Review Innovations
CE Review for April 2024 – Strength of Materials 3
Helical Spring
Shear and Moment in Beams
Maximum Shear Stress:
16𝑃𝑅
𝑑
𝑆𝑠 =
(1 + )
3
𝜋𝑑
4𝑅
Where:
P = load on spring
R = mean radius of spring
d = diameter of spring material (wire)
PROBLEM 4
A.M. Wahl’s Formula:
16𝑃𝑅 4𝑚 − 1 0.615
𝑆𝑠 =
(
+
)
𝜋𝑑 3 4𝑚 − 4
𝑚
Where:
m = 2R/d (spring index)
(4m – 1)/(4m – 4) – Wahl factor
Deformation/deflection:
𝛿=
Where:
64𝑃𝑅3 𝑛
𝐺𝑑 4
A beam ABCD is loaded and supported as shown.
1. Determine the reaction at A.
2. Determine the maximum shear in the beam.
3. Determine the maximum moment in the beam.
PROBLEM 5:
The raft shown in the figure supports the load w1 and w2. For
this problem, w1 = 145 kN/m, w2 = 290 kN/m, L1 = 3 m, L2 =
6 m, and L3 = 3 m.
L1
w1
n = number of turns
G = Modulus of Rigidity
RAFT
PROBLEM 1:
A helical spring is made by wrapping wire 20 mm in diameter
around a forming cylinder 150 mm in diameter. A compressive
load of 2.4 kN is then applied to the spring. The total number
of turns is 18 and G = 83 GPa.
1. Determine the maximum shearing stress due to
torsion alone.
2. Find the shearing stress on the innermost fiber of the
spring.
3. Calculate the deformation of the spring.
PROBLEM 2:
A rigid plate of negligible mass rests on a spring system
consisting of five vertical springs: four identical outer springs
in a square formation with a bigger spring at the center. The
central spring is 7 mm shorter than the outer springs. Each of
the outer springs consists of 18 turns of 12 mm wire on a mean
diameter of 100 mm. The central spring has 24 turns of 20mm wire on a mean diameter of 150 mm. A load P = 5 kN is
now applied to the plate. G = 83 GPa.
1. Determine the load carried by the central spring.
2. Find the maximum shearing stress on the central
spring.
3. Calculate the deformation of the outer spring.
PROBLEM 3:
For the assembly shown, the spring with a spring constant of
10.2 kN/m is subjected to an initial tension of 200 N. Two
forces P are applied at points B and D of the wires and directed
away from each other until B & D are 0.60 m apart.
L3
w2
L2
L
1.
2.
3.
4.
5.
What is the length “L” of the raft so that the resulting
upward pressure is uniform?
What is the shear at 3 m from the left end?
At what distance from the left end will the shear in the
raft be zero?
Determine the maximum negative moment.
Determine the maximum positive moment.
FOR PRACTICE:
PROBLEM 6:
A prestressed concrete section has dimensions as shown in
the figure:
1200
250
100
600
Ap
Note: All dimensions
are in millimeters.
Varies
450
1.
2.
3.
Determine the area of the section.
Find the distance of the centroid from the top of the
section.
Calculate the moment of inertia of the section.
Ans. 330,000 mm2, 290.91 mm, and 17.173x109 mm4
PROBLEM 7:
1.
2.
3.
Determine the force F acting on the spring.
Calculate the force acting on each wire.
Find the value of each load P.
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 (02) 8735-9161
0919-227-9194
A precast pile 12 m long is to be lifted by a crane from a
horizontal position to a vertical position by means of a single
sling attached to a point on the pile a distance “x” from the
nearer end of the pile. The pile weighs 4.8 kN per linear meter.
1. If x = 3 m, determine the maximum positive moment
developed in the pile during the lifting procedure.
2. If x = 3 m, determine the maximum negative moment
developed in the pile during the lifting procedure.
3. Determine the value of “x” so that the maximum
moment in the pile is minimum.
Ans. 38.4 kN-m, 21.6 kN-m, and 3.515 m
Davao FB: Review Innovations Davao Branch
 (082) 221-1121 0930-256-0998
CE Review for April 2024 – Strength of Materials 3
Review Innovations
PROBLEM 8: CE Board November 2010
PROBLEM 12:
Two uniform loads of 112 kN/m are acting downward on the
concrete pad shown. The pressure “q” under the pad is
uniform.
The cross-section of a steel beam is composed of a Z-section
with a steel plate welded on top. The dimensions are given
below:
112 kN/m
112 kN/m
q
1.0m
1.
2.
3.
1.5m
2.0m
1.5m
1.0m
Determine the maximum shear.
Determine the max. moment.
Determine the distance from the left where the
flexural stress is zero.
1.
2.
3.
Ans. 48 kN, 42 kN-m, and 3.5 m
PROBLEM 9:
Given the shear diagram for the beam ABCDE.
12 kN
A
B
C
1m 1m
Ans. 21,600 mm2, 52.5 mm, and 106 × 106 mm4
2°
9 kN
D
Find the area of the composite section.
̅ to the centroid C.
Determine the distance 𝒚
Calculate the moment of inertia 𝑰𝒙𝑪 with respect to an
axis through C and parallel to the x axis.
PROBLEM 13:
E
Given the plane area as shown,
-18 kN
1.5 m
3m
1. Determine the maximum intensity of the triangular load.
2. Determine the magnitude of the couple at point B of the
beam.
3. Calculate the moment at critical points.
4. Identify the location of all points of inflection.
Ans. 12 kN/m, 24 kN-m, (12, -12, 7.2, and -9 kN -m), and 4.4 m
1.
PROBLEM 10:
2.
A beam AB 15 m long is simply supported at points A and B. It
carries a uniformly varying load ranging from 230 N/m at A to
500 N/m at B.
500 N/m
230 N/m
A
B
15 m
1.
2.
3.
Determine the maximum shear in the beam.
Find the distance of the point of maximum moment
from A.
Calculate the maximum moment in the beam.
3.
Calculate the moment of inertia of the area about the
x axis.
Determine the coordinate 𝑦̅ of the centroid of the
plane area.
Calculate the moment of inertia of the area with
respect to the centroidal x axis.
Ans. 55.37 × 106 mm4, 86.13 mm, and 7.71 × 106 mm4
PROBLEM 14: CE Board May 2001
A wooden beam 150 mm wide by 300 mm deep is loaded as
shown. The maximum flexural stress developed is 8 MN/m2.
P
w
3m
2m
3m
Ans. 3,075 N, 7.957 m, and 10.30 kN-m
PROBLEM 11:
1.
The shear diagram for a beam is shown below. Reconstruct
the load diagram and draw the moment diagram. Indicate the
location of inflection points.
2.
3.
Compute the maximum moment the beam section can
resist.
Determine the maximum value of the uniform load w
in kN/m.
Calculate the maximum value of the concentrated
load P.
Ans: 18 kN-m, 9 kN/m, and 18 kN
PROBLEM 15:
A material’s increase in strength beyond its yield point before
its ultimate strength is reached is called:
a) Elasticity
c) Strain hardening
b) Rupture
d) Yielding
Manila FB: @ReviewInnovationsOfficial
 (02) 8735-9161
0919-227-9194
Davao FB: Review Innovations Davao Branch
 (082) 221-1121 0930-256-0998