Structural Analysis and Stability

Introduction

Steel structures are assembled from members whose strength and stiffness depend not only on their material properties and cross-sections, but also on how the members are connected and restrained. Structural analysis converts loads into internal forces, displacements, and support reactions. Stability analysis determines whether the structure can maintain equilibrium when subjected to those actions, including the effects of imperfections and deformation.

For construction technologists, stability is particularly important because steel is strong and relatively light. These advantages can produce slender columns, flexible frames, and members that are sensitive to second-order effects. A frame may satisfy a simple strength check and still be unsafe because it loses stability, undergoes excessive lateral displacement, or experiences force amplification caused by deformation.

This chapter develops idealized models for determinate and indeterminate steel frames, distinguishes first-order from second-order behavior, and explains buckling, effective length, imperfections, and frame stability. The discussion is based on classical structural mechanics. In practice, design must also comply with the governing standard, such as the applicable national steel design code, and must be coordinated with connection, fire, construction-stage, and foundation design.

Learning objectives

By the end of this chapter, a student should be able to:

1. Idealize a steel frame for structural analysis and identify its load paths.
2. Distinguish statically determinate from statically indeterminate structures.
3. Calculate reactions and member actions for simple determinate systems.
4. Explain the assumptions and limitations of first-order and second-order analysis.
5. Describe elastic and inelastic buckling and use Euler’s critical-load relationship appropriately.
6. Determine the significance of effective length and end restraint.
7. Explain the structural effects of initial crookedness, residual stress, and connection imperfections.
8. Evaluate basic frame stability using drift, sway, and stability indicators.
9. Select appropriate analysis procedures and communicate assumptions clearly.

1. Structural idealization and load paths

A real steel building contains three-dimensional members, semi-rigid connections, floor diaphragms, cladding, foundations, and construction tolerances. Analysis begins by replacing this physical system with an idealized model. The model must preserve the behavior relevant to the design question.

A typical building frame may be idealized as:

• Nodes, where members meet and forces or displacements are related;
• Members, represented by their centerlines and section properties;
• Supports, representing restraints supplied by foundations or adjacent structure;
• Loads, applied as concentrated forces, distributed loads, moments, or prescribed displacements;
• Connections, idealized as pinned, rigid, or semi-rigid;
• Diaphragms and bracing, which distribute and resist horizontal actions.

The load path is the continuous route by which an action travels from its point of application to the ground. For example, gravity load may pass from a roof sheet to purlins, rafters, columns, base plates, foundations, and soil. Wind load may pass through cladding and floor diaphragms to braced bays or moment-resisting frames. An incomplete load path is a fundamental stability defect, regardless of the apparent strength of individual members.

A model is useful only when its assumptions are stated. A pin-jointed truss model, for example, assumes forces are transferred through ideal pins and that loads are applied at nodes. If significant bending is introduced by eccentric connections or distributed loading, the truss model may be inappropriate.

2. Equilibrium, compatibility, and constitutive behavior

Structural analysis rests on three complementary requirements.

Equilibrium requires that the resultant forces and moments acting on the structure balance. For a planar structure:

Here ∑F_x and ∑F_y are the sums of horizontal and vertical forces, and ∑M_O is the sum of moments about any point O. Forces are commonly expressed in kN and moments in kN·m.

Compatibility requires that connected parts deform consistently. A joint cannot have two different displacements at the same location, and a continuous beam cannot normally develop a displacement discontinuity at an unbroken section.

Constitutive behavior relates force and deformation. For a prismatic axial member in the elastic range:

where N is axial force, E is Young’s modulus, A is cross-sectional area, and ε is axial strain. Since ε is dimensionless, EA has units of force. For bending, the corresponding elastic relationship is commonly expressed as:

where M is bending moment, I is the second moment of area about the relevant bending axis, and κ is curvature. For small deflection, κ is approximately the second derivative of the deflection curve with respect to the member axis.

