Plate Girders - Theory & Concepts

Plate Girders

Analysis and design of built-up steel plate girders, focusing on web slenderness, proportioning limits, and tension field action.

When standard rolled W-shapes are not large enough to carry the required loads or span the required distances (e.g., in highway bridges, railway bridges, or large industrial buildings carrying heavy cranes), engineers design plate girders. A plate girder is a built-up I-shaped section made by welding individual steel plates together to form the web and flanges.

Why Use Plate Girders?

The advantages and common applications of using built-up plate girders.

Advantages of Plate Girders

Customization: The designer can optimize the cross-section by choosing specific web depth, web thickness, flange width, and flange thickness independently to precisely match the required moment and shear capacities at every point along the span.
Efficiency: Material is placed exactly where it is needed. For example, thicker or wider flanges can be used at midspan where bending moments are highest, while thinner flanges are used near supports. The web can be kept extremely thin to save weight.
Long Spans: Plate girders can span distances far beyond the capability of standard rolled sections (e.g., up to 200 feet or more).

Web Slenderness

The fundamental challenge of plate girder design: the impact of slender webs on performance and design.

To minimize weight and material cost, the primary objective in plate girder design is to make the web as deep as possible (to maximize the moment arm between the flanges, thus minimizing the required flange area) and as thin as possible.

This makes the web highly susceptible to local buckling under bending (flexure) and shear long before the material reaches its yield strength.

Web Slenderness Ratio ($h/t_w$)

The central parameter governing plate girder behavior is the web slenderness ratio:

\lambda = \frac{h}{t_w}

Where

h

is the clear distance between flanges and

t_w

is the web thickness.

Unlike standard W-shapes, which are almost universally "compact," plate girders almost always have "non-compact" or "slender" webs according to AISC Table B4.1. This drastically changes how they are designed compared to standard beams.

Flexural Strength of Plate Girders

Determining the bending capacity when the web is slender.

Because the web is typically slender, it will buckle locally under the compressive stresses caused by the primary bending moment before the flanges can reach their full yield stress. AISC specifications (Chapter F4 or F5) account for this by either:

Reducing the allowable flexural strength of the compression flange: A web plastification factor ( $R_{pc}$ ) or a reduction factor ( $R_g$ ) is applied to limit the maximum stress in the flange, acknowledging that the buckled web is shedding compressive stress to the stiffer flange.
Ignoring a portion of the web in compression (Effective Width Method): Only a small effective width of the web adjacent to the compression flange is considered active, and the section properties ( $I$ , $S$ ) are recalculated based on this reduced area.

Shear Strength and Tension Field Action

How plate girders resist massive shear forces through remarkable post-buckling behavior.

The most significant unique behavior of plate girders is how they resist shear. A very thin, deep web is terrible at resisting shear forces without buckling.

Pre-Buckling Shear

The initial shear resistance mechanism before the web buckles diagonally.

Initially, the thin web resists shear through a pure shear stress state (which can be resolved into principal diagonal tension and compression stresses acting at

45^\circ

). However, because the web is thin, it will buckle diagonally under the principal compressive stress at a relatively low load. For a typical beam, this would be failure.

Post-Buckling Strength: Tension Field Action (TFA)

The secondary load-carrying mechanism that develops after the web buckles.

If transverse stiffeners are provided at appropriate intervals along the span, the web does not fail immediately after it buckles. Instead, a new, highly stable load-carrying mechanism develops called Tension Field Action (TFA).

How Tension Field Action Works

Once the web buckles diagonally in compression (forming wrinkles), it completely loses its ability to carry any more compressive stress. However, it can still carry diagonal tensile stress across the wrinkles.

The web essentially acts like a series of diagonal tension ties (like the cables in a bridge). The transverse stiffeners, which are rigid plates welded vertically to the web, act as vertical compression struts. The top and bottom flanges act as the top and bottom chords.

The plate girder literally transforms its internal load path into a Pratt truss! This post-buckling strength is often far greater than the initial buckling strength.

Plate Girder: Tension Field Action

Load: 10 kN

Apply Shear Load

0Theoretical Buckling LimitMax Capacity

Simulation Status:

Current Load: 10 kN
State: Elastic Shear (Unbuckled)

Nominal Shear Strength with TFA

Calculating the total shear capacity when Tension Field Action is permitted.

