Question 15.CS.1: The steel-frame building under construction shown in CS Phot...
The steel-frame building under construction shown in CS Photo 15.1 illustrates the floor systems typically used in such structures, where each floor consists of a network of steel beams and girders that support a reinforced concrete deck. CS Photo 15.2 details such floor framing prior to the placement of the concrete deck, showing that the beams are supported by the girders, and in turn the girders are supported by the columns. Because the beams are web-connected only, these members are normally treated as being simply supported (see Case Study 4.2). For this case study, we will assume that the beams are W410 × 60, 8.8 m in length, and evenly spaced at 2.2 m. In design, apportioned to each beam will be that part of the concrete deck having a width equaling the beam spacing; this is the portion of the deck that is considered to be tributary to the beam. For the assumed beam spacing of 2.2 m, this tributary width is depicted in CS Fig. 15.1. (Normally concrete floors like this are cast upon a ribbed metal deck that supports the concrete until it hardens; for the purposes of this case study, it is assumed that the metal deck does not alter the following considerations.)
In designing any system or structure, it is of primary importance to provide for the adequate support of all anticipated loads, and to do so in an economical fashion. But there will often be some ancillary requirements to satisfy as well. These could include certain functional or operational conditions, referred to as serviceability requirements. For example, in building structures like the one considered here, there will be limits on the live load deflection of floors to ensure that nonstructural elements (such as plaster ceilings) are not damaged or adversely affected by the deflections. It might also be necessary to ensure that user comfort, or perhaps even the operation of highly sensitive equipment, is not impaired by the excessive vibration of floors under ordinary usage. The design guide Vibrations of Steel-Framed Structural Systems Due to Human Activity† presents methods for making such assessments, and a key parameter used in these evaluations is the natural frequency of beams, f_b (in cycles/s), where each beam supports a tributary portion of the concrete deck. This design guide provides the following approximation for determining the beam natural frequency:
f_b=0.18\sqrt{\frac{g}{\Delta }} (1)
Where g = acceleration due to gravity
Δ = midspan static deflection of simply supported beam due to its actual supported weight
From Concept Application 15.2, this midspan deflection Δ was found to be
\Delta =\frac{5wL^4}{384E_s I_t} (2)
where, in the situation of a steel beam supporting a tributary portion of concrete deck,
w = uniformly distributed weight per unit length of the actual dead and live load
L = beam span
E_s = modulus of elasticity of steel
I_t = moment of inertia of the transformed section (converting concrete to steel)
Using the assumed floor geometry and beam section, and also assuming that the modulus of elasticity for the concrete and steel used is E_c = 25 GPa and E_s = 200 GPa, respectively, and that the actual loads acting on each beam are w_{\text{dead}} = 7 kN/m and w_{\text{live}} = 1 kN/m, let’s determine the natural frequency of vibration for the beams.
STRATEGY: Following the process given in Sec. 11.3, we can transform the concrete and steel cross section given in CS Fig. 15.1 into an equivalent section of all steel. Then, using the transformed section and the given approximation for the natural frequency of vibration of a beam, we can determine this natural frequency.
†See T. M. Murray, D. E. Allen, E. E. Ungar, and D. B. Davis, Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2nd ed., Steel Design Guide 11, American Institute of Steel Construction, Chicago, IL, 2016.
MODELING:
Transformed Section. First compute the ratio
n=\frac{E_c}{E_s}=\frac{25 GPa}{200 GPa}=\frac{1}{8}(Note that because we are transforming the concrete into an equivalent amount of steel, the concrete modulus is placed in the numerator and the steel modulus is placed in the denominator.) The width of the steel that replaces the original concrete portion is obtained by multiplying the original width by n:
n (2.2 m)= \frac{1}{8} (2.2 m)= 275 mm
Neutral Axis. CS Fig. 15.2 shows the transformed section of all steel. From Appendix D, the W410 × 60 has the properties
d = 406 mm A = 7610 mm² I_x = 216 × 10^6 mm^4
The neutral axis passes through the centroid of the transformed section, and is located at
\bar{Y}=\frac{\Sigma \bar{y} A}{\Sigma A}=\frac{(253 mm)(275 mm \times 100 mm)+0}{(275 mm \times 100 mm)+\left(7610 mm^2\right)}=198.2 mmCentroidal Moment of Inertia of Transformed Section. Using CS Fig. 15.2 and the parallel-axis theorem,
I_t=\frac{1}{12}(275)(100)^3+(275 \times 100)(253-198.2)^2+216 \times 10^6+(7610)(198.2)^2I_t=620.4 \times 10^6 mm^4
ANALYSIS:
Midspan Static Deflection of Beam. Using Eq. (2) and the given data,
w=w_{\text {dead }}+w_{\text {live }}=7 kN / m+1 kN / m=8 kN / m\Delta=\frac{5 w L^4}{384 E_s I_t}=\frac{5\left(8 \times 10^3 N / m\right)(8.8 m)^4}{384\left(200 \times 10^9 N / m^2\right)\left(620.4 \times 10^{-6} m^4\right)}=5.035 \times 10^{-3} m
Beam Natural Frequency. Using Eq. (1) and g = 9.81 m/s²,
f_b=0.18 \sqrt{\frac{g}{\Delta}}=0.18 \sqrt{\frac{9.81 m / s^2}{5.035 \times 10^{-3} m}}f_b = 7.95 cycles/s
REFLECT and THINK: We have just estimated the natural frequency of the beams, but the natural frequency of the overall floor will also be affected by the vibration characteristics of the girders that support the beams. Following the methods given in the referenced design guide, the girder natural frequency f_g can be determined in a manner somewhat similar to that used here to evaluate the beam natural frequency f_b. The floor natural frequency of vibration f can then be estimated through the relationship†
\frac{1}{f^2}=\frac{1}{f^{2}_{b}}+\frac{1}{f^{2}_{g}}†See T. M. Murray, D. E. Allen, E. E. Ungar, and D. B. Davis, Vibrations of Steel-Framed Structural Systems Due to Human Activity, 2nd ed., Steel Design Guide 11, American Institute of Steel Construction, Chicago, IL, 2016.
