Problem: A structural engineer is designing a simple industrial warehouse. Recommend and justify the most appropriate structural steel shapes for:

1. The main roof beams spanning $30\text{ feet}$ ($9.1\text{ m}$).
2. The diagonal bracing in the walls to resist lateral wind loads.
3. A heavy column in a multi-story section supporting major axial gravity loads.

Solution:

1. Roof Beams: A W-Shape (Wide Flange) is most appropriate. Its I-shaped cross-section concentrates material in the flanges, maximizing the moment of inertia ($I$) and section modulus ($S$) about the strong axis, making it highly efficient at resisting flexural (bending) stresses.
2. Diagonal Bracing: An L-Shape (Angle) or a square/rectangular HSS (Hollow Structural Section) is recommended. Bracing carries primary axial forces (tension or compression). Angles are economical and easy to connect via gusset plates, while HSS provides excellent resistance to buckling in all directions.
3. Axial Column: A deep W-Shape (such as a W12 or W14 profile with thick flanges and web) or a heavy HSS section is recommended. Columns carry heavy axial compression, where resistance to buckling about both the strong and weak axes is critical.

Problem: A structural steel floor girder is subjected to a service dead load bending moment ($M_D$) of $15\text{ kip}\cdot\text{ft}$ and a service live load bending moment ($M_L$) of $30\text{ kip}\cdot\text{ft}$. Calculate the required flexural strength based on:

1. Allowable Strength Design (ASD) load combinations.
2. Load and Resistance Factor Design (LRFD) load combinations using the standard load factor combination $1.2D + 1.6L$.

Solution:

Problem: A structural steel roof girder is subjected to the following service gravity loads:

• Dead Load ($D$) = $20\text{ kips}$
• Roof Live Load ($L_r$) = $15\text{ kips}$
• Snow Load ($S$) = $25\text{ kips}$

Determine the governing required axial strength using both ASD and LRFD methodologies.

Solution:

Problem: A W14x90 steel column ($F_y = 50\text{ ksi}$) has a gross cross-sectional area $A_g = 26.5\text{ in}^2$. It is subjected to a factored axial compressive load $P_u = 850\text{ kips}$. The nominal compressive strength $P_n$ is determined to be $1050\text{ kips}$. Verify if the column is adequate to resist the load using LRFD.

Solution:

Problem: A batch of structural steel plates has the following chemical composition by weight:

• Carbon ($C$) = $0.18\%$
• Manganese ($Mn$) = $0.90\%$
• Chromium ($Cr$) = $0.05\%$
• Molybdenum ($Mo$) = $0.02\%$
• Vanadium ($V$) = $0.01\%$
• Nickel ($Ni$) = $0.04\%$
• Copper ($Cu$) = $0.15\%$

Using the standard International Institute of Welding (IIW) carbon equivalent formula: $\text{CE} = C + \frac{Mn}{6} + \frac{Cr + Mo + V}{5} + \frac{Ni + Cu}{15}$

Calculate the Carbon Equivalent ($\text{CE}$). If a $\text{CE}$ above $0.40\%$ indicates poor weldability requiring preheating, state whether preheating is required.

Solution:

Problem: A continuous structural steel bridge girder of length $L = 120\text{ feet}$ ($36.6\text{ m}$) is installed at an ambient temperature of $60^\circ\text{F}$ ($15.6^\circ\text{C}$). The extreme temperature range expected at the site is from $-10^\circ\text{F}$ to $110^\circ\text{F}$.

Using a coefficient of thermal expansion for structural steel $\alpha = 6.5 \times 10^{-6} /^\circ\text{F}$, calculate:

1. The maximum thermal contraction from the installation temperature.
2. The maximum thermal expansion from the installation temperature.
3. The total range of movement the expansion joints must accommodate.

Solution:

Problem: A set of structural drawings lists the following steel sections:

1. $W18 \times 50$
2. $L6 \times 4 \times 1/2$
3. $HSS8 \times 6 \times 3/8$

Explain exactly what each designation represents including all nominal dimensions and weights.

Solution:

1. $W18 \times 50$: Represents a Wide Flange (W-shape) section. The nominal depth of the member is $18\text{ inches}$, and the weight of the member is $50\text{ lbs per linear foot}$.
2. $L6 \times 4 \times 1/2$: Represents an Angle (L-shape) with unequal legs. The longer leg is $6\text{ inches}$, the shorter leg is $4\text{ inches}$, and the thickness of the legs is $1/2\text{ inch}$.
3. $HSS8 \times 6 \times 3/8$: Represents a Hollow Structural Section (HSS). It is a rectangular tube with outer dimensions of $8\text{ inches}$ by $6\text{ inches}$, and a wall thickness of $3/8\text{ inch}$.

Problem: A W16x31 floor beam spans $24\text{ feet}$ and is spaced at $8\text{ feet}$ on center. It supports a service live load of $80\text{ psf}$ ($3.83\text{ kPa}$). The moment of inertia of a W16x31 about its strong axis is $I_x = 375\text{ in}^4$, and the Modulus of Elasticity is $E = 29,000\text{ ksi}$.

If the building code limits the service live load deflection to $L/360$, verify if the beam satisfies this serviceability limit state.

Solution:

Scenario: A design firm is tasked with selecting the primary structural material for a new 40-story office tower in a highly active seismic zone. The building requirements specify maximizing open leasable floor space, achieving rapid speed of construction, and providing excellent ductility under seismic loading.

Solution: Structural steel is selected over reinforced concrete due to the following engineering justifications:

1. High Strength-to-Weight Ratio: Steel's high strength-to-weight ratio allows columns to have significantly smaller cross-sectional areas compared to concrete columns carrying the same load. This directly maximizes the usable/leasable floor area within the tower.
2. Construction Velocity: Prefabricated steel components are fabricated under tight tolerances in shops and quickly bolted or welded on-site, drastically accelerating the overall project schedule.
3. High Ductility: In seismic events, structural steel members exhibit exceptional ductility, allowing them to undergo large plastic deformations to absorb and dissipate seismic energy without catastrophic failure.

Scenario: An industrial chemical processing plant is being designed using exposed structural steel framing. The facility will house flammable organic solvents, and local building codes require a strict 2-hour fire resistance rating for all load-bearing structural frames to prevent catastrophic building collapse.

Solution: Although structural steel is non-combustible, it rapidly loses strength and stiffness at elevated temperatures (losing approximately $50\%$ of its room-temperature yield strength around $1,100^\circ\text{F}$ or $593^\circ\text{C}$).

The engineering team specifies intumescent paint as the passive fire protection system for all exposed structural steel columns and beams. Intumescent coatings provide the following benefits:

1. Aesthetic & Space Saving: It is applied thin, behaving like regular paint under normal conditions.
2. Reaction to Heat: When exposed to extreme temperatures during a fire, the coating undergoes a chemical reaction that causes it to swell/expand into a thick, highly insulating carbonaceous char layer.
3. Insulation: This expanded layer insulates the underlying steel from the direct heat, delaying temperature rise in the steel core and successfully maintaining its load-bearing capacity for the required 2 hours.