1. Beam Design Beams are designed to safely support the design loads.
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Design
Beams are designed to safely support the design loads.
Beams are primarily designed for bending and shear.
Beam deflection must be checked.
Beams are sized to minimize material.
Bending is the combination of tension and deflection that occurs along the beam when it is loaded.
Shear is a force that acts perpendicular to the axis of the member, causing the internal particles of the member to slide against each other.
Deflection is the amount of movement of a structural member under loading. Beam deflection is typically measured in inches. Deflection is limited by building codes. The maximum deflection is based on what the beam supports.
For building materials, cost is generally directly related to the amount of material used. The more material (in other words, the more weight), the higher the cost. Designers typically attempt to reduce the weight of the construction materials to reduce cost.
Deflection
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2. Steps in Beam Design Establish the design loads Analyze the beam
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Steps in Beam Design
Establish the design loads
Analyze the beam
Select the preliminary member
Evaluate the preliminary design
Redesign (if needed) – Repeat the above steps as necessary to achieve a safe and efficient design
Design and detail the structural component
Structural design is an iterative process.
You have already learned how engineers determine design loads, and you have analyzed beams for maximum shear and bending moment.
Next, you need to select a preliminary member that you anticipate will provide adequate bending and shear strength with the least cost.
This preliminary member must be evaluated to ensure that it will provide a safe and serviceable design. You may find that the chosen member does not provide enough shear strength or enough stiffness to adequately resist deflection.
So you must revise your choice, always keeping the cost in mind. This will change the section properties used and may affect your loading calculations, which can require recalculation, reanalysis, and redesign. This process is repeated until you find a cost effective member that will be both safe and serviceable.
The last step in the process is to document your design with construction drawings.
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3. Axial Stress Tension and compression Axial stress represented by
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Axial Stress
Tension and compression
Axial stress represented by
Axial stress results from a force that acts along the length of the member. Tension (or pulling) and compression (or pushing) are axial forces or axial stresses.
Tension stress and compression stress are represented by the Greek letter sigma.
Where F = applied force
A = cross sectional area resisting load
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4. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Strain
The change in size or shape of a material caused by the application of external forces
Axial strain,
Strain is also referred to as deformation.
Axial strain is represented by the Greek letter epsilon.
Strain is unitless or is recorded as inches/inch.
Where = change in length
L = original length
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5. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Stress and Strain
In many materials, stress is directly related to strain up to a certain point.
If you have taken POE, you will recognize this stress-strain curve.
We will design steel structural members. This graph shows the typical relationship between tensile stress (vertical axis) and strain for steel.
As the tensile stress increases, strain increases. In fact, the relationship is linear until the material reaches its yield stress. At this point, the material begins to elongate much more quickly with respect to an increase in stress.
The maximum stress that a material can carry is called the ultimate stress of the material. If a member is loaded beyond the ultimate stress, the material will begin to stretch very quickly (called necking) and eventually break at the fracture point.
The slope of the linear portion of the graph is called the modulus of elasticity, or Young’s modulus, and is represented by a capital E. Each material has a unique modulus of elasticity. For steel, E = 29 million psi.
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6. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Stress and Strain
Elastic behavior – material will return to its original shape when unloaded
Plastic behavior – material will retain some deformation when unloaded
Yield Point
Structural members are designed to act elastically during the service life of a building
The portion of the curve to the left of the yield point is the ELASTIC REGION. Material will behave elastically if the stress does not exceed the yield strength of the material.
The portion of the curve to the right of the yield point is the PLASTIC REGION. Material will behave plastically if the stress exceeds the yield strength of the material.
Although a member will not necessarily fail once it begins to act plastically, the excessive deformation is undesirable and can be dangerous.
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7. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Factor of Safety
The ratio of the maximum safe load to the maximum allowable design load
Magnitude of the factor of safety varies depending on the loading conditions and type of forces induced
When designing a structure, the safety of the occupants is of the utmost importance, as well as protection of property.
Many factors affect the behavior of a structure. As a structural engineer, you have little control over many of these factors
For example, although you design the structure for a given load, there is no guarantee that the structure will not experience higher loads.