Equilibrium alone is sufficient to solve a statically determinate structure. In an indeterminate structure, compatibility and member stiffness are also required.

3. Determinate and indeterminate steel frames

A structure is statically determinate when all reactions and internal actions can be obtained from equilibrium equations alone. A structure is statically indeterminate when additional equations based on compatibility and stiffness are required.

For a planar rigid-jointed structure, a convenient qualitative measure of indeterminacy is the number of unknown force components minus the number of independent equilibrium equations. Exact formulas depend on the structural arrangement and whether mechanisms or redundant constraints exist, so a simple count must be checked against stability and connectivity.

A determinate structure has practical advantages: member forces do not depend on elastic stiffness in the ideal model, and support settlement or uniform temperature change generally does not create secondary forces when no restraint prevents free movement. However, determinate structures may have less redundancy and may be more vulnerable to the loss of a single member.

Indeterminate frames distribute actions among several members. They can provide reserve strength and reduced deflection, but their force distribution depends on E, I, connection stiffness, support conditions, and construction sequence. Differential support settlement, temperature change, fabrication tolerances, and locked-in deformations can generate significant forces.

Worked example 1: determinate portal reaction

Consider a simply supported, 8 m span beam carrying a uniformly distributed load of 20 kN/m over the full span. The supports are a pin at A and a roller at B. The total load is:

The load acts at midspan. Taking moments about A:

Therefore:

Vertical equilibrium gives:

There is no horizontal reaction because no horizontal load is applied. The maximum sagging moment occurs at midspan, where shear changes sign:

These results follow from equilibrium alone. They would not, however, describe a continuous or rigid-jointed frame without additional compatibility and stiffness analysis.

4. Analysis of indeterminate frames

Indeterminate frames are commonly analyzed by the force method, displacement method, matrix stiffness method, or suitable computer software. The underlying principles are the same.

Force method

The force method selects redundant reactions or member forces, removes them to create a determinate primary structure, and imposes compatibility. For a redundant force X, the compatibility equation can be written conceptually as:

where δ_0 is the displacement caused by the applied loading in the released structure, f is the flexibility coefficient, and δ_required is the prescribed displacement, often zero. The method is useful for understanding indeterminacy but becomes cumbersome as the number of redundants increases.

Displacement and stiffness methods

The displacement method takes joint translations and rotations as primary unknowns. In matrix form, the linear elastic relationship is:

where [K] is the global stiffness matrix, {u} is the vector of unknown nodal displacements and rotations, and {P} is the vector of applied nodal loads. Units must be consistent: for example, forces in kN, lengths in m, moments in kN·m, and stiffness terms in compatible units.

The matrix is assembled from member stiffness relationships after transforming local member axes into global coordinates. Boundary conditions are then imposed and the equations solved. Member end forces are recovered from the calculated displacements.

The stiffness method is the basis of most structural analysis software. Its reliability depends on model quality. A computer can solve the equations of an unsuitable model with great numerical precision and still produce an unsafe engineering conclusion.

Modeling decisions that influence results

Important decisions include:

• whether a connection is pinned, rigid, or semi-rigid;
• whether a floor acts as a rigid or flexible diaphragm;
• whether members are prismatic or tapered;
• whether shear deformation is significant;
• whether releases are physically present and correctly assigned;
• whether supports restrain translation, rotation, or both;
• whether bracing is continuous and connected to the foundation;
• whether load combinations represent realistic simultaneous actions.

A rigid connection transfers axial force, shear, and moment while maintaining approximately the angle between connected members. A pinned connection transfers force but ideally no moment. Actual connections are often semi-rigid, and their rotational stiffness may affect frame drift and moment distribution.

5. First-order and second-order behavior

In first-order analysis, equilibrium is written using the original, undeformed geometry. Internal forces are assumed not to change because of structural displacement. This approximation is suitable when deflections are small and their influence on equilibrium is negligible.

