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Consulting Structural Engineering Scenario 3 – Functions, Function Addition,
Choosing the Right Beam
Prepared By: W.J. Cichanski, PE, KPFF Consulting Engineers, Inc.
Introduction to Math Teachers
This scenario will:
Introduce students to mathematical applications in the structural engineering field
Give practice in plotting functions
Give practice in graphic addition of functions
Give practice in converting units
Give practice in interpreting plots of functions
GLEs
1.1.1
2.1.1
1.1.2
2.2.1
1.1.3
2.2.2
1.1.4
2.2.4
College Readiness Standards
1.1
1.2
1.3
2.1
8.2
8.3
8.4
1.1.6
3.1.1
1.2.3
3.2.1
1.4.5
3.3.1
1.5.1
4.2.2
1.5.2
4.2.3
1.5.4
5.3.1
1.5.6
5.3.2
2.2
2.3
3.1
3.3
4.2
7.1
8.1
The problem requires the student to graph several functions given the algebraic
expression for each function, and add the individual functions graphically to determine
which of three steel beams should be used to solve a design problem.
The student will be provided with a schematic diagram of the idealized beam as it will be
supported in the building. The loads that the beam must carry will also be shown in this
schematic drawing. The student will also be provided with mathematical expressions that
allow one to calculate the bending that will occur in the beam as a result of each
individual load. Each function will be unique, and the student will be asked to graph
each function using the loads provided so that each graphed function will represent the
quantity of beam bending as a function of the length of the beam. After the individual
functions are graphed, the student must algebraically add the functions and determine the
magnitude of the maximum bending value and where it is located along the length of the
beam.
Once this combined graphical solution is obtained, the student will be provided with an
algebraic expression for calculating the maximum bending “stress” in the beam. Other
aids will also be provided for solving that expression.
Once the maximum bending stress is calculated for each of the three candidate beams, the
student will be provided with a value for the maximum bending stress that is allowed by
the current building code, and will be required to select which of the three candidate
beams should be selected for the project.
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Introduction
You are representing your design firm at a critical meeting that has been called during the
construction of a building that your firm has designed. Although construction has been
progressing well, the owner’s representative has just announced that she wishes to add a
storage area over a workshop. She has provided a description of the items to be stored
there, and when you review this description, you immediately realize that the current
beam design over the workshop will not be strong enough. You alert the owner
representative that the current beam design must be changed.
She accepts your information and directs you to proceed immediately to calculate the
new beam design requirements. But she also asks you to determine if the new beam can
be one of the three steel beams that the contractor already has on site so as to avoid
ordering additional materials.
The Structural Engineering Industry
A Structural Consulting Engineering firm is generally classified as a “service” business.
The company not only consults with its clients in the initial development of a
construction project, it provides the design documents required for the project’s
completion. The design must meet all client specifications, building codes and other
requirements. Clients may be building owners, private real estate developers,
governmental agencies, professional architects, professional engineers, construction
contractors, and others.
Students interested in careers in this business must develop proficiency in mathematics as
well as facility in communications skills. Such skills are especially important in a
consulting engineering firm, as not only must engineers have the ability to solve
problems using his knowledge in math and technical skills, he must be able to
communicate to his clients clearly his reasoning and solution.
Educational requirements at the entry level in this career include a minimum of a
Bachelor or Science degree in Engineering. Depending upon the type of structures being
designed by specific engineering firms, a Master of Science degree may also be required.
Accredited universities and colleges offering these degrees commonly require students to
have demonstrated mathematics proficiency in courses such as calculus, analytical
geometry, differential equations, and probability theory.
Participants in a Typical Structural Consulting Engineering Design Project
Scenario
A typical consulting engineering design scenario includes a number of interacting
participants as follows:
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Principal in Charge (PIC) -- Under Washington State law, the PIC must be a licensed
professional engineer who is responsible for the entire design and must assure that the
design meets all applicable building codes. This is the person who personally affixes
his or her professional engineering “seal” to the design documents. By State law this
is the individual who is labeled “In Responsible Charge” of the design.
Project Manager (PM) – The PM is also a licensed professional engineer who serves
in the role of lead technical design manager in the office on a day-to-day basis. The
remainder of the design team reports directly to the Project Manager. Project
Managers typically have ten or more years of design experience. In addition to
leading the activities of the design group, the Project Manager will also be
responsible for all communications between the design firm and the client (including
contract management and billings, and marketing consulting firm services).
Design Engineer (DE) – A typical design team assigned to the Project Manager will
have several design engineers. The level of work related experience within the group
of design engineers varies from none (a recent degreed graduate) to perhaps as much
as 30 years or more (some individuals prefer to remain in the role of design engineer
rather than assume the management duties of a Project Manager).
CAD Technician – The CAD (Computer Aided Design) Technician is primarily
responsible for creating the actual engineering drawings that define the design and
construction requirements for the project. The CAD Technician also provide a wide
variety of support to assure that the geometry of the structure depicted on the
drawings matches the survey requirements for the project.
The typical process in a design project includes the following tasks:
 A completed structure generally consists of many individual parts. The end
product consists of design drawings, contract specifications, and an engineers’
estimate of the probable construction cost. The Principal in Charge (PIC) and the
Project Manager (PM) are responsible for the entire design, whereas the Design
Engineer (DE) is primarily responsible for the design of parts of the structure.
The CAD Technician is responsible for completing the computer aided design
drawings that clearly depict the design to be constructed.
 The PIC and the PM develop all of the information needed by the DEs for their
design work, such as the overall size of the completed structure, how much load it
is required to carry, and by what building code it is to be designed.
