#### Installation — business terrible - 1 part

September 8th, 2015

Typical (economical) steel gravity floor framing systems implement simply-connected beams and girders. This system, theoretically, has no inherent robustness because if an interior column is lost, there are no moment resisting connections to help span the compromised column and a theoretical mechanism immediately forms. However, the scenario generated by an interior column being rendered ineffective results in activation of two-way membrane action in the composite-steel concrete floor framing system and two-way flexure/catenary grillage action in the structural steel framing. There is a synergy between these two component systems that has only recently been studied in relation to fire (Allam et al. 2000; Bailey et al. 2000; Burgess et al. 2001; Cai et al. 2002; Huang et al. 2000a; Huang et al. 2000b; Huang et al. 2003a; Huang et al. 2003b). In addition, the beams and girders do indeed have connections at their ends that support not only tension forces (Owens and Moore 1992), but bending moments as well (Astaneh-Asl et al. 1989b; Liu and Astaneh-Asl 2000a; Liu and Astaneh-Asl 2000b; Owens and Moore 1992; Wales and Rossow 1983).

In the present analysis, a deformation compatibility approach is used in conjunction with two separate static analyses: the first considering two-way membrane action in the slab; and the second considering two-way-grillage catenary/flexure action in the steel framing. These two analysis components are described in the schematics in Figures 2 and 3.

Figure 2 Two-Way Membrane Action Resulting from Ineffective Interior Column. |

Ineffective Interior Column

Two-Way Catenary and Flexural Forces in Beams and Girders

Figure 3 Two-Way Catenary/Flexure Action Resulting from Ineffective Interior Column.

As the interior column is rendered ineffective, the slab and grillage of steel members are forced to deform in a compatible manner and they both resist vertical deformation to the extent that their strength allows. The two-way membrane behavior in the slab is assumed to follow the theory described previously. Two way grillage (catenary/flexure) behavior in the steel framing can be computed using nonlinear structural analysis. These two theories can be used together to evaluate the robustness present in the typical interior 30-ft by 30-ft simple structural steel framing bay.

The framing connections that are assumed are considered flexible and are most commonly fabricated as double web angles (and sometimes referred to as web cleats). In order to assess the capabilities of double angle connections in facilitating the 3D grillage behavior, the web cleat moment capacity, tension capacity, and shear capacity needed to be determined. This process can be started by looking at the web cleat connection as being composed of bolt elements as shown in Figure 4.

Figure 4 Web-Cleat to Bolt Element Transformation. |

Researchers have been studying methodologies for determining pure-moment and tension capacities of bolted angle connections for quite some time (Astaneh-Asl et al. 1989a; Astaneh-Asl et al. 2002; Astaneh-Asl et al. 1989b; DeStefano and Astaneh-Asl 1991; DeStefano et al. 1991; DeStefano et al. 1994; Liu 2003; Liu and Astaneh-Asl 2000a; Liu and Astaneh-Asl 2000b; Shen and Astaneh-Asl 1999; Shen and Astaneh-Asl 2000; Wales and Rossow 1983).

The present study uses the approach of Shen and Astaneh-Asl (2000) and Liu and Astaneh-Asl (2000b) to develop nonlinear tension and compression behavior for bolt elements. These bolt elements can then be assembled to form web cleats whereupon moment-rotation behavior or tension/compression response of the connection can be developed. A trilinear tension-deformation response for the bolt element is derived using the procedure suggested by Liu and Astaneh-Asl (2000a); Liu and Astaneh-Asl (2000b); Shen and Astaneh-Asl (1999) and Shen and Astaneh-Asl (2000) with slight modifications. The compression and tension response models are shown in Figure 5.

Figure 5 Double Angle Bolt Element Response: (a) Tension and (b) Compression. |

Three characteristic points on the tension response are generated using procedures recommended by Shen and Astaneh-Asl (2000) with slight modification. Point (PT VST 1) is defined using the yield moment in the legs of the angle. The initial stiffness, KTJ, is essentially the linear elastic stiffness of the bolt element considering bending of the legs perpendicular to the beam web and the axial extension of the leg parallel to the beam web. Point (PTZ,8T2) corresponds to the plastic mechanism capacity of the angle legs perpendicular to the beam web. The post-yield mechanism stiffness is defined as KT2. The final point on the tension-deformation response is (PTU, STU). This point corresponds to the ultimate loading for the bolt element exclusive of bolt tension rupture or bolt shear rupture. It is defined through consideration of the angle legs perpendicular to the beam web forming catenary tension between the bolts in the support and the legs parallel to the beam web. The tension in the catenary at this ultimate loading is taken to be the loading corresponding to fracture on the net area through the angle leg perpendicular to the beam web. The final stiffness in the response is defined as KT3.

