⋅
:=
The vertical displacement at A is
∆A_y ∆A1 ∆A2+:= ∆A_y 1.875 mm= Ans
The horizontal displacement at A is
∆A_x
Px L
3⋅
3E Iy⋅
:= ∆A_x 6.250 mm= Ans
Problem 12-102
The framework consists of two A-36 steel cantilevered beams CD and BA and a simply supported
beam CB. If each beam is made of steel and has a moment of inertia about its principal axis of Ix =
46(106) mm4, determine the deflection at the center G of beam CB.
Given: P 75kN:= I 46 106( )⋅ mm4:=
E 200GPa:= LDC 4.8m:=
LBC 4.8m:= LAB 4.8m:=
Solution:
Consider beam BC :
Support Reactions : By symmetry, B=C=R
+ ΣFy=0; 2R P− 0= R 0.5P:=
Method of Superposition :
Using the table in Appendix C, the required slope
and displacement are,
Consider beam DC : ∆C
R LDC
3⋅
3E I⋅
:=
Due to symmetry, ∆B ∆C:=
Consider beam BC : ∆'G
P LBC
3⋅
48E I⋅
:=
Hence, the displacement at G is
∆G ∆C ∆'G+:=
∆G 169.04 mm= Ans
Problem 12-103
Determine the reactions at the supports A and B, then draw the moment diagram. EI is constant.
Problem 12-104
Determine the reactions at the supports A and B, then draw the shear and moment diagrams. EI is
constant. Neglect the effect of axial load.
Problem 12-105
Determine the reactions at the supports A, B, and C; then draw the shear and moment diagrams. EI is
constant.
Problem 12-106
Determine the reactions at the supports, then draw the shear and moment diagram. EI is constant.
Problem 12-107
Determine the moment reactions at the supports A and B. EI is constant.
Problem 12-108
Determine the value of a for which the maximum positive moment has the same magnitude as the
maximum negative moment. EI is constant.
Problem 12-109
Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.
Problem 12-110
The beam has a constant E1I1 and is supported by the fixed wall at B and the rod AC. If the rod has a
cross-sectionalarea A2 and the material has a modulus of elasticity E2 , determine the force in the rod.
Problem 12-111
Determine the moment reactions at the supports A and B, and then draw the shear and moment
diagrams. Solve by expressing the internal moment in the beam in terms of Ay and MA. EI is constant.
Problem 12-112
Determine the moment reactions at the supports A and B. EI is constant.
Problem 12-113
Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams.
EI is constant.
Problem 12-114
Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams.
EI is constant.
Problem 12-115
Determine the reactions at the supports. EI is constant.
b 1.8m:=Given: P 25kN:= a 1.2m:=
Solution: L a b+:=
Due to symmetry, the slope at support
B is zero. Hence, θB = 0 and tanθB = 0.
Moment-Area Theorems :
Ans
Set EI 1kN m2:= A 1kN:=
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, L..:=
For point load P:
M'1 x1( ) 0( )
1
EI
⋅:= M'2 x2( ) P− x2 a−( )⋅⎡⎣ ⎤⎦
1
EI
⋅:=
For reaction A :
M''1 x1( ) A x1⋅( )
1
EI
⋅:= M''2 x2( ) A x2⋅( )
1
EI
⋅:=
0 1 2 3
50
0
Distance (m)
M
/E
I M'1 x1( )
M'2 x2( )
x1 x2,
0 1 2 3
0
5
Distance (m)
M
/E
I M''1 x1( )
M''2 x2( )
x1 x2,
Deflection :
tA.B_1
1
2
M'2 L( )⋅ b⋅ a
2 b⋅
3
+⎛⎜
⎝
⎞
⎠
⋅:=
tA.B_2
1
2
M''2 L( )⋅ L⋅
2L
3
⎛⎜
⎝
⎞
⎠
⋅:=
tA.B tA.B_1 tA.B_2
Ay
A
⎛
⎜
⎝
⎞
⎠
⋅+=
Compatibility : tA.B 0:=
0 tA.B_1 tA.B_2
Ay
A
⎛
⎜
⎝
⎞
⎠
⋅+= Ay A−
tA.B_1
tA.B_2
⎛
⎜
⎝
⎞
⎠
⋅:= Ay 10.80 kN= Ans
Support Reactions : Due to symmetry, Cy Ay:= Cy 10.80 kN= Ans
+ ΣFy=0; Ay By+ Cy+ 2P− 0= By 2P Ay− Cy−:= By 28.40 kN=
Problem 12-116
Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.
