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۱۴۰۵/۰۴/۰۸ Civil & Structural Engineering ۱۸ دقیقه زمان مطالعه

Plastic Collapse Theorems: 3 Fundamental Principles for Frame Analysis (2026 Guide)

Table of Contents

۱. Introduction to Plastic Collapse Theorems in Structural Engineering

Modern limit-state structural design relies fundamentally on our ability to predict the ultimate load-carrying capacity of ductile structural frameworks beyond their initial elastic limit. The classical elastic method evaluates structures based on the stress state at the most heavily stressed fiber reaching yield stress $sigma_y$. However, ductile steel and reinforced concrete structures possess reserve strength through moment redistribution. The analytical foundation governing this post-elastic capacity is established by the plastic collapse theorems.

Understanding plastic collapse theorems provides structural engineers with rigorous mathematical bounds to bracket the true collapse load factor $lambda_c$. By establishing safe lower bounds through static equilibrium and kinematically admissible upper bounds through postulated failure mechanisms, engineers can rapidly assess structural safety under severe overload, seismic excitation, and accidental actions.

Plastic analysis evaluates the structure at the point of incipient collapse, transforming an indeterminate framework into an unstable kinematic mechanism through the localized formation of plastic hinges. Whether evaluating continuous bridge girders or multi-story moment-resisting frames, applying plastic collapse theorems guarantees that structural collapse predictions rest upon rigorous variational principles rather than trial-and-error approximations.

[Internal Link: Read our foundational guide on Plastic Analysis of Structures]

۲. Foundational Assumptions of Limit Analysis

To apply plastic collapse theorems with absolute theoretical consistency, structural mechanics establishes several classical idealizations regarding material constitutive laws and structural kinematic behavior.

۲.۱ Rigid-Plastic Material Idealization

The primary assumption underlying the plastic collapse theorems is the rigid-perfectly plastic material model (often termed the standard Prandtl-Reuss material without hardening). Elastic strains are assumed negligible compared to plastic strains developed at collapse:

$$
varepsilon = varepsilon_e + varepsilon_p approx varepsilon_p quad text{as } lambda to lambda_c
$$

Under pure flexure, cross-sections develop a fully plastic bending moment $M_p = Z_p sigma_y$, where $Z_p$ represents the plastic section modulus and $sigma_y$ is the material yield strength. Once a cross-section reaches $M_p$, an idealized plastic hinge forms. The hinge sustains constant moment $M_p$ while undergoing indefinite plastic rotation $theta_p ge 0$, provided local and lateral-torsional buckling are restrained.

[Visual Suggestion: Stress-strain and moment-curvature idealization curves comparing elastic-perfectly plastic and rigid-plastic cross-section behavior – Alt Text: plastic collapse theorems]

۲.۲ Small Deformation Geometry

The classic plastic collapse theorems assume first-order structural geometry. The equilibrium equations are formulated with respect to the undeformed geometry of the structure. Consequently, geometric non-linearities, secondary $P$-$Delta$ moments, and membrane tensile stiffening are excluded from basic limit analysis.

The structure must have sufficient ductility. The rotational capacity $R = theta_u / theta_p$ of each plastic hinge must be large enough to permit the formation of a complete collapse mechanism without premature brittle fracture or localized web/flange buckling.

+-------------------------------------------------------------------------------+
|                      BASIC PREREQUISITES FOR LIMIT ANALYSIS                   |
+-----------------------------+-------------------------------------------------+
| Prerequisite Parameter      | Physical Requirement in Structural Framework   |
+-----------------------------+-------------------------------------------------+
| Material Ductility          | Ductile plateau with $varepsilon_u ge 15%$   |
| Section Compactness         | Class 1 Plastic Cross-sections (AISC/Eurocode 3)|
| Geometric Stability         | Adequate lateral-torsional bracing at hinges   |
| Shear Capacity              | $V_{Ed} le 0.5 V_{pl,Rd}$ (Negligible M-V drop)|
| Axial Load Ratio            | $N_{Ed} / N_{pl,Rd} < 0.15$ (Pure flexure model)|
+-----------------------------+-------------------------------------------------+

۳. The Three Fundamental Plastic Collapse Theorems

The entire framework of limit analysis rests upon three classical theorems formulated by Prager, Hodge, Drucker, and Greenberg: the lower bound theorem, the upper bound theorem, and the uniqueness theorem.