In second-order analysis, equilibrium is written using the displaced geometry. Axial forces acting through lateral or out-of-straightness displacements create additional moments. For a member with axial compression P and lateral displacement Δ, the additional moment is approximately:

The total moment can therefore be represented as:

where M_1 is the first-order moment. The expression is a conceptual approximation; rigorous analysis accounts for the displacement shape, member stiffness, end conditions, and changes in axial force.

Two effects are commonly distinguished:

• P–Δ effect: amplification caused by translation of a member or story, such as frame sway.
• P–δ effect: amplification caused by curvature or local bowing within an individual member.

Second-order effects become important when axial compression is high, members or stories are slender, lateral stiffness is low, or imperfections are significant. They are especially relevant in unbraced frames, tall buildings, slender columns, and frames with large drift.

A useful elastic amplification concept for a member is:

where P_cr is the relevant elastic critical load. This idealized expression applies to a simplified elastic member under conditions for which the assumed moment shape and boundary conditions are appropriate, and only while P is below P_cr. It demonstrates that amplification increases rapidly as P approaches P_cr. Design codes use more detailed procedures and limits.

6. Column buckling

Buckling is a stability failure in which a member under compression changes from a nearly straight equilibrium configuration to a laterally displaced configuration. It may occur before the average compressive stress reaches the yield stress.

For an ideal, perfectly straight, axially loaded, pin-ended prismatic column in elastic behavior, Euler’s critical load is:

where:

• P_cr = elastic critical axial load, in N or kN;
• E = Young’s modulus, in Pa or kN/m²;
• I = second moment of area about the buckling axis, in m⁴;
• L = actual unsupported length, in m;
• K = effective-length factor, dimensionless.

The corresponding critical stress is:

where r = √(I/A) is the radius of gyration and KL/r is the slenderness ratio. The formula assumes a uniform member, ideal end conditions, concentric loading, elastic material behavior, and no initial imperfection or residual stress. Real columns depart from these assumptions, so Euler load is a reference stability quantity, not automatically an allowable design resistance.

The effective length is:

It is the length of an equivalent pin-ended column having the same elastic critical load as the actual member. Typical idealized K values include approximately 1.0 for pin-ended, 0.5 for fixed-ended, and 2.0 for a cantilever with one fixed and one free end. These values are idealizations. Real frame columns often have rotational and translational restraints between these limiting cases, and sway conditions strongly influence K.

Because P_cr varies with 1/(KL)², doubling effective length reduces the elastic critical load to one quarter, all else being equal. Buckling about the axis with the smaller radius of gyration is usually critical.

Local and global buckling

Global buckling involves lateral deflection of the member as a whole. Local buckling involves plate elements of the cross-section, such as a flange or web, buckling between their boundaries. Local slenderness limits are therefore needed before assigning a section its full plastic or compact resistance.

Other modes include torsional buckling and flexural–torsional buckling, particularly in open or unsymmetrical sections. A design must consider the relevant mode rather than assuming simple weak-axis flexural buckling is always governing.

Inelastic buckling and design resistance

Euler’s relationship applies only in the elastic range. Short, stocky columns may reach yield or develop plastic resistance before elastic buckling. Intermediate columns experience interaction between yielding, residual stresses, initial crookedness, and instability. Steel design standards account for this behavior using column curves, reduction factors, or equivalent imperfection approaches.

A generic design format is:

where N_Ed is the design axial force and N_b,Rd is the code-based buckling resistance. The exact expression for N_b,Rd depends on the adopted standard, steel grade, cross-section classification, buckling curve, partial factors, and loading conditions. Values must not be transferred between design codes without checking their definitions and safety formats.

7. Effective length and frame restraint

Effective length is not merely a property of a column. It is a property of the column and its surrounding structural system. Restraint may be supplied by beams, bracing, floor diaphragms, adjacent columns, foundations, or cladding, but only if these components have adequate stiffness and strength and are properly connected.

