6.1 Structural Loadings and Load Paths

Structural Loadings and Load Paths

In structural engineering, the accurate determination of design loads and the understanding of how these loads travel through a structure are the most critical steps in ensuring safety and serviceability. This section covers the gravity and lateral load paths, standard design philosophies (Allowable Strength Design (ASD) vs. Load and Resistance Factor Design (LRFD)), and details the calculations of dead loads, live loads, snow loads, wind loads, and earthquake loads based on the Minimum Design Loads and Associated Criteria for Buildings and Other Structures (ASCE 7) standard.

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Gravity and Lateral Load Paths

A structural load path refers to the continuous, uninterrupted path that forces follow from their point of application to their final resistance in the ground. If a load path is interrupted, structural failure will occur at the point of discontinuity.

1. Gravity Load Path

Gravity loads act downward and are transferred through horizontal elements to vertical elements, and then to the foundation. A typical gravity load path flows as follows:

$$\text{Floor/Roof Slab or Deck} \rightarrow \text{Secondary Beams or Joists} \rightarrow \text{Primary Girders} \rightarrow \text{Columns or Bearing Walls} \rightarrow \text{Footings/Foundations} \rightarrow \text{Soil/Rock}$$

• Tributary Area ($A_T$): The surface area of a slab or deck that contributes load directly to a specific structural member. For a simply supported floor beam, the tributary area is bounded by the span length and the sum of half the distance to the adjacent beams on either side: $$A_T = L \times \left( \frac{S_1 + S_2}{2} \right)$$ Where $L$ is the member span, and $S_1, S_2$ are the spacings to adjacent members.
• Influence Area ($A_I$): The area over which an applied load can affect the force in a given member. For a beam, the influence area is twice the tributary area ($A_I = 2A_T$), representing the full panel area on both sides. For a column, it is four times the tributary area ($A_I = 4A_T$), representing the four surrounding panels.

2. Lateral Load Path

Lateral loads (wind and seismic forces) act horizontally and must be transferred to the foundation via the lateral force-resisting system (LFRS). The lateral load path flows as follows:

$$\text{Exterior Cladding/Facade} \rightarrow \text{Girts or Studs} \rightarrow \text{Floor/Roof Diaphragms} \rightarrow \text{Lateral Resisting Elements (Shear Walls/Braced Frames/Moment Frames)} \rightarrow \text{Foundations} \rightarrow \text{Soil/Rock}$$

• Diaphragms: Roof and floor slabs act as horizontal deep beams that distribute lateral forces to the vertical lateral force-resisting elements based on their relative stiffness. Rigid diaphragms distribute loads based on the relative rigidity of the vertical resisting elements, whereas flexible diaphragms distribute loads based on tributary width.

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Design Philosophies: ASD vs. LRFD

Structural design uses two primary methods to ensure that a member's strength exceeds the acting forces: Allowable Strength Design (ASD) and Load and Resistance Factor Design (LRFD).

ASD Load Combinations (ASCE 7)

ASD evaluates structures under nominal (unfactored) service loads. Key basic combinations include:

1. $D$
2. $D + L$
3. $D + (L_r \text{ or } S \text{ or } R)$
4. $D + 0.75L + 0.75(L_r \text{ or } S \text{ or } R)$
5. $D + 0.6W$
6. $D + 0.75L + 0.75(0.6W) + 0.75(L_r \text{ or } S \text{ or } R)$
7. $0.6D + 0.6W$
8. $D + 0.7E$
9. $0.6D + 0.7E$

LRFD Load Combinations (ASCE 7)

LRFD applies load factors to account for the probability and variation of ultimate loads. Key basic combinations include:

1. $1.4D$
2. $1.2D + 1.6L + 0.5(L_r \text{ or } S \text{ or } R)$
3. $1.2D + 1.6(L_r \text{ or } S \text{ or } R) + (L \text{ or } 0.5W)$
4. $1.2D + 1.0W + L + 0.5(L_r \text{ or } S \text{ or } R)$
5. $1.2D + 1.0E + L + 0.2S$
6. $0.9D + 1.0W$
7. $0.9D + 1.0E$

Note: $D$ = Dead, $L$ = Live, $L_r$ = Roof Live, $S$ = Snow, $R$ = Rain, $W$ = Wind, $E$ = Earthquake.

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Dead Loads (D)

Dead loads are static gravity loads consisting of the permanent weight of the structure, including structural members, walls, floors, roofs, ceilings, stairways, and fixed service equipment.

