SECTION C DERIVATION OF EQUATIONS FOR CODE REQUIRED LOADING ON INTERMEDIATE RAILING COMPONENTS

Transcription

1 Ultra-tec Cale Railing Systems C-1 SECTION C DERIVATION OF EQUATIONS FOR CODE REQUIRED OADING ON INTERMEDIATE RAIING COMPONENTS C.1 OAD-DEFECTION REATIONSHIP FOR AN EXTENSIBE, FEXIBE CABE UNDER A PARTIA UNIFORM OAD Symols and Notations Deflection of cale under uniform load, q ef. q ef R x R y s s 1 T o T 1 T avg ength of partial uniform load, in. Spacing etween intermediate supports, in. Partial uniform load required to produce deflection, plf. In-plane end reaction, due to deflection of cale, ls. Out-of-plane end reaction, due to deflection of cale, ls. ength of the curved segment of cale under the partial uniform load, in. ength of the straight segment of cale etween the partial uniform load and the support, in. Tension load in cale at point of maximum deflection, ls. Tension load in straight segment of cale, ls. Average tension load in curved segment of cale, ls. Ojective Given a mid-span deflection,, determine the partial uniform load required to produce that deflection. Determination of Reactions Out-of-plane end reactions, R y, can e calculated y taking the sum of moments aout one of the support points: Σ M : q ef R y q ef R y Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

2 Ultra-tec Cale Railing Systems C- R y q ef (1) The in-plane end reaction, R x, can e calculated y taking the sum of the moments aout the mid-point of the cale, using forces to the right of the applied load: Σ M : Strain Compatiility q ef R x R y 4 R x R x R x R y q ef 8 q ef q ef 8 q ef q ef q ef () The deflected shape of the cale under the uniform load is a paraola of the form: y q x T o where the origin (x and y) is the point of maximum deflection. The mid-span tension in the cale, T o, can e calculated y taking the sum of the moments aout the right-hand end point of the cale, using forces to the right of the applied load: Σ M : q ef T o 4 T o which is equal to R x, otained in Eqn. (). q ef (3) Sustituting Eqn. (3) into the paraola equation gives us: q ef x y q ef Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

3 Ultra-tec Cale Railing Systems C-3 x y (4) The length of the paraola from midpoint to the end of the partial uniform load is given y: dy s 1 (5) where dy/ represents the incremental change in y, given an incremental change in x, which is the derivative of Eqn (4): dy d x dy x (6) Sustituting Eqn. (6) ack into Eqn. (5) yields: s 1 x The cale deflection at the point where the partial uniform load ends can e calculated using Eqn. (4): y The length of the straight segment of cale etween the uniform load and the support can e calculated using the Pythagorean theorum: s 1 s 1 The cale extension, δ, over the original length,, is: ( ) δ s s 1 Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

4 Ultra-tec Cale Railing Systems C-4 x δ 1 (7) The tension in the straight segment of cale, etween the end of the uniform load and the support, T 1, can also e computed using the Pythagorean theorum: T 1 R x R y Sustituting R y and R x from Eqns (1) and () yields: T 1 q ef q ef T 1 q ef 1 (8) To determine an equation for tension in the cale at any point along the loaded paraola, we turn once again to the Pythagorean theorum: T R x R y Recall that T o is equal to R x, therefore this equation ecomes: q ef T T o Sustituting x for /, gives us an equation for tension in the cale at any point along the loaded paraola: ( ) T( x) T o q ef x (9) The average tension in the cale along the paraola can e otained y integrating Eqn. (9) and then dividing y the original length of the segment, /: T avg T o ( q ef x) Sustituting Eqn. (3) into the aove equation and rearranging yields: T avg q ef ( q ef x) Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

5 Ultra-tec Cale Railing Systems C-5 T avg q ef x The total elongation is a result of the cale stretch due to the tension in the cale over two distinct regions, the paraolic segment under the load and the straight segments etween the load and the supports. The total elongation is: Simultaneous Equations δ δ T avg E A E A q ef T 1 ( ) E A x E A q ef 1 q ef δ x ( ) 1 (1) E A Since the total elongation given y Eqns. (7) and (1) must e the same, we now have two equations which can e used to solve for one variale in terms of the other. Sustituting Eqn. (7) for δ in Eqn. (1) and rearranging to solve for q ef gives us: 1 x... q ef E A ( ) x... 1 q ef E A 1 x x ( ) 1 Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

6 Ultra-tec Cale Railing Systems C-6 Mathcad Function: ( ) : A q ef, D,, π D 4 1 ft E eff A ft x x ( ) 1 Example 1: Given a 3/8" diameter 1x19 wire rope supported at 4", calculate the 1 foot length uniform load required to cause a maximum deflection of 1": D : : : :.375 in 4 in 1 ft 1 in ( ) plf q ef, D,, Example : Given a 3/8" diameter 1x19 wire rope supported at 4", calculate the 1 foot length uniform load required to cause a maximum deflection of.5": D : : : :.375 in 4 in 1 ft.5 in ( ) 19.9 plf q ef, D,, Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

7 Ultra-tec Cale Railing Systems C-7 C. OAD-DEFECTION REATIONSHIP IN FEXURA BENDING Symols and Notations D E eff Deflection of cale under uniform load, q, in. ength of partial applied load, in. Diameter of wire rope cale, in. Effective Modulus of Elasticity for wire rope cale, ksi. I Moment of Inertia, in 4. I 1x19 Moment of inertia of 1x19 wire rope, in 4. q Ojective Spacing etween intermediate supports, in. Partial uniform load required to produce deflection, ls. Given a deflection,, determine the partial uniform load required to produce that deflection due to flexural ending. Flexural Bending The deflection of a simply-supported eam 3.5 feet long under a partial uniform load of length 1 foot was empirically computed to e: q 3 5 E I This equation can e rearranged to calculate the load necessary to cause a deflection of : q 5 E I 3 Mathcad Function: ( ) q, D,, : 5 E eff I 1x19 ( D) 3 Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

