Beams: Shear and Serviceability

Beams: Shear and Serviceability

Analysis and design of steel beams for shear capacity and serviceability limits like deflection, ponding, and floor vibration.

While flexure (bending moment) often governs the size of steel beams, they must also be explicitly checked for shear capacity and serviceability requirements. Shear failure can occur near supports or under heavy concentrated loads. Serviceability involves ensuring the beam performs well under everyday use without excessive deflection, vibration, or localized damage.

Shear Strength

The ability of the beam's web to resist shear forces without yielding or buckling.

In a typical I-shaped beam, the shear force is primarily resisted by the web. The flanges carry the bending moment, and the web carries the shear. The shear capacity depends on the web's area (

A_w = d \times t_w

) and its susceptibility to shear buckling.

Shear Yielding vs. Shear Buckling

Shear Yielding: Occurs in compact, thick webs where the entire web cross-section reaches its shear yield stress ( $\tau_{yield} \approx 0.60 F_y$ ). The web deforms plastically in shear. This is the dominant failure mode for almost all standard rolled W-shapes.
Shear Buckling: Occurs in slender webs (like plate girders) before the material yields. The web buckles diagonally out-of-plane due to the principal compressive stresses induced by the shear force. Once buckled, the web's capacity to carry shear drops significantly unless tension field action (TFA) is permitted.

V_n

Calculating the fundamental shear capacity of the section.

The nominal shear strength of a steel beam web (without Tension Field Action) is given by AISC Chapter G:

Nominal Shear Strength

Calculates the shear capacity of a steel beam web.

$$
V_n = 0.60 F_y A_w C_v
$$

C_v

Determining the reduction in shear capacity due to the potential for web shear buckling.

The value of

C_v

depends on the web's slenderness ratio (

h/t_w

, where

h

is the clear distance between flanges).

Determining Cv for Unstiffened Webs (AISC G2.1)

The web slenderness determines which buckling regime controls the shear capacity.

Case 1: Plastic Shear (Yielding) If

h/t_w \le 2.24\sqrt{E/F_y}

C_v = 1.0

Most standard W-shapes with $F_y \le 50$ ksi fall into this category. The web yields completely before buckling.

Case 2: Inelastic Shear Buckling If

2.24\sqrt{E/F_y} < h/t_w \le 1.10\sqrt{k_v E/F_y}

C_v = \frac{1.10\sqrt{k_v E/F_y}}{h/t_w}

The web begins to buckle locally while some portions are yielding.

Case 3: Elastic Shear Buckling If

h/t_w > 1.10\sqrt{k_v E/F_y}

C_v = \frac{1.51 k_v E}{(h/t_w)^2 F_y}

The web buckles elastically before any yielding occurs. Common for slender plate girders without transverse stiffeners.

Where

k_v = 5.0

for unstiffened webs with

h/t_w < 260

. If transverse stiffeners are provided (e.g., in plate girders),

k_v

is calculated based on the aspect ratio (

a/h

) of the web panel, and Tension Field Action (TFA) may be utilized to significantly increase the nominal shear strength (

V_n

) beyond these basic limits (see Plate Girders module).

\phi_v V_n

Applying resistance factors for LRFD design.

AISC provides a special, more lenient resistance factor for the vast majority of rolled W-shapes because shear yielding is a very ductile, predictable failure mode.

For LRFD:

For webs of rolled I-shaped members with $h/t_w \le 2.24\sqrt{E/F_y}$ (which is almost all of them): $\phi_v = 1.00$
For all other webs (built-up sections, slender webs, channels, etc.): $\phi_v = 0.90$

Concentrated Forces (Web Yielding & Web Crippling)

Checking the beam web for failure under heavy, localized loads.

When a beam is subjected to heavy concentrated loads (like a column bearing on the top flange) or at reaction points (like the beam resting on a support pad), the highly localized compressive force can crush or buckle the thin web beneath it. These are collectively known as Local Web Failures. If the web fails these checks, bearing stiffeners (vertical plates welded to the web beneath the load) must be added.

Local Web Yielding (AISC J10.2)

Occurs when the concentrated compressive load causes the web to yield at the flange-to-web junction (the fillet toe). It assumes the load spreads out at a 2.5:1 slope.

Formula (Interior Load, acting far from the end):

R_n = F_{yw} t_w (5k + l_b)

Formula (Edge Load, near support):

R_n = F_{yw} t_w (2.5k + l_b)

Where:

$F_{yw}$ = Web yield stress
$t_w$ = Web thickness
$k$ = Distance from outer face of flange to the web toe of the fillet
$l_b$ = Length of bearing (the physical length of the load applied to the flange parallel to the beam axis)
$\phi = 1.00$ (LRFD)

Web Crippling (AISC J10.3)

Occurs when the web buckles locally (wrinkles) under the compression of a concentrated load, treating the web like a short, slender column.

