Shape Factor For I Section: Derivation, Proof and Solved Examples (2026)

1. Introduction to I-Beam Flexural Plasticity

Rolled steel wide-flange beams and universal I-sections represent the primary workhorses of modern structural steel framing. Their cross-sectional efficiency stems from concentrating large flange areas at maximum distance from the bending axis. However, this optimal material placement significantly alters how plasticity develops across the section. Determining the shape factor for i section is vital for calculating plastic moment capacity, conducting plastic frame analysis, and designing compact steel members.

                  TYPICAL I-SECTION GEOMETRY (STRONG AXIS)
                 +--------------------------------------+  ^
                 |           Top Flange (bf x tf)       |  | tf
                 +------------------+-------------------+  v
                                    |                   |  ^
                                    | Web (tw x hw)     |  | hw = h - 2tf
                                    |                   |  v
                 +------------------+-------------------+  ^
                 |         Bottom Flange (bf x tf)      |  | tf
                 +------------------+-------------------+  v
                 |<---------------- bf ---------------->|

The general shape factor $S$ quantifies the reserve capacity between first yield $M_y$ and full cross-sectional plastification $M_p$:

$$S = \frac{M_p}{M_y} = \frac{Z_p}{Z_e}$$

For standard rolled steel I-sections bent about their strong axis ($x-x$), the major-axis shape factor typically ranges between $1.10$ and $1.18$ (with a common benchmark average of $1.14$). In contrast, when an I-section is bent about its weak axis ($y-y$), the shape factor increases to between $1.50$ and $1.60$.

The major-axis flexural reserve is much lower than that of a solid rectangular section ($S = 1.50$) because $80\%$ to $85\%$ of an I-beam’s material already resides in the extreme outer flanges. When the flanges reach yield stress, only a modest elastic reserve remains within the thin central web.

2. Cross-Sectional Geometry and Notation of I-Sections

Consider a doubly symmetric I-section defined by standard geometric parameters: overall depth $h$, flange width $b_f$, flange thickness $t_f$, web thickness $t_w$, clear web depth $h_w = h – 2 t_f$, and total area $A = 2 b_f t_f + h_w t_w$.

          STRONG AXIS STRESS PROFILE          WEAK AXIS STRESS PROFILE
             +---------------+                    +---------------+
             |===============| +fy                |=======|=======|
             +---------------+                    +---------------+
                     |                                    |
                     | +fy                                |
             --------+-------- PNA                --------+-------- PNA
                     |                                    |
                     | -fy                                |
             +---------------+                    +---------------+
             |===============| -fy                |=======|=======|
             +---------------+                    +---------------+

Because the cross-section is doubly symmetric about both the horizontal $x-x$ axis and vertical $y-y$ axis, the elastic neutral axis (ENA) and plastic neutral axis (PNA) coincide at the geometric centroid for pure bending.

3. Mathematical Derivations of Strong-Axis ($x-x$) Properties

3.1 Strong-Axis Elastic Section Modulus ($Z_{e,x}$)

The second moment of area $I_x$ about major axis $x-x$ is calculated using the subtraction method:

$$I_x = \frac{b_f h^3 – (b_f – t_w) h_w^3}{12}$$

The maximum distance to outermost fiber is $y_{\max} = h/2$. The strong-axis elastic section modulus $Z_{e,x}$ is:

$$Z_{e,x} = \frac{I_x}{h/2} = \frac{b_f h^3 – (b_f – t_w) h_w^3}{6 h}$$

3.2 Strong-Axis Plastic Section Modulus ($Z_{p,x}$)

In the fully plastic state under strong-axis flexure, the entire area above the horizontal centerline is in uniform compression ($+f_y$) and the area below is in uniform tension ($-f_y$):

$$Z_{p,x} = 2 \left( A_{\text{ flange}} \cdot \bar{y}_{\text{ flange}} + A_{\text{half-web}} \cdot \bar{y}_{\text{half-web}} \right)$$

Substituting geometric terms:

$$Z_{p,x} = 2 \left[ (b_f t_f) \left( \frac{h – t_f}{2} \right) + \left( t_w \frac{h_w}{2} \right) \left( \frac{h_w}{4} \right) \right] = b_f t_f (h – t_f) + \frac{t_w h_w^2}{4}$$

Fully plastic moment capacity is $M_{p,x} = f_y Z_{p,x}$.