The catenary tension force may or may not be able to form as a result of limits states being exceeded and therefore, a third point (PT3,ST3) is defined. The loading, PT3, is defined through consideration of the following bolt-element limit states;

• catenary tension fracture in the angle legs perpendicular to the beam web;

• tear-out bearing failure of the bolts in the beam web;

• tear-out bearing failure of the bolts in the angles;

• tension fracture of the bolts including prying action (Thornton 1985);

• tension fracture of the bolts excluding prying (superfluous);

• shear fracture of the bolts.

The yield point on the bolt element compression-deformation response (PC1,SC1) is defined by considering three strength limit states;

• yield in the angle legs parallel to the beam web;

• yielding in the beam web;

• shear fracture of the bolts.

The ultimate loading capacity of the bolt element in compression is defined through consideration of the following strength limit states; [6]

Figure 6 Bolt Element Tension and Compression Response for L4x3.5 Double Angles: (a) W18x35; and (b) W21x68. |

The tension-deformation response varies considerably with beam shape and angle thickness. This is a byproduct of the varying limit states considered in the computations. For example, when thin angles are considered, the catenary tension action is allowed to form and rupture of the angle legs is the controlling limit state. However, as the angles get thicker, other limit states control the behavior. This is indicated by the “capping” of the tension forces in the 5/16, 3/8, and 1/2-inch angle thickness in the W18x35 beam shape and the 3/8 and 1/2-inch angle thickness with the W21x68 girder shape. The compression – deformation response is consistent indicating that the limit states controlling strength are consistent as well.

The bolt element ultimate strengths can be used to contribute to the determination of the tension capacity of the double angle connections through simple summation of the bolt element tension strengths in any given connection. The tensile capacity of the double-angle connection is determined through consideration of two additional limit states beyond those assumed for the bolt elements (Foley et al 2006):

• shear rupture of the bolts;

• tension fracture of the bolts including prying;

• block shear rupture in the angle legs parallel to the beam web;

• block shear rupture in the beam web;

• bearing tear-out failure in the angle legs parallel to the beam web;

• bearing tear-out failure in the beam web;

• catenary tension rupture in the angle legs perpendicular to the beam web.

The pure moment capacity of the double web-angle connection is determined using the bolt element tension – and compression-deformation response parameters described previously. The pure moment condition is defined by the deformation compatibility and internal equilibrium. The process for determining the pure moment capacity of the connection begins with defining the tension and compression response for each bolt element in the connection. A controlling state of deformation in the extreme tension angle, or extreme compression angle is assumed. These deformations are taken from the appropriate angle force-deformation curves. The connection rotation angle is then varied until the summation of all forces determined using the bolt element response curves sum to zero. This corresponds to the pure moment capacity of the connection. It should be noted that this process is iterative and the compression or tension deformation limit states may control the behavior. Details of the procedure can be found in (Foley et al. 2006).

The shear strength of the double angle connection given the beam shape chosen can be determined using the AISC Manual (AISC 2001). It should be noted that unfactored strengths were utilized and therefore, all manual-obtained strengths were divided by 0.75. The shear strengths for the double angles and beam shapes considered assume: Lev -1.5′; Leh -1.5′; ф~ 1.0; and 3/4” A325N bolts in STD holes. No consideration of expected strengths of the material in defining the shear strength was given.

The beams in the grillage are assumed to be W18x35’s and the girders are W21x68’s. From the AISC-LRFDM (AISC 2001), the W18 sections can support 3-5 rows of bolts, while the W21 sections can support 4-6 bolt rows with traditional spacing and end distances. Therefore, only these numbers of bolt rows were considered. Double-angle connections alone have a tensile capacity that ranges from 0.1-0.30 of the squash load of the cross-section (Foley et al. 2006). These are fairly significant tensile capacities (if taken as cumulative over all beam and girder members within the 3D system. The loading capacities are consistent with those found in testing by (Owens and Moore 1992). The moment capacities are very low, however (Foley et al. 2006). They range from 0.05 – 0.20 of the plastic moment capacity of the beam cross-section. This is consistent with the strength portion of the definition of a flexible connection (AISC 2005).