x1 0 0.01 a⋅, a..:= x3 L 1.01 L⋅, L b+( )..:=
x2 a 1.01 a⋅, L..:= x4 L b+( ) 1.01 L b+( )⋅, 2L( )..:=
V1 x1( ) Ay( )
1
kN
:= V3 x3( ) Ay P− By+( )
1
kN
⋅:=
V2 x2( ) Ay P−( )
1
kN
⋅:= V4 x3( ) Ay P− By+ P−( )
1
kN
⋅:=
M1 x1( ) Ay x1⋅( )
1
kN m⋅
:=
M2 x2( ) Ay x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M3 x3( ) Ay x3⋅ P x3 a−( )⋅− By x3 L−( )⋅+⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M4 x4( ) Ay x4⋅ P x4 a−( )⋅− By x4 L−( )⋅+ P x4 L− b−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 1 2 3 4 5 6
20
0
20
Distance (m)
Sh
ea
r (
kN
)
V1 x1( )
V2 x2( )
V3 x3( )
V4 x4( )
x1 x2, x3, x4,
0 1 2 3 4 5 6
20
0
20
Distance (m)
M
 (k
N
-m
)
M1 x1( )
M2 x2( )
M3 x3( )
M4 x4( )
x1 x2, x3, x4,
Problem 12-117
Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.
Support B is a thrust bearing.
Problem 12-118
Determine the reactions at the supports. EI is constant.
Problem 12-119
Determine the value of a for which the maximum positive moment has the same magnitude as the
maximum negative moment. EI is constant.
Problem 12-120
Determine the moment reactions at the supports A and B, then draw the shear and moment diagrams.
EI is constant.
Problem 12-121
Determine the reactions at the supports A and B. EI is constant.
Problem 12-122
Determine the reactions at the bearing supports A, B, and C of the shaft, then draw the shear and
moment diagrams. EI is constant. Each bearing exerts only vertical reactions on the shaft.
Given: a 1m:= P 0.4kN:=
b 2m:=
Solution: L 4a:=
Set EI kN m2:=
Method of Superposition :
Using the table in Appendix C, the required displacements are,
∆B1
P a⋅
6 EI⋅
b
L
⎛⎜
⎝
⎞
⎠
⋅ L2 a2− b2−( )⋅:=
∆B2
B L3⋅
48EI
−=
The compatibity condition at B requires
2∆B1 ∆B2+ 0=
∆B2 2− ∆B1=
B L3⋅
48E I⋅
2∆B1=
B
48 EI⋅
L3
⎛
⎜
⎝
⎞
⎠
2∆B1( )⋅:= B 0.550 kN= Ans
Support Reactions : Given
+ ΣFy=0; A B+ C+ 2P− 0= (1)
 
ΣΜC=0; A L⋅ P 3a( )⋅− B b( )⋅+ P a( )⋅− 0= (2)
Solving Eqs. (1) and (2): Guess A 1kN:= C 1kN:=
A
C
⎛
⎜
⎝
⎞
⎠
Find A C,( ):=
A
C
⎛
⎜
⎝
⎞
⎠
0.125
0.125
⎛
⎜
⎝
⎞
⎠
kN= Ans
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, 2a( )..:= x3 2a( ) 1.01 2a( )⋅, 3a( )..:= x4 3a( ) 1.01 3a( )⋅, L..:=
V1 x1( ) A( )
1
kN
:= V2 x2( ) A P−( )
1
kN
⋅:= V3 x3( ) A P− B+( )
1
kN
⋅:=
V4 x4( ) A P− B+ P−( )
1
kN
⋅:=
M1 x1( ) A x1⋅( )
1
kN m⋅
