CELE Structural Theory & Analysis — Deflections of StructuresRevision Notes

Formulas

Exam Tips

• Memorize the most tested NSCP 2015 deflection limits: L/360 (live load, floor members supporting brittle finishes), L/240 (total load), L/180 (roof, not supporting brittle finishes).
• When a problem asks 'does the beam satisfy deflection requirements?' always compute δ, then compare with L/limit.
• The elastic curve equation (EI d²y/dx² = M) is rarely tested directly; the board favors the energy methods.

Key Points

• Deflections are checked against NSCP 2015 Table 406.2.1 allowable limits (e.g., L/240 for total load, L/360 for live load on floors supporting non-structural elements).
• ACI 318-19 Section 24.2 limits immediate and long-term deflections of concrete members to protect attached non-structural elements.
• Energy stored in a deformed elastic structure is called strain energy U; it equals the work done by the applied loads (W_ext = U for linear elastic systems).
• For a linear elastic structure, the external work done by a slowly applied load P moving through displacement δ is W = Pδ/2.
• Deflection methods based on energy are general — they apply equally to trusses, beams, and frames without re-deriving separate equations.
• The principle of superposition applies when material is linearly elastic and deformations are small (both are standard assumptions in licensure exam problems).

Definitions

Section Title

1. Why Deflections Matter — Serviceability and Compatibility

Common Mistakes

• Applying deflection formulas to structures with large deformations or inelastic behavior where superposition is invalid.
• Confusing the deflection limit with the span: L/360 means deflection ≤ span/360, not 360 mm.
• Using NSCP live-load deflection limits for total-load checks — always identify which limit applies to which load combination.

Formulas

Exam Tips

• Always set up a tabular solution for board exams: columns for Member | L | A | E | N | n | nNL/AE. This organizes work and prevents arithmetic errors.
• For symmetric trusses with symmetric loading, n and N will be equal for symmetric members — compute once and multiply by 2.
• If the answer is negative, the deflection is opposite to the assumed direction of the unit load. State this explicitly in your answer.
• The most commonly tested truss deflection in board exams is the vertical deflection at a loaded joint of a simple Pratt or Warren truss.

Key Points

• The unit-load method is derived from the principle of virtual work: virtual external work = virtual internal work.
• Procedure: (a) Solve the real structure for member forces N under the actual loads. (b) Remove actual loads and apply a unit load (1 kN or 1 kN·m) at the point and direction of the desired deflection. (c) Solve for member forces n under the unit load. (d) Apply the truss deflection formula.
• The unit load must be in the same direction as the desired deflection (vertical unit load → vertical deflection; horizontal unit load → horizontal deflection; unit moment → rotation).
• A positive result means the deflection is in the same direction as the unit load; negative means opposite direction.
• For an L-shaped truss or any statically determinate truss, method of joints or sections gives both N and n.
• All members must use consistent units: if AE is in kN, then N in kN, L in m, δ in m.

Definitions

Section Title

2. Virtual Work Method (Unit-Load Method) — Trusses

Common Mistakes

• Applying the unit load in the wrong direction: a horizontal unit load computes horizontal deflection, not vertical.
• Using mixed units — N in kN with L in mm and AE in N produces a nonsensical answer. Always standardize (N in kN, L in m, AE in kN).
• Forgetting members with zero N or zero n — they contribute zero to the sum, but must not be omitted from the tabulation (to show awareness).
• Sign error: tension is (+), compression is (−). Like signs multiply to give a positive contribution (deflection in the direction of the unit load).
• Confusing n (unit-load forces) with N (real forces) in the summation table.