                           +-------------------------------------+
                           |      PLASTIC COLLAPSE THEOREMS      |
                           +------------------+------------------+
                                              |
                     +------------------------+------------------------+
                     |                                                 |
+--------------------+-------------------+   +-------------------------+--------------------+
|       LOWER BOUND (STATIC) THEOREM     |   |      UPPER BOUND (KINEMATIC) THEOREM       |
+----------------------------------------+   +----------------------------------------------+
| • Satisfies Equilibrium: $sum F=0$    |   | • Postulates Kinematic Mechanism             |
| • Yield Condition: $|M(x)| le M_p$    |   | • Equates Internal Work to External Work     |
| • Yields SAFE load factor:             |   | • Yields UNSAFE load factor:                 |
|   $lambda_s le lambda_c$            |   |   $lambda_k ge lambda_c$                  |
+--------------------+-------------------+   +-------------------------+--------------------+
                     |                                                 |
                     +------------------------+------------------------+
                                              |
                                              v
                           +-------------------------------------+
                           |         UNIQUENESS THEOREM          |
                           |   $lambda_s = lambda_c = lambda_k$ |
                           |       Exact Collapse Load           |
                           +-------------------------------------+

۳.۱ The Static Theorem (Lower Bound Theorem)

The lower bound theorem (also designated the Static Theorem or Safe Theorem) is formulated as follows:

Statement: If a distribution of internal bending moments $M(x)$ throughout the structure can be found that satisfies both structural equilibrium with the applied loads $lambda P_i$ and the yield condition $|M(x)| le M_p$ at every cross-section, then the load factor $lambda_s$ is less than or equal to the true collapse load factor $lambda_c$:

$$
lambda_s le lambda_c
$$

The lower bound theorem represents a remarkably powerful design tool. If an engineer constructs any statically admissible bending moment diagram that does not exceed $M_p$ anywhere in the frame, the structure will not collapse under that load. The static theorem guarantees safety and forms the conceptual foundation for conservative design codes.

[Internal Link: Review the mathematical derivation in Plastic Hinge Formation and Moment Redistribution]

۳.۲ The Kinematic Theorem (Upper Bound Theorem)

The upper bound theorem (also known as the Kinematic Theorem or Unsafe Theorem) approaches collapse from structural kinematics:

Statement: If a compatible collapse mechanism is defined with plastic hinges formed at locations of maximum moment, and the work done by the external loads during a virtual displacement is equated to the internal plastic work dissipated in the hinges, the resulting load factor $lambda_k$ is greater than or equal to the true collapse load factor $lambda_c$:

$$
lambda_k ge lambda_c
$$

The upper bound theorem states that a structure will find the weakest possible failure path. If an engineer guesses an incorrect collapse mechanism, the calculated collapse load will always overestimate the true capacity. Therefore, kinematic limit analysis involves searching for the critical kinematic mechanism that minimizes the computed collapse load factor.

۳.۳ The Uniqueness Theorem: Exact Collapse State

The uniqueness theorem synthesizes both bounds into an exact criterion:

Statement: A load factor $lambda$ corresponds to the exact, unique plastic collapse load $lambda_c$ if and only if there exists a bending moment distribution that simultaneously satisfies:
۱. Equilibrium Condition: The internal bending moments are in static equilibrium with the applied loads $lambda_c P$.
۲. Yield Condition: The bending moment nowhere exceeds the fully plastic capacity: $|M(x)| le M_p$.
۳. Mechanism Condition: Sufficient plastic hinges exist with $M = pm M_p$ to transform the structure (or a sub-assemblage) into a kinematically single-degree-of-freedom mechanism.

$$
lambda_s le lambda_c le lambda_k implies text{When } lambda_s = lambda_k = lambda_c, text{ the solution is unique and exact.}
$$

۴. Derivation and Mathematical Formulation via Virtual Work

The rigorous mathematical proof of the plastic collapse theorems is derived from the principle of virtual work. Consider an arbitrary framed structure subject to proportional external nodal and distributed loads $lambda mathbf{F}$.