A braced frame has a separate system that resists most lateral translation. Column ends may then have appreciable rotational restraint, and the frame’s sway component is limited. In an unbraced or sway frame, lateral translation is part of the buckling mode. Small changes in beam-to-column stiffness, connection stiffness, or bracing continuity can substantially change the critical load.

Effective-length methods are convenient for member design but can be misleading when applied mechanically to complex frames. A direct second-order analysis or eigenvalue analysis may better represent the interaction of members and frame sway. Even then, eigenvalue results describe an idealized critical state and do not by themselves include imperfections, yielding, residual stress, connection slip, or construction sequence.

8. Imperfections and realistic stability assessment

No fabricated steel frame is perfectly straight, perfectly plumb, or perfectly stress-free. Important imperfections include:

• initial member crookedness;
• frame out-of-plumbness and lack of verticality;
• residual stresses from rolling, welding, cutting, and cooling;
• connection gaps, slip, and tolerance effects;
• load eccentricity;
• foundation movement and differential settlement.

An initially crooked compression member experiences bending when loaded. A simple representation is:

where y_0(x) is an assumed initial imperfection, y_m is its maximum amplitude, x is the coordinate along the member, and L is the member length. The sine shape is an idealization selected for analytical convenience; actual crookedness may have a different shape.

Frame imperfections can be represented by an initial sway angle, equivalent horizontal forces, or explicit geometric imperfections in a nonlinear model. Equivalent horizontal forces are useful for hand calculations and code procedures, whereas explicit imperfections are useful in advanced numerical analysis. The chosen representation must be consistent with the design standard and must not double-count the same imperfection in multiple ways.

Residual stresses may cause portions of a cross-section to yield earlier than a uniform-stress model predicts. They also influence inelastic buckling. Welding and restrained cooling can produce particularly significant residual stress patterns.

9. Frame stability and drift

A frame is stable when it has a unique equilibrium configuration for the considered loading and boundary conditions, and when small disturbances do not produce uncontrolled displacement. Stability may be lost through member buckling, a sway mechanism, connection failure, foundation movement, or a mechanism caused by inadequate bracing.

Story drift is the relative horizontal displacement between two adjacent floor levels. Excessive drift can damage cladding and partitions, impair building serviceability, increase P–Δ effects, and create uncomfortable movement. Drift limits are normally prescribed by codes or project criteria and depend on occupancy, nonstructural systems, wind or seismic action, and analysis assumptions.

A useful qualitative stability indicator compares gravity load with lateral stiffness. In a story-level approximation, a stability coefficient may be written as:

where P_story is the gravity load carried by the story, Δ_story is its interstory drift, V_story is the story shear, and h_story is story height. This dimensionless ratio measures the relative importance of second-order gravity effects. Its precise interpretation and limiting values are code-dependent; it is not a universal acceptance criterion.

If second-order effects are small, a first-order model may be adequate subject to the governing code. If they are moderate, amplified first-order forces may be permitted. If they are large, a geometrically nonlinear analysis, possibly including material nonlinearity and imperfections, may be required.

10. Analysis procedures and nonlinear behavior

A practical hierarchy of analysis is:

1. Linear first-order analysis for stable, sufficiently stiff systems where deformation effects are demonstrably small.
2. Elastic second-order analysis when P–Δ and P–δ effects are important but stresses remain within the elastic range.
3. Geometrically and materially nonlinear analysis when instability, yielding, plastic hinges, connection nonlinearity, or collapse mechanisms must be represented.
4. Advanced analysis with imperfections and residual stresses for critical structures or where permitted by the design standard.

A linear eigenvalue buckling analysis solves an idealized stability problem and estimates load multipliers at which the tangent stiffness becomes singular. It is valuable for identifying likely buckling modes, but it normally overestimates real resistance because it assumes perfect geometry and elastic behavior.

