Chapter 8 - Bending Members
© 2006, 2007, 2008, 2011 T. Bartlett Quimby
As presented in Chapter 1, the homework problems involve the design of elements of three different structures plus some unrelated details. Please see the relevant links below for each structure. When completing the problems, consider both ASD and LRFD design philosophies unless otherwise specified by the instructor. Consider all limit states presented in this and prior chapters. Consider developing a generic spreadsheet that you can apply to similar problems.
The complete design of flexure members involves the investigation of flexure, shear, deflection as well as the design of the connection(s) to supporting and supported structural elements.
Problems M8.1: A W18x35 section is used as a simply supported beam with a span of 20 ft. Determine the nominal moment capacity/capacities, Mn, along the beam for the following conditions. Assuming that the beam supports a uniformly distributed load, also determine the magnitude of the distributed loads that it can support for each case without violating the flexural requirement. Assume that the load is 1 part dead load (including self weight) and 2 parts live load.
Problem M8.2: Select the lightest W section for the condition shown in MISCDET_STL 2/S5.2. The beam is simply supported and has full lateral support of the compression flange. The distributed load consists of 1 k/ft dead load (not including self weight!) + 1.5 k/ft floor live load. The live load is to be placed where it will cause maximum effect. In other words, it is necessary to compute the envelope of possible moments that beam will see as the live load is strategically placed. It is also necessary to decide where to place the load for maximum deflection. Influence lines are useful in determining where to place the live load so as to achieve maximum effect.
Problem M8.3: Select the lightest W section for the transfer beam shown in MISCDET_STL 3/S5.2. In addition to it's own weight, the beam supports two concentrically loaded columns. The leftmost column supports 50 k dead load, 75 k live load, and 20 k snow load. The column at the end of the cantilever supports 25 k dead load, 60 k live load, and 15k snow load. The beam has lateral support only at points of vertical support and at points of load application (i.e. where the columns are). Ignore self weight of the beam.
Problems M8.4: Select the lightest W section that can support a uniform load that consists of 1.2 k/ft dead load and 1.5 k live load on a 25 ft span without violating the applicable limit states. The beam has full lateral support of the compression flange. Provide some discussion on how the steel type influences the final selection.
Problem M8.5: Create a spreadsheet that will compute the nominal moment capacity, Mn, of any I shaped section (i.e. W, M, S, or HP) in the AISC inventory for any ASTM available steel type and on a given laterally unsupported length, Lb. The spreadsheet is to use values of Cb provided by the user. The spreadsheet is also to graph the moment capacity (Mn) vs. laterally unsupported length (Lb) diagram for the selected member over a range of Lb equal to zero to the maximum of 60 ft, three times Lr, or 1.1*Lb.
Problems M8.6: A W18x35 section is used as a simply supported beam with a span of 20 ft. Determine the nominal shear capacity, Vn. Assuming that the beam supports a uniformly distributed load, also determine the magnitude of the distributed loads that it can support without violating the shear requirement. Assume that the load is 1 part dead load (including self weight) and 2 parts live load.
Problem M8.7: A series of parallel beams are used to support a pedestrian bridge that spans 25 ft. The beams have full lateral support. The inner beams support 50% more load than do the outer (or edge) beams. Considering only flexure, select the lightest W section for the outer beams if the uniform distributed load consist of 400 lbs/ft dead load (not including self wt) and 1,000 lbs/ft live load. Using the same size section, add cover plates to the top and bottom of the beams so that the beams can support the extra load. Specify the width, thickness and extent of the cover plates (use the same cover plate for the top and bottom of the beams). Neatly sketch a scaled drawing of the beam profile.
Problem M8.8: Given the continuous beam shown in MISCDET_STL 1/S5.3, select a W section that will provide sufficient strength to meet the positive moment flexural strength requirement on the middle span. The beam has full lateral support of the compression flange. Design cover plates to add moment capacity to the beam where needed. The cover plates should be the same top and bottom at each location where needed. Provide a neatly drawn, to scale, moment capacity vs. moment envelope diagram in addition to a neatly drawn (to scale) beam profile with appropriate dimensions. The beam is subject to a uniform dead load of 1 k/ft and a uniform live load (which must be strategically placed so as to cause maximum effect) of 1.5 k/ft.