[Visual Suggestion: Free-body diagram and virtual displacement kinematically admissible field for a multi-span frame – Alt Text: Virtual work plastic collapse theorems mechanism]

۴.۱ Virtual Work Equations for a Generic Collapse Mechanism

Let a postulated collapse mechanism produce virtual hinge rotations $theta_j$ at $m$ plastic hinge locations, accompanied by virtual displacements $Delta_i$ under external load components $P_i$. The Principle of Virtual Work equates external work $W_{ext}$ to internal work $W_{int}$:

$$
W_{ext} = lambda sum_{i=1}^{n} P_i Delta_i
$$

$$
W_{int} = sum_{j=1}^{m} M_{p,j} |theta_j|
$$

Equating $W_{ext} = W_{int}$ gives the kinematic collapse load factor:

$$
lambda_k = frac{sum_{j=1}^{m} M_{p,j} |theta_j|}{sum_{i=1}^{n} P_i Delta_i}
$$

۴.۲ Mathematical Proof of the Lower Bound Property

Let $mathbf{M}^s(x)$ be a statically admissible moment field satisfying equilibrium with load factor $lambda_s$ such that $|M^s(x)| le M_p$. Let $mathbf{u}^c, boldsymbol{theta}^c$ be the actual kinematic displacement and rotation fields corresponding to the true collapse mechanism under the true collapse load factor $lambda_c$.

By virtual work, applying the statically admissible stress field $mathbf{M}^s$ over the true collapse kinematic deformation field gives:

$$
lambda_s sum P_i Delta_i^c = sum M_j^s theta_j^c
$$

Similarly, applying the actual collapse moment field $mathbf{M}^c$ (where $M_j^c = M_p operatorname{sgn}(theta_j^c)$ at all active hinges) over the true collapse mechanism yields:

$$
lambda_c sum P_i Delta_i^c = sum M_j^c theta_j^c = sum M_{p,j} |theta_j^c|
$$

Subtracting the two equations gives:

$$
(lambda_c – lambda_s) sum P_i Delta_i^c = sum left( M_{p,j} |theta_j^c| – M_j^s theta_j^c right)
$$

Because $|M_j^s| le M_{p,j}$, we have $M_j^s theta_j^c le M_{p,j} |theta_j^c|$, which guarantees:

$$
M_{p,j} |theta_j^c| – M_j^s theta_j^c ge 0 quad forall j
$$

Since the external work $sum P_i Delta_i^c > 0$ for positive loading, it directly follows that:

$$
lambda_c – lambda_s ge 0 implies lambda_s le lambda_c
$$

This completes the formal proof of the lower bound theorem within the plastic collapse theorems.

External Link (Followed): Read the AISC Specification for Structural Steel Buildings (AISC 360-22 Appendix 1 on Inelastic Analysis)

۴.۳ Mathematical Proof of the Upper Bound Property

Now consider an arbitrary postulated kinematically admissible collapse mechanism with virtual displacements $Delta_i^k$ and hinge rotations $theta_j^k$. Equating external and internal virtual work under the kinematic load factor $lambda_k$ yields:

$$
lambda_k sum P_i Delta_i^k = sum M_{p,j} |theta_j^k|
$$

Now, apply the true collapse stress field $mathbf{M}^c$ (which is in static equilibrium with $lambda_c mathbf{P}$ and satisfies $|M^c(x)| le M_p$) to this postulated virtual displacement field $Delta_i^k, theta_j^k$:

$$
lambda_c sum P_i Delta_i^k = sum M_j^c theta_j^k
$$

Subtracting this from the kinematic virtual work expression gives:

$$
(lambda_k – lambda_c) sum P_i Delta_i^k = sum left( M_{p,j} |theta_j^k| – M_j^c theta_j^k right)
$$

Since $|M_j^c| le M_{p,j}$ everywhere, the right-hand term $M_{p,j} |theta_j^k| – M_j^c theta_j^k ge 0$ is unconditionally non-negative. Therefore:

$$
lambda_k – lambda_c ge 0 implies lambda_k ge lambda_c
$$

This proves the upper bound theorem and confirms that any postulated kinematic mechanism provides an unsafe or exact estimate of the collapse capacity.

۵. Step-by-Step Worked Example: Fixed-Base Portal Frame Analysis

To illustrate how plastic collapse theorems operate in engineering practice, we analyze a single-story, single-bay rectangular portal frame with fixed column bases.