Nonlinear analysis requires careful control of load increments, convergence tolerances, element formulation, boundary conditions, and imperfections. A nonconvergent solution may indicate a physical instability, but it may also result from poor numerical settings or an inadequate model. Engineering judgment remains essential.

11. Worked example 2: elastic buckling comparison

Two pin-ended steel columns have the same E and cross-sectional area. Column A has length 3 m and second moment of area I. Column B has length 6 m and the same I. Compare their Euler critical loads.

Using:

for pin-ended columns:

Thus, Column B has one quarter of the elastic critical load of Column A. The result does not mean that Column B’s code design resistance is necessarily exactly one quarter, because real design resistance also depends on yielding, imperfections, residual stresses, section classification, and safety factors. It does demonstrate the strong influence of unsupported length.

12. Applied case: stability review of a warehouse frame

Background

The following is an illustrative case, not a project-specific design. A single-storey steel warehouse uses rigid portal frames at regular spacing. The roof carries gravity loads, while wind produces transverse frame sway. The columns are relatively slender, and the roof cladding is assumed to provide restraint only where positively detailed and verified.

Problem

A preliminary first-order analysis gives acceptable member strength ratios. However, the calculated roof drift is significant, and the columns carry substantial compression. The design team must determine whether the first-order results are reliable.

Analysis

The team first checks the idealization. Column bases are modeled according to their actual rotational stiffness rather than automatically as fixed. The rafters and columns are assigned correct flexural stiffness, and releases are checked. The roof diaphragm and any longitudinal bracing are traced to the foundations to confirm a complete lateral load path.

A second-order elastic analysis is then performed. Initial frame sway is included in accordance with the applicable standard. The analysis shows increased column moments caused by P–Δ action. A separate member check considers P–δ effects and buckling about both principal axes. The team also verifies that purlins, eaves ties, and bracing can provide the assumed restraint.

Approach and solution

The design is revised in two ways: a longitudinal bracing line is added to improve global stability, and selected columns are increased in weak-axis stiffness. Connection stiffness and base assumptions are updated using the connection designer’s information. The model is rerun with governing load combinations, realistic imperfections, and construction-stage conditions.

Outcome

The revised frame has lower drift and lower second-order moment amplification. The final design is not justified merely by a lower member utilization ratio; it is justified by a continuous load path, verified restraints, acceptable drift, stable analysis behavior, and documented assumptions.

Learning points

1. First-order strength checks can conceal instability sensitivity.
2. Bracing is effective only when it is continuous, connected, and strong enough.
3. Effective length depends on the frame, not solely on the isolated column.
4. Model assumptions must be communicated to the connection and construction teams.
5. Construction-stage stability may govern before the completed building provides its intended restraint.

13. Professional practice, safety, and technology

Stability design is a safety-critical activity. Engineers and technologists must identify the governing code, document load combinations, record assumptions, and obtain independent checking for significant or unusual structures. Altering a bracing member, connection, or temporary support without reassessment can change the entire stability system.

Construction sequencing deserves explicit attention. A frame may be stable when complete but unstable while columns are being erected, before diaphragm action develops, or before bolts and welds are fully installed. Temporary bracing, erection tolerances, lifting forces, and wind during construction must be addressed in the temporary works design.

Structural software, BIM models, automated code checks, and digital surveying can improve coordination. Artificial intelligence may assist with model checking, anomaly detection, and sensitivity studies, but it cannot replace engineering responsibility. AI-generated or automatically assembled models require verification of geometry, units, releases, restraints, load combinations, and code settings. A visually complete digital model is not evidence of structural correctness.

Sustainable design also has a stability dimension. Efficient use of steel should not produce excessive slenderness, difficult erection, or dependence on uncertain restraint from nonstructural components. Robustness, inspectability, repairability, and safe deconstruction should be considered alongside minimum material quantity.