Problem M8.9: Design the bearing plates on the tops of the concrete walls for the transfer beam shown in MISCDET_STL 3/S5.2. Assume that the beam is a W27x84 made of ASTM A992 steel. The concrete has a 28-day compressive strength, f'c, of 3,000 psi. Use the same thickness for the two plates. Neglecting its own weight, the beam supports two concentrically loaded columns. The leftmost column supports 50 k dead load, 75 k live load, and 20 k snow load. The column at the end of the cantilever supports 25 k dead load, 60 k live load, and 15k snow load. The bearing plate dimension "N" should not exceed the width of the walls that they sit on. If this is insufficient, recommend design changes to fix the situation. Consider only LRFD.
There are lots of beams with a wide variety of loading configurations in the dormitory building! You can use your structural analysis skills to determine the loading on the statically determinate portions of the framework, but the moment frames will required some form of indeterminate analysis. Note that all the beams receive full lateral support from either the floor or roof system.
Problem D8.1: On the floor framing plan (DORM S1) you will notice several regions that have beams in a repetitive pattern. These beams are referred to as joists. The floor plan has four different joist specifications: J201, J202, J203, and J204. The joists are all simply supported single span beams with uniformly distributed loadings. Each joist supports a width of floor equal to the spacing of the joists, so the distributed load on the 2D beam diagram, w, equals the spacing of the joists (i.e. the tributary width) times the magnitude uniform pressure on the floor. To designate a joist design, it is necessary to state the size and spacing of the member. For this problem, select the size and spacing of the I shaped section (W, M, S, C, or MC) that results in the least weight framing system. Due to limitations on the floor slab, the maximum joist spacing is limited to 6 ft. Consider only ASD. Complete the following table:
Problem D8.2: Select the lightest section for the three floor beams along grid C between grids 3 and 11 on drawing DORM S1. Note that the region between grids B & C is corridor space and the region between grids C & E are dorm room space.
Problem D8.3: Select the least weight members to use as joists/rafters on the roof framing plan (DORM S2). These members support roof dead load and snow load. Complete the following table:
Problem D8.4: Select the lightest section for the three roof beams along grid C between grids 3 and 11 on DORM S2.
For extra practice, choose any other statically determinate beam in the floor or roof plan and select a least weight section.
Being an axial force truss system, the tower does not provide us with opportunity to compute flexural forces and size members to meet them. The building and bridge are better choices for problems in this chapter.
The principle flexural members in the bridge are the deck beams, girders and end girders. All are statically determinate. They are also relatively short allowing a single wheel to be the defining vehicle load. For the purposes of these problems, treat the vehicle loading as a live load in the ASCE 7 load combinations.
Problem B8.1: Select the lightest W section for the deck beams. Since all deck beams are to be the same size, we design for the worst case scenario. The interior deck beams support more deck than do the outer ones, so that is the one that we will use. The interior deck beams support 7'-6" width of 6" thick (75 sqft of surface area) concrete deck plus their own weight in addition to a 16 kip vehicle point load that can be placed so as to cause maximum effect. The beams have full lateral support from the concrete slab.
Problem B8.2: Select the lightest W section for a typical girder. The girder supports the dead load reactions from the deck beams from two sides where they connect to the girder plus two sets of wheel loads as shown in TBRDG 3/S2. Each wheel load is 16 kips. Assume that the concrete deck and the joists (NOT including girder weight!) weigh an average of 80 lbs per square foot of deck surface. For deflection, consider only that the vehicle related deflection is to be limited to 1/500th of the beam span. Note that the maximum moment and maximum shear come from two different load arrangements.
Problem B8.3: Select the lightest W section for an end girder. The end girder supports the dead load reactions from the deck beams from one side only where they connect to the girder plus two sets of wheel loads as shown in TBRDG 3/S2. Each wheel load is 16 kips.
Problem B8.4: Redesign the deck beam based on the loading for the outer deck beams. Using the same size beam for the inner deck beams, add flexural cover plates (if necessary) so that the beams meet the flexural requirement. The outer beams support themselves plus a tributary width of 5'-2" of 6" thick reinforced concrete (75 psf) deck plus a 16 k vehicle wheel load placed to cause maximum moment.