Formulas

Exam Tips

• Memorize the four standard formulas: cantilever-tip-point (PL³/3EI), cantilever-tip-UDL (wL⁴/8EI), SS-midspan-point (PL³/48EI), SS-midspan-UDL (5wL⁴/384EI). These appear in roughly 60% of beam deflection board exam problems.
• When integrating ∫mM/EI dx for linearly varying m and parabolic M, use the tabulated integral formulas (e.g., area of parabola = 2/3 × base × height) to avoid long integration.
• Check units at the end: the result must be in meters (if L is in m) or millimeters (if L in mm). A deflection of 0.007 m = 7 mm is reasonable; 700 m is not.
• For frames, sketch the m-diagram first (it is always simpler — usually triangular or trapezoidal) before drawing M.

Key Points

• For beams and frames, bending is the dominant internal action; shear and axial contributions are neglected unless stated otherwise.
• The procedure mirrors the truss method: (a) Compute M(x) due to real loads. (b) Apply a unit load at the point/direction of interest; compute m(x). (c) Integrate mM/EI over each member.
• The integration ∫mM/EI dx must be done member by member, with x measured consistently (usually from a free end or left support).
• For rotation, apply a unit couple (1 kN·m) instead of a unit force; the result is in radians.
• Graphical (moment-area) integration can be used when m is linear and M is parabolic or linear — use the M·A table (area of M-diagram times ordinate of m at centroid).
• For frames, each member contributes a separate integral; sum all contributions.
• EI may vary between members of a frame — use the correct EI for each member's integral.
• Results match classical formulas: cantilever end deflection = PL³/(3EI); simply supported midspan = PL³/(48EI) for central load, 5wL⁴/(384EI) for UDL.

Definitions

Section Title

3. Virtual Work Method (Unit-Load Method) — Beams and Frames

Common Mistakes

• Setting up M(x) with inconsistent sign conventions between the real and virtual systems — use the same sign convention throughout (positive sagging or positive hogging) for both M and m.
• Forgetting to integrate over ALL members of a frame, not just the loaded member.
• Using mm for L and kN for loads without adjusting E and I units — this is the single most common arithmetic error; use a consistent SI set (kN, m, kN·m²) or (N, mm, N·mm²).
• Applying the midspan formula PL³/48EI to a beam that is not simply supported or where the load is not at midspan.
• Confusing deflection (mm) with rotation (radians) when reading the question — 'find the slope' means compute θ, not δ.

Formulas

Exam Tips

• Whenever you see a beam/frame deflection problem with no load at the target point, think Castigliano + dummy load immediately.
• The fastest approach for standard beams (SS, cantilever) is to use the memorized formulas. Use Castigliano only when the problem geometry or loading is non-standard.
• In multi-span frames, the dummy load Q must be placed correctly — wrong placement changes ALL the M expressions.
• Castigliano and unit-load always give identical answers. If you solve by both methods and get different numbers, there is an arithmetic error.
• Board exams often ask: 'Using Castigliano's theorem, find the deflection at the free end.' This is a signal to show the ∫[M/EI × ∂M/∂P]dx procedure explicitly.

Key Points

• Castigliano's Second Theorem states: the partial derivative of the total strain energy U with respect to a load P_i equals the deflection δ_i at the point and in the direction of P_i.
• Similarly, the partial derivative of U with respect to an applied moment M_i equals the rotation θ_i at that point.
• For bending-dominated structures: δ = ∫[M/EI × ∂M/∂P] dx — this is algebraically identical to the unit-load method because ∂M/∂P plays the exact role of m.
• Dummy Load Technique: If no real load acts where the deflection is needed, introduce a fictitious dummy load Q at that point, compute ∂M/∂Q, integrate, then set Q = 0. This is the practical advantage of Castigliano over unit-load.
• Castigliano applies to any elastic structure — trusses, beams, frames, curved members.
• For trusses: δ = Σ[NL/(AE) × ∂N/∂P].
• The theorem is only valid for linear elastic structures (Hooke's Law obeyed, small deformations).