14. Common errors and best practices

Common errors include treating a semi-rigid connection as perfectly pinned or rigid without justification; omitting weak-axis buckling; assuming cladding provides restraint without positive attachment; using an inappropriate effective-length factor; ignoring second-order effects; applying loads to the wrong member axis; and accepting software output without checking equilibrium and deformed shape.

Best practice is to:

• sketch the physical load path before building the analytical model;
• perform independent equilibrium checks for reactions and total load;
• inspect undeformed and deformed shapes;
• compare simple members with hand calculations;
• test sensitivity to stiffness, supports, and connection assumptions;
• check both completed and construction-stage conditions;
• consider global, member, local, torsional, and connection stability;
• record units, code edition, load combinations, imperfections, and analysis type;
• arrange independent review for critical assumptions.

Chapter Summary

Structural analysis idealizes a steel structure so that equilibrium, compatibility, and constitutive behavior can be applied. Determinate structures can be solved from equilibrium alone, whereas indeterminate frames require stiffness and compatibility relationships. The matrix stiffness equation [[M:[K]{u} = {P}]] provides the foundation for most computer-based linear analysis.

First-order analysis uses the original geometry. Second-order analysis includes the effect of deformation on equilibrium; the additional moment is conceptually related to [[M:PΔ]]. Member P–δ effects and frame P–Δ effects can substantially amplify moments in slender or lightly braced systems.

Euler’s equation, [[M:P_cr = π²EI ÷ (KL)²]], describes ideal elastic buckling of a uniform pin-ended column and shows the strong influence of effective length and stiffness. Real steel members are affected by yielding, residual stresses, local buckling, initial crookedness, eccentricity, and connection behavior. Therefore, code-based buckling resistance and realistic stability assessment are required.

Frame stability depends on a complete load path, adequate bracing, member and connection stiffness, realistic supports, acceptable drift, and consideration of imperfections and construction stages. Computer analysis is valuable, but its results are only as reliable as the structural model and engineering assumptions.

Key Terms

Buckling: Loss or change of equilibrium of a compression member or structural system.

Compatibility: Requirement that connected structural components undergo consistent deformation.

Constitutive relationship: Relationship between stress and strain or force and deformation.

Determinate structure: Structure whose reactions and internal forces can be obtained from equilibrium alone.

Effective length: Length of an equivalent pin-ended member having the same elastic critical load.

Euler buckling: Ideal elastic buckling theory for a slender, initially straight column.

First-order analysis: Analysis using the undeformed structural geometry.

Flexural buckling: Buckling involving lateral bending of a member.

Global buckling: Instability involving the member or frame as a whole.

Indeterminate structure: Structure requiring compatibility and stiffness relationships in addition to equilibrium.

Initial imperfection: Deviation from the ideal geometry before loading.

Local buckling: Buckling of a plate element within a cross-section.

P–Δ effect: Second-order effect caused by axial load acting through overall frame or member translation.

P–δ effect: Second-order effect caused by axial load acting through local member curvature.

Residual stress: Stress remaining in a member after fabrication or cooling without external load.

Second-order analysis: Analysis in which equilibrium is formulated using the displaced geometry.

Sway frame: Frame whose lateral translation is a significant part of its response or buckling mode.

Stability: Ability of a structure to maintain equilibrium under disturbances and design actions.

Story drift: Relative lateral displacement between adjacent levels.

Stiffness matrix: Matrix relating nodal displacements and rotations to applied nodal forces and moments.

Review Questions

Short Answer Questions

1. What are the three fundamental requirements used in structural analysis?
2. Distinguish a determinate structure from an indeterminate structure.
3. What is the difference between a P–Δ effect and a P–δ effect?
4. State the assumptions underlying Euler’s elastic buckling equation.
5. Why can a first-order analysis be unconservative for a slender unbraced frame?