3.3 Major-Axis Flexural Proof ($S_x$)

Dividing plastic modulus $Z_{p,x}$ by elastic modulus $Z_{e,x}$ gives the theoretical ratio for the major axis:

$$S_x = \frac{Z_{p,x}}{Z_{e,x}} = \frac{b_f t_f (h – t_f) + \frac{t_w h_w^2}{4}}{\frac{b_f h^3 – (b_f – t_w) h_w^3}{6 h}}$$

For typical rolled wide-flange shapes, $S_x \approx 1.12 \text{ to } 1.18$.

4. Weak-Axis ($y-y$) Plastic Properties and Flange Behavior

4.1 Weak-Axis Elastic Section Modulus ($Z_{e,y}$)

When an I-section is bent about its minor axis $y-y$:

$$I_y = \frac{2 t_f b_f^3 + h_w t_w^3}{12}, \quad Z_{e,y} = \frac{I_y}{b_f/2} \approx \frac{t_f b_f^2}{3}$$

4.2 Weak-Axis Plastic Section Modulus ($Z_{p,y}$)

About the vertical axis $y-y$, each flange forms two rectangular blocks of width $b_f/2$ and thickness $t_f$:

$$Z_{p,y} = \frac{t_f b_f^2}{2} + \frac{h_w t_w^2}{4}$$

4.3 Minor-Axis Reserve Ratio Evaluation ($S_y$)

Evaluating weak-axis ratio: $S_y = Z_{p,y} / Z_{e,y} \approx 1.50 \text{ to } 1.60$. Flanges behave essentially as solid rectangular plates bent across their widths.

5. Plastic Hinge Length and Moment-Curvature Response

5.1 Moment-Curvature Relationship and Curvature Ductility

The moment-curvature response of an I-section shows a steep initial linear branch up to $M_y$, followed by a brief elasto-plastic knee as the web yields, reaching the plastic plateau $M_p$. Because $S_x \approx 1.15$, full plastification occurs rapidly with minimal post-yield strain hardening required.

5.2 Hinge Length Spread Under Major-Axis Point Loading

Under a central concentrated load on simple span $L$, plastic hinge spread length $L_p$ evaluates to:

$$L_p = L \left( 1 – \frac{1}{S_x} \right) = L \left( 1 – \frac{1}{1.15} \right) \approx 0.130 L$$

Because $S_x$ is compact ($1.15$), plastic deformation concentrates tightly within $13\%$ of the span, demanding high rotational ductility from the material.

6. Comprehensive Step-by-Step Worked Numerical Calculations

6.1 Section Dimensions and Steel Material Properties

Consider a standard universal steel beam profile UB 457x152x67:

• Depth $h = 457.0\text{ mm}$, Flange width $b_f = 152.0\text{ mm}$

• Flange thickness $t_f = 15.0\text{ mm}$, Web thickness $t_w = 9.0\text{ mm}$

• Clear web height $h_w = 427.0\text{ mm}$, Steel Grade S355 ($f_y = 355\text{ MPa}$)

• Span $L = 8.0\text{ m}$ (Simply supported)

6.2 Strong-Axis ($x-x$) Calculations: $Z_{e,x}$, $Z_{p,x}$, and $S_x$

Step 1: Moment of Inertia and Elastic Modulus
$$I_x = \frac{152(457)^3 – (152 – 9)(427)^3}{12} = 2.80354 \times 10^8\text{ mm}^4$$
$$Z_{e,x} = \frac{2.80354 \times 10^8}{228.5} = 1.22693 \times 10^6\text{ mm}^3 = 1.22693 \times 10^{-3}\text{ m}^3$$
$$M_{y,x} = 355\text{ MPa} \times 1.22693 \times 10^{-3}\text{ m}^3 = 435.56\text{ kN}\cdot\text{ m}$$

Step 2: Plastic Modulus and Moment Capacity
$$Z_{p,x} = 152(15)(457 – 15) + \frac{9(427)^2}{4} = 1{,}007{,}760 + 410{,}240.25 = 1.41800 \times 10^6\text{ mm}^3$$
$$M_{p,x} = 355\text{ MPa} \times 1.41800 \times 10^{-3}\text{ m}^3 = 503.39\text{ kN}\cdot\text{ m}$$
$$S_x = \frac{1{,}418{,}000.25}{1{,}226{,}930} = 1.15573 \approx 1.156$$