Definitions

Section Title

4. Castigliano's Second Theorem

Common Mistakes

• Differentiating U with respect to the wrong variable — you must differentiate with respect to the load AT the point of interest, not with respect to any load.
• Forgetting to set Q = 0 after differentiating when using the dummy load technique — leaving Q in the answer makes it a function of Q, not a number.
• Confusing Castigliano's First Theorem (∂U/∂δ = P, force from energy) with the Second Theorem (∂U/∂P = δ, deflection from energy).
• Not recognizing that ∂M/∂P is simply the m-diagram — students who see this connection solve problems much faster.
• Forgetting the factor of 2 when differentiating U = N²L/(2AE): ∂U/∂N = NL/(AE), then apply chain rule ∂N/∂P.

Formulas

Exam Tips

• Write all four fundamental formulas at the top of your scratch paper at the start of any board exam session — saves time during the test.
• Ratio trick: a fixed-fixed beam is 4× stiffer than SS (PL³/192 vs PL³/48), and a propped cantilever is intermediate. Useful for quick sanity checks.
• If EI is given in N·mm², convert to kN·m² by dividing by 10¹²: 2×10¹³ N·mm² = 2×10¹³/10¹² = 20 kN·m².

Key Points

• These formulas are derived by integration or virtual work; they must be memorized for board exam efficiency.
• All formulas assume homogeneous, prismatic beams (constant EI) with small, linear elastic deformations.
• For beams with varying EI, the formulas do not apply directly — use virtual work integration.
• The 'maximum deflection' in a simply supported beam under a non-central load occurs near but not exactly at midspan.
• For composite loading, use superposition: δ_total = δ_from_load1 + δ_from_load2 + ...
• NSCP 2015 requires that deflections be checked after construction and under service loads (not factored loads).

Definitions

Section Title

5. Key Standard Deflection Formulas Reference

Common Mistakes

• Using PL³/48EI when the load is NOT at midspan — this formula is only for a central load on a simply supported beam.
• Applying the SS beam formula to a fixed-end beam (δ_max for fixed-fixed, central load is PL³/192EI — four times stiffer).
• Superposing deflections for loads in different problems without verifying same span, same E, same I.

Exam Tips

• In 4-choice MCQ format (typical board exam), use process of elimination: wrong units, negative magnitudes, or values outside the 1–50 mm range for typical structural spans are immediately suspect.
• If the problem specifies E and I separately (e.g., E = 200 GPa, I = 450×10⁶ mm⁴), compute EI = 200,000 MPa × 450×10⁶ mm⁴ = 9×10¹³ N·mm² = 90,000 kN·m² before substituting.
• Time management: standard deflection formula problems (types 5.1–5.6 above) should take under 2 minutes each. Unit-load integration problems may take 5–8 minutes — budget accordingly.
• For frame problems with right-angle joints: treat each member separately; choose x-origin at the free end or joint to simplify m(x) expressions.

Key Points

• For unit-load (virtual work) method on beams: draw FBDs, write M(x), apply unit load, write m(x), integrate ∫mM/EI dx.
• For unit-load on trusses: find reactions and member forces for real loads (N), then for unit load (n), tabulate, sum nNL/AE.
• For Castigliano on beams: write M as a function of x AND P, differentiate ∂M/∂P, integrate ∫(M/EI)(∂M/∂P)dx.
• For Castigliano with dummy load: insert Q at target point, differentiate ∂M/∂Q, integrate, set Q=0.
• Always state units and direction of deflection in the final answer.
• Check against known formulas or order of magnitude estimates (mm range for typical spans is reasonable).

Section Title

6. Step-by-Step Procedures and Board Exam Strategy

Common Mistakes

• Skipping the free body diagram — without a proper FBD, moment equations M(x) will be wrong.
• Not checking whether EI is constant — if it varies, you cannot factor it out of the integral.
• Solving for deflection in kN·m (energy units) instead of m (deflection units) — unit check catches this: [kN·m × kN·m] / [kN·m²] = m ✓
• In Castigliano with dummy Q: differentiating M with respect to real load P instead of Q.