## Section Aa

FIGURE 5.41 Wall-bearing beam anchored with government anchor.

group of fasteners transmits the shear from the splice plates to the web on that side. This shear acts through the center of gravity of that fastener group. The two transmitted shears, then, form a couple 2Ve, where e is half the distance between the shears. This moment must be taken by the fasteners in both groups. In the design of a splice, however, it generally is simpler to work with only one side of the joint. Hence, when the forces on one fastener group are computed, the fasteners should be required to carry the shear V plus a moment Ve. (In a symmetrical splice, e is the distance from O to the splice.)

The classical elastic method assumes that in resisting the moment, each fastener in the group tends to rotate about O. As a consequence:

The reaction Pb of each fastener acts normal to the radius vector from O to the fastener

The magnitude of Pb is proportional to the distance r from O.

The resisting moment provided by each fastener is proportional to its distance from O.

The applied moment equals the total resisting moment, the sum of the resisting moments of the fasteners in the group,

These consequences result in the relationship

where M = applied moment, in-kips

Pb = load due to M on outermost fastener in group, kips J = sum of squares of distances of fasteners in group from center of gravity of group, in2 (analogous to polar moment of inertia) c = distance of outermost fastener from center of gravity, in

Hence, when the moment applied to a fastener group is known, the maximum stress in the fasteners can be computed from Eq. (5.13).

This stress has to be added vectorially to the shear on the fastener (Fig. 5.42). The shear (kips) is given by

where n is the number of fasteners in group. The resultant stress must be less than the allowable capacity of the fastener.

Depending on the fastener pattern, the largest resultant stress does not necessarily occur in the outermost fastener. Vectorial addition of shear and bending stresses may have to be performed for the most critical fasteners in a group to determine the maximum.

In computing the fastener stresses, designers generally find that the computations are simpler if the forces and distances are resolved into their horizontal and vertical components. Advantage can be taken of the fact that

where Iy = sum of squares of distances measured horizontally from centrr of gravity to fasteners, in2

Ix = sum of squares of distances measured vertically from center of gravity to fasteners, in2

### Ix and Iy are analogous to moment of inertia.

Fatigue need not be considered when calculating bolt stresses but should be taken into account in designing splice plates subjected to bending moments.

Example—AASHTO ASD. The 48 X 5/16 in A36 steel web of a plate girder in a highway bridge is to be field spliced with 7/s-in-diameter A325 bolts (Fig. 5.43). Maximum shear is 95 kips, and maximum bending moments are 700 and - 200 ft-kips. The moment of inertia Ig of the gross section of the girder is 14,670 in4. Design the web splice for A36 steel. With an allowable stress of 12 ksi, the shear capacity of the web is

One design criterion, then, is 0.75Vc = 0.75 X 180 = 135 kips. A second design criterion is the average shear:

. ABOUT NEUTRAL t FIGURE 5.43 Example of design of a girder-web splice.

This governs for the bolts in the web splice.

The gross moment of inertia of the web is 2,880 in4. From this should be subtracted the moment of inertia of the holes. With the bolts arranged as shown in Fig. 5.43 and area per hole = 1 X 5/i6 = 0.313 in2,

Ix = 2[(5)2 + (10)2 + (15)2 + (20)2]0.313 = 1500 X 0.313 = 470 in4

Hence the net moment of inertia of the web is 2880 - 470 = 2410 in4. Allowable bending stress for the girder is 20 ksi.

The maximum bending stress for a moment of 700 ft-kips and Ig = 14,670 in4 is, for the gross section,

700 X 12 X 24,875 ^ , . ft =-14670-= 14.2 ksi < 20 ksi—OK

and the average stress for design of the splice is Fb = (14.2 + 20)12 = 17.1 ksi > 0.75 X 20. Hence the girder design moment is 700 X 17.1/14.2 = 843 ft-kips. The net moment carried by the web then is 843 X 2880/14,670 = 166 ft-kips, to which must be added the moment due to the 3.25-in eccentricity of the shear. The total web moment thus is 166 X 12 + 137.5 X 3.25 = 2439 in-kips.

For determination of the maximum stress in the bolts due to this moment, J is needed for use in Eq. (5.13). It is obtained from Eq. (5.15) and bolt-hole calculations:

By Eq. (5.13) then, the load on the outermost bolt due to moment is

The vertical component of this load is

And the horizontal component is

By Eq. (5.14), the load per bolt due to shear is Pv = 137.5/18 = 7.63 kips. Consequently, the total load on the outermost bolt is the resultant

R = V(1.20 + 7.63)2 + (16.04)2 = 18.31 kips The allowable capacity of a 7/8-in A325 bolt in double shear is

2 x 0.601 x 15.5 = 18.6 kips > 18.31 kips The bolts and bolt arrangement are satisfactory.

For determination of the maximum stress in the splice plates, note that they are 43 in deep and must carry the moment Mw = 2439 in-kips. Try two plates, each 5/i6 in thick. For the gross section, the moment of inertia is

and for the net section,

I = 4141 - 0.3125 X 1(52 + 102 + 152 + 202)4 = 3203 in4

The stress on the gross section is

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Renewable energy is energy that is generated from sunlight, rain, tides, geothermal heat and wind. These sources are naturally and constantly replenished, which is why they are deemed as renewable. The usage of renewable energy sources is very important when considering the sustainability of the existing energy usage of the world. While there is currently an abundance of non-renewable energy sources, such as nuclear fuels, these energy sources are depleting. In addition to being a non-renewable supply, the non-renewable energy sources release emissions into the air, which has an adverse effect on the environment.

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