CELE Structural Theory & Analysis — Deflections of StructuresSummary

Strain energy is the elastic energy stored internally when a structure deforms under load. For axial members, U_axial = N²L/(2AE), where N is the axial force, L is length, A is area, and E is Young's modulus. For bending members, U_bending = ∫(M²/2EI)dx, where M is bending moment and I is moment of inertia. The total strain energy equals the external work done by loads on an elastic structure: W_external = U_internal. This equality is the cornerstone of energy methods.

The unit-load method calculates a single deflection by applying a fictitious unit load (or unit moment for rotations) at the point and direction of interest. The real structure experiences real loads with internal forces (N, M); the virtual load system produces virtual internal forces (n, m). The deflection is found from: δ = Σ(nNL/AE) for trusses, or δ = ∫(mM/EI)dx for beams/frames. The unit load can be a concentrated force (for linear deflection) or a concentrated moment (for angular deflection/rotation). This method is direct, does not require equilibrium equations to be solved for unknowns, and works uniformly across all structure types.

Castigliano's second theorem states that the deflection (or rotation) at a point equals the partial derivative of the total strain energy with respect to the load (or moment) applied at that point: δ_i = ∂U/∂P_i, θ_i = ∂U/∂M_i. For bending, this expands to δ = ∫(M/EI)(∂M/∂P)dx. If no real load exists at the deflection target, apply a dummy load Q, differentiate, then set Q = 0. Algebraically, Castigliano's theorem is equivalent to the unit-load method—the partial derivative ∂M/∂P plays the role of the virtual moment m. This theorem elegantly connects strain energy to deflections and is particularly powerful for finding deflections under multiple loads.

For truss analysis, each member carries an axial force N (found from joint equilibrium or method of sections under real loads). The unit-load method requires applying a unit load at the joint of interest and finding the member forces n in this virtual system. The joint deflection is then: δ = Σ(nNL/AE) summed over all members. Member length L and axial stiffness AE must be consistent in units (typically m, and kN respectively in SI). Sign convention: tension is positive, compression negative; products of like signs yield positive contributions to deflection. Common truss deflection problems involve triangular trusses, Warren trusses, or planar grids.

For bending structures (beams, frames, portal frames), the virtual work method uses the bending moment formulation: δ = ∫(mM/EI)dx, integrated over the length of all bending members. M(x) is the real bending moment under the applied loads; m(x) is the bending moment due to the unit virtual load. Both M and m vary along the span; numerical integration or segment-wise analytical integration is required. For frames, the integral extends over all members. For a rotation at a point, apply a unit moment instead of a unit force, and m becomes a couple. The product mM must account for sign conventions (tension on one side is positive bending).

The NSCP 2015 Chapter 5 (Code Provisions for Deflections) specifies maximum permissible deflections to protect structural and non-structural elements. Common limits include: floor beams under live load ≤ L/240, roof beams ≤ L/180, cantilevers ≤ L/60 (where L is span). These limits prevent cracking of partitions, misalignment of doors/windows, visible sag, and vibration problems. Deflections must be calculated at service loads (not factored loads used in strength design), typically combining dead load and sustained live load components. NSCP Section 5.2.5 provides a table of limits for different member types and load cases. Compliance with deflection limits is a prerequisite for code approval in the Philippines.

Statically indeterminate structures have more unknown reactions/internal forces than equilibrium equations can solve. The missing equations come from geometric compatibility: deformations of different parts must fit together consistently. A classic example: a continuous beam over two spans has three reactions but only two equilibrium equations. The compatibility condition states that the deflection at the interior support must be zero (it is supported). This gives the third equation. Deflection analysis provides these compatibility conditions, making it possible to solve for redundant reactions. Energy methods elegantly formulate these conditions. Understanding this connection is essential for advancing to indeterminate structural analysis, covered in subsequent chapters.

In virtual work, the product nN (or mM) must be computed carefully. For trusses: tension (N > 0) and tension (n > 0) multiply positively; compression members (N < 0, n < 0) also multiply positively; mixed signs (one tension, one compression) yield negative products. The physical interpretation: if both the real and virtual loads cause the same type of stress (both pull or both push), the deflection contribution is in the direction of the virtual load. For beams: bending moment sign convention (often positive = sagging) must be consistent. If M and m have the same sign throughout an element, the product is positive. The sum/integral of all products gives the total deflection. Careless sign errors are a common exam mistake.

When finding a deflection or rotation at a point where no real load is applied, introduce a fictitious dummy load Q at that point (in the direction of the desired deflection). Solve the structure with Q included, compute the strain energy U as a function of Q, then differentiate: δ = ∂U/∂Q, and finally set Q = 0. Alternatively, using the unit-load method, apply a unit dummy load, compute n or m with this unit value, then multiply by N or M (which do not depend on the dummy load). This technique is standard in Castigliano's method and essential for solving practical problems where deflections are sought at unloaded points.

Strain energy U is the stored elastic energy in a deformed structure. The virtual work principle states that the external work done by a unit virtual load equals the virtual strain energy it causes, yielding δ = ∫(mM/EI)dx. Thus, virtual work is the practical extraction of a specific deflection from the global strain energy.