• Unit Weights of Materials: Designers use standard material weights to compute dead loads. Reinforced concrete is typically assumed to weigh $150 \text{ pcf}$ ($23.6 \text{ kN/m}^3$), steel is $490 \text{ pcf}$ ($77 \text{ kN/m}^3$), and timber ranges from $35$ to $50 \text{ pcf}$.
• Calculation Methodology: Dead loads are computed as: $$w_{\text{dead}} = \sum (\text{Material Density} \times \text{Thickness}) + \text{Allowance for MEP/Finishes}$$

Step-by-Step Example:

Calculate the design dead load for a floor system consisting of a $5\text{-inch}$ reinforced concrete slab, steel beams spaced at $8\text{ ft}$ on center (estimated beam weight is $35\text{ plf}$), and a ceiling/MEP allowance of $12\text{ psf}$.

1. Concrete Slab Dead Load: $$w_{\text{slab}} = \frac{5\text{ in}}{12\text{ in/ft}} \times 150\text{ pcf} = 62.5\text{ psf}$$
2. Distributed Beam Dead Load: $$w_{\text{beam}} = \frac{35\text{ plf}}{8\text{ ft}} = 4.375\text{ psf}$$
3. Ceiling/MEP Allowance: $$w_{\text{ceiling}} = 12\text{ psf}$$
4. Total Dead Load: $$D = 62.5 + 4.375 + 12 = 78.875\text{ psf}$$

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Live Loads (L)

Live loads are transient gravity loads produced by the occupancy and use of the building. These exclude environmental loads (wind, snow, earthquake, rain) and dead loads.

1. Minimum Uniformly Distributed Live Loads

ASCE 7 Table 4.3-1 defines minimum design live loads ($L_0$) for various occupancies:

• Residential (bedrooms): $30\text{ psf}$
• Offices (first-floor lobbies): $100\text{ psf}$; (upper floors): $50\text{ psf}$ plus a $20\text{ psf}$ partition allowance
• School Classrooms: $40\text{ psf}$; Corridors: $80\text{ psf}$ (first floor) or $100\text{ psf}$ (upper floors)
• Assembly Areas (fixed seats): $60\text{ psf}$; (lobbies/moveable seats): $100\text{ psf}$
• Heavy Manufacturing: $250\text{ psf}$

2. Live Load Reduction

Members supporting large areas are permitted to be designed for a reduced live load ($L$) because it is highly unlikely that the entire area will be subjected to the maximum design live load simultaneously. The ASCE 7 formula is:

$$L = L_0 \left( 0.25 + \frac{15}{\sqrt{K_{LL} A_T}} \right)$$

Where:

• $L$ = reduced design live load per square foot.
• $L_0$ = unreduced design live load per square foot.
• $K_{LL}$ = live load element factor (reflects member configuration).
• $A_T$ = tributary area in square feet.

Values of $K_{LL}$:

• Interior columns, exterior columns, and corner columns = $4$
• Interior beams and edge beams without cantilever slabs = $2$
• Cantilever beams = $1$
• One-way slabs = $2$
• Two-way slabs = $4$

Live Load Reduction Limits:

• No reduction is allowed if the influence area ($A_I = K_{LL} A_T$) is less than $400\text{ ft}^2$.
• The reduced live load $L$ cannot be less than $0.50 L_0$ for members supporting one floor.
• The reduced live load $L$ cannot be less than $0.40 L_0$ for members supporting two or more floors.
• No reduction is allowed for live loads exceeding $100\text{ psf}$ (except under strict rules for multiple floors), passenger vehicle garages, or public assembly areas.

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Snow Loads (S)

Snow loads are gravity loads acting downward on the roof of a structure. They are based on ground snow loads ($p_g$) and roof characteristics.

1. Flat Roof Snow Load ($p_f$)

For flat roofs (slopes $\le 5^\circ$), the design snow load is:

$$p_f = 0.7 C_e C_t I_s p_g$$

Where:

• $p_g$ = ground snow load (obtained from ASCE 7 geographic maps).
• $C_e$ = exposure factor (depends on terrain categories B, C, D and exposure of the roof; ranges from $0.7$ to $1.2$).
• $C_t$ = thermal factor (depends on the thermal state of the building; e.g., $1.0$ for heated buildings, $1.1$ for unheated structures, $1.2$ for open unheated structures).
• $I_s$ = importance factor for snow (based on Risk Category; e.g., $0.8$ for Category I, $1.0$ for Category II, $1.1$ for Category III, $1.2$ for Category IV).

Minimum Flat Roof Snow Load ($p_{f,\min}$):

• If $p_g \le 20\text{ psf}$, then $p_{f,\min} = I_s p_g$.
• If $p_g > 20\text{ psf}$, then $p_{f,\min} = 20 I_s$.