Engineers can’t control the quality of the building materials used. What if a beam has a flaw that is not identified before it is installed?
What if a construction worker fails to install all of the bolts in a connection and the error is not identified during inspection?
A factor of safety is used to provide a comfort zone that can allow a margin of error for unforeseen circumstances that can adversely affect a building’s performance.
The maximum safe load may be different depending on the design method used. We will use the allowable strength design method so that the maximum safe load is considered the load at which the member will reach the yield point.
In other words, the maximum allowable load is set to a level well below the maximum safe load by a factor of safety. If the factor of safety is two, this means that the structure element must be designed so that the member is twice as strong as it theoretically needs to be in an ideal situation.
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8. Allowable Strength Design (ASD)
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Allowable Strength Design (ASD)
Strength is related to stress
Strength indicates internal force
Maximum safe load = nominal strength
Internal force that causes yielding across the entire cross section
Maximum allowable load = allowable strength
Strength is another way to indicate stress. Strength is related to stress in that strength indicates an internal force, while stress indicates an internal force per unit area.
Note: Two different members that have the same internal stress will not have the same internal force due to the differences in the cross section of the members.
There are two standard structural design methods: Allowable Strength Design (ASD) and Load Resistance Factor Design (LRFD). Depending on the material, codes and standards may specify one or the other method. In some cases, which is true for structural steel, the engineer may choose the preferred method.
We will use ASD for structural design in this class.
ASD limits the maximum internal force that results from the design load combinations within a structural member.
In ASD the maximum safe load is referred to as the nominal strength of a member. The nominal strength is the internal load that causes yielding across the entire cross section of the member.
The allowable strength is the maximum internal force that can be induced in a member based on the design loads.
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9. Allowable Strength Design (ASD)
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Allowable Strength Design (ASD) OR
Where = factor of safety
= nominal strength
= internal force due to design loads
= allowable strength
This is a generic formula that applies to all kinds of internal forces, including tension, compression, shear, and bending moment. The second formula is simply a manipulation of the first.
We will focus on bending moment and shear strength when designing beams.
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10. Allowable Bending Strength
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Allowable Bending Strength OR
Where = 1.67 = factor of safety for bending
= nominal bending moment strength
= internal bending moment due to design loads
= allowable bending strength
The two equations are equivalent. Note the factor of safety for bending moment is 1.67.
Again, the nominal bending moment is the maximum safe bending moment in a steel member.
Ma is the actual internal bending moment that the member feels under a given design load. This is typically taken as the maximum bending moment from the beam analysis. That is Ma = Mmax.
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11. Nominal Bending Strength
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Nominal Bending Strength
Where = yield stress of steel
= plastic section modulus
In general, most structural steel is high strength with a yield stress Fy= 50 ksi or 50,000 psi. However, other yield stresses are available in certain structural steel shapes.
The plastic section modulus is a section property of the member. A section property depends on the shape of the cross section of the member.
Just a note. In this class we will assume that the beams are always laterally supported along their compression flange, that means that the compression flange (usually the top flange) is held in position so that it can not move side-to-side. This is generally true when a beam supports a floor if the floor is attached to the beam. If a beam is not laterally supported (as with a balance beam), the beam can buckle under the load before it reaches maximum strength.
NOTE: We will assume that every beam and girder is laterally supported along its length so that it will not buckle under loading. If a beam is not laterally supported, buckling must be checked.
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12. Plastic Section Modulus, Z
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Plastic Section Modulus, Z
Section property
Indicates the moment carrying capacity of a member
Available in tabular form in design manuals
The plastic section modulus, represented by capital Z, gives an indication of how well a member can carry bending loads – the larger the plastic section modulus, the larger the maximum bending moment before failure.
This website provides section properties and dimensions for structural steel sections. You will use this site extensively for the next activity. Write it down and save it as a favorite in your web browser. You will using this information often.
[You may want to access the website and demonstrate the process for locating the structural steel wide flange dimensions and properties.]
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13. Plastic Section Modulus, Z
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Plastic Section Modulus, Z
Flange
I-shaped members are excellent choices for beams
Structural steel wide flange
Designation W (depth)
Example: W12 x 58(weight)
Stronger when bending about the x – x axis, Zx
Web
In general, the more cross sectional area further away from the neutral axis, the larger the section modulus.
