7.3 Beams, Columns, and Frames

Beams, columns, and frames are the primary load-carrying elements in most structures. Understanding how each one behaves under different loading conditions is the foundation of structural design: you need to predict bending, buckling, and force distribution before you can size any member safely.

This section covers beam stress and deflection, column buckling capacity, frame analysis techniques, and two classical methods for solving indeterminate structures.

Beam Behavior Under Loads

Bending and Shear Analysis

When a beam carries a transverse load, it develops internal bending moments and shear forces along its length. Shear force and bending moment diagrams are the primary tools for visualizing how these internal forces vary from one end of the beam to the other.

The flexure formula connects bending stress to the internal moment:

$\sigma = \frac{Mc}{I}$

where $M$ is the bending moment at the section, $c$ is the distance from the neutral axis to the point of interest, and $I$ is the moment of inertia of the cross-section. Stress is zero at the neutral axis and reaches its maximum at the outermost fibers.

Shear stress also varies across the cross-section:

• For symmetrical sections (like a rectangle or I-beam), the maximum shear stress occurs at the neutral axis, not at the edges.
• The general shear formula is $\tau = \frac{VQ}{It}$, where $V$ is the internal shear force, $Q$ is the first moment of area above (or below) the point of interest, and $t$ is the width of the section at that point.

When a beam experiences multiple load types at once (bending, shear, torsion), you can use the principle of superposition to combine the individual stress contributions and find the overall stress state at any point.

Torsional Effects and Deflection

Torsional loads twist a beam and create shear stresses:

• For circular sections, the shear stress varies linearly with distance from the center of twist, reaching a maximum at the outer surface.
• For non-circular sections, the stress distribution is more complex and doesn't follow a simple linear pattern.

Several methods exist for calculating beam deflection. Each has its strengths depending on the problem:

1. Integration of the elastic curve equation — You integrate the moment equation twice to get the deflection function. Works well for simple loading but gets tedious for complex cases.
2. Moment-area method — Uses the area under the $M/EI$ diagram to find slopes and deflections. Particularly handy when you only need the deflection at a specific point.
3. Conjugate beam method — Replaces the real beam with a fictitious "conjugate" beam loaded with the $M/EI$ diagram. The shear and moment in the conjugate beam give you the slope and deflection of the real beam.

Under combined loading, failure criteria help predict when a beam will yield or fracture:

• Maximum normal stress theory — Failure occurs when the largest normal stress reaches the material's yield strength. Best suited for brittle materials.
• Maximum shear stress theory (Tresca) — Failure occurs when the maximum shear stress reaches the shear yield strength. More appropriate for ductile materials.

Column Capacity Determination

Buckling Analysis

Columns are compression members, and their most distinctive failure mode is buckling, where the column deflects laterally instead of simply crushing. Euler's formula gives the critical buckling load for a long, slender column:

$P_{cr} = \frac{\pi^2 EI}{(KL)^2}$

Here, $E$ is the modulus of elasticity, $I$ is the moment of inertia about the weaker axis, $L$ is the column length, and $K$ is the effective length factor that accounts for end conditions.

Common $K$ values to know:

• Pinned-pinned: $K = 1.0$
• Fixed-fixed: $K = 0.5$
• Fixed-pinned: $K \approx 0.7$
• Fixed-free (cantilever): $K = 2.0$

The slenderness ratio $KL/r$ (where $r$ is the radius of gyration) determines which failure mode governs. High slenderness ratios mean the column is likely to buckle elastically (Euler buckling). Low slenderness ratios mean the column will fail by material yielding or crushing before it ever buckles.

For columns in the intermediate range, the secant formula gives a more accurate prediction because it accounts for initial eccentricities and material nonlinearity that Euler's formula ignores.

Combined Loading and Cross-Section Considerations

Columns rarely experience pure axial compression. Eccentric loads or lateral forces introduce bending alongside compression. Interaction diagrams plot the combinations of axial load and bending moment that a column can safely resist. Any load combination that falls inside the curve is acceptable; anything outside means failure.