Formula (Interior Load):

R_n = 0.80 t_w^2 \left[ 1 + 3\left(\frac{l_b}{d}\right)\left(\frac{t_w}{t_f}\right)^{1.5} \right] \sqrt{\frac{E F_{yw} t_f}{t_w}}

Similar, more conservative formulas exist for edge loads depending on the ratio of

l_b/d

Where:

$d$ = Overall beam depth
$t_f$ = Flange thickness
$\phi = 0.75$ (LRFD) due to the brittle nature of local buckling

Web Sidesway Buckling (AISC J10.4)

Occurs when the compression flange is restrained against rotation but the tension flange is not (e.g., a beam resting on a bearing pad but not bolted down, while the top flange is restrained by a concrete slab).

If the beam is subjected to a heavy concentrated load, the entire web can buckle sideways, shifting laterally.

The nominal capacity $R_n$ depends on the ratio of $h/t_w$ and the ratio of $L_b/b_f$ .
If the web fails this check, it must be braced laterally at the load point, or bearing stiffeners must be installed.

Serviceability: Deflection

Ensuring the beam does not bend excessively under normal service conditions.

Serviceability limits ensure the structure is functional, aesthetically pleasing, and comfortable for occupants. Excessive deflection can damage non-structural elements (cracking drywall, breaking glass partitions, jamming doors) and cause a feeling of instability. Deflection is often the governing design criterion for long-span beams, rather than flexural strength.

Deflection Limits (IBC / AISC)

Deflection limits are not hard failure limits but rather building code requirements specified as a fraction of the beam's span (

L

). Common limits include:

Live Load Only: $L/360$ (typical for floors supporting plaster ceilings).
Total Load (Dead + Live): $L/240$ .
Roof Members (Total Load): $L/180$ or $L/240$ depending on the roofing material (e.g., rigid vs. flexible).

Calculating Deflection

Methods and formulas for determining the expected deflection of a beam.

Deflection (

\Delta

) is calculated using elastic beam theory (virtual work, moment-area, or standard tables). The formula depends on the load and support conditions.

Crucially, service (unfactored) loads ( $D, L, W$ ) are ALWAYS used for deflection calculations, never LRFD factored loads ( $1.2D+1.6L$ ).

Common Deflection Formulas

Uniformly Distributed Load ( $w$ ) on a Simply Supported Beam:

\Delta_{max} = \frac{5 w L^4}{384 E I}

Point Load ( $P$ ) at Midspan of a Simply Supported Beam:

\Delta_{max} = \frac{P L^3}{48 E I}

Where:

$E$ = Modulus of elasticity ( $29,000$ ksi for steel)
$I$ = Moment of inertia about the axis of bending ( $I_x$ typically)
$L$ = Span length

Ensure consistent units! Multiply by 1728 to convert $L^3$ (ft $^3$ ) to in $^3$ , or $L^4$ (ft $^4$ ) by 172812.*

Beam Deflection Check

Uniform Load ($w$): 5.0 kips/ft

Span ($L$): 30 ft

Moment of Inertia ($I_x$): 510 in$^4$

\\Delta_{max}

Actual Deflection ( $\\Delta_{max}$ ):6.161 in
Allowable Deflection ($L/360$):1.000 in
Status: FAILS - Excessive Deflection

Serviceability: Ponding and Floor Vibrations

Additional limit states that can severely impair the function or comfort of a structure.

Beyond simple deflection, steel beams must be checked for specific serviceability issues, particularly in specialized lightweight floor and roof systems.

Ponding in Roof Systems

Flat roofs are uniquely susceptible to ponding, a progressive instability. When rain falls, the roof framing deflects. This depression allows more water to accumulate, increasing the load, which causes further deflection, leading to more water accumulation. This positive feedback loop can lead to sudden roof collapse.

Prevention: The primary defense is providing adequate roof slope (minimum 1/4 inch per foot) and proper primary and secondary overflow drainage to prevent water accumulation entirely.
Design Check (AISC Appendix 2): If sufficient slope is not provided, the entire roof system (deck, secondary purlins, and primary girders) must be explicitly checked for stiffness. AISC provides specific stiffness index formulas ( $C_p$ and $C_s$ ) to ensure the framing is stiff enough to resist the cascading weight of accumulated water.

Floor Vibrations (AISC Design Guide 11)

Long-span, lightweight steel floors can experience unacceptable vibrations caused by human activity (walking, dancing, aerobics, machinery). While perfectly safe structurally, the bounce can make occupants nauseous or uncomfortable.

Natural Frequency ( $f_n$ ): The key parameter is the floor system's natural frequency. For typical office floors, a natural frequency greater than 3 to 5 Hz is often recommended to avoid resonance with the walking excitation frequency ( $1.5 - 2.5$ Hz).
Acceleration Limit ( $a_p/g$ ): Human perception is sensitive to acceleration. AISC DG11 limits the peak acceleration to fractions of gravity (e.g., $0.5\% g$ for offices, $5.0\% g$ for active spaces).
Damping ( $\beta$ ): Adding damping (partitions, heavy furniture, specialized ceiling systems) helps dissipate the vibrational energy faster. Bare steel floors have very low damping ( $\sim 1\%$ ), making vibration a common issue.
Solution: Increasing the stiffness (moment of inertia, $I$ ) of the floor beams and girders is the primary way to increase the natural frequency and reduce peak acceleration.