Bilinear moment-rotation response and axial load-extension response curves can be generated for the double angle connections using the bolt element response shown in Figure 5. Compression response characteristics are only used for defining moment rotation response. The connections in the grillage are not expected to go into compression in the ineffective column scenario considered. Tension secant stiffness for the bolt element, kBE , is defined using point (ST2,PT2) on the tension-deformation response. The tensile capacity of each bolt element in the double-angle connection then contributes to the tensile and moment capacity of the connection. The bilinear tension-deformation response of the bolt element is then characterized by the secant stiffness and the bolt element tensile capacity, PT 3 .

The rotational and axial stiffness of the web-cleat connections are estimated using the magnitudes of the bolt element secant stiffness. In the case of axial tension, the axial stiffness of the double angle connection is simply the sum of the stiffness of each bolt element in the web cleat,

n

kBE, (5)

/=1

In general, if the bolt element stiffness, kBE, is known and there is nb bolt elements in the web cleat connection, the rotational stiffness can be computed as (Foley et al. 2006),

nb-1

*,=£ /2 ‘{kBE ■ s2 ) (6)

1=1

where s is the pitch of the bolt elements (taken as a constant value of 3 inches).

The axial stiffness and flexural stiffness of the web cleat connections can be defined as a function of the axial rigidity and flexural rigidity of the connected member. This is mathematically defined as,

AE

Ks = a, s— (7)

FI

Ke=ae— (8)

The rotational stiffness of the web-cleat connections are well below the stiffness limit corresponding to flexible connections (AISC 2005) given by ae – 2 . The majority of the rotational stiffness multipliers are in the range; 0.05 <ae< 1.50 (Foley et al. 2006). One exception is the 5 bolt arrangement in a W18x35 beam member. The axial stiffness multiplier for the majority of the connection arrangements lies in the range 0.10 <ae< 1.8 With the 5-bolt connection in the W18x35 member giving as~ 2.3.

The analysis begins by computing the capacity of the concrete-steel composite slab system acting as a two-way membrane using equations (1) through (4). The steel deck is assumed to be 2VLI22 (Vulcraft 2005) and 40% of the cross-sectional area is assumed to be effective as tensile reinforcement (Foley et al.

2006). Welded-wire-mesh is assumed in the concrete deck: 6×6-W1.4xW1.4 (shrinkage and temperature reinforcement). When the interior column looses effectiveness, the concrete slab panel is 60-feet by 60- feet. The membrane capacity of the concrete slab-steel deck system is approximately 50-psf at 26.2 inches of vertical deflection at the center of the panel (Foley et al. 2006). This magnitude of vertical deflection corresponds to an approximate rotational demand of 0.073 radians, which is well below the limit of 0.21 radians (GSA 2003). It should also be noted that the rotation computed here is a total rotation (elastic plus plastic components). Therefore, the magnitude computed is conservative. The tension force in the steel deck running perpendicular to the in fill beams is approximately 566 lbs/in (Foley et al. 2006) along one edge of the panel.

The capacity of the steel grillage is then computed. A structural model for the steel floor framing system was developed for use in MASTAN2 (Zieman and McGuire 2000). A schematic of the analytical model is shown in Figure 7.

Figure 7 Steel Grillage Model Schematic (System 1) Illustrating Axial and Moment Connection Modeling for Nonlinear Analysis. |

All members are modeled using multiple elements: in-fill beams are modeled using 10 elements and girders are modeled using 9 elements. The in-fill beams were modeled using 4 analytical segments. Two segments (i. e. 1/2 of the beam length) were centered on the beam mid-span. The end 1/4 lengths of beam were subdivided into 4 additional segments to facilitate connection modeling. Therefore, all in-fill beams contain end segments that are 1/16th of their span. The end segments in the girders (at column supports and interior column location) were broken down into 4 segments yielding end connection segments of 1/12th the girder span.

The end connections were modeled in the analytical segments of the beams and girders located immediately adjacent the fixed supports, the supporting girders, and the interior column. The connection rotational stiffness, Kg, was input using the built-in capability. The connection moment capacity was interjected into the analytical model by adjusting the beam or girder’s plastic moment capacity to ■ Zx.

