Learning Goal:

To calculate a maximum live load for a beam

and the minimum size for its bearing plate

assuming that the average shear stress in the

beam and the average normal stress on the

foundation are the limiting design conditions,

while using the limit state design method.

The design procedure for a structure is a bit

more complicated than just determining the

loads, picking a geometry, and then verifying

that the design can carry the applied loads.

There is a fairly large amount of uncertainty in

the loads applied to a structure. These variable

loads come from things like wind, snow, people,

and equipment. There also is uncertainty in the

material properties.

When using limit state design, the uncertainties

in the loads and material properties are

considered separately. The design requirements

are expressed as an inequality: $P_n{}_i{R}_i,$

where $$ is a parameter to account for factors

like the material strength uncertainty and the

failure modes, $P_n$ is the nominal strength, $R_i$ is

an applied load, and ${}_i$ is a factor for the

uncertainty of a load. Two common load types

used are dead loads with ${}_D=1\mathrm{.}4$ and live

loads with ${}_L=1\mathrm{.}6.$

As shown, a portion of a structure contains a

$W6412$ wide flange beam bearing on a

concrete foundation with a bearing plate. Only

the thin vertical part $($web$)$ of the beam is

considered when sizing a beam to carry a shear

load. A $W6412$ beam has a web area of

$A_w=8\mathrm{.}874c{m}^2.$

$($Figure $1)$

Part A $-$ The maximum allowed live load

Suppose the beam is carrying a known shear load of $R_D=21kN.$ In this particular situation, the resistance

factor is $=0\mathrm{.}9$ for the shear load, and the nominal shear stress is $204$MPa. What is the magnitude of the

maximum live load $($in addition to $R_D)$ that can be supported in shear by this beam?

Express your answer to three significant figures with the appropriate units.

View Available Hint$($s$)$

Correct

Part B $-$ The minimum required plate size

The beam$-$plate system is resting on a concrete foundation so that the beam load is converted to a normal load

on the foundation. Typically, a bearing plate is required to distribute the load from the beam onto the foundation

so that the foundation does not crack due to locally high stresses. This situation can be modeled as a rigid plate

applying a uniform stress to the concrete. In this case, the resistance factor is $=0\mathrm{.}6$ and the maximum

allowed average compressive stress on the concrete is $21\mathrm{.}25$MPa. Use the maximum load found in Part A to

determine the minimum area required for the bearing plate.

Express your answer to three significant figures with the appropriate units.

Figure