Two additional cross-section considerations:

• Local buckling can occur in thin-walled or slender elements within the column profile (like the flanges of a wide-flange section) even if the column as a whole is stable. This is why design codes specify width-to-thickness limits.
• The radius of gyration $r = \sqrt{I/A}$ relates the cross-section's moment of inertia to its area. A larger $r$ means a more efficient section for resisting buckling, which is why hollow tubes and wide-flange shapes are preferred over solid rectangles for columns.

Simple Frame Design

Structural Analysis Techniques

A frame is a structure made of beams and columns connected at rigid or pinned joints. Before analyzing a frame, you need to assess its statical determinacy:

• A determinate frame can be solved using equilibrium equations alone ($\sum F_x = 0$, $\sum F_y = 0$, $\sum M = 0$).
• An indeterminate frame has more unknowns than equilibrium equations and requires additional methods (compatibility, moment distribution, etc.).

Once you know the analysis approach, you'll draw three types of internal force diagrams for each member:

• Axial force diagrams — show tension or compression along the member
• Shear force diagrams — show transverse internal forces
• Bending moment diagrams — show how the moment varies along the member

The method of joints and method of sections (familiar from truss analysis) can also be applied to frame members. For finding displacements and rotations, virtual work principles are a powerful tool: you apply a fictitious unit load at the point of interest and use energy methods to calculate the real displacement there.

Compatibility and Load Effects

At every joint in a frame, the connected members must deform consistently. These compatibility conditions ensure that rotations and displacements match where members meet. For indeterminate frames, compatibility equations supplement equilibrium to provide enough equations for a solution.

Influence lines are useful when a frame or beam is subject to moving loads (think bridge girders as vehicles cross). An influence line shows how a specific reaction, shear, or moment at a fixed point changes as a unit load moves across the structure. They help you identify the critical load position that produces the maximum effect.

The degree of static indeterminacy tells you exactly how many extra equations (beyond equilibrium) you need. This number directly guides your choice of analysis method.

Moment Distribution and Slope-Deflection Methods

These are two classical techniques for solving indeterminate beams and frames. Both are still taught because they build strong intuition about how moments and rotations relate in a structure.

Moment Distribution Technique

Moment distribution is an iterative method developed by Hardy Cross. Rather than solving a system of equations all at once, you progressively balance unbalanced moments at each joint until the structure reaches equilibrium.

The process works in steps:

1. Lock all joints — Assume every joint is fixed. Calculate the fixed-end moments (FEMs) for each loaded member using standard formulas (these are tabulated for common load cases).
2. Calculate distribution factors — At each joint, determine how the unbalanced moment splits among the connected members. The distribution factor for each member equals its relative stiffness ($EI/L$) divided by the sum of all member stiffnesses at that joint.
3. Release one joint at a time — Unlock a joint, distribute the unbalanced moment to the connected members according to the distribution factors, then re-lock it.
4. Apply carry-over factors — When you distribute a moment at one end of a member, half of that moment (for prismatic members) carries over to the far end. The carry-over factor is typically $0.5$.
5. Repeat — Continue releasing, distributing, and carrying over at each joint. The unbalanced moments get smaller with each cycle. Stop when they're negligibly small.
6. Sum all moments — Add up the FEMs, distributed moments, and carry-over moments at each member end to get the final moments.

Slope-Deflection Analysis

The slope-deflection method takes a different approach. Instead of iterating, it expresses the moment at each end of a member as a function of the end rotations ($\theta$) and relative displacements ($\Delta$):

$M = \frac{2EI}{L}(2\theta_{near} + \theta_{far} - 3\psi) + FEM$

where $\psi = \Delta/L$ is the chord rotation and $FEM$ is the fixed-end moment from applied loads. The term $EI/L$ represents the member stiffness.

You then write equilibrium equations at each joint (the sum of moments at a joint must equal zero) and solve the resulting system of simultaneous equations. The number of unknowns equals the degree of kinematic indeterminacy (the number of independent joint rotations and translations).

Both methods give the same final answers. Moment distribution is often faster for hand calculations on simpler structures, while slope-deflection sets up more naturally for computer-based solutions.