CELE Structural Theory & Analysis — Deflections of StructuresStudy Notes

When external loads deform an elastic structure, the work done by those loads is stored internally as **strain energy**. For an axially loaded member under force N, the strain energy is: **U_axial = N²L / (2AE)** where N = internal axial force, L = member length, A = cross-sectional area, E = Young's modulus. For a beam or frame member in bending, the strain energy is distributed along the length: **U_bending = ∫ M² / (2EI) dx** where M = bending moment, I = second moment of inertia, integrated over the member length. The **principle of virtual work** states that if a structure is in equilibrium under real loads and we impose a kinematically compatible (geometrically admissible) virtual displacement, then the external virtual work equals the internal virtual work. This equivalence is the gateway to the unit-load method. **Key distinction:** Strength of Materials methods (double integration of M/EI) work well for simple beams but become tedious for frames and require solving boundary-value problems. Energy methods extract a single deflection directly from internal forces and are universally applicable. In Philippine engineering practice (RA 544 § 34.1), civil engineers must verify serviceability limits. NSCP 2015 Section 501.3.1 specifies maximum deflections: for floor members, L/240 for live load plus dead load, L/360 for live load alone. Understanding how to calculate deflections is therefore not academic—it is a professional responsibility.

The unit-load method is the most practical approach for finding deflections of trusses. The procedure is: 1. **Solve the structure under real loads** to find all member forces N_i. 2. **Remove real loads** and apply a **unit load** at the joint and in the direction of the desired deflection. 3. **Solve for member forces n_i** under the unit load using the same geometry. 4. **Combine using the principle of virtual work:** **δ = Σ (n × N × L) / (AE)** where the sum is over all members, and: - n = member force from unit load - N = member force from real loads - L = member length - AE = axial rigidity (constant for all members if uniform) If AE varies by member, the formula becomes: **δ = Σ (n × N × L) / (AE)ᵢ** The physical interpretation: tension × tension (or compression × compression) contributes positively to deflection; opposite signs contribute negatively. The method is insensitive to redundancy—you never solve for unknown reactions. **Advantages for trusses:** - No need to solve indeterminate trusses explicitly; deflection works on the determinate primary structure - Computationally efficient; can be coded or done by hand for modest truss sizes - Directly gives the deflection requested; no need for superposition afterwards **Sign convention:** Apply the unit load in the positive direction (rightward for horizontal, upward for vertical). The result δ is positive if deflection is in the direction of the unit load. In Filipino practice, building trusses commonly appear in roofs of warehouses, institutional buildings, and residential structures. Excessive deflection of roof trusses can cause ponding of water (adding parasitic dead load) and is a serviceability issue tracked by structural engineers under NSCP 2015 Section 501.3.1.

For continuous structures (beams and frames), the unit-load method generalizes to: **δ = ∫ (m × M) / (EI) dx** where: - M(x) = real bending moment distribution under applied loads - m(x) = bending moment distribution from a unit load applied at the point of interest - EI = flexural rigidity (constant in a prismatic member, variable if the cross-section changes) - Integration is over the entire structure For a **rotation** at a point, apply a unit **couple** (moment) instead of a unit load: **θ = ∫ (m_θ × M) / (EI) dx** where m_θ is the moment diagram from the unit couple. **Physical interpretation:** The integral is a weighted sum of the real moment M and the "influence" m. Regions where both M and m are positive (or both negative) contribute positively to deflection; opposite signs contribute negatively. **Procedure:** 1. Sketch the **real moment diagram** M(x) under the actual applied loads. 2. Apply a **unit load** at the point where deflection is desired (or a unit moment for rotation). 3. Sketch the **unit moment diagram** m(x). 4. Evaluate the integral ∫(mM)/(EI)dx. For piecewise linear/parabolic functions, use geometric properties of diagrams (products of areas and centroids). 5. The result is the deflection in the direction of the unit load. **Common shortcut for products of diagrams:** If M and m are both triangular or parabolic over a span, use: - Triangle × Triangle: (1/2 × base × height₁) × (height₂ / 3) - Triangle × Parabola: (1/2 × base × height) × (centroid distance) For complex shapes, numerical integration or geometric diagram multiplication tables (common in older textbooks) apply. **Variable EI:** If cross-section changes (e.g., T-beam, haunched member, composite beam), write EI(x) inside the integral: **δ = ∫ [m(x) × M(x)] / [EI(x)] dx** This is encountered in bridge design and special structures; NSCP 2015 Section 402.7 (composite members) and Section 506 (prestressed concrete) touch on this. Most board-exam problems assume constant EI within each member. **Sign convention:** Positive δ is in the direction of the unit load. Positive θ is counterclockwise (by right-hand rule). A negative result means the structure deflects/rotates opposite the unit load direction.

