Which structural analysis principle states that the total deflection of a
linear elastic structure subjected to several loads is equal to the
algebraic sum of the deflections caused by each load acting
separately?
A. Compatibility principle
B. Principle of superposition
C. Virtual work principle
D. Minimum potential energy principle
Correct Answer: B
Rationale:
The principle of superposition applies to structures that behave linearly
elastically, meaning the relationship between load and response
remains proportional and material behavior stays within the applicable
elastic range. Under these conditions, the structural response produced
by several loads can be determined by analyzing each load separately
and then algebraically adding the individual responses. For example, if a
beam experiences dead load, live load, and a concentrated load, its
total moment, shear, or deflection can be obtained by summing the
corresponding responses from each load case. This principle is
fundamental to structural analysis because it allows complicated
loading conditions to be broken into simpler, more manageable cases.
Page 1 of 79
,A simply supported beam spans 20 ft and carries a uniformly
distributed load of 2.0 kip/ft over its entire span. What is the
maximum bending moment in the beam?
A. 50 kip-ft
B. 75 kip-ft
C. 100 kip-ft
D. 150 kip-ft
Correct Answer: C
Rationale:
For a simply supported beam subjected to a uniformly distributed load
over the entire span, the maximum positive bending moment occurs at
midspan and is calculated using Mmax=wL2/8M_{max}=wL^2/8.
Substituting the given values gives
Mmax=(2.0)(202)/8=100M_{max}=(2.0)(20^2)/8=100 kip-ft. The result
represents the largest internal bending moment developed in the beam
under the specified loading. Recognizing standard beam-loading
relationships is important on the PE exam because they allow rapid
determination of reactions, shear, and moment without constructing
the complete differential-equation solution.
A steel tension member carries an axial tensile force of 120 kips. If its
gross cross-sectional area is 6.0 in², what is its average tensile stress?
A. 20 ksi
B. 24 ksi
C. 30 ksi
D. 72 ksi
Correct Answer: A
Page 2 of 79
,Rationale:
Average axial stress is calculated by dividing the applied axial force by
the gross cross-sectional area: σ=P/A\sigma=P/A. Using 120 kips and 6.0
in² gives 120/6=20120/6=20 ksi. This calculation assumes the force is
distributed uniformly enough across the gross section for an average
stress to be meaningful. In actual steel-member design, the engineer
must also consider the applicable limit states, including yielding of the
gross section, fracture of the net section, block shear, connection
behavior, and the applicable resistance or allowable-strength
provisions.
A rectangular reinforced-concrete beam is subjected primarily to
positive bending. Where is the primary flexural tension reinforcement
normally placed?
A. Near the neutral axis
B. Near the compression face
C. Uniformly throughout the web
D. Near the tension face
Correct Answer: D
Rationale:
For a conventionally reinforced concrete beam subjected to positive
bending, the lower portion of the beam is generally in tension while the
upper portion is in compression. Because concrete has relatively low
tensile strength and is not relied upon to resist the majority of the
flexural tension after cracking, reinforcing steel is placed near the
tension face to provide the required tensile resistance. The
reinforcement develops tensile force while the concrete compression
zone develops compressive force, allowing the internal couple to resist
Page 3 of 79
, the applied bending moment. Proper reinforcement placement and
development are therefore central to reinforced-concrete flexural
design.
A column has an effective length of 15 ft and a radius of gyration of
3.0 in. What is its slenderness ratio KL/rKL/r, assuming K=1.0K=1.0?
A. 45
B. 60
C. 75
D. 90
Correct Answer: D
Rationale:
The slenderness ratio is calculated as KL/rKL/r, with all lengths
expressed in the same units. The effective length is 15 ft, or 180 in. With
K=1.0K=1.0 and r=3.0r=3.0 in., the slenderness ratio is
180/3=60180/3=60. Therefore, the correct numerical result is 60. The
slenderness ratio is important because it influences column buckling
behavior. As a compression member becomes more slender, its
susceptibility to instability increases, and its nominal axial capacity is
generally reduced relative to a short, stocky member.
A structural steel beam is laterally unsupported over a long portion of
its span. Which failure mode becomes particularly important in
evaluating its flexural capacity?
