Module 8: Steel Beams (Flexural Members) - Examples & Applications

Module 8: Steel Beams (Flexural Members) - Examples & Applications

Bending Stresses and Plastic Moment Capacity

Basic: Calculating Plastic Moment Capacity

Determine the nominal plastic moment capacity (

M_p

) of a W16x36 beam of A992 steel (

F_y = 345 \text{ MPa}

Given Section Properties:

Plastic Section Modulus ( $Z_x$ ): $1.05 \times 10^6 \text{ mm}^3$

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Intermediate: Moment Capacity of a Laterally Supported Beam

Determine the LRFD design flexural strength (

\phi_b M_n

) of the same W16x36 beam (

M_p = 362.25 \text{ kN}\cdot\text{m}

). The beam is laterally supported continuously by a concrete floor deck. Assume the section is compact.

LRFD Factor: $\phi_b = 0.90$ for flexure.

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Laterally Unsupported Beams (LTB)

Advanced: Calculating Inelastic LTB Capacity (Zone 2)

Determine the nominal moment capacity (

M_n

) of a W14x90 beam of A992 steel (

F_y = 345 \text{ MPa}

) with an unbraced length

L_b = 6.0 \text{ m}

. Assume the moment gradient yields a bending coefficient

C_b = 1.14

Given Section Properties and Limits:

Plastic Moment ( $M_p$ ): $600 \text{ kN}\cdot\text{m}$
Yield Moment modified for residual stress ( $0.7 F_y S_x$ ): $420 \text{ kN}\cdot\text{m}$
Limiting unbraced length for plastic yielding ( $L_p$ ): $4.0 \text{ m}$
Limiting unbraced length for inelastic buckling ( $L_r$ ): $12.0 \text{ m}$

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Conceptual: The Effect of Unbraced Length on Moment Capacity

An engineer designs a roof using standard W-shape steel beams spaced 3 meters apart. Initially, the metal roof deck was planned to be directly fastened to the top flanges of the beams, providing continuous lateral support. However, to save money, the contractor proposes removing the direct fastening and only bracing the beams at their ends (12 meters apart). Explain the structural consequences of this change regarding the beam's moment capacity.

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Shear Strength and Deflection

Basic: Checking Shear Strength

A W24x68 steel beam (

F_y = 345 \text{ MPa}

) must support a maximum factored shear force (

V_u

) of

450 \text{ kN}

. Determine if the beam has adequate shear capacity. Assume

C_v = 1.0

(no web buckling) and

\phi_v = 1.00

for LRFD shear.

Given Section Properties:

Overall depth ( $d$ ): $603 \text{ mm}$
Web thickness ( $t_w$ ): $10.5 \text{ mm}$

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Intermediate: Checking Deflection Limits

A simply supported floor beam spans

8.0 \text{ meters}

. Under service (unfactored) loads, the immediate elastic deflections are calculated as:

\Delta_{\text{Dead}} = 12 \text{ mm}

and

\Delta_{\text{Live}} = 18 \text{ mm}

. The building code restricts live load deflection to

L/360

and total load deflection to

L/240

Verify if the beam satisfies serviceability requirements.

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Combined Axial and Bending Stresses

Advanced: Checking a Beam-Column (AISC Interaction Equation)

A W14x90 column is subjected to a factored axial compressive load of

P_u = 1200 \text{ kN}

and a factored bending moment of

M_{ux} = 250 \text{ kN}\cdot\text{m}

Given Capacities:

Design Compressive Strength ( $\phi_c P_n$ ): $3000 \text{ kN}$
Design Flexural Strength ( $\phi_b M_{nx}$ ): $550 \text{ kN}\cdot\text{m}$
Moments about the y-axis are zero ( $M_{uy} = 0$ ).

Determine if the member satisfies the AISC interaction criteria.

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