The axial loading characteristics were included in a slightly different manner. MASTAN2 does not allow axial spring characteristics to be directly modeled. The cross-sectional areas of the beam or girder in the end connection segments were defined to be r/P ■ Ag. This reduction in cross-sectional area also

created implied linear spring stiffness in this isolated region of the beam equal to as ■ AE/L = r/P ■ AE/L.

The method of modeling connections creates a “stub member” that has an axial capacity and a moment capacity that is the same as the connection intended. Three systems with varying connection characteristics were considered (Foley et al. 2006):

System 1 System 2 System 3

rjM = 0.10 a0 = 0.50 rjM = 0.50 a0 = 5.0 rjM = 0.30 a0 = 2.0

rjP = 0.20 as = 0.20 rjP = 0.30 as = 0.30 rjP = 0.30 = 0.30

System 1 has strength and stiffness characteristics typical of web-cleat connections used in structural steel floor framing systems. System 2 has strength and stiffness characteristics typical of partially restrained beam-to-girder connections (Rex and Easterling 2002) and there is long-standing use of partially restrained girder-to-column connections. The axial strength and stiffness were increased slightly from that of System 1. A third system was considered. This system had a better balance between axial capacity and moment capacity than system 2. The axial strength and stiffness for the connections in system 3 were left the same as those in system 2. The bending strength and stiffness of the connections were reduced to a level slightly above that in System 1 and below that in system 2. The moment and axial strength characteristics are consistent with web cleat connections that are relatively thick (compared to the typical thickness used) and a number of bolt rows that fills up the beam and girder web (Foley et al. 2006).

MASTAN2 then uses these pieces of information to create an interaction (yield) surface of the form shown in Figure 8 for three systems considered.

Figure 8 Member and Connection Interaction Surfaces for Connected Member and Three Grillage Systems (connection characteristics vary). |

It should be noted that minor-axis bending is assumed to have a connection capacity that is equal to the minor axis moment capacity of the members and the connection stiffness in the minor-axis direction is infinite relative to the flexural rigidity of the connected beam (i. e. the connection is fully-restrained).

Each floor system is evaluated independently under the assumption that it carries its own loading. The slab system was determined previously to be capable of supporting approximately 50 psf through membrane action. The total unfactored live loading used for design of the system is: 80-psf dead loading; and 50 psf office occupancy live loading. The total point in time live loading that can be assumed present at the time a column is rendered ineffective can be computed as (GSA 2003);

qp i t. = 1.0(80 psf) + 0.25(50 psf) = 93 psf

The steel grillage will then be required to carry the following superimposed loading (with a deformation that is compatible with the slab membrane);

qgrillage = Pdyiam ■ (l-0D + 0.25L) – 50 = Piynam ■ (93psf) – 50psf

At pseudo-static loading levels (fldynam = 2.0) prescribed in the GSA Guidelines (GSA 2003), the

grillage will need to support a uniformly distributed loading of 136 psf. However, this assumes that the supporting column is “vaporized”. Furthermore, former studies (Liu et al. 2005; Marchand and Alfawakhiri 2004; Powell 2005) and relatively recent research (Foley et al. 2006) have shown that the multiplier commonly used to simulate dynamic loading can vary considerably. If the supporting column is not “vaporized”, but simply compromised (i. e. it still has a fraction of its initial load capacity), then one might argue that the point-in-time loading alone needs to be carried [p<ynum = 1.0) without dynamic multiplication. Therefore, in this case, the grillage must support 43 psf superimposed loading.

The MASTAN2 model shown in Figure 7 was analyzed using 2nd order inelastic analysis and a reference superimposed loading on the steel grillage of 108 psf. The load deformation response for the three systems is shown in Figure 9.