Long Answer Questions

1. Explain how a steel building is idealized for analysis, including nodes, members, supports, connections, and load paths.
2. Describe the matrix stiffness method and explain the meaning of [K], {u}, and {P}.
3. Discuss the physical meaning of effective length and explain why it depends on the surrounding frame.
4. Explain how initial imperfections, residual stresses, and construction tolerances affect column and frame stability.
5. Compare linear first-order, elastic second-order, and geometrically/materially nonlinear analysis.

Analytical / Application Questions

1. A pin-ended column has E = 200 GPa, I = 8.0 × 10⁶ mm⁴, and length 4.0 m. Calculate its Euler critical load. State whether the result can be used directly as a design resistance and justify your answer.
2. A two-storey unbraced frame shows a large increase in column moment when second-order analysis is used. Identify at least four possible causes and propose practical design or modeling checks.
3. A warehouse frame is stable when complete but becomes flexible during erection. Develop a stability review procedure addressing temporary bracing, construction sequence, wind, connections, foundations, and verification of the final analytical model.

Multiple Choice Questions with Answers

Question 1

Which set of equations expresses planar static equilibrium?

A. [[M:∑F_x = 0]], [[M:∑F_y = 0]], and [[M:∑M_O = 0]]

B. [[M:∑F_x = 1]], [[M:∑F_y = 1]], and [[M:∑M_O = 1]]

C. Only [[M:∑M_O = 0]]

D. Only the compatibility equations

Correct Answer: A. [[M:∑F_x = 0]], [[M:∑F_y = 0]], and [[M:∑M_O = 0]]

Question 2

What additional requirements are needed for an indeterminate structure after equilibrium equations are written?

A. Only a larger load factor

B. Compatibility and constitutive or stiffness relationships

C. Only a reduction in steel grade

D. No additional requirements

Correct Answer: B. Compatibility and constitutive or stiffness relationships

Question 3

In a second-order analysis, equilibrium is formulated using which geometry?

A. The undeformed geometry only

B. The displaced geometry

C. The cross-sectional geometry only

D. The geometry of the foundation only

Correct Answer: B. The displaced geometry

Question 4

For an axial force P acting through a lateral displacement Δ, the conceptual second-order moment is:

A. P ÷ Δ

B. P + Δ

C. PΔ

D. P − Δ

Correct Answer: C. PΔ

Question 5

For an ideal pin-ended column, Euler’s critical load is proportional to which quantity?

A. [[M:EI ÷ L²]]

B. [[M:EI × L²]]

C. [[M:L² ÷ EI]]

D. [[M:E ÷ I L²]]

Correct Answer: A. [[M:EI ÷ L²]]

Question 6

If the effective length of an ideal elastic column doubles while all other properties remain constant, its Euler critical load becomes:

A. Twice as large

B. Unchanged

C. Half as large

D. One quarter as large

Correct Answer: D. One quarter as large

Question 7

Which statement about Euler buckling load is correct?

A. It automatically equals the code design resistance of every steel column

B. It assumes a perfect elastic member and is a reference stability quantity

C. It applies only to beams in bending

D. It includes all fabrication imperfections automatically

Correct Answer: B. It assumes a perfect elastic member and is a reference stability quantity

Question 8

Which condition most directly increases frame P–Δ effects?

A. Low axial load and zero drift

B. High lateral stiffness and short columns

C. Large gravity load combined with significant lateral drift

D. A perfectly rigid diaphragm with no gravity load

Correct Answer: C. Large gravity load combined with significant lateral drift

Question 9

Which is an example of an initial imperfection?

A. A perfectly straight theoretical member

B. Member crookedness before loading

C. A reaction calculated from equilibrium

D. Young’s modulus of steel

Correct Answer: B. Member crookedness before loading

Question 10

Why should a construction-stage stability check be performed?

A. Completed-building bracing and diaphragm action may not yet exist

B. Steel has no strength during construction

C. Euler’s equation is prohibited during erection

D. Construction loads are always zero

Correct Answer: A. Completed-building bracing and diaphragm action may not yet exist