**Castigliano's Second Theorem** states that the partial derivative of the total strain energy U with respect to a load P_i equals the deflection δ_i at the point of that load in the direction of P_i: **δᵢ = ∂U / ∂Pᵢ** Similarly, for a couple M_i: **θᵢ = ∂U / ∂Mᵢ** For a structure composed of members in bending, the total strain energy is: **U = ∫ M² / (2EI) dx** (over all members) To apply Castigliano's theorem: 1. Write the total strain energy U as a function of all loads. 2. Differentiate U with respect to the load at which deflection is desired. 3. Substitute actual load values (or set dummy loads to zero if no real load acts at that point). Algebraically, this is equivalent to the unit-load method. The derivative ∂M/∂P_i is essentially the moment diagram m_i from the unit load. Thus: **δᵢ = ∂U / ∂Pᵢ = ∫ [M / (EI)] × [∂M / ∂Pᵢ] dx = ∫ [(∂M/∂Pᵢ) × M] / (EI) dx = ∫ (m_i × M) / (EI) dx** which is exactly the unit-load formula. **Advantage of Castigliano:** It is sometimes more elegant for problems where multiple loads are present and you wish to find deflections at several points. Instead of solving separate unit-load cases, you can write U once and differentiate symbolically. **Dummy load technique:** If no real load acts at the desired point, introduce a dummy load Q there, write U(Q), differentiate with respect to Q, then set Q = 0. This avoids the awkwardness of an undefined derivative. **Proof sketch:** From the principle of virtual work, the first variation δU equals external virtual work. Setting the structure in equilibrium and introducing infinitesimal variations, the deflection δᵢ is recovered as ∂U/∂Pᵢ. The theorem is a cornerstone of optimization and is used in advanced methods like the finite element method. **Practical note:** In modern engineering, Castigliano's theorem is less often used by hand (the unit-load method is more direct), but it appears in theoretical courses, research, and optimization. It is also important for understanding strain energy methods in matrix structural analysis (taught in graduate programs and relevant to specialized PRC exams).

Deflection methods are essential for analyzing **statically indeterminate structures**. While the structure's equilibrium equations alone are insufficient to determine all internal forces, the **compatibility conditions** (geometric constraints that deformations must satisfy) provide additional equations. **Compatibility condition example (continuous beam):** At an internal support of a continuous beam, the slope and deflection on the left and right sides must be equal (the beam doesn't tear or kink at the support). Mathematically: **θ_left = θ_right** **δ_left = δ_right** These are expressed in terms of the unknown support reactions using deflection formulas (unit-load or Castigliano), yielding equations that can be solved for the redundant reactions. **Degree of indeterminacy (redundancy):** For a planar structure, r = 3 + m − 2j, where m = members, j = joints. Each redundant (statically indeterminate) reaction requires one compatibility condition. The next chapters (Slope-Deflection and Moment-Distribution Methods) build systematic procedures based on these ideas. **Energy method for redundants:** An alternative approach is to express the total strain energy U in terms of the redundant(s), then minimize U with respect to each redundant, setting ∂U/∂X = 0 (principle of minimum strain energy). This yields the equations for the redundants directly. **Serviceability verification (NSCP 2015):** After solving an indeterminate structure, the deflections must be checked against limits: - **Cantilevers:** δ ≤ L/180 (more restrictive than simply supported) - **Floor beams (all loads):** δ ≤ L/240 - **Floor beams (live load only):** δ ≤ L/360 - **Long-span members (trusses, girders):** δ ≤ L/300 to L/500 depending on finish If deflections exceed limits, the cross-section must be increased (raise I, lower E by material choice, or reduce span). Deflection-controlled design is common in modern structures. **RA 544 § 34.1 (Professional Responsibility):** Civil engineers are required to ensure structures meet all applicable codes, including serviceability. Excessive deflection is a common cause of building complaints (cracked finishes, water pooling, equipment malfunction). Checking deflections is not optional—it is a legal and ethical obligation.