6.3 Weak-Axis ($y-y$) Calculations: $Z_{e,y}$, $Z_{p,y}$, and $S_y$

$$I_y = \frac{2(15)(152)^3 + 427(9)^3}{12} = 8.80562 \times 10^6\text{ mm}^4$$
$$Z_{e,y} = \frac{8.80562 \times 10^6}{76} = 115{,}863.42\text{ mm}^3, \quad M_{y,y} = 41.13\text{ kN}\cdot\text{ m}$$
$$Z_{p,y} = \frac{15(152)^2}{2} + \frac{427(9)^2}{4} = 181{,}926.75\text{ mm}^3, \quad M_{p,y} = 64.58\text{ kN}\cdot\text{ m}$$
$$S_y = \frac{181{,}926.75}{115{,}863.42} = 1.570$$

6.4 Beam Span Collapse Load Evaluation

For simply supported span ($L = 8.0\text{ m}$) under central load $P$:

• $P_{y,x} = 4 M_{y,x} / L = 217.78\text{ kN}$

• $P_{u,x} = 4 M_{p,x} / L = 251.70\text{ kN}$

• Ratio: $P_{u,x} / P_{y,x} = 1.156$

7. Compactness Classifications and Local Buckling Limits

To mobilize full plastic moment capacity, members must prevent premature local plate buckling:

1. AISC 360-22 Specification: Under [AISC 360-22 Specification for Structural Steel Buildings at AISC], Table B4.1b classifies members. Flange limit $\lambda_f = b_f / (2 t_f) \le 0.38 \sqrt{E/f_y}$ and web limit $\lambda_w = h_w / t_w \le 3.76 \sqrt{E/f_y}$. Both compact in our example.
2. Eurocode 3 (EN 1993-1-1): In [Eurocode 3 EN 1993-1-1 Steel Design Standard at Eurocodes Building the Future], Class 1 cross-sections are required for plastic analysis ($c/t_f \le 9\epsilon$, $d/t_w \le 72\epsilon$).
3. Lateral-Torsional Buckling: Members require unbraced length $L_b \le L_p$ to prevent out-of-plane buckling before plastic mechanism formation.

8. Summary of I-Section Plastic Behavior

The shape factor for i section illustrates why wide-flange profiles dominate modern structural engineering. While the strong-axis shape factor ($S_x \approx 1.15$) offers a modest $15\%$ plastic reserve, it reflects the optimal structural efficiency of placing material far from the neutral axis. Meanwhile, the weak-axis shape factor ($S_y \approx 1.57$) delivers substantial reserve capacity during out-of-plane bending. Proper verification of compactness guarantees that these plastic capacities translate reliably into safe structures.

References & Standards Cited

1. AISC 360-22: Specification for Structural Steel Buildings, American Institute of Steel Construction, Chicago, IL, 2022.
2. EN 1993-1-1:2005+A1:2014: Eurocode 3: Design of steel structures – Part 1-1: General rules and rules for buildings, CEN, Brussels, 2014.
3. Salmon, C. G., Johnson, J. E., & Malhas, F. A. (2008). Steel Structures: Design and Behavior, 5th Edition, Pearson, Upper Saddle River, NJ.
4. Horne, M. R. (1979). Plastic Theory of Structures, 2nd Edition, Pergamon Press, Oxford.
5. Bruneau, M., Uang, C. M., & Sabelli, R. (2011). Ductile Design of Steel Structures, 2nd Edition, McGraw-Hill, New York.

Frequently Asked Questions (FAQ)

Why is the major-axis plastic reserve for an I-section lower than a solid rectangle?

An I-section has most of its area concentrated in the outer flanges. When flanges yield, only a small portion of area remains in the thin web to provide reserve strength, giving $S approx 1.12 - 1.18$.

What is the typical range for the major-axis shape factor of standard W-beams?

For standard AISC W-beams and European UB/HE sections, major-axis shape factor $S_x$ falls between $1.12$ and $1.18$, with $1.14$ being average.

Why is the minor-axis shape factor of an I-beam close to 1.55?

In weak-axis bending, two wide flanges act like solid rectangular plates bent across their widths, producing a shape factor around $1.50 - 1.60$.

What happens if an I-section fails compactness requirements?

If an I-section exceeds compactness limits, local plate buckling occurs before reaching full plastic moment $M_p$.

Does the shape factor of an I-beam change with steel grade?

No. The shape factor is purely a geometric ratio ($Z_p / Z_e$).