:= M2 x2( ) A x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M3 x3( ) A x3⋅ P x3 a−( )⋅− B x3 2a−( )⋅+⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M4 x4( ) A x4⋅ P x4 a−( )⋅− B x4 2a−( )⋅+ P x4 3a−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 2 4
0
Distance (m)
Sh
ea
r (
kN
)
V1 x1( )
V2 x2( )
V3 x3( )
V4 x4( )
x1 x2, x3, x4,
0 2 4
0.1
0
0.1
Distance (m)
M
 (k
N
-m
)
M1 x1( )
M2 x2( )
M3 x3( )
M4 x4( )
x1 x2, x3, x4,
Problem 12-123
The A-36 steel beam and rod are used to support the load of 40 kN. If it is required that the allowable
normal stress for the steel is σallow = 125 MPa, and the maximum deflection not exceed 1.25 mm,
determine the smallest diameter rod that should be used. The beam is rectangular, having a height of
125 mm and a thickness of 75 mm.
Given: σallow 125MPa:= Lb 1.2m:= Lr 1.5m:=
δallow 1.25mm:= b 75mm:= h 125mm:=
E 200GPa:= P 40kN:=
Solution:
Compatibility at B : δr δb=
Ib
b h3⋅
12
:=
Fr Lr⋅
E Ar⋅
P Fr−( ) Lb3⋅
3E Ib⋅
=
Strength criterion: Assume rod reaches its maximum stress.
σallow
Fr
Ar
=
σallow Lr⋅
E
P Fr−( ) Lb3⋅
3E Ib⋅
=
Fr P
3σallow Lr⋅ Ib⋅
Lb
3
−:=
Fr 36.03 kN=
Check Bending Stress : c 0.5 h⋅:=
Mmax P Fr−( ) Lb⋅:= σmax
Mmax c⋅
Ib
:= σmax 24.41 MPa=
< σallow =125 MPa (O.K.!)
Check Deflection :
δb
P Fr−( ) Lb3⋅
3E Ib⋅
:= δb 0.94 mm= < δallow =1.25mm (O.K.!)
Hence, from σallow
Fr
Ar
= Ar
π do
2⋅
4
=
σallow
4Fr
π do
2⋅
=
do
4Fr
π σallow⋅
:=
do 19.16 mm= Ans
Problem 12-124
Determine the reactions at the supports A, B, and C, then draw the shear and moment diagrams. EI is
constant.
Given: a 2m:= P 60kN:= w 50
kN
m
:=
b 4m:=
Solution: L 2a b+:=
Set EI 1kN m2:=
Method of Superposition :
Using the table in Appendix C, the required displacements are,
∆B1
5w L4⋅
768EI
:=
∆B2
P a⋅
6 EI⋅
b
L
⎛⎜
⎝
⎞
⎠
⋅ L2 a2− b2−( )⋅:=
∆B3
B L3⋅
48EI
−=
The compatibity condition at C requires
∆B1 ∆B2+ ∆B3+ 0=
∆B3 ∆B1 ∆B2+( )−=
B L3⋅
48E I⋅
∆B1 ∆B2+=
B
48 EI⋅
L3
⎛
⎜
⎝
⎞
⎠
∆B1 ∆B2+( )⋅:= B 166.25 kN= Ans
Support Reactions : Given
+ ΣFy=0; A B+ C+ P− w b⋅− 0= (1)
 
ΣΜC=0; A L⋅ P a b+( )⋅− B b⋅+ 0.5 w b
2⋅⋅− 0= (2)
Solving Eqs. (1) and (2): Guess A 1kN:= C 1kN:=
A
C
⎛
⎜
⎝
⎞
⎠
Find A C,( ):=
A
C
⎛
⎜
⎝
⎞
⎠
11.88
81.88
⎛
⎜
⎝
⎞
⎠
kN= Ans
x1 0 0.01 a⋅, a..:= x2 a 1.01 a⋅, 2a( )..:= x3 2a( ) 1.01 2a( )⋅, L..:=
V1 x1( ) A( )
1
kN
:= V2 x2( ) A P−( )
1
kN
