Beams

7 tools in the Structures area.

Section Properties

Cross-section properties of a beam from a section drawn on the screen: click points on the grid to lay out the outer boundary, drag or type the coordinates to adjust them, and add holes for hollow sections. The tool reports the area, centroid, second moments of area about the drawing axes and the centroid, principal moments and principal axis angle, radii of gyration, extreme fibre distances and elastic section moduli about the centroidal and principal axes, plastic section moduli and plastic neutral axes with the shape factors, first moments and widths at the centroidal axes for shear stress, perimeter, the Saint-Venant torsion constant J from Prandtl's stress function solved on a grid (hollow sections included), and the shear centre and warping constant C_w from the warping function. Signed-in analysts can save a section to their profile and load it into this and later beam tools.

1 worked example reproduced, listed on the tool page.

Beam Stress

Fiber stresses and strains of a straight beam from the section resultants (axial force, two shears, torque and two bending moments) on a section drawn on the screen or loaded from My Sections. The loads can be entered in any of four published sign conventions (vector right-hand rule, the aerospace fiber rule of Bruhn and Peery, Nastran CBAR / CBEAM element forces, the civil beam-diagram rule) and lines from a Nastran .f06 can be pasted straight in. Several load cases run at once; the page writes out every equation for one chosen case and tabulates the critical fibers, strains, shear stresses and margins of safety of all of them.

1 worked example reproduced, listed on the tool page.

Beam Deflection

Reactions, shear and moment diagrams, slopes and deflections of a single-span beam by the classical closed forms: simply supported, cantilever, fixed at both ends, propped cantilever and simply supported with an overhang, under point loads, uniform and partial uniform loads, triangular loads and end moments (eighteen catalogue cases), or any number of loads on a simply supported or cantilever beam added by superposition. Every case is solved exactly with singularity functions, so the shear and moment equation of each segment is written with the substituted numbers and the maximum moment, the maximum deflection and its position, the end slopes, the deflection at a chosen point and the stiffness P/δ follow; the textbook formulas of AISC Table 3-23 and the Gere and Hibbeler appendices are printed beside the values they reproduce, and a margin of safety is written against an allowable deflection (L/360, L/240, L/180 or entered). Four diagrams (load, shear, moment, deflection). The default is a W12x26 on a 20 ft simple span under 80 lb/in, whose mid-span deflection 5wL^4/(384EI) = 0.584 in the page reproduces.

1 worked example reproduced, listed on the tool page.

Continuous Beam and Moment Distribution

Continuous beams of up to eight spans on pinned, fixed, free (cantilever) or spring supports, with support settlements, under uniform, partial, triangular, point and couple loads: reactions, shear and moment diagrams with the maxima located, slopes and deflections, pattern live loading (every combination of loaded spans, or the alternate and adjacent-span patterns of the building codes, as an envelope), a moving axle group stepped along the beam, influence lines, moment redistribution and a plain comparison with an allowable moment and a span-over-n deflection limit. The beam is solved exactly by the stiffness method with Hermite beam elements and closed-form fixed-end moments, and the same case is worked by Hardy Cross moment distribution as the handbook table (stiffness factors, distribution factors, fixed-end moments, the balancing and carry-over cycles to an entered tolerance, the final moments) and by Clapeyron's three-moment equation. The default is three equal 20-ft spans of a W12x26 under 1.0 kip/ft dead and 1.5 kip/ft patterned live load (illustrative), for which the page reproduces the classical coefficients: support moment 0.1 w L squared, end-span maximum 0.08 w L squared, end reaction 0.4 w L.

1 worked example reproduced, listed on the tool page.

Curved Beam

Stresses in a beam curved in the plane of bending, for a section drawn on the screen: the section constants of the Winkler (curved-beam) theory are computed exactly for the polygon by Green's theorem, the circumferential stress at the inner and outer fibres and at any radius follows from the normal force and bending moment, the radial stress and the transverse shear stress are found from the equilibrium of the part of the section between the inner fibre and each radius and their maxima are located, and the straight-beam formulas are shown alongside for comparison. The results page draws the section with the centroid and neutral-axis radii and the two stresses across the depth. Saved sections from the Section Properties tool can be loaded.

Curved Beam Thin-Walled

Stresses in a curved beam of I, T or channel section built from plate dimensions, with Bleich's correction for the flanges: in a curved beam the radial component of the circumferential stress bends the flange outstands towards the web, so the flange tips carry less stress than the plane-section theory assumes and the stress at the web-flange junction is higher. The tool computes Bleich's effective-width ratio α and transverse-stress factor β for each flange from his flange model, replaces each flange by its effective width, and runs the same Winkler analysis as the drawn-section Curved Beam tool: circumferential, radial and shear stresses across the depth with tables and charts, plus the transverse bending stress in each flange and a comparison with the uncorrected section.

Lateral-Torsional Buckling

Lateral-torsional buckling strength of a doubly symmetric I-beam or a channel bent about its strong axis: the elastic critical moment of Timoshenko (warping and Saint-Venant torsion), the AISC 360 limiting lengths L_p and L_r, the nominal flexural strength M_n in the plastic, inelastic and elastic ranges with the moment-gradient factor C_b, and a margin of safety against the applied moment. The section comes from plate dimensions (area, moments of inertia, plastic modulus, torsion constant and warping constant computed exactly from the drawn outline), from a saved section, or from properties typed in; a saved MIL-HDBK-5J material fills E, G and F_y.

1 worked example reproduced, listed on the tool page.

Other Structures sections

