Study guide
Elastic Deflection Methods
PE Civil: Structural Exam
Track your progress as you read
Finished sections get a check mark.
1 Exam Identity and Reference Workflow
2026 PE Civil: Structural Identity, Format, and Scoring Electronic Handbook, Design Standards, Units, and Version Control
2 Chapter 2: Gravity and Construction Loads
2.1 Dead Loads and Self-Weight 2.2 Live Loads, Reduction, and Load Placement 2.3 Construction-Stage Loads
3 Environmental Loads
Wind Loads and Pressure Paths Seismic Loads and Lateral Force Distribution Snow, Rain, and Ice Loads
4 Moving, Impact, and Lateral Soil Loads
Vehicular, Crane, and Other Moving Loads Impact and Dynamic Allowances Earth Pressure and Surcharge Loads
5 Chapter 5: Load Paths and Combinations
5.1 Tributary Areas and Complete Vertical/Lateral Load Paths 5.2 Strength and Service Load Combinations
6 Reactions, Diagrams, and Basic Axial Effects
Free-Body Diagrams, Reactions, Shear, and Moment Qualitative Diagrams, Influence Lines, and Critical Placement Axial Tension, Compression, and Deformation
7 Chapter 7: Axial Systems and Shear Effects
7.1 Trusses, Braces, and Column Axial Force 7.2 Shear Stress, Shear Flow, and Built-Up Sections 7.3 One-Way, Two-Way, Punching, and Interface Shear Effects
8 Flexure
Section Properties and Elastic Flexural Stress Beam and Slab Flexural Load Effects Biaxial and Unsymmetrical Flexure
9 Combined Stress and Deflection
Combined Axial and Flexural Stress Principal Stress, Mohr's Circle, and Combined Shear Elastic Deflection Methods
10 Chapter 10: Serviceability, Torsion, and Stability
10.1 Serviceability: Deflection, Drift, Vibration, and Ponding 10.2 Torsion and Warping Effects 10.3 Buckling, Effective Length, and Second-Order Stability
11 Fatigue, Robustness, Thermal, and Bearing Effects
Fatigue and Progressive-Collapse Concepts Thermal Deformation and Bearing/Contact Stress
12 Inspections, Submittals, and Adjacent Facilities
Special Inspections and Verification Duties Shop Drawings, Submittals, RFIs, and Delegated Design Construction Effects on Adjacent Facilities
13 Chapter 13: Temporary Works and Safety
13.1 Formwork, Falsework, Scaffolding, Shoring/Reshoring, Bracing, and Anchorage 13.2 Construction, Roadside, and Work-Zone Safety
14 Soils and Core Concrete Properties
Soil Phase Relationships and Classification Soil Strength, Permeability, and Compressibility Fresh and Hardened Concrete Properties
15 Chapter 15: Prestressed Concrete and Steel Properties
15.1 Reinforced, Precast, Pretensioned, and Post-Tensioned Concrete Materials 15.2 Structural and Cold-Formed Steel Properties 15.3 Steel Toughness, Weldability, Corrosion, and Temperature Effects
16 Timber, Masonry, and Material Conformance
Timber Properties and ASD Design Values Masonry, CMU, Mortar, Grout, and Brick Veneer Properties Concrete and Masonry Test Methods Steel, Timber, and Other Material Specification Conformance
17 Horizontal Members I
Steel Beams and Girders Reinforced Concrete Beams and Slabs Timber and Masonry Horizontal Members
18 Chapter 18: Diaphragms and Vertical Members
18.1 Diaphragms, Collectors, Struts, and Composite Slabs 18.2 Steel Columns and Beam-Columns 18.3 Concrete Columns and Bearing Walls
19 Shear Walls, Trusses, and Frames
Shear Walls and Vertical Lateral Members Trusses and Braced Systems Moment Frames and Stability Systems
20 Composite/Bridge Systems and Bolted Connections
Composite Construction and Structural System Selection Bridge Systems and Load Distribution Bolted Steel Connections
21 Chapter 21: Welded, Embedded, and Anchored Connections
21.1 Welded Steel Connections 21.2 Bearing, Embedded, and Precast Connections 21.3 Anchors, Post-Installed Anchors, and Wood/Masonry Connections
22 Shallow Foundations
Isolated, Spread, and Strip Footings Combined and Strap Footings Mat Foundations and Foundation Slabs
23 Deep Foundations
Driven Piles: Axial Capacity and Settlement Drilled Shafts, Piers, and Caissons Pile Groups, Caps, and Lateral Deep-Foundation Response
24 Chapter 24: Retaining Structures
24.1 Lateral Earth/Water Pressure and Drainage 24.2 Sliding, Overturning, Bearing, and Global Stability 24.3 Concrete Cantilever, Gravity, and Basement Walls 24.4 Sheet-Pile, Anchored, and Mechanically Stabilized Earth Walls
Elastic Deflection Methods
Key Takeaways Explain This Topic AI
Slope is the rotation theta = dv/dx, while deflection v is translational displacement; one can be zero when the other is not
Use curvature M/(EI) with the declared sign convention and enforce the displacement and rotation conditions supplied by the actual supports
Tabulated formulas are valid only when load pattern, support conditions, evaluation location, and stiffness assumptions match
Superposition requires linear response and identical geometry, boundary conditions, stiffness model, coordinates, and response direction
A unit force in virtual work extracts a displacement, while a unit moment extracts a rotation
Check deflection dimensions, symmetry, support values, and deformation shape before accepting a numerical answer
Last updated: July 2026
Elastic Deflection Methods
For July 2026 problems, use the beam formulas and mechanics relationships in the current NCEES PE Civil Reference Handbook supplied for the test date . A deflection formula is not identified only by its load. Support conditions, evaluation point, stiffness, and sign are part of the formula.