2. Sloped Roof Snow Load ($p_s$)

On roofs with slopes $> 5^\circ$, the snow load is adjusted using a slope factor ($C_s$):

$$p_s = C_s p_f$$

The factor $C_s$ depends on the slope angle, whether the roof is warm or cold, and whether the surface is slippery (e.g., metal or glass) to facilitate sliding.

3. Snow Drifts

Snow drifting occurs at steps in roof elevations, parapet walls, and other roof projections. Wind blows snow across the roof, causing it to accumulate in aerodynamic shadows.

• Leeward Drift: Snow blowing from the higher roof accumulates on the lower roof against the step wall.
• Windward Drift: Snow blowing across the lower roof is caught against the step wall. The maximum height of the drift ($h_d$) is calculated using the length of the higher roof ($l_u$) for leeward drift, or the length of the lower roof for windward drift. The drift width is typically taken as $w = 4 h_d$. The drift load is represented as a triangular surcharge added to the flat roof snow load.

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Wind Loads (W)

Wind loads are lateral and uplift pressures caused by the kinetic energy of moving air on structural surfaces.

1. Wind Velocity Pressure ($q_z$)

The pressure exerted by the wind at a height $z$ above ground level is:

$$q_z = 0.00256 K_z K_{zt} K_d K_e V^2$$

Where:

• $q_z$ = velocity pressure in $\text{psf}$.
• $V$ = basic wind speed in $\text{mph}$ (based on risk category and geographic maps).
• $K_z$ = velocity pressure exposure coefficient (depends on height $z$ and terrain exposure B, C, or D).
• $K_{zt}$ = topographic factor (accounts for wind speed-up over hills, ridges, or escarpments).
• $K_d$ = wind directionality factor (accounts for the probability of the wind blowing from the worst-case direction; typically $0.85$ for main frames).
• $K_e$ = ground elevation factor (accounts for change in air density with elevation).

2. Wind Pressure Design Methods

• Main Wind Force Resisting System (MWFRS): Designed to resist wind loads acting on the structural system as a whole (e.g., shear walls, braced frames, moment frames).
• Components and Cladding (C&C): Designed for localized wind pressures acting on individual elements (e.g., windows, studs, purlins, cladding panels). C&C pressures are significantly higher than MWFRS pressures due to localized turbulence and edge vortices.

The net design wind pressure ($p$) for MWFRS is typically:

$$p = q G C_p - q_i (GC_{pi})$$

Where $G$ is the gust effect factor, $C_p$ is the external pressure coefficient, and $GC_{pi}$ is the internal pressure coefficient.

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Earthquake Loads (E)

Earthquake loads are inertial lateral forces generated by the acceleration of the building mass during a seismic event.

1. Equivalent Lateral Force Procedure

The Equivalent Lateral Force (ELF) procedure is the most common method for seismic design on the PE Civil exam.

• Seismic Base Shear ($V$): The total design lateral force at the base of the structure:

  $$V = C_s W$$

  Where $W$ is the effective seismic weight of the structure (dead load plus permanent equipment, and a portion of live/snow loads under certain conditions), and $C_s$ is the seismic response coefficient:

  $$C_s = \frac{S_{DS}}{\left( \frac{R}{I_e} \right)}$$

  Where:

  • $S_{DS}$ = design spectral response acceleration parameter at short periods.
  • $R$ = response modification coefficient (indicates the ductility of the lateral system; e.g., $R = 8.0$ for special reinforced concrete shear walls, $R = 3.0$ for ordinary steel moment frames).
  • $I_e$ = seismic importance factor (depends on Risk Category).

  Limits on $C_s$:

  $$C_s \le \frac{S_{D1}}{T \left( \frac{R}{I_e} \right)} \quad \text{for } T \le \text{Transition Period } T_L$$ $$C_s \ge 0.044 S_{DS} I_e \ge 0.01$$

  Where $S_{D1}$ is the design spectral response acceleration at a 1-second period, and $T$ is the fundamental period of the structure.

2. Vertical Distribution of Seismic Forces

The total base shear $V$ is distributed to each floor level $x$ using:

$$F_x = C_{vx} V = \frac{w_x h_x^k}{\sum_{i=1}^n w_i h_i^k} V$$

Where:

• $w_x, w_i$ = portion of effective seismic weight assigned to level $x$ or $i$.
• $h_x, h_i$ = height from the base to level $x$ or $i$.
• $k$ = exponent related to the structure period $T$ ($k = 1.0$ for $T \le 0.5\text{ s}$, $k = 2.0$ for $T \ge 2.5\text{ s}$, and linearly interpolated between $1.0$ and $2.0$ for intermediate periods).