Example Stayed beam with two pylons

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1 Example Stayed beam with two pylons A roof structure is a stayed beam. The roof span is 300 ft. Stay vertical run is 20 ft. The deck is weighs 12 PSF. Beams have a transverse spacing equal to 40 feet. The roof carries a transient snow load equal to 20 PSF. Main cables and suspenders are bridge strand furnished by Bethlehem. Beams are compact W shapes in A992 steel. Pylons are HSS in 50ksi steel. Pylons are braced at roof deck. AISC specifications are followed for beam selection Figure 1 Six, of eight, stays support loads from the roof deck. For roof deck load k 24 k 24 k 24 k 24 k 18 k Figure 2 For roof snow load k 40 k 40 k 40 k 40 k 30 k Figure 3

2 Preliminary design We know the roof deck load, 0.48 klf on the beam. We don t know self weight of beam, pylons or stay cables yet. We will work with the roof deck load and the snow load to estimate forces and to make initial selections of stays, beams and pylons Tension in stays. Selection. The vertical components of tension in stays 1, 2 and 3 are computed from end reactions of beams; that is when beams are assumed to behave like a set of simply support spans. We get horizontal components of tension in stays using the rise and run of each stay. Then we can compute tension. Stay 0 is different. We will compute the force in Stay 0 in a later step. For dead load Table Stay Dx ft Dy ft Fz k Fy k ?? ?? ?? For dead plus snow load Table 2 Stay Dx ft Dy ft Fz k Fy k ?? ?? ?? Beams Moment and Thrust. Selection. Moments in beams are estimated for simple spans. Thrusts in beams are from horizontal components of tension in stays. Unbraced length Lb is taken as 10 ft. Length for major axis buckling is 50 ft.

3 For dead load Table 3 Beam M kft P k ? 2 150? 3 150? For dead plus snow load Table 4 Beam M kft P k Select Pallow k Mallow kft int W21x Shape kl/r x kl/r y kl/r Fe ksi Fcr ksi Pallow k Lp ft Mallow kft W21X Thrust in Pylon. Selection. The pylons carry vertical components of tension from stays. For stays 1, 2, and 3, vertical components come directly from loads imposed at the roof deck. Stays 0 are there to provide horizontal equilibrium at top of pylons. So there is an additional vertical force that we have to include. Consider a free body diagram that includes a pylon and stay 0. Figure 4 shows the forces from stays 1, 2 and 3 acting at the top of the pylon for the reference state. Compression in the pylon and tension in stay 0 must provide equilibrium. Figure 5 shows the forcers from stays 1, 2 and 3 for dead load plus snow load. For dead loads: Figure 4

4 Solve joint equilibrium equations H0 k P k External k horiz -1 15/40 15 vert -20/ Use row reduction to find H0 =? P =? For dead plus snow loads: Figure 5 Solve joint equilibrium equations H0 k P k External k horiz -1 15/40 40 vert -20/ Use row reduction to find H0 =? P =? Selection for Pylon Buckling length equals 21.4 ft, along the member with rise equal to 20 ft and run equal to 7.5ft. Select HSS9X9X3/8 A = 11.8 in2, r = 3.51 in, Pallow KL/r Fe ksi Fcr ksi k

5 Selection for Stay 0 Ratio of hypotenuse to horizontal projection is sqrt(20^2+75^2) / 75 =? T =? Tu =? Select 1-1/2 strand Help for reference state adjustment We also want an expression to help us balance the reference state. We expect that we will adjust H0 to get horizontal equilibrium at the top of pylon. We expect that we will adjust P to get vertical equilibrium at the top of pylon. This one we have a closer look at (Figure 6). Figure 6 Use joint equilibrium equations H0 k P k External k horiz -1 15/40 0 vert -20/ Find H0 =? P =?

6 Model for Cable Structure App Nodes for a model are mostly a straight mapping of the diagrams we have developed for this roof. To note: Beams begin and end at nodes that allow y-direction translation. Stays 0 end at nodes that are fixed for all 3 translations. Table 5 ID X ft Y ft Z ft sx sy sz left abut True True Roller support for beam at left end left side Beam connection to Stay 1, left left Beam connection to Stay 2, left mid left Beam connection to Stay 3, left mid right Beam connection to Stay 3, right right Beam connection to Stay 2, right right side Beam connection to Stay 1, right right abut True True Roller support for beam at right end left pylon top right pylon top left pylon base True True True right pylon base True True True left abut stay True True True Pin support for Stay 0, left right abut stay True True True Pin support for Stay 0, right Components Four components for stay. Four components for beam. One component for pylon. Stays There are four types of stays. They differ in size and in force at reference state. Lets collect these: Table 6 H (horiz) k H (vert) k Tu tn Strand Dia Stay 0 - Stay 1 - Stay 2 - Stay 3 - Beams We select a single W shape, as shown above. Pylon We select a single HSS, as shown above. Loads There are two distributed external loads.

7 Deck = k/ft vertical Snow = -0.8 k/ft vertical Balance the reference state Try this. The model file, BigRoof.csxml is posted at the course website. The forces for components in the file are for deck loads only. Your first look at equilibrium in reference state will be out of balance due to self weight of components. Balance the reference state. Report your results: ID H kip Beam 0 0 Beam 1? Beam 2? Beam 3? Pylon? Stay 0? Stay 1? Stay 2? Stay 3? Analysis for Snow Load. Use 1.0 load factor for beams. ID P k Vix k Viy k Mix kft Miy kft Miz kft Mjx kft Mjy kft Mjz kft Beam Beam Beam Beam Beam Beam Beam Pylon Pylon Compute max moments in beams ID x+ ft M+ kft Mmax kft Pallow k Mallow kft int Beam Beam Beam Beam Beam Beam Beam W21x111 is OK

8 Check pylon Pmax = -242 k, Pallow = 239 k, Fails! Update to HSS9x9x1/2, Pallow = 305k Analysis for Snow Load. Use 2.2 load factor for strand. ID P k Tu tn Stay /2 OK Stay Stay /16" OK Stay Stay /8 NG -> 1-1/4" Stay Stay /16 OK Stay

9 Update Selections & Reference State ID H kip Beam 0 0 Beam Beam Beam Pylon Stay Stay Stay Stay Analysis for Snow Load. Use 1.0 load factor for beams. ID P k Vix k Viy k Mix kft Miy kft Miz kft Mjx kft Mjy kft Mjz kft Beam Beam Beam Beam Beam Beam Beam Pylon Pylon Compute max moments in beams ID x+ ft M+ kft Mmax kft Pallow k Mallow kft int Beam Beam Beam Beam Beam Beam Beam W21x111 is OK Check pylon Pmax = -242 k, Pallow = 305 k, OK

10 Analysis for Snow Load. Use 2.2 load factor for strand. ID P k Tu tn Stay /2 OK Stay Stay /16" OK Stay Stay /4 OK Stay Stay /16 OK Stay

Suspended Beam Roof with Pylons

Cable Supported Structures George.Hearn@colorado.edu 25 Suspended Beam Roof with Pylons A roof structure is a suspended beam. The roof span is 200 ft. Main cable sag is 20 ft. Suspender length varies.