I-shaped members, such as steel wide flange sections, are especially good at carrying bending moment. The illustration here shows the cross section of a structural steel wide flange member. The flanges, which constitute a large portion of the cross sectional area, are located away from the neutral axis.
The designation for wide flange members is the letter W followed by the depth x weight per foot of the member. For example, a W 12 x 58 has an approximate depth of 12 inches and weighs approximately 58 pounds per foot of length.
Wide flange sections are stronger when bent about the x – x axis (called the strong axis); therefore, the best orientation for a beam installation is that shown. If the beam were rotates 90 degrees (so that it looked like an H), it would carry less bending moment. Be careful to use the correct plastic section modulus – Zx when bent about the strong axis.
Flange
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14. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Design Example
Choose the lightest wide flange steel section available to support a live load of 230 psf and a dead load of 133 psf over a simple span of 18 feet and a supporting width of 3’. Assume the beam will support a plaster ceiling. Use Fy = 50 ksi.
Plf stands for pounds per linear foot. These loads represent uniform loads along the beam.
Copy this example into your notebook.
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15. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Design Example
ω= (LL + DL) * Supporting width ( ) * 3’ = 1090psf
Mmax= ω(L)2 8
= 1090(18’)2
First analyze the beam to determine the maximum shear and bending moment.
The shear and bending moment diagrams are shown in the MD Solid analysis OR can be calculated using beam formula OR can be found by sketching shear and moment diagrams.
Note: The maximum shear force is 9810 pounds, and the maximum moment is ft-lb. The units are displayed in the lower left corner of each diagram.
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16. Bending Strength Select beam based on bending. = 1.67 where
Beam Design
Bending Strength
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Select beam based on bending.
where
= 1.67
Always design for moment first because it will almost always control beam design over shear.
Ma is the maximum moment due to the design loads.
The factor of safety for bending is 1.67.
Therefore, the nominal moment of the selected beam must be greater than or equal to 73,722 ft-lb.
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17. Bending Strength & Section Modulus
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Bending Strength & Section Modulus
Since
Zx= Mn Fy
Check tables of section properties to choose the lightest wide flange section that has a Zx greater than 17.7 in^3.
Use section property tables or to investigate section properties.
W10 x 17 works Z x = 18.7 in.3 > in.3
But a W 12 x 16 weighs less with Z x = 20.1 in.3 > in.3
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18. Shear Strength Since and W12 x 16 works For a W 12 x 16
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Shear Strength
Since and
For a W 12 x 16
d = in. t w =0.220 in.
Find the nominal shear strength and compare it to the required minimum of 14, 715 lb.
Use section property tables or to investigate section properties.
Unless the beam is unusually short and heavily loaded, shear typically will not control but should be checked.
79,134 lb ≥ 14,715 lb
W12 x 16 works
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19. Shear Strength Check shear strength. where Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Check shear strength.
where
Va is the maximum shear due to the design loads.
The factor of safety for shear is 1.5.
Therefore, the nominal shear strength of the selected beam must be greater than or equal to 14,715 lb.
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20. Allowable Shear Strength
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Allowable Shear Strength OR
Where = 1.5 = factor of safety for shear
= nominal shear strength
= internal shear force due to design loads
= allowable shear strength
Shear strength is calculated in a manner similar to bending strength. Note that the factor of safety for shear is 1.5 compared to the factor of safety for bending of 1.67.
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21. Nominal Shear Strength
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Nominal Shear Strength
Where = nominal shear strength = yield stress of steel
Aw = area of the web
Nominal shear strength is based on the area of the web of the structural steel shape since the web will generally carry the shear force.
The web area for a wide flange is the product of the member depth (d) and the thickness of the web (tw). The depth and web thickness are tabulated for steel sections in design manuals.
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22. Beam Deflection Deflection limit supporting plaster ceilings
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Deflection
Deflection limit supporting plaster ceilings
L/240 for Dead + Live Loads
L/360 of Live Load
Deflection limit supporting non-plaster ceilings
L/180 for Dead + Live Loads
L/240 of Live Load
WHY?