The nominal shear strength (

V_n

) of a plate girder with transverse stiffeners is the sum of the shear buckling strength of the web (

V_{web}

, before buckling) and the post-buckling tension field strength (

V_{tfa}

, after buckling).

TFA Shear Strength Equation (AISC G2.2)

When

h/t_w

is slender and Tension Field Action is permitted (based on stiffener spacing and panel aspect ratio):

V_n = 0.60 F_y A_w \left[ C_v + \frac{1 - C_v}{1.15 \sqrt{1 + (a/h)^2}} \right]

Where:

$C_v$ = Web shear coefficient (the fraction of the theoretical shear yield capacity reached before buckling occurs).
$a$ = Clear distance between transverse stiffeners.
$h$ = Clear distance between flanges.
$(a/h)$ = Panel aspect ratio.
The first term inside the brackets ( $C_v$ ) represents the pre-buckling strength.
The second term represents the massive additional capacity gained entirely from Tension Field Action.

Types of Stiffeners

Plates welded to the web to prevent buckling, distribute loads, and enable TFA.

Because plate girder webs are so thin, they almost always require stiffening. There are three primary types of stiffeners used.

1. Bearing Stiffeners (AISC J10)

These are required at unframed ends (supports) and at locations of heavy concentrated loads. They prevent local web yielding and crippling under direct compression.

Must extend the full depth of the web and bear tightly against the loaded flange.
Usually placed in pairs (one on each side of the web).
Designed as a fictitious column with an effective length of $0.75h$ , consisting of the stiffener plates plus a small effective width of the central web (typically $12 t_w$ to $25 t_w$ ).

2. Intermediate Transverse Stiffeners (AISC G2)

These are placed vertically at intervals along the span to increase the shear buckling strength of the web and to enable Tension Field Action (TFA).

Act as the vertical compression struts in the TFA "Pratt truss" model.
Spacing ( $a$ ) is determined by the designer based on the required shear capacity vs. the un-stiffened capacity. Typical $a/h$ ratios range from 1.0 to 3.0.
If TFA is permitted, intermediate stiffeners must have sufficient area and moment of inertia to resist the substantial compression forces induced by the diagonal tension field anchoring to them.

3. Longitudinal Stiffeners

Rarely used in building construction, but common in massive bridge girders. Placed horizontally along the web (usually in the compression zone) to significantly increase the bending buckling strength of the web.

Flange-to-Web Weld Design

Connecting the built-up components.

The flanges must be welded to the web to form the I-shape. These welds are continuous fillet welds (or sometimes CJP groove welds). The size of the weld is governed by the horizontal shear flow (

q

, kips/inch) at the interface between the flange and the web.

Shear Flow Formula

Calculates the shear flow (force per unit length) required for connection design.

$$
q = \\frac{V Q}{I}
$$

The required weld size must resist this horizontal shear flow in addition to any local vertical wheel loads (e.g., in a crane girder). Furthermore, plate girders subjected to many loading cycles (bridges, cranes) must have these continuous welds rigorously checked for fatigue, which often dictates a much larger weld size than required for static strength.

Key Takeaways

Plate girders are highly optimized built-up sections used for very long spans and heavy loads where standard rolled shapes are insufficient.
To save immense weight and cost, plate girder webs are typically extremely slender ( $h/t_w$ is large), making them highly susceptible to local buckling under both flexure and shear.
Bearing stiffeners are required at supports and under heavy concentrated point loads to prevent the web from crippling or crushing locally.
Intermediate transverse stiffeners are used to artificially reduce the shear panel aspect ratio ( $a/h$ ), drastically increasing shear capacity and enabling post-buckling strength.
Tension Field Action (TFA) is a remarkable post-buckling mechanism where the buckled web acts like diagonal tension ties and the stiffeners act as vertical compression struts (like a truss), significantly increasing the girder's total shear capacity beyond its buckling limit.
The continuous welds connecting the flanges to the web are designed primarily to resist horizontal shear flow ( $VQ/I$ ) and fatigue.

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