Structural Analysis

1. Understanding and Calculating Structural Loads

Determine the resultant force caused by the dead load and the live load.

Loads are fundamental. Structural analysis begins with identifying and quantifying the loads a structure must support. These include dead loads (structure's weight), live loads (occupancy, furniture), and environmental loads (wind, snow, seismic). Accurate load determination is crucial for safe and economical design.

Load types vary. Different materials and uses result in different load intensities. For example:

• Stone concrete dead load: ~12 lb/ft²/in.
• Lightweight concrete dead load: ~8 lb/ft²/in.
• Office live load: 50 lb/ft²
• Heavy storage live load: 250 lb/ft²

Environmental loads are complex. Wind and snow loads depend on location, terrain, building height, and shape. Wind creates pressure (windward) and suction (leeward), while snow loads vary with ground snow load, roof shape, and exposure. These loads are often specified by building codes.

2. Classifying Structures: Determinate, Indeterminate, Stable, Unstable

Statically determinate.

Structural classification matters. Structures are classified based on their ability to be analyzed using only the equations of static equilibrium. Statically determinate structures can be solved directly, while indeterminate structures require considering material properties and deformation. Unstable structures cannot maintain equilibrium under load.

Determinacy criteria. For planar structures, determinacy is often assessed by comparing the number of unknown reactions (r) and internal releases (if any) to the number of equilibrium equations (3 per rigid body or joint).

• Determinate: r = 3n (for rigid bodies) or b + r = 2j (for trusses)
• Indeterminate: r > 3n or b + r > 2j (degree of indeterminacy = r - 3n or b + r - 2j)
• Unstable: r < 3n or b + r < 2j, or if reactions are parallel or concurrent.

Stability is paramount. An unstable structure will collapse under load. Instability can arise from insufficient supports or improper arrangement of members or supports, leading to mechanisms or uncontrolled movement.

3. Analyzing Statically Determinate Beams and Frames for Reactions

By = 48.0 kN

Equilibrium is the key. For statically determinate beams and frames, support reactions can be found by applying the three equations of static equilibrium: sum of forces in x-direction = 0, sum of forces in y-direction = 0, and sum of moments about any point = 0.

Support types dictate reactions. Different supports provide different constraints and thus different reaction components:

• Pin: Resists horizontal and vertical force (2 reactions).
• Roller: Resists force perpendicular to the rolling surface (1 reaction).
• Fixed: Resists horizontal force, vertical force, and moment (3 reactions).

Compound structures require segmentation. Structures with internal pins or hinges can be broken down into multiple rigid bodies. Equilibrium equations are applied to each segment and/or the entire structure to solve for all unknown reactions.

4. Analyzing Statically Determinate Trusses

FCD = 780 lb (C)

Trusses are efficient. Trusses are lightweight structures composed of slender members connected at their ends by pins. Members are assumed to carry only axial force (tension or compression). Analysis determines the force in each member.

Methods of analysis:

• Method of Joints: Apply equilibrium equations (sum Fx=0, sum Fy=0) at each pin joint. Start at joints with few unknown members.
• Method of Sections: Cut through the truss to isolate a section. Apply equilibrium equations to the section to solve for forces in the cut members. Useful for finding forces in specific members quickly.

Zero-force members exist. Some truss members carry no load under a specific loading condition. Identifying these members simplifies analysis and design. They often occur at joints with only two non-collinear members or three members where two are collinear and no external load is applied at that joint.

5. Determining Internal Forces: Shear and Moment Diagrams

MC = 0.667 kN # m

Internal forces resist loads. Beams and frames develop internal forces (normal force, shear force, and bending moment) to resist external loads. These forces vary along the length of the member.

Diagrams visualize variation. Shear and moment diagrams plot the variation of shear force (V) and bending moment (M) along the member's axis. These diagrams are essential for structural design, as they show the maximum values and locations of internal forces.

Relationships govern diagrams. The diagrams are related by calculus:

• The slope of the shear diagram at any point equals the distributed load intensity at that point (dV/dx = w).
• The change in shear between two points equals the area under the load diagram between those points.
• The slope of the moment diagram at any point equals the shear force at that point (dM/dx = V).
• The change in moment between two points equals the area under the shear diagram between those points.

6. Analyzing Cables and Arches

TCD = 6.41 kN(Max)

Cables carry tension. Flexible cables support loads by developing tension along their length. Under distributed vertical loads, a cable forms a parabolic shape. Under concentrated loads, it forms a series of straight line segments.

Arches carry compression. Arches are curved structures that support loads primarily through axial compression. Three-hinged arches are statically determinate and can be analyzed using equilibrium equations applied to segments separated by hinges.

Funicular shape is ideal. A funicular shape is the shape a cable takes under a given load. If an arch is built in the funicular shape for its dead load, it will ideally carry that load purely in compression, minimizing bending stresses.

7. Using Influence Lines for Moving Loads

(MC) max = 141.6 kN # m

Influence lines show load effect. An influence line is a graph showing how a specific internal force (reaction, shear, or moment) at a point in a structure varies as a unit load moves across the structure.

Purpose for moving loads. Influence lines are crucial for determining the maximum effect (shear, moment, reaction) caused by moving loads, such as vehicles on a bridge or cranes on a beam. By placing the actual loads at positions corresponding to the peaks of the influence line, the maximum value of the function is found.

Müller-Breslau principle simplifies. This principle states that the influence line for a force or moment at a point is proportional to the deflected shape of the structure when a unit displacement corresponding to that force or moment is introduced at that point. This provides a quick way to sketch the shape of influence lines.