       W = 2P (Vertical load at mid-span of beam)
           |
           v
    B +----+----+ C
      |    E    |
      |         |  H = P (Lateral sway load)
      |         |  --> applied at B
    h |         |
      |         |
    A ///     /// D
        Span L = 2h

۵.۱ Frame Geometry and Applied Loading

  • Column Height: $h = 4.0text{ m}$
  • Beam Span: $L = 2h = 8.0text{ m}$
  • Plastic Moment Capacity: Uniform cross-section throughout, $M_p = 180text{ kN}cdottext{m}$
  • Applied Vertical Load: $V = 2P$ applied at beam midspan $E$
  • Applied Horizontal Load: $H = P$ applied at knee joint $B$
  • Degree of Static Indeterminacy: $DSI = 3(1) – 0 = 3$.
  • Number of Possible Plastic Hinge Locations: $N = 5$ (Points $A, B, C, D, E$).
  • Number of Independent Mechanisms: $I = N – DSI = 5 – 3 = 2$ (Beam mechanism and Sway mechanism).

[Visual Suggestion: Detailed kinematic hinge diagrams for Beam, Sway, and Combined collapse mechanisms of portal frame – Alt Text: Plastic collapse theorems portal frame mechanisms]

۵.۲ Beam Mechanism Evaluation

Assume plastic hinges form at joints $B, E,$ and $C$. Columns remain vertical; only the beam sways downward at center $E$ by virtual displacement $delta_v = frac{L}{2} theta = 4.0 theta$.

Hinge rotations:
- Hinge at B: theta
- Hinge at E: 2theta
- Hinge at C: theta

  1. External Virtual Work ($W_{ext}$):
    The horizontal load $H$ does zero work since column heads do not displace laterally.
    $$
    W_{ext} = (2P) cdot delta_v = 2P(4.0 theta) = 8.0 P theta
    $$

  2. Internal Virtual Work ($W_{int}$):
    $$
    W_{int} = M_p |theta| + M_p |2theta| + M_p |theta| = 4 M_p theta
    $$

  3. Kinematic Load Calculation:
    $$
    W_{ext} = W_{int} implies 8.0 P theta = 4 M_p theta implies P_1 = frac{4 M_p}{8.0} = 0.500 frac{M_p}{1.0} = 0.500 left(frac{180}{1}right) = 90.0text{ kN}
    $$

By the upper bound theorem, the true collapse parameter $P_c le 90.0text{ kN}$.

[Internal Link: Explore our worked examples on Collapse Load Analysis of Continuous Beams]

۵.۳ Sway Mechanism Evaluation

Assume lateral displacement $Delta = h theta = 4.0 theta$. Plastic hinges develop at column bases $A, D$ and column tops/joints $B, C$. Beam moves purely horizontally without relative vertical displacement.

Hinge rotations:
- Hinge at A: theta
- Hinge at B: theta
- Hinge at C: theta
- Hinge at D: theta

  1. External Virtual Work ($W_{ext}$):
    $$
    W_{ext} = H cdot Delta = P(4.0 theta) = 4.0 P theta
    $$

  2. Internal Virtual Work ($W_{int}$):
    $$
    W_{int} = M_p(theta + theta + theta + theta) = 4 M_p theta
    $$

  3. Kinematic Load Calculation:
    $$
    ۴.۰ P theta = 4 M_p theta implies P_2 = 1.000 M_p = 180.0text{ kN}
    $$

The sway mechanism gives an upper bound of $P le 180.0text{ kN}$.

۵.۴ Combined Mechanism and Lower Bound Equilibrium Verification

Superimposing beam and sway mechanisms eliminates the hinge at joint $B$ because the counter-clockwise rotation from the sway mechanism cancels the clockwise hinge rotation from the beam mechanism ($theta – theta = 0$).

Hinges develop at four locations: $A, E, C,$ and $D$.