⋅:= V3 x3( ) A P− B+ w x3 2a−( )⋅−⎡⎣ ⎤⎦
1
kN
⋅:=
M1 x1( ) A x1⋅( )
1
kN m⋅
:= M2 x2( ) A x2⋅ P x2 a−( )⋅−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
M3 x3( ) A x3⋅ P x3 a−( )⋅− B x3 2a−( )⋅+ 0.5 w⋅ x3 2a−( )2−⎡⎣ ⎤⎦
1
kN m⋅
⋅:=
0 5
200
0
200
Distance (m)
Sh
ea
r (
kN
) V1 x1( )
V2 x2( )
V3 x3( )
x1 x2, x3,
0 5
100
0
100
Distance (m)
M
 (k
N
-m
) M1 x1( )
M2 x2( )
M3 x3( )
x1 x2, x3,
Problem 12-125
Determine the reactions at support C. EI is constant for both beams.
Problem 12-126
Determine the reactions at A and B. Assume the support at A only exerts a moment on the beam. EI is
constant.
Problem 12-127
Determine the reactions at the supports A and B. EI is constant.
Problem 12-128
Each of the two members is made from 6061-T6 aluminum and has a square cross section 25 mm X
25 mm. They are pin connected at their ends and a jack is placed between them and opened until the
force it exerts on each member is 2.5 kN. Determine the greatest force P that can be applied to the
center of the top member without causing either of the two members to yield. For the analysis neglect
the axial force in each member. Assume the jack is rigid.
Given: b 25mm:= h 25mm:=
Ro 2.5kN:= L 4m:=
E 68.9GPa:= σallow 255MPa:=
Solution:
Compatibility at B : δE δF=
P R Ro+( )−⎡⎣ ⎤⎦ L
3⋅
48E I⋅
R Ro+( ) L3⋅
48E I⋅
=
P R Ro+( )− R Ro+( )=
R 0.5P Ro−=
Criterion: Maximum moment occursat center of each member.
Top member:
Mmax
1
2
P R Ro+( )−⎡⎣ ⎤⎦
L
2
⋅= Mmax
P L⋅
8
=
Bottom member:
Mmax
1
2
R Ro+( )
L
2
⋅= Mmax
P L⋅
8
=
Both members will yieldat the same time.
I
b h3⋅
12
:= c 0.5 h⋅:= σallow
Mmax c⋅
I
=σallow
P L⋅( ) c⋅
8I
=
P
8I( ) σallow⋅
L c⋅
:=
P 1.33 kN= Ans
Problem 12-129
Determine the reactions at the supports, then draw the shear and moment diagrams. EI is constant.
Problem 12-130
The beam is supported by a pin at A, a spring having a stiffness k at B, and a roller at C. Determine the
force the spring exerts on the beam. EI is constant.
Problem 12-131
The beam AB has a moment of inertia I = 200(106) mm4 and rests on the smooth supports at its ends.
A 18-mm-diameter rod CD is welded to the center of the beam and to the fixed support at D. If the
temperature of the rod is decreased by 80°C, determine the force developed in the rod. The beam and
rod are both made of A-36 steel.