Torsion

Behavior of steel members subjected to twisting moments.

While standard W-shapes are highly efficient for flexure (bending), they perform exceptionally poorly when subjected to torsion (twisting) because they are "open" sections.

Torsional Behavior

Torsion in steel members is resisted by two entirely different internal stress mechanisms:

Pure (St. Venant) Torsion: Resisted by shear stresses forming continuous closed loops within the cross-section. The torsional capacity depends entirely on the torsional constant ( $J$ ). Closed shapes (like HSS pipes or rectangular tubes) have massive $J$ values and are highly efficient in pure torsion. Open shapes (W-shapes) have tiny $J$ values.
Warping Torsion: When an open section (like a W-shape) twists, its cross-section distorts out of its original plane (the flanges try to bend in opposite directions). If this warping is restrained (e.g., by fixed supports or stiffeners), it induces massive longitudinal normal stresses (warping stresses) in the flanges, acting similarly to bending stresses. The capacity depends on the warping constant ( $C_w$ ).

Design Rule: For members subjected to significant torsion (e.g., spandrel beams supporting heavy eccentric facades, curved bridge girders), HSS (Hollow Structural Sections) or box girders are strongly preferred over open I-shapes. If a W-shape must be used, extensive torsional bracing is required to prevent twisting.

Key Takeaways

Beam shear capacity ( $V_n$ ) is primarily provided by the web area ( $d \times t_w$ ) and is reduced by the $C_v$ factor if the web is slender enough to buckle.
Most standard W-shapes have compact webs ( $h/t_w \le 2.24\sqrt{E/F_y}$ ) that fail by ductile shear yielding, allowing a shear coefficient $C_v = 1.0$ and a resistance factor $\phi_v = 1.00$ .
Slender webs (common in plate girders) can fail by shear buckling, requiring a reduction in shear capacity ( $C_v < 1.0$ ) unless tension field action is designed for.
Heavy, concentrated point loads (like column bases or supports) can cause local web yielding or brittle web crippling, which may necessitate bearing stiffeners.
Serviceability requirements, particularly live load and total load deflection, often dictate the required moment of inertia ( $I_x$ ) for long-span beams.
Deflection calculations must always use service (unfactored) loads, not ultimate (factored LRFD) loads.
Flat roofs must be checked for ponding instability, and lightweight, long-span floors must be checked for uncomfortable human-induced vibrations.
Open W-shapes are terrible at resisting torsion; closed HSS shapes should be used whenever significant twisting forces are present.

PreviousBeams: Flexure - Examples & Applications

Quiz Me

NextBeams: Shear and Serviceability - Examples & Applications

Shear Strength

Shear Yielding vs. Shear Buckling

Nominal Shear Strength ( $V_n$ )

Nominal Shear Strength

Web Shear Coefficient ( $C_v$ )

Determining Cv for Unstiffened Webs (AISC G2.1)

Design Shear Strength ( $\phi_v V_n$ )

Concentrated Forces (Web Yielding & Web Crippling)

Local Web Yielding (AISC J10.2)

Web Crippling (AISC J10.3)

Web Sidesway Buckling (AISC J10.4)

Serviceability: Deflection

Deflection Limits (IBC / AISC)

Calculating Deflection

Common Deflection Formulas

Beam Deflection Check

Serviceability: Ponding and Floor Vibrations

Ponding in Roof Systems

Floor Vibrations (AISC Design Guide 11)

Torsion

Torsional Behavior

Beams: Shear and Serviceability

Shear Strength

Shear Yielding vs. Shear Buckling

Nominal Shear Strength (VnV_nVn​)

Nominal Shear Strength

Web Shear Coefficient (CvC_vCv​)

Determining Cv for Unstiffened Webs (AISC G2.1)

Design Shear Strength (ϕvVn\phi_v V_nϕv​Vn​)

Concentrated Forces (Web Yielding & Web Crippling)

Local Web Yielding (AISC J10.2)

Web Crippling (AISC J10.3)

Web Sidesway Buckling (AISC J10.4)

Serviceability: Deflection

Deflection Limits (IBC / AISC)

Calculating Deflection

Common Deflection Formulas

Beam Deflection Check

Serviceability: Ponding and Floor Vibrations

Ponding in Roof Systems

Floor Vibrations (AISC Design Guide 11)

Torsion

Torsional Behavior

Nominal Shear Strength ( $V_n$ )

Web Shear Coefficient ( $C_v$ )

Design Shear Strength ( $\phi_v V_n$ )