in one horizontal direction compared to their span lengths in the perpendicular direction.
Recalling from frame analysis that flexural stiffness is inversely related to span length, it is clear that the slab panels shown in Fig. 5-1 would be much stiffer in their shorter span direction than in the longer span direction. Thus, for any load applied to floor panels similar to those in Figs. 5-1a and 5-1c, a higher percentage of the load would be carried in the short span direction as compared to the long span direction. In a concrete floor system where the ratio of the longer to the shorter span length is greater than or equal to 2, it is common practice to provide flexural reinforcement to resist the entire load in the short direction and only provide minimum steel for temperature and shrinkage effects in the
Floor beams
Floor beams
(c) Floor plan with two intermediate floor beams.
(a) Floor plan with one intermediate floor beam.
(b) Section A–A.
Fig. 5-1
Typical one-way floor systems.
long direction. Such slabs are referred to as one-way slabsbecause they are designed to carry applied loads in only one direction. A floor system consisting of one-way slabs and supporting beams, as shown in Fig. 5-1, is referred to as a one-way floor system.
If the floor systems in Fig. 5-1 were modified such that the only beams were those that spanned between the columns, the remaining slab panel would have a long span to short span ratio of less than 2. For such a case, flexural reinforcement would be provided in the two principal horizontal directions of the slab panel to enable it to carry applied loads in two directions. Such slabs are referred to as two-way slabs. The analysis and design of two-way floor systemswill be discussed in Chapter 13.
Load Paths in a One-Way Floor System
Consider the idealized one-way floor system shown in Fig. 5-2. The floor system is not realistic, because it does not have openings for stairwells, elevators, or other mechanical systems. However, this floor system will be useful as a teaching tool to discuss load paths and the analysis of bending moments and shear forces in the various structural members.
To study load paths in a one-way floor system, assume a concentrated load is applied at the point pin the central slab panel of the floor system shown in Fig. 5-2. This concen- trated load could represent part of a uniformly distributed live load or dead load acting on a specified portion (e.g., 1 ft by 1 ft) of the floor area. The one-way slab panel is assumed to initially carry the concentrated load in the north–south direction to the points mandnon the two adjacent floor beamssupporting the one-way slab. The floor beamsthen carry the loads in the east–west direction to the points h, i, j, and kon the girdersthat support the floor beams.Girderis the name given to a primary support member (beam) that spans from col- umn to column and supports the floor beams. A schematic sketch of a slab, floor beam and girder system is given in Fig. 5-3. Girders normally have a total member depth that is greater than or equal to the depth of the floor beams that it supports. The final step on the load path for the floor system in Fig. 5-2 is the transfer of loads from the girders to the
24 ft
Girders Floor beams
N
W
Z Y
X
24 ft
28 ft 24 ft 28 ft
8 ft 8 ft 8 ft
p n
i j k
h m
Fig. 5-2
Load paths in a one-way floor system.
Fig. 5-3
Slab, beam, and girder floor system.
columns at W, X, Y, and Z. It should be noted that some of the floor beams in a typical floor system will connect directly to columns, and thus, they transfer their loads directly to those columns, as is the case for the floor beam between the columns at WandX.
Tributary Areas, Pattern Loadings, and Live Load Reductions
Floor systems in almost all buildings are designed for uniformly distributed dead and live loads, normally given or calculated in unit of pounds per square foot (psf). The symbol qwill be used to represent these loads with subscripts Lfor live load and Dfor dead load. Total dead load normally is composed of dead loads superimposed on the floor system as well as the self-weight of the floor members. Typical live load values used in design of various types of structures were given in Table 2-1. The analysis procedure for concentrated loads will be presented later in this section.
Floor beams typically are designed to resist area loads acting within the tributary areafor that beam, as shown by the shaded regions in Fig. 5-4. As discussed in Chapter 2, the tributary area extends out from the member in question to the lines of zero shear on ei- ther side of the member. The zero shear lines normally are assumed to occur halfway to the next similar structural member (floor beam in this case). Thus, the width of the tribu- tary area for a typical floor beam is equal to the sum of one-half the distances to the adja- cent floor beams. For a floor system with uniformly spaced floor beams, the width of the tributary area is equal to the center-to-center spacing between the floor beams. Unless a more elaborate analysis is made to find the line of zero shear in an exterior slab panel, the width of the tributary area for an edge beam is assumed to be one-half the distance to the adjacent floor beam, as shown in Fig. 5-4. After the tributary width has been established,
D A
E H
12 ft 12 ft
24 ft
24 ft 6 ft
6 ft 6 ft
6 ft 6 ft
28 ft 24 ft 28 ft
F G
B C
Fig. 5-4
Tributary areas for floor beams.
the area load, q, is multiplied by the tributary width to obtain a line load,w (lbs/ft or kips/ft), that is applied to the floor beam. This will be demonstrated in Example 5-1.