Unit used: °C deg:=
Given: rc 18mm:= Ib 200 10
6( )⋅ mm4:=
Lc 1.25m:= Lb 3m:=
∆T 80°C:= α 12 10
6−( )⋅ 1
°C
:=
E 200GPa:=
Solution:
Section Property: Ac
π
4
rc
2⋅:=
Method of Superposition :
Using the table in Appendix C, the required displacements are,
vC
Fc Lb
3⋅
48E Ib⋅
=
Using the axial force formula,
δF
Fc Lc⋅
Ac E
−=
The thermal contraction is,
δT α ∆T( )⋅ Lc⋅:=
The compatibity condition at C requires
vC δF δT+= vC δF− δT=
Fc Lb
3⋅
48E Ib⋅
Fc− Lc⋅
Ac E
− δT=
Fc
δT
Lb
3
48E Ib⋅
Lc
Ac E
+
:=
Fc 31.07 kN= Ans
Problem 12-132
Determine the deflection at the end B of the clamped A-36 steel strip. The spring has a stiffness of
k = 2 N/mm. The strip is 5 mm wide and 10 mm high. Also, draw the shear and moment diagrams for
the strip.
P 50N:=Given: L 200mm:=
h 10mm:= k 2
N
mm
:=
t 5mm:= E 200GPa:=
Solution:
Section Property : I
t h3⋅
12
:=
Method of Superposition :
Using the table in Appendix C, the required displacements are,
∆B1
P L3⋅
3 E⋅ I⋅
:=
∆B2
F L3⋅
3 E I⋅( )
−= F k ∆B⋅=
The compatibity condition at B requires
∆B ∆B1 ∆B2+=
∆B
P L3⋅
3 E I⋅( )⋅
k ∆B⋅ L
3⋅
3 E I⋅( )
−=
∆B
P L3⋅
3 E I⋅( )⋅ k L3⋅+
:= ∆B 1.50 mm= Ans
Support Reactions :
B k ∆B⋅ P−:= A B−:= MA B L⋅:=
B 46.992− N= A 46.992 N= MA 9.40− N m⋅=
x1 0 0.01 L⋅, L..:= V1 x1( )
A
N
:= M1 x1( ) MA A x1⋅+( )
1
N m⋅
:=
0 0.1
0
50
Distance (m)
Sh
ea
r (
N
)
V1 x1( )
x1
0 0.1
10
0
Distance (m)
M
 (N
-m
)
M1 x1( )
x1
Problem 12-133
The beam is made from a soft elastic material having a constant EI. If it is originally a distance ∆ from
the surface of its end support, determine the distance a at which it rests on this support when it is
subjected to the uniform load w0 , which is great enough to cause this to happen.
Problem 12-134
The box frame is subjected to a uniform distributed loading w along each of its sides. Determine the
moment developed in each corner. Neglect the deflection due to axial load. EI is constant.
Problem 12-135
Determine the equation of the elastic curve for the beam. Specify the slope and displacement at A. EI is
constant.
Problem 12-136
The wooden beam is subjected to the loading shown. Assume the support at A is a pin and B is a roller.
Determine the slope at A and the displacement at C. Use the moment-area theorems. EI is constant.
Problem 12-137
Determine the maximum deflection between the supports A and B. EI is constant. Use the method of
integration.
Problem 12-138
If the bearings at A and B exert only vertical reactions on the shaft, determine the slope at B and the
deflection at C. EI is constant. Use the moment-area theorems.
Problem 12-139
The W200 X 36 simply supported beam is subjected to the loading shown. Using the method of
superposition, determine the deflection at its center C. The beam is made of A-36 steel.
Given: a 2.4m:= Mo 7.5kN m⋅:=
E 200GPa:= w 100
kN
m
:=
Solution: L 2a:=
Use W 200x36 : I 34.4 106( )⋅ mm4:=
Elsastic Curve : The elastic curves for the uniform load and couple
moment are drawn separately as shown. 
Method of Superposition :
Using the table in Appendix C, the required displacement are,
∆C1
5w L4⋅
768E I⋅
:=
∆C2
Mo
6E I⋅
a
L
⎛⎜
⎝
⎞
⎠
a2 3a L⋅− 2L2+( )⋅:=
Hence, the displacement at C is
∆C ∆C1 ∆C2+:=
∆C 51.80 mm= Ans
Problem 12-140
The shaft is supported by a journal bearing at A, which exerts only vertical reactions on the shaft, and
by a thrust bearing at B, which exerts both horizontal and vertical reactions on the shaft. Draw the
bending-moment diagram for the shaft and then, from this diagram, sketch the deflection or elastic
curve for the shaft's centerline. Determine the equations of the elastic curve using the coordinates x1
and x2 . EI is constant.