 The DEs are responsible for completing the detailed design for each of the
structural parts using the criteria established by the PIC and PM.
 The DEs are also required to determine the strength requirements for the materials
used in constructing the structural parts that they design. In general, higher
strength materials will increase the resulting construction cost, so care must be
taken to provide the most cost effective structure at all times.
 The design of a structure is generally an iterative or repetitive process, because
each part must meet all the requirements of the whole finished structure in terms
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of client requirements like strength of materials, cost of the project, building
codes, etc. Because of this iterative process, all participants on the design team
must understand each other clearly and have good communication skills as well as
technical skills.
Choosing the Right Beam
This problem involves choosing a simple steel beam intended for use in a building
construction project. In buildings, beams are horizontal members used to support
elevated floors and roofs. The horizontal beam is typically supported at both ends by
walls or columns. One of the critical issues related to a proper beam design is to limit the
amount of bending “stress” that occurs in the beam when it is loaded by the floor above.
Loads from the floor may include the weight of the floor, the weight of the furniture or
machinery resting on the floor, and the weight of the people who use the floor.
As a beam is loaded by the floor above, it deflects downward and assumes a shape that is
characteristic of the applied loading. This deflected shape is the result of the bending in
the beam. Figures 1(a) and 1(b) illustrate this condition. For the beam shown in Figure
1, this downward deflection will create tension stresses in the bottom of the beam and
compression stresses in the top of the beam. For various safety reasons, the building
code requires designers to select beams for various loading applications that will not
exceed a specified maximum bending stress. The stress that is created in the beam when
loaded is a function of its dimensions and shape. Each of the three beams to be
considered in this problem has different dimensions although they all have approximately
the same shape. Therefore, it follows that each of the three beams will have a different
resulting bending stress when subjected to the same loading.
You will be provided with expressions that define the bending induced in the beam for
each load to be considered. These expressions define how the bending magnitude varies
as a function of beam length for any given load. The first task will be to graph each
function (bending as a function of length) for the applied loads, then add them
graphically to define the total bending in the beam as a function of length. You need to
determine the magnitude of the maximum beam bending value and where it occurs along
the beam length. Using the maximum bending magnitude you need to compute the
resulting bending stress in each of the three beams using an expression for calculating the
stress and the critical dimension needed to calculate the bending “stress”. The proper
beam to use is determined by comparing the three different “stress” magnitudes with the
building code allowable stress and selecting the beam that is in conformance with the
building code.
Your client, in the hopes of saving costs, has offered you a choice of 3 beams that you
may use in constructing a structure that was not part of the original plan. You meet with
the contractor who provides you with the information you need about the three beams
that are already on site. You know that all three beams are called “wide flange beams,”
and you have calculated the magnitude of the bending that will result in the storage area
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above the workshop. The contractor provides you with a table of beam cross section
dimensions of the beams; the table also tells you the “section modulus” of each beam.
The section modulus is a property of a beam needed to determine the bending stress once
you have calculated the magnitude of the bending that will result from the storage area
load above the workshop. You are now prepared to determine which of the three beams
should be used.
Figure 2 illustrates the beam and its loading. There are three design loads that need to be
considered:
Loads P1 and P2 - These are concentrated loads that are caused by heavy storage rack
units that will be installed above the workshop. The magnitudes of these loads are 5000
lbs. and 6000 lbs. respectively. P1 and P2 are located 10 feet and 6 feet respectively
from the left wall support.
Load W ll - This is what is commonly called a uniform load that is spread across the
entire length of the beam. This load is caused by the items that are randomly stored on
the floor above the workshop. Based on the intended use of the building, the owner has
decided that the magnitude of this distributed load will be 400 lbs. per foot of beam
length (400 lb./ft.)
The beam is supported on two walls and is 20 feet long. Beam length is designated as
‘L’.
Simple beam bending load cases are illustrated in the information provided. Load Case 7
applies to Load P1, and Load Case 8 applies to Load P2. The expressions for bending as
a function of beam length are Mx = P (x) / 2 for x < L /2, and Mx = Pbx / L for x > a,
where L is the beam length, b is the distance to the point of load from the right support
wall, and x is any point on the beam. (Mx) bending units are ft. – lbs.
Note to instructor: For x > L / 2 in load case 7, M(x) = P(L-x) / 2, and for x> a in load
case 8, M(x) = P(a)(L-x) / L. Depending upon the skill level of the student, these
additional functions can either be given to the student as an aid in problem solving, or
the student can be required to develop these functions by careful study of the function
shapes in the load case information provided.
Load Case 1 applies to the uniform distributed load W ll. The expression for bending as
a function of beam length is Mx = W(x) (l - x) / 2 where l is the beam length and x is any
point on the beam. (Mx) bending units are ft. – lbs.
1. Graph the bending functions for each load using the information given.
2. Graphically add the three functions to determine the magnitude and location on
the beam where the bending will be maximum.
3. The bending value calculated above will have units of ft. – lbs. Convert these to
units of inch – lbs. (12 inches = 1 foot).
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4. The bending tension stress in the beam can be computed using the following
equation: Bending Stress = Bending Value (inch – lbs.)
Beam Section Modulus (inches3)
Review the tabulated values for the beam section moduli for the three beams. The
appropriate section modulus information from these tables is identified as Sxx.
5. The building code limits the bending stress in a building steel beam to:
23,760 lbs. / inches2
The beams available at the job site are marked in the beam properties tables
provided. Which of the three beams should be used so that the it will have just
enough strength to keep the resulting tensions stress below 23, 760 lbs. / in.2?