Figure 9 Load Deformation Response of Three Grillage Systems Considered. |

The load deformation response of system 1 indicates that there is a very early transition from flexural behavior to catenary behavior in the grillage. The connection strengths and stiffness result in the crosssections at the ends of the members reach the yield surfaces very early in the response and the large displacements result in catenary tension in the grillage forming. This transition is exhibited by the shallow yield plateau-like response and subsequent stiffening behavior. The applied load ratio that results in deformations compatible with the membrane displacement computed earlier (26 inches) is 0.46. This indicates that the capacity of the system (both slab and grillage) is;

qcap = 0.46 (108) + 50 «100 psf

Therefore, System 1 can definitely support the point in time live loading and there is some reserve for dynamic amplification: fldynam = 100/93 = 1.08 . If one were to assume that the system could continue to

deflect without membrane reinforcement in the slab rupturing, or the anchorage of this reinforcement being compromised (e. g. deflection to approximately 30 inches), the membrane capacity would increase and the catenary capacity of the grillage could increase. This increase is shown in Figure 9 at ALR = 0.52. This would result in the system capacity moving upward to,

qcap = 0.52 (108) + 50 * 106 psf

and the dynamic multiplier would naturally increase as well to /3dynam = 106/93 = 1.14 . One should note

that shrinkage and temperature welded wire fabric reinforcement was assumed as well as 22-gauge steel deck. Greater capacities can likely be attained if thicker deck is used and mild-steel reinforcement rather than welded wire mesh (Foley et al. 2006).

At 26 inches of vertical displacement, the total rotation over the beam and girder span of 30 feet was computed previously as approximately 0.07 radians. This is very close to the plastic rotational limit of

0. 06 radians recommended for web-angle connections (FEMA 2000a). However, the present rotational demand is “total” and the plastic demand will likely align itself close to this limit. Therefore, the rotational demands at the level of loading considered are not likely to cause rupture of the connections.

The same reference loading was applied to the steel grillage of system 2. The load deformation response of the grillage system 2 is also shown in Figure 9. It is interesting to note that the catenary (stiffening) response is not present. The reason for this is that a plastic mechanism (flexural) forms at an applied load ratio of 0.58 with vertical deformation slightly less than 5 inches. This amount of vertical deformation is not sufficient to activate the geometric stiffness for the members in the floor system. In other words, analytically, catenary action is not allowed to form and the system numerically “fails”. It is understood that there will be a conversion to catenary action once the mechanism forms, but the structural analysis is not able to consider this transformation because the tangent stiffness matrix of the system is singular at the instant this group of beam mechanisms forms.

The number of hinges that form in System 2 and the loading range over which they form is much less than that of System 1. One would like to have a system where there is a significant number of hinges forming so that full advantage of the structural indeterminacy and load redistribution is taken. When the hinges form over very short loading ranges, there is less redundancy and toughness in the system. The significantly smaller deformation in System 2 at the formation of the collapse mechanism would indicate that the grillage will form a bending moment collapse mechanism first with subsequent reliance on backup capacity catenary action after significant vertical deformation.

Experimental rotations attained by Rex and Easterling (2002) for the partially-restrained beam-to – girder connections were reported to be on the order of 0.05 radians. If one were to rely on catenary action after the flexural mechanisms occurs, the vertical deformations in the system would likely rapidly increase to those found in the first system (approximately 26 inches). As a result, even though the flexural mechanism forms early at 5 inches of deformation there will need to be an additional 21 inches of deformation in the grillage needed to activate catenary action. As a result, the rotational demands on these connections are likely to be on the order of 0.07 radians. It is unclear if the PR beam-to-girder connection can support his level of rotational demand without fracture.

The axial stiffness and strength of the connections in system 3 are consistent with those of system 2 and therefore, it is expected that the catenary behavior will be the same in the two systems once it is activated. The load deformation response of system 3 is shown in Figure 9. After the formation of the flexural collapse mechanism in system 2, it is likely that the steel grillage will need to abruptly accumulate an additional 20 inches of deflection in order to reach the catenary tension stiffening that comes from the contribution of geometric stiffness. This behavior is analogous to snap-through behavior in arches and is schematically indicated in Figure 9. It should be noted that the response of system 3 indicates that system 2 will indeed be able to reach the same load carrying capacity of system 3, but it is not economically advantages to provide additional bending moment capacity and stiffness when there is no enhancement in load carrying capacity. Furthermore, dynamic snap-through behavior may have adverse ramifications with regard to system integrity and toughness.

System 3 is capable of supporting an applied load ratio of 0.65 at 26 inches of vertical deformation, which is compatible with the deformations needed for the slab system membrane to support 50 psf. This reveals that a relatively economical (simple-framing) system can support the following superimposed floor loading;

qcap = 0.65(108) + 50 * 120 psf

This magnitude of loading reveals that this system can allow for a dynamic amplification factor of Pdynam = 120/93 = 1.30.