Given: L 0.3m:= a 0.1m:= P 400N:=
Solution:
Support Reactions : Lo 2L:=
Due to anti-symmetry, 
 
B A−=
ΣΜB=0; A 2L( )⋅ P 2a( )⋅− 0=
A
a
L
⎛⎜
⎝
⎞
⎠
P:= B A−:=
A 133.33 N= B 133.33− N=
Moment Function : M1 x1( ) A x1⋅= M2 x2( ) B x2⋅=
Slope and Elastic Curve :
E I⋅
d2 v1⋅
dx1
2
⋅ M x1( )= E I⋅
d2 v2⋅
dx2
2
⋅ M x2( )=
E I⋅
d2v1
dx1
2
⋅ A x1⋅= E I⋅
d2v2
dx2
2
⋅ B x2⋅=
E I⋅
dv1
dx1
⋅
A x1
2⋅
2
C1+= (1) E I⋅
dv2
dx2
⋅
B x2
2⋅
2
C3+= (3)
E I⋅ v1⋅
A x1
3⋅
6
C1 x1⋅+ C2+= (2) E I⋅ v2⋅
B x2
3⋅
6
C3 x2⋅+ C4+= (4)
Boundary Conditions : v1=0 at x1=0 and v2=0 at x2=0.
From Eq. (2): 0 0 0+ C2+= From Eq. (4): 0 0 0+ C4+=
C2 0:= C4 0:=
Continuity Condition 1 :
dv1
dx1
dv2
dx2
−= at x1=L and at x2=L.
From Eqs. (1) and (3),
A L2⋅
2
C1+
B L2⋅
2
− C3−= C1 C3−= (5)
Continuity Condition 2 : v1 v2= at x1=L and at x2=L.
From Eqs. (2) and (4),
A L3⋅
6
C1 L⋅+
B L3⋅
6
C3 L⋅+= C1
B A−( ) L2⋅
6
C3+= (6)
Solving Eqs. (5) and (6),
C1
B A−( ) L2⋅
12
:= C1 2.00− N m
2⋅= Ans
C3 C1−:= C3 2.00 N m
2⋅= Ans
Hence, Co
A
6
:= Co 22.22 N= Ans
v1
1
EI
Co x1
3⋅ C1 x1⋅+⎛⎝
⎞
⎠= Ans
v2
1
EI
Co− x2
3⋅ C3 x2⋅+⎛⎝
⎞
⎠= Ans
M/EI diagram :
Set EI 1kN m2⋅:=
x1 0 0.01 L⋅, L..:= x2 L 1.01 L⋅, 2L( )..:=
M'1 x1( ) A x1⋅( )
1
EI
⋅:= M'2 x2( ) A x2⋅ P 2a( )⋅−⎡⎣ ⎤⎦
1
EI
⋅:=
0 0.2 0.4
0
Distance (m)
M
/E
I (
1/
m
)
M'1 x1( )
M'2 x2( )
x1 x2,
Problem 12-141
The rim on the flywheel has a thickness t, width b, and specific weight γ. If the flywheel is rotating at
a constant rate of ω, determine the maximum moment developed in the rim. Assume that the spokes do
not deform. Hint: Due to symmetry of the loading, the slope of the rim at each spoke is zero. Consider
the radius to be sufficiently large so that the segment AB can be considered as a straight beam fixed at
both ends and loaded with a uniform centrifugal force per unit length. Show that this force is w =
btγω2r/g.
Problem 12-142
Determine the moment reactions at the supports A and B. Use the method of integration. EI is
constant.
Problem 12-143
Using the method of superposition, determine the magnitude of M0 in terms of the distributed load w
and dimension a so that the deflection at the center of the beam is zero. EI is constant.
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