A. Lateral-torsional buckling
B. Punching shear
C. Bearing failure of concrete
D. Local crushing of reinforcing steel
Page 4 of 79
linear elastic structure subjected to several loads is equal to the
algebraic sum of the deflections caused by each load acting
separately?
A. Compatibility principle
B. Principle of superposition
C. Virtual work principle
D. Minimum potential energy principle
Correct Answer: B
Rationale:
The principle of superposition applies to structures that behave linearly
elastically, meaning the relationship between load and response
remains proportional and material behavior stays within the applicable
elastic range. Under these conditions, the structural response produced
by several loads can be determined by analyzing each load separately
and then algebraically adding the individual responses. For example, if a
beam experiences dead load, live load, and a concentrated load, its
total moment, shear, or deflection can be obtained by summing the
corresponding responses from each load case. This principle is
fundamental to structural analysis because it allows complicated
loading conditions to be broken into simpler, more manageable cases.
Page 1 of 79
,A simply supported beam spans 20 ft and carries a uniformly
distributed load of 2.0 kip/ft over its entire span. What is the
maximum bending moment in the beam?
A. 50 kip-ft
B. 75 kip-ft
C. 100 kip-ft
D. 150 kip-ft
Correct Answer: C
Rationale:
For a simply supported beam subjected to a uniformly distributed load
over the entire span, the maximum positive bending moment occurs at
midspan and is calculated using Mmax=wL2/8M_{max}=wL^2/8.
Substituting the given values gives
Mmax=(2.0)(202)/8=100M_{max}=(2.0)(20^2)/8=100 kip-ft. The result
represents the largest internal bending moment developed in the beam
under the specified loading. Recognizing standard beam-loading
relationships is important on the PE exam because they allow rapid
determination of reactions, shear, and moment without constructing
the complete differential-equation solution.
A steel tension member carries an axial tensile force of 120 kips. If its
gross cross-sectional area is 6.0 in², what is its average tensile stress?
A. 20 ksi
B. 24 ksi
C. 30 ksi
D. 72 ksi
Correct Answer: A
Page 2 of 79
,Rationale:
Average axial stress is calculated by dividing the applied axial force by
the gross cross-sectional area: σ=P/A\sigma=P/A. Using 120 kips and 6.0
in² gives 120/6=20120/6=20 ksi. This calculation assumes the force is
distributed uniformly enough across the gross section for an average
stress to be meaningful. In actual steel-member design, the engineer
must also consider the applicable limit states, including yielding of the
gross section, fracture of the net section, block shear, connection
behavior, and the applicable resistance or allowable-strength
provisions.
A rectangular reinforced-concrete beam is subjected primarily to
positive bending. Where is the primary flexural tension reinforcement
normally placed?
A. Near the neutral axis
B. Near the compression face
C. Uniformly throughout the web
D. Near the tension face
Correct Answer: D
Rationale:
For a conventionally reinforced concrete beam subjected to positive
bending, the lower portion of the beam is generally in tension while the
upper portion is in compression. Because concrete has relatively low
tensile strength and is not relied upon to resist the majority of the
flexural tension after cracking, reinforcing steel is placed near the
tension face to provide the required tensile resistance. The
reinforcement develops tensile force while the concrete compression
zone develops compressive force, allowing the internal couple to resist
Page 3 of 79
, the applied bending moment. Proper reinforcement placement and
development are therefore central to reinforced-concrete flexural
design.
A column has an effective length of 15 ft and a radius of gyration of
3.0 in. What is its slenderness ratio KL/rKL/r, assuming K=1.0K=1.0?
A. 45
B. 60
C. 75
D. 90
Correct Answer: D
Rationale:
The slenderness ratio is calculated as KL/rKL/r, with all lengths
expressed in the same units. The effective length is 15 ft, or 180 in. With
K=1.0K=1.0 and r=3.0r=3.0 in., the slenderness ratio is
180/3=60180/3=60. Therefore, the correct numerical result is 60. The
slenderness ratio is important because it influences column buckling
behavior. As a compression member becomes more slender, its
susceptibility to instability increases, and its nominal axial capacity is
generally reduced relative to a short, stocky member.
A structural steel beam is laterally unsupported over a long portion of
its span. Which failure mode becomes particularly important in
evaluating its flexural capacity?
A. Lateral-torsional buckling
B. Punching shear
C. Bearing failure of concrete
D. Local crushing of reinforcing steel
Page 4 of 79