For one-way slabs, the width of the tributary area is set equal to the width of the analysis strip, which is commonly taken as 1 foot. Thus, the cross-hatched area in Fig. 5-5 represents both the tributary area and the width of the analysis strip for the continuous one- way slab portion of this floor system. The effective line load,w, is found by multiplying the area load,q, times the width of the analysis strip (usually 1 ft).
Pattern Loadings for Live Load
The largest moments in a continuous beam or a frame occur when some spans are loaded with live load and others are not. Diagrams, referred to as influence lines,often are used to determine which spans should and should not be loaded. An influence line is a graph of the variation in the moment, shear, or other effect at one particular pointin a beam due to a unit load that moves across the beam.
Figure 5-6a is an influence line for the moment at point C in the two-span beam shown in Fig. 5-6b. The horizontal axis refers to the positionof a unit (1 kip) load on the beam, and the vertical ordinates are the moment at Cdue to a 1-kip load acting at the point in question. The derivation of the ordinates at B, C,andEis illustrated in Figs. 5-6c to 5-6e.
When a unit load acts at B, it causes a moment of 1.93 k-ft at C(Fig. 5-6c). Thus, the ordi- nate at Bin Fig. 5-6a is 1.93 k-ft. Figure 5-6d and e show that the moments at Cdue to loads at C and Eare 4.06 and respectively. These are the ordinates at C andE in Fig. 5-6a and are referred to as influence ordinates. If a concentrated load of Pkips acted at point E, the moment at C would be P times the influence ordinate at E, or If a uniform load of wacted on the span A–D, the moment at Cwould bewtimes the area of the influence diagram from AtoD.
Figure 5-6a shows that a load placed anywhere between AandDwill cause positive moment at point C, whereas a load placed anywhere between DandFwill cause a nega- tive moment at C. Thus, to get the maximum positive moment at C, we must load span A–Donly.
Two principal methods are used to calculate influence lines. In the first, a 1-kip load is placed successively at evenly spaced points across the span, and the moment (or shear) is calculated at the point for which the influence line is being drawn, as was done M = - 0.90P k-ft.
-0.90 k-ft,
N
1 ft.
12 ft 12 ft
24 ft
24 ft
28 ft
28 ft 24 ft
Fig. 5-5
Width of analysis strip and tributary area for one-way slab strip.
(kip-ft)
Fig. 5-6
Concept of influence lines.
in Figs. 5-6c to 5-6e. The second procedure, known as the Mueller-Breslau principle[5-1], is based on the principle of virtual work, which states that the total work done during a virtu- al displacement of a structure is zero if the structure is in equilibrium. The use of the Mueller-Breslau principle to compute an influence line for moment at Cis illustrated in Fig. 5-6f. The beam is broken at point Cand displaced, so that a positive does work by acting through an angle change Note that there was no shearing displacement at C, so does not do work. The load,P, acting at Bwas displaced upward by an amount and hence did negative work. The total work done during this imaginary displacement was
so
(5-1) Mc = Pa¢B
ucb Mcuc - P¢B = 0
¢B
Vc
uc.
Mc
where is the influence ordinate at B. Thus, the deflected shapeof the structure for such a displacement has the same shape and is proportional to the influence line for moment at C. (See Figs. 5-6a and 5-6f.)
The Mueller-Breslau principle is presented here as a qualitative guideto the shape of influence lines to determine where to load a structure to cause maximum moments or shears at various points. The ability to determine the critical loading patterns rapidly by using sketches of influence lines expedites the structural analysis considerably, even for structures that will be analyzed via computer software packages.
Influence lines can be used to establish loading patterns to maximize the moments or shears due to live load. Figure 5-7 illustrates influence lines drawn in accordance with the Mueller-Breslau principle. Figure 5-7a shows the qualitative influence line for moment at B.
The loading pattern that will give the largest positive moment at Bconsists of loads on all spans having positive influence ordinates. Such a loading is shown in Fig. 5-7b and is referred to as an alternate span loading or a checkerboard loading. This is the common loading pattern for determining maximum midspan positive moments due to live load.
The influence line for moment at the support Cis found by breaking the structure at Cand allowing a positive moment, to act through an angle change The resulting deflected shape, as shown in Fig. 5-7c, is the qualitative influence line for The max- imumnegativemoment at Cwill result from loading all spans having negative influence ordinates, as shown in Fig. 5-7d. This is referred to as an adjacent spanloading with alternate span loading occurring on more distant spans. Adjacent span loading is the common loading pattern for determining maximum negative moments at supports due to live load.
Qualitative influence lines for shear can be drawn by breaking the structure at the point in question and allowing the shear at that point to act through a unit shearing dis- placement,¢,as shown in Fig. 5-8. During this displacement, the parts of the beam on the
Mc. uc. Mc,
¢B/uc
A
A B
B
B C
C
C
C
D E
E
F G H J K L
Qualitative influence line for moment at C.
Fig. 5-7
Qualitative influence lines for moments and loading patterns.