Hinge rotations:
- Base A: theta
- Midspan E: 2theta
- Joint C: 2theta (theta from sway + theta from beam)
- Base D: theta

  1. External Virtual Work ($W_{ext}$):
    $$
    W_{ext} = P(h theta) + 2Pleft(frac{L}{2} thetaright) = P(4.0 theta) + 2P(4.0 theta) = 12.0 P theta
    $$

  2. Internal Virtual Work ($W_{int}$):
    $$
    W_{int} = M_p(theta + 2theta + 2theta + theta) = 6 M_p theta
    $$

  3. Kinematic Upper Bound Solution:
    $$
    ۱۲.۰ P theta = 6 M_p theta implies P_3 = frac{6 M_p}{12.0} = 0.500 M_p = 0.500(180) = 90.0text{ kN}
    $$

+-------------------------------------------------------------------------------+
|                 MECHANISM COLLAPSE LOAD COMPARISON SUMMARY                    |
+---------------------+-------------------+-------------------+-----------------+
| Mechanism Candidate | External Work     | Internal Work     | Collapse Load P |
+---------------------+-------------------+-------------------+-----------------+
| Pure Beam Mechanism | $8.0 P theta$    | $4.0 M_p theta$  | $P = 90.0text{ kN}$ |
| Pure Sway Mechanism | $4.0 P theta$    | $4.0 M_p theta$  | $P = 180.0text{ kN}$|
| Combined Mechanism  | $12.0 P theta$   | $6.0 M_p theta$  | $P = 90.0text{ kN}$ |
+---------------------+-------------------+-------------------+-----------------+

Both the beam mechanism and combined mechanism yield $P = 90.0text{ kN}$. To verify this by the lower bound theorem, we evaluate the bending moment at non-hinge joint $B$. In the combined mechanism, equilibrium gives $M_B = 0 < M_p = 180text{ kN}cdottext{m}$.

Because $|M(x)| le M_p$ everywhere across all members while satisfying global and local static equilibrium, the conditions of the uniqueness theorem are fully met. The exact plastic collapse load is precisely:

$$
P_c = 90.0text{ kN} quad (text{with } V_c = 180.0text{ kN}, H_c = 90.0text{ kN})
$$

۶. Engineering Failure Modes and Forensic Insights

Forensic engineering investigations frequently reveal structural failures caused by misinterpreting structural plastic behavior. Plastic analysis assumes that structural members possess sufficient plastic rotation capacity before rupture.

       M
       ^
   M_p |-----------------------+ (Ideal Rigid-Plastic Plateau)
       |                      / 
       |    Ductile Section  /    Local Web/Flange Buckling (Premature Collapse)
       |                    /     
       |                   /       v
       +------------------+--------------------> Rotation theta
                          theta_p   theta_u

Key Mechanisms That Invalidate Limit Theorems

  1. Premature Local Flange and Web Buckling: If thin-walled structural members (Class 3 or Class 4 cross-sections under Eurocode 3) are used, plate buckling occurs before the cross-section reaches the fully plastic moment $M_p$. The moment capacity drops sharply, preventing the formation of subsequent plastic hinges required to activate the mechanism.
  2. Brittle Connection Fracture: Welded moment connections with low notch-toughness weld metal or unreinforced beam-to-column joints can undergo sudden brittle fracture at moderate rotations (as observed in the 1994 Northridge earthquake). This disrupts the moment redistribution path required by the plastic collapse theorems.
  3. Second-Order $P$-$Delta$ Instability: In flexible tall structures or soft-story frames, lateral sway produces substantial secondary overturning moments ($P cdot Delta$). This creates a negative post-yield stiffness slope, causing frame destabilization prior to full kinematic mechanism development.

External Link: Review the NIST Technical Investigation on Structural Failure Modes and Performance

۷. Structural Code Provisions and Modern Limit State Standards

International design codes incorporate plastic collapse theorems under strict qualifying criteria to guarantee ductility and structural robustness.

[Visual Suggestion: Summary flowchart of Eurocode 3 / AISC 360 plastic analysis compliance and cross-section ductility classification – Alt Text: Plastic collapse theorems design flowchart]

International Standards Comparison

  • AISC 360-22 (Chapter 1 & Appendix 1): Permits direct plastic design for steels with $F_y le 450text{ MPa}$ ($65text{ ksi}$). Flange and web width-to-thickness ratios $lambda = b/t$ must not exceed compact limits $lambda_{pd}$. The unbraced length $L_b$ adjacent to plastic hinge locations must satisfy:
    $$
    L_{pd} le left[ 0.12 + 0.076 left(frac{M_1}{M_2}right) right] left(frac{E}{F_y}right) r_y
    $$
  • Eurocode 3 (EN 1993-1-1 Section 5.4): Requires Class 1 (plastic) cross-sections at all prospective plastic hinge zones. Class 1 sections can form a plastic hinge with the rotation capacity required for plastic analysis without reduction of resistance.
  • ASCE 7-22 Minimum Design Loads: Implements plastic mechanism principles in seismic overstrength factor $Omega_0$ determinations and collapse prevention limit state verifications.