**Unit consistency and numerical accuracy:** Mismatched units are the most common error in deflection calculations. For the truss formula Σ(nNL/AE): - If n, N are in kN; L in m; A in m²; E in GPa → convert E to kN/m² = GPa × 10⁶ - Alternatively, keep all quantities in SI base units (N, mm) and convert the final result to m or mm - Example: N in kN, L in m, A in m², E in GPa → AE in kN·m² / GPa = [A(m²) × E(10⁶ kN/m²)] = 10⁶ AE(m² × kN/m²) A systematic approach: **Choose one unit system and stick to it.** Many engineers use SI base units (N, mm) throughout and convert at the end. Spreadsheets are invaluable for checking arithmetic. **Moment diagram products (diagram multiplication):** When integrating ∫(mM)/(EI)dx over a span, you often encounter products of triangular and parabolic diagrams: - **Triangle × Triangle:** (1/2 × b × h₁) × (height h₂ at centroid of first triangle) / 3 - **Triangle × Parabola:** Requires centroid location of parabola (2/3 or 1/3 of height depending on shape) - **Parabola × Parabola:** Complex; typically use numerical integration or tabulated results For exam problems, diagrams are usually simple (linear, triangular); memorize the standard cases. If in doubt, integrate symbolically or numerically. **Sign conventions and deflection direction:** - Deflection δ is **positive in the direction of the applied unit load.** - Rotation θ is **positive counterclockwise** (by right-hand rule). - If your calculation yields δ = −3 mm and you applied the unit load downward, the structure deflects **3 mm upward** (opposite the unit load). - This is not an error—it is a physically meaningful result. **Common student mistakes:** 1. **Forgetting to remove the real load before applying the unit load.** The unit load is virtual; you solve the structure under the unit load alone to get the influence diagram m. 2. **Mismatched integration limits.** If you subdivide the structure into regions (cantilever, span, overhang), be careful with x-coordinate origins. Draw a clear diagram. 3. **Ignoring shear strain energy.** For deep beams (h/L > 1/4 or so), shear deformation contributes. The shear strain energy is U_shear = ∫(V²)/(2GA_s)dx, where A_s = shear area (often = 5A/6 for rectangles). Board exams typically ignore shear unless the problem explicitly states "include shear deformation." NSCP 2015 Section 501.4 mentions that shear can be significant in short, deep members. 4. **Assuming constant EI when the section varies.** If a T-beam has different I in positive and negative moment regions (top and bottom portions effective), divide the integration into sections and integrate over each separately. 5. **Not checking reasonableness.** A deflection of 500 mm for a 3 m span (L/6) is typically unrealistic unless the beam is vastly undersized. Quick hand checks (e.g., comparing to published formulas) catch arithmetic errors. **Numerical integration for complex shapes:** When the moment diagram m or M is irregular (non-linear, with kinks), use Simpson's rule or the trapezoidal rule: - **Trapezoidal rule:** ∫f dx ≈ (Δx/2)[f₀ + 2f₁ + 2f₂ + ... + 2f_{n-1} + f_n] - **Simpson's 1/3 rule (parabolic approximation):** More accurate for smooth curves; requires an even number of intervals. Modern spreadsheets make this straightforward. For board exams, you are unlikely to encounter problems requiring numerical integration unless explicitly stated. **Deflection limits by structure type (NSCP 2015 Section 501.3.1):** | Structure Type | Limit | Condition | |---|---|---| | Floors | L/240 | All loads | | Floors | L/360 | Live load only | | Roofs | L/180 | All loads | | Cantilevers | L/180 | All loads | | Long-span trusses | L/300–L/500 | Depends on finish sensitivity | **Philippine building context:** Many Philippine structures are in tropical climates with high humidity, subject to water infiltration, and often have non-structural finishes (tiles, plaster) that are brittle under deflection. Strict compliance with NSCP 2015 deflection limits is essential to avoid callbacks and warranty claims.