h z (Draft, 010) Bracing for Earthquae Resistant Design 1 September 18, 00 (010 update) Rigid Roof Idealization and Column Stiffness Relative to the columns, the roof structural system might be quite rigid, resulting in the classical deformed shape shown in the Figure. u In this deformed configuration, the column stiffness is dictated by a state of fixed-fixed boundary conditions. In order to define the lateral column stiffness () for this fixed-fixed state, we start with the beam Equation of equilibrium and the appropriate boundary conditions (see next page for derivations): h EI w = 0 w(0) = w (0) = w (h) = 0, and w(h) = u Resulting in: w = ( (z /h ) - (z /h ) ) u u F With shear force Q = - EI w, the shear force F at h becomes: F = -EIw (h) = (1EI / h ) u w F = u and (column) is therefore, = (1EI / h ) 1
h = 1 Bending beam Equation EI w = 0 (case of zero pressure acting along the beam length) EI w = c 1 EI w = c 1 z + c EI w = c 1 z / + c z + c (slope) EI w = c 1 z /6 + c z / + c z + c (displacement) w (0) = 0 results in c = 0 w (0) = 0 results in c = 0 w (L) = 0 results in c = - ( c 1 / h ) w (L) = u results in c 1 = - ( 1 EI / h ) u Therefore w = ( (z /h ) ( z /h ) ) u Note: Moment (M) = - EI w Shear force = M = - EI w Bracing Building is supported laterally by: 0 1) Bending stiffness of I-beams in the and directions ) Axial stiffness of slender rod braces in the direction Mass Tae weight (w)of roof as 0 lb/ft and calculate the mass (m) East-West Direction () North-South direction () w 00 0 m 6.6lb-sec /in = 0.066 ip-sec /in g 86. Note: Acceleration of gravity (g) = 86. in/sec
h = 1 First version: September 18, 00 (010 update) E s = 9,000 si (Steel Young s Modulus) 0 Steel I-Beam columns (Section a-a) x y (W8x Steel I-beam) I x = 8.8 in I y = 18. in Note that I y is much smaller than I x Brace (1 in diameter circular bar) Cross sectional area A = 0.785 in Rigid roof a a a a a a a a East-West Direction () x y North-South direction () In -direction 1EI h 1 9x10 8.8 1x1 x 8.58 ips in 5 September 18, 00 (010 update) -direction Equation of motion: m u u cu u 0 where or u g c ζc cr ζ m u z u u 0 g and z Maybe 1.-1.5% for steel where Note: and n() = sqrt (8.58 / 0.066) = 8.76 radians/sec f n() = 8.76 / ( x.18) =.58 Hz (cycles/sec) T n() = 1/.58 = 0. seconds 6
In -direction 1EI h 1 9x10 18. 1x1 y 8.5 ips As can be seen, lateral stiffness of the columns in the direction is much lower than that in the direction. The braces will change this situation dramatically. Thus, we will rely on brace stiffness since column stiffness is relatively small and not intended for lateral support (only to carry vertical load). Laterally, f s = brace u, or brace = f s / u in From geometry, f s = p cos q and u = (d / cos q resulting in brace = (p/d) cos q In the brace, axial stress is related to axial strain by brace q u p f s (p/a) = E s (d/l) (A is brace cross-sectional area) so that (p/d) = (AE s /L), and therefore brace = ( AE s / L ) cos q q From the building geometry cosθ 0 1 0 0.8575 10 0.8575 59.8 L = sqrt (0 + 1 ) =. ft 0.785 9 and brace ips in.1 Braces are slender in this case, and therefore only provide added stiffness when subjected to tensile force (the braces sag or bucle when in compression. As such, only two braces will be providing lateral stiffness at any given time. As such, (bracing) 59.8 119.6 ips u u cu u 0 g in Since the stiffness due to bracing is much larger than that due to the columns, we will only rely on the bracing for stiffness in this direction: m or, u u z u u 0 g where Note: and now, n() = sqrt (119.6 / 0.066) = 50.6 radians/sec f n() = 50.6 / ( x.18) = 8.06 Hz (cycles/sec) T n() = 1/ 8.06 = 0.1 seconds 8
Question: Will all braces in the frame be effective at the same time? Euler Bucling Load http://en.wiipedia.org/wii/bucling : EI Ncr L, L = L = Effective Bucling Length πr π 0.5" I I 0.09 in 1 L.ft.ft 79.6 in 1 N cr 9000si 0.09in 79.6in (79.6in) 0.179 ips 179lbs Answer: When the brace experiences compression, bucling load is minimal. This is why, we used only two braces at a time in our calculations (of the installed braces) 9 Bucling Restrained Braces (BRBs) http://jiano.typepad.com/photos/un categorized/brb_0.jpg 5
Bracing Systems (a) Diagonal, b) Chevron, and c) V-Braced 11 1 6
Testing http://peer.bereley.edu/~yang/neeszipper/summary.html http://nees.buffalo.edu/projects/zipperframes/collaboratoryresearchgt-ucb-ucb-ub-usf.pdf 7
Energy Dissipation Devices Viscous Dampers From Publication by Alessandro Martelli and Massimo Forni 8