Castigliano's second theorem (δ = ∂U/∂P) and the unit-load method (δ = ∫(mM/EI)dx) are mathematically identical. Differentiating U with respect to P yields ∂M/∂P, which is exactly the virtual moment m from applying a unit load. Thus, both methods calculate the same deflection; the choice is pedagogical or computational.

Indeterminate structures require compatibility equations (equal number to the degree of static indeterminacy). Deflection methods provide these compatibility equations by enforcing that displacements are consistent with supports (e.g., deflection at a fixed support = 0). Thus, mastering deflection analysis is the prerequisite to solving indeterminate problems.

Under linear elasticity, the total deflection from multiple loads is the algebraic sum of deflections from each load separately. Combined with virtual work, this allows engineers to analyze complex loading patterns by decomposing them into simpler cases and superposing results.

NSCP 2015 specifies maximum allowable deflections to ensure structures remain functional and comfortable for occupants. These limits are independent of strength (ultimate-load) design and must be checked separately at service load levels. Compliance requires calculating deflections using virtual work or other analytical methods.

Applying a unit load at a point in a specific direction yields the deflection component in that direction. To find horizontal deflection, apply a horizontal unit load; for vertical deflection, apply a vertical unit load. For rotation about an axis, apply a unit moment about that axis. The direction of the virtual load determines the component calculated.

Deflections are inversely proportional to stiffness: δ ∝ 1/(AE) for axial deformation, δ ∝ 1/(EI) for bending. Higher E (stiffer material) or larger A or I (larger cross-section) reduces deflection. This relationship guides design decisions: to limit deflection, increase member stiffness or reduce span length.

An office building floor system in Manila must satisfy NSCP 2015 deflection limits (L/240 for floor beams under live load). An engineer designs a steel I-beam span of 6 m. Using virtual work, the engineer calculates the midspan deflection under the combined dead load (beam self-weight) and the design live load. If deflection exceeds L/240 = 25 mm, the beam must be increased in size (larger moment of inertia I) or the span reduced. This iterative design process balances strength, serviceability, and cost. Real-world example: a 6 m span office beam in the Philippines often uses a 450 mm deep I-beam (ASTM A36 steel) to satisfy deflection limits while also meeting strength requirements.

A three-span continuous bridge beam carries traffic loads. The reactions at the interior supports (redundants) are found by solving compatibility equations: the deflection at each interior support is zero (it is supported). Virtual work is applied at each support to write these deflection equations, which are then solved simultaneously with equilibrium equations. The result is the complete distribution of moments and shears. This is essential for checking bearing design and ensuring no uplift at supports.

A steel roof truss spanning a warehouse in Cebu carries a concentrated load (perhaps a hanging crane hook or HVAC unit). Using the unit-load method, engineers calculate the joint deflection at the load point to ensure the truss does not sag excessively or interfere with equipment below. For a typical Warren truss with 20 m span and a 50 kN load, deflections are often on the order of 10–50 mm, which must be acceptable for the function of the building. Excessive deflection can misalign suspended equipment or affect the performance of roller-based systems.

A reinforced concrete portal frame (warehouse or factory structure) in Davao is designed to support equipment racks. The horizontal deflection of the frame under wind load must be limited to ensure stability and to prevent interference with internal bracing or equipment. Virtual work is applied with a horizontal unit load at the roof level to find this lateral deflection. If the calculated deflection exceeds tolerable limits (sometimes 1/500 of the frame height), the columns must be increased in size (larger I) or moment-resisting connections must be stiffened.

If a support of a long-span bridge settles by a known amount Δ (e.g., 50 mm due to foundation consolidation), the reaction redistribution must be calculated. Using Castigliano's theorem with a dummy load at the settled support, engineers compute how the settlement induces additional moments and shears throughout the structure. This is critical for assessing safety and planning maintenance. In the Philippines, typhoons and heavy rains can cause localized foundation movement; the ability to compute settlement-induced forces is essential for bridge safety assessment.

A steel frame may experience temperature gradients or may be slightly out of square due to fabrication tolerances. These effects can be modeled by introducing 'dummy' displacements or rotations and using Castigliano's method to find the induced stresses. For example, if a member is fabricated 2 mm too short, the frame must deform to accommodate this discrepancy, inducing additional bending and axial forces. Calculating these forces ensures the frame is still safe and that design stresses are not exceeded.

A truss with internal members (one more member than the minimum for static determinacy) is indeterminate by degree 1. The redundant member force is found by requiring that the relative deflection between its end joints, under the combined action of all loads and the redundant force, is consistent with its length. Virtual work equations set up this compatibility condition, enabling solution for the redundant force and all other internal forces. This is a precursor to more complex indeterminate analyses.

Modern structures are often instrumented with displacement sensors. Calculated deflections (using virtual work) are compared against measured deflections to detect anomalies—degradation, damage, or overloading. If measured deflections exceed calculated values, the structure may have lost stiffness (e.g., cracked concrete, corroded reinforcement) and must be investigated. This application bridges analysis, monitoring, and maintenance in the facility management phase.