Ceiling cracks in plaster
Roof ponding (flat roofs)
Visual or psychological reasons
Designer’s judgment
Building codes and design standards restrict deflection of structural members.
Excessive deflection, although not normally a safety issue, can adversely affect the performance of a building. Too much movement can cause cracks in walls and ceilings, roof ponding, misalignment of building systems, and incorrect operation of equipment. In addition, excessive deflection can give people an uncomfortable “motion sickness” feeling, or can simply make them feel uncomfortable about the potential safety of the building.
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23. Beam Selection Process
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Selection Process
Select beam based on bending moment
Check shear strength
Check deflection
Revise beam selection as necessary
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24. Check Deflection Deflection limit supporting plaster ceilings
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Check Deflection
Deflection limit supporting plaster ceilings
L/240 for Dead + Live Loads
L/360 of Live Load
Based on building codes and design standards, the deflection limits are L/240 for dead + live load and L/360 for live load alone.
For a simple beam with a uniform load, the maximum deflection will occur at center span and can be calculated using this equation.
Be sure to convert every quantity to pounds and inches so that units will correctly cancel.
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25. Check Deflection Maximum deflection due to design loads
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Check Deflection
Maximum deflection due to design loads
Dead + Live Load Deflection
The deflection formulas are given in Activity Beam Analysis Short Cuts.
For a simple beam with a uniform load, the maximum deflection will occur at center span and can be calculated using this equation.
The total load is a dead load of 300 plf plus a live load of 790 plf = 1090 plf.
The moment of inertia, another section property, is tabulated in design manuals. I for the W12 x 14 is 88.6 in^4.
Be sure to convert every quantity to pounds and inches so that units will correctly cancel. Notice that the last unit cancelation (purple) leaves inches in the denominator of the denominator. Dividing by a fraction is equivalent to multiplying by the reciprocal of the fraction – so the result is inches.
W 12 x 16 will work
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26. Maximum deflection due to design loads
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Check Deflection
Maximum deflection due to design loads
Live Load Deflection
The applied live load is 790 plf. The deflection due to live load is 0.62 inches, which is greater than the code specified L/360 = 0.6 in.
The W12 x 16 will not work. You must try another beam with a larger I.
W 12 x 16 will not work
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27. Evaluation and Redesign Check Deflection
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Evaluation and Redesign Check Deflection
Try W 12 x 19
Dead + Live Deflection
You need a beam with a larger moment of inertia in order to reduce the deflection. Look at the beam tables. A W 12 x 19 is slightly heavier, but has a much larger I = 130. The lightest W14 is 22 lb/ft. Try the W12 x 19.
The W 12 x 19 works for live load deflection.
W 12 x 19 OK
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28. Evaluation and Redesign Check Shear and Bending Moment
Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Evaluation and Redesign Check Shear and Bending Moment
By inspection, the plastic section modulus and web area for the W12 x 19 are larger than those for the W12 x 16 and are therefore sufficient to safely support the bending moment and shear.
Use a W12 X 19
Revisit your bending moment and shear calculations to make sure the plastic section modulus and web area are sufficient.
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29. Beam Design Beam Design Deflection Stress Evaluation and Redesign
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Beam Design
Beam Design
Deflection
Stress
Evaluation and Redesign
Axial Stress
Strain
Factor of Safety
Bending Stress
Shear Stress
Beam Selection
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30. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Stress
A measure of the magnitude of the internal forces acting between particles of the member resulting from external forces
Expressed as the average force per unit area
When a beam is loaded, the internal fibers of the member must carry and transfer the resulting internal shear and bending moment to the beam supports.
These internal forces result in internal stress in the member.
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31. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Axial Stress
Example: Find the tensile stress of a 2 in. x 3 in. tension member subjected to a 45 kip axial load.
1 kip = 1000 pounds
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32. Beam Design
Civil Engineering and Architecture
Unit 3 – Lesson 3.2 – Structures
Strain
Example: Calculate the strain if a 16 ft long structural member elongates 1.5 in. when subjected to a tensile load of 67 kips.
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