[Internal Link: Review our structural guide on Shape Factor Calculation for Structural Sections]

۸. Synthesis and Engineering Wrap-Up

Mastering plastic collapse theorems bridges the gap between theoretical continuum plasticity and practical structural engineering. By establishing rigorous bounds through static equilibrium and kinematic virtual work, engineers can determine the true capacity of complex indeterminate frames. Relying on the upper bound alone risks catastrophic unconservative errors, while depending solely on crude elastic distributions leaves substantial material efficiency unharvested. The true elegance of structural mechanics is found when static equilibrium and kinematic admissibility converge, establishing a unique state of structural equilibrium and safety.

Frequently Asked Questions (FAQs)

۱. Why is the lower bound theorem considered inherently safe for structural design?

The lower bound theorem states that any internal moment distribution that satisfies static equilibrium with the applied loads and nowhere violates the cross-section yield condition ($|M| le M_p$) yields a load factor $lambda_s le lambda_c$. Because the structure will redistribute loads to find at least this capacity without collapsing, designing via lower bound equilibrium guarantees that the applied design load will not exceed the collapse threshold.

۲. What distinguishes the upper bound theorem from the lower bound theorem?

The upper bound theorem is kinematic: it postulates an assumed collapse mechanism and calculates the load factor $lambda_k$ by equating external work to internal plastic energy dissipation. This calculation is unsafe ($lambda_k ge lambda_c$) because an incorrect mechanism fails to identify the weakest failure path. In contrast, the lower bound theorem is static, satisfies equilibrium, and provides a safe bound ($lambda_s le lambda_c$).

۳. How does strain hardening affect the validity of plastic collapse theorems?

Standard plastic collapse theorems assume an elastic-perfectly plastic or rigid-perfectly plastic material without strain hardening. Real structural steel exhibits strain hardening after yielding, meaning the actual ultimate load supported by a frame is slightly higher than the theoretically calculated $lambda_c$. Thus, ignoring strain hardening introduces an additional conservative reserve margin into limit analysis.

۴. Can plastic analysis be applied to reinforced concrete frameworks?

Yes, but with strict limitations on rotational capacity. Unlike structural steel, concrete is a brittle material in tension and has limited compressive strain capacity ($varepsilon_{cu} approx 0.0035$). Plastic analysis is permitted in reinforced concrete (as recognized by ACI 318 and Eurocode 2) only if members are under-reinforced, ductile reinforcement steel is utilized, and moment redistribution percentages are capped (typically $le 20text{–}30%$).

۵. Why are Class 3 and Class 4 steel sections prohibited in plastic design?

Class 3 (semi-compact) and Class 4 (slender) steel cross-sections undergo local plate buckling before or shortly after reaching the elastic yield moment $M_y$. Because they cannot sustain the full plastic moment $M_p$ through large inelastic rotations without local buckling, they cannot form stable plastic hinges, invalidating the foundational kinematic assumptions of limit analysis.

References & Standards Cited

  1. AISC (2022). Specification for Structural Steel Buildings (ANSI/AISC 360-22), American Institute of Steel Construction, Chicago, IL.
  2. CEN (2005). Eurocode 3: Design of steel structures — Part 1-1: General rules and rules for buildings (EN 1993-1-1:2005), European Committee for Standardization, Brussels.
  3. Baker, J. F., & Heyman, J. (1969). Plastic Design of Frames: Volume 1, Fundamentals, Cambridge University Press.
  4. Horne, M. R. (1979). Plastic Theory of Structures, ۲nd Edition, Pergamon Press, Oxford.
  5. Drucker, D. C., Prager, W., & Greenberg, H. J. (1952). “Extended Limit Design Theorems for Continuous Media,” Quarterly of Applied Mathematics, Vol. 9, No. 4, pp. 381–۳۸۹.
  6. ASCE (2022). Minimum Design Loads and Associated Criteria for Buildings and Other Structures (ASCE/SEI 7-22), American Society of Civil Engineers, Reston, VA.

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