- Slope and displacement are different responses
- Let transverse deflection be v(x) . For small rotations, beam slope is
θ(x) = dv/dx .
Slope is a rotation, measured in radians and dimensionless in calculations. Deflection is a translation with length units. At the center of a symmetrically loaded simple beam, slope is zero by symmetry while deflection is usually maximum, not zero. At a simple support, deflection is zero but slope can be nonzero. Confusing these conditions corrupts integration constants.
For an Euler-Bernoulli beam under a consistent sign convention, curvature is related to bending moment by a form of
d²v/dx² = M(x)/(EI) .
Some sign conventions place a minus sign on one side. Either is workable when the load, moment, curvature, and deflection signs are consistent. E is elastic modulus and I is the section moment of inertia about the bending axis. Use the stiffness specified for the problem; do not silently alternate among gross, cracked, transformed, or effective inertia.
Method 1: integrate the moment-curvature equation
Find reactions, write M(x) for every interval, then integrate:
EI θ(x) = ∫M(x) dx + C1
EI v(x) = ∫∫M(x) dx dx + C1x + C2 .
Determine constants from actual boundary and continuity conditions. Typical geometric conditions are:
- Location Geometric condition
- Pin or roller support v = 0
- Fixed support v = 0 and θ = 0
- Interior point with no hinge or gap v and θ are continuous
- Symmetry line of symmetric deformation θ = 0
An internal hinge has zero bending moment but generally not zero deflection. At a free end, deflection and slope are not prescribed as zero; end shear and moment conditions follow from loading. Piecewise moment functions require enough constants and continuity or compatibility equations.
Method 2: handbook cases and superposition
Tabulated formulas save time when the diagram exactly matches. Confirm whether the listed quantity is maximum deflection, deflection at a named point, or end slope, and confirm whether the load is total force or force per length. Powers of span are a strong clue: a point-load beam deflection commonly scales with PL³/(EI) , while uniform-load deflection scales with wL⁴/(EI) .
Linear-elastic responses can be superposed:
vtotal(x) = Σvi(x) and θtotal(x) = Σθi(x) .
Add signed responses at the same coordinate and in the same direction. Every component case must use the same structure, supports, span, coordinate system, and stiffness model. One cannot add a simply supported formula to a cantilever formula and claim the result describes either beam. Superposition also becomes unreliable after yielding, contact changes, large-displacement behavior, or other nonlinear response.
Worked combined loading
A simply supported steel beam spans 16 ft and has constant E = 29,000 ksi and I = 400 in⁴ . It carries a full-span uniform load w = 1.0 kip/ft and a 12-kip point load at midspan. Find elastic midspan deflection. Convert L = 192 in and w = 1/12 kip/in . Both handbook cases have the same beam and boundary conditions, so superposition is valid.
- For the uniform load,
- δw = 5wL⁴/(384EI)
= 5(1/12)(192⁴)/[384(29,000)(400)] = 0.1271 in downward .
- For the midspan point load,
- δP = PL³/(48EI)
= 12(192³)/[48(29,000)(400)] = 0.1525 in downward .
The signed responses act in the same direction, so
δtotal = 0.1271 + 0.1525 = 0.2796 in downward .
By symmetry, the midspan slope is zero even though the deflection is the largest downward displacement. At each simple support, deflection returns to zero. An answer of zero midspan deflection obtained from θ = 0 would confuse rotation with translation.
Method 3: virtual work and a unit load
Virtual work is efficient when only one displacement is required. For bending deformation of a linear-elastic beam,
δ = ∫ M(x)m(x)/(EI) dx ,
where M is the moment from real loads and m is the moment from a unit force applied at the desired displacement location in the desired direction. The integral covers every member and interval contributing to response. Its sign reports displacement relative to the chosen unit-force direction.
To calculate a rotation instead, apply a unit moment at the desired point and direction; the corresponding virtual moment diagram replaces m . A unit force extracts translation, while a unit moment extracts rotation. For a cantilever of length L with a real tip load P , measured from the fixed end, M = -P(L-x) and the same-direction unit-load diagram is m = -(L-x) . Then
δtip = ∫0^L P(L-x)²/(EI) dx = PL³/(3EI) ,
which agrees with the standard case. If axial deformation matters, a corresponding ∫Nn/(AE) dx term may be added; include only deformation modes required by the model.
Choose and verify
Use integration when the moment function and boundary conditions are straightforward, tables when the case matches exactly, superposition when several compatible linear cases combine, and virtual work when one response in a larger system is sought. Then perform four checks:
- dimensions reduce to length for deflection and to a dimensionless value for slope;
- supports and symmetry satisfy their geometric conditions;
- curvature sense follows the bending-moment sign and the deflected shape is physically plausible;
load scaling is sensible—doubling a load doubles linear-elastic response, while doubling span can increase deflection by a cubic or fourth-power factor.
These checks distinguish a correct formula lookup from a correct structural solution.