8 Ultra-tec Cale Railing Systems C-8 Example: Given a 3/8" diameter 1x19 wire rope supported at 4", calculate the uniform load required to cause a deflection of 1" due to pure ending: D : : : :.375 in 4 in 1 ft 1 in ( ) plf q, D,, C.3 EFFECTS OF CABE PRESTRESSING The effect of cale prestressing is to provide a force to alance an applied load. This alancing force is directly related to the geometry of the cale and the prestressing force. Symols and Notations q ps R y Deflection of cale at mid-span, in. ength of partial applied load, in. Spacing etween intermediate supports, in. Applied prestressing force, ls. Partial uniform alance load due to prestressing, ls. Out-of-plane end reaction, ls. Ojective Given an initial prestress force,, and a mid-span deflection,, determine the resulting alancing force, q ps. Balancing Force The end reaction, R y, can e found y taking the sum of the moments aout the other end point: Σ M : q ps R y q ps R y Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

9 Ultra-tec Cale Railing Systems C-9 R y q ps Taking the sum of the moments at mid-span and considering forces to the right, we can compute the alance force q ps : Σ M : q ps 4 R y q ps q ps 4 8 q ps 8 Our applied load is equal to the magnitude of q ps, ut opposite in sign. Therefore, in the context of our applied load, the equation for q ps ecomes: q ps 8 Mathcad Function: ( ) q ps,,, : 8 Example: Given a 3/8" diameter 1x19 wire rope supported at 4", calculate the 1 foot long alancing load with a 4 l prestress force and a deflection of 1": D : : : : :.375 in 4 in 1 ft 4 lf 1 in ( ) 76. plf q ps,,, Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

10 Ultra-tec Cale Railing Systems C-1 C.4 PUTTING IT A TOGETHER Symols and Notations Deflection of cale under uniform load, q. ength of applied partial uniform load, in. D Diameter of wire rope cale, in. q q q ef q ps Applied prestressing force, ls. Spacing etween supports, in. Partial uniform load required to produce deflection, plf. Component of uniform load, q, resisted y flexural ending, plf. Component of uniform load, q, resisted y stretching of cale, plf. Component of uniform load, q, resisted y cale prestressing, plf. Comined oad-deflection Relationship The effects of cale stretch, flexural ending, and prestressing force comine to create a composite relationship etween the applied load and the deflection of the cale. That is, for a given uniform load, the cale will deflect until the load is alanced y the sum of the reactions due to cale stretch, flexure, and prestressing force. Recall the load-deflection relationships previously derived: Extensile, Flexile Cale: q ef E A 1 x x ( ) 1 5 E I Flexural Bending: q 3 8 Presstressing: q ps Strain compatiility laws tell us that when a load is applied to the cale, the deflection in each of the aove cases must the same. Therefore, for a given deflection, the applied load required to cause that deflection is the sum of the three components: q q ef q q ps Mathcad Function: ( ) : q ef D q, D,,, ( ) (,,, ) q (, D,, ) q ps,,, Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

11 Ultra-tec Cale Railing Systems C-11 Example: Given a 3/8" diameter 1x19 wire rope supported at 4" and with a prestress load of 4 ls., calculate the 1 foot long uniform load required to cause a mid-span deflection of 1": D : : : : :.375 in 4 in 1 ft 4 lf 1 in ( ) 4.6 plf q, D,,, C.5 BUIDING CODE OAD REQUIREMENTS The 6 International Building Code and 7 California Building Code require intermediate rails "to withstand a horizontally applied normal load of 5 pounds on an area equal to 1 square foot, including openings and space etween rails" (IBC / CBC ). To meet this requirement, the end reactions, R x and R y, caused y the 5 ls over 1 square ft load must e determined, so that they may e included in the railing system frame calculations. Symols and Notations Deflection of cale under uniform load, q app. D q app R x R y End Reactions ength of applied load q app, in. Diameter of wire rope cale, in. Applied prestressing force, ls. Spacing etween supports, in. Applied uniform load, plf. In-plane support reaction, ls. Out-of-plane support reaction, ls. Recall, from Section C.1, that R x and R y can e found using the sum of the moments aout one of the supports and aout the middle of the cale, respectively. Sustituting the applied partial load, q app, for the partial load, q ef, and noting that the deflection represents the comined effects of the extensile-flexile cale, ending in the cale and prestressing, Eqns. (1) and () may e rewritten: q app R y (11) R x q app (1) Determining the out-of-plane reaction, R y, is straightforward; however, calculating the in-plane reaction, R x, requires knowing the deflection in the cale,. Given the applied uniform load, q app, we can calculate the deflection using the Mathcad root() function introduced earlier: Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.

12 Ultra-tec Cale Railing Systems C-1 ( ( ) q app, ) root q, D,,, Knowing the value for, the reactions can e determined, as shown in the Mathcad functions elow. Mathcad Functions: R y ( q app, ) : q app ( ) :.1 in root( q(, D,, ) q app, ) R x D,,, q app, q app, Example. Given 3/8" wire rope cales spaced at 3.11", supported at 4", and with a 4 l prestress force in each cale, determine the reactions for a single cale when sujected to the code-required 5 psf loading. D : s : : : q app :.375 in 3.11 in 4 in 1 ft 4 lf : ( 5 psf) s q app plf R y ( q app, ) 6.5 lf ( ) 46.3 lf R x D,,, q app, END OF SECTION C Copyright 3-8, Ultra-tec Corporation, Carson City, Nevada. All rights reserved.