8. Calculating Structural Deflections

vc = - PL3 / 6EI

Deflection is crucial for serviceability. While strength ensures a structure doesn't break, stiffness ensures it doesn't deflect excessively under load, which could cause cracking, vibration, or aesthetic issues. Deflection analysis predicts structural deformation.

Various methods exist. Several techniques are available to calculate beam and frame deflections:

• Integration Method: Integrate the moment equation twice (EI d²v/dx² = M) to find the elastic curve equation.
• Moment-Area Theorems: Relate the slope and deflection between two points to the area and moment of the M/EI diagram.
• Conjugate-Beam Method: Analyze a fictitious "conjugate" beam loaded with the M/EI diagram to find deflections (as moments) and slopes (as shears) in the real beam.
• Virtual Work/Castigliano's Theorem: Apply a virtual unit load or use strain energy derivatives to find displacement.

EI is a key property. Flexural rigidity (EI), the product of the material's modulus of elasticity (E) and the member's moment of inertia (I), directly influences stiffness and deflection. Higher EI means less deflection.

9. Analyzing Statically Indeterminate Structures: The Force Method

By = 7 w L / 128

Indeterminacy requires more. Statically indeterminate structures have more reactions or internal forces than can be solved by equilibrium alone. The force method (or flexibility method) addresses this by treating redundant supports or members as unknowns.

Steps of the force method:

1. Identify the degree of indeterminacy and choose redundant forces/moments.
2. Remove the redundants to create a statically determinate "primary" structure.
3. Calculate the displacement/rotation at the location of each redundant in the primary structure due to the applied loads.
4. Calculate the displacement/rotation at the location of each redundant due to each unit redundant force/moment.
5. Write compatibility equations stating that the total displacement/rotation at the redundant locations must match the actual conditions (usually zero displacement/rotation at supports).
6. Solve the compatibility equations for the redundant forces/moments.
7. Use equilibrium to find the remaining reactions and internal forces.

Compatibility is key. The core principle is ensuring that the deformations of the structure are compatible with the support conditions and member connections.

10. Analyzing Statically Indeterminate Structures: Displacement Methods

MBA + MBC = 0

Displacement methods focus on joint rotations/displacements. Unlike the force method, displacement methods (like Slope-Deflection and Moment Distribution) treat joint rotations and displacements as the primary unknowns.

Slope-Deflection Method: Relates member end moments to joint rotations, relative joint displacements (settlements/sidesway), and fixed-end moments.

• Write slope-deflection equations for each member end moment.
• Write equilibrium equations at each joint (sum of moments = 0) and for the structure (shear equilibrium for sidesway).
• Solve the system of equations for the unknown joint rotations and displacements.
• Substitute these values back into the slope-deflection equations to find the member end moments.

Moment Distribution Method: An iterative process that distributes unbalanced moments at joints until equilibrium is achieved.

• Calculate fixed-end moments for each member assuming all joints are fixed.
• Calculate distribution factors (DF) at each joint based on member stiffness (K).
• Release joints one by one, distributing the unbalanced moment (sum of FEMs) to connected members based on DFs.
• Carry over half the distributed moment to the far end of each member (carry-over factor, COF).
• Repeat until moments converge.

Fixed-end moments are starting points. Both methods rely on pre-calculated fixed-end moments for standard loading cases. Stiffness (K) and carry-over factors (COF) depend on member properties (EI, L) and end conditions (fixed, pinned).

11. Approximate Analysis of Indeterminate Structures

FBH = 12.1 k (T)

Simplify for quick estimates. For complex indeterminate structures, approximate methods provide rapid estimates of forces and moments, useful for preliminary design or checking detailed analysis results. These methods introduce simplifying assumptions to make the structure statically determinate.

Assumptions vary by structure type:

• Trusses: Assume diagonals carry tension or compression, or that shear in a panel is distributed equally among diagonals.
• Portal Frames (Lateral Loads): Assume inflection points at mid-height of columns and mid-span of girders, or distribute lateral shear among columns based on stiffness (portal method) or area (cantilever method).

Portal vs. Cantilever Method:

• Portal Method: Assumes interior columns carry twice the shear of exterior columns. Suitable for low-rise frames.
• Cantilever Method: Assumes axial stress in columns is proportional to their distance from the frame's centroid. Suitable for tall frames.

Results are approximate. These methods provide reasonable estimates but are not exact. They are based on assumed behavior rather than rigorous analysis.

12. Matrix Stiffness Method for Truss Analysis

K = k1 + k2 + k3

Matrix methods automate analysis. The stiffness method is a powerful, systematic approach suitable for computer implementation. It assembles a global stiffness matrix (K) relating nodal forces (Q) to nodal displacements (D) via the equation Q = KD.

Steps for truss analysis:

1. Define global coordinate system and number nodes and degrees of freedom (DOF) at each node.
2. For each member, determine its local stiffness matrix (k') relating local forces to local displacements.
3. Transform each local stiffness matrix (k') into a global stiffness matrix (k) using transformation matrices based on member orientation (lx, ly).
4. Assemble the global stiffness matrix (K) by summing the contributions of each member's global stiffness matrix (k) based on shared DOF codes.
5. Partition the global stiffness equation (Q = KD) based on known (support) and unknown (free) displacements.
6. Solve for the unknown displacements using the known forces (applied loads).
7. Calculate member forces using member stiffness and nodal displacements.

Assembly is key. The power of the method lies in systematically assembling the global stiffness matrix from individual member contributions, allowing for the analysis of large, complex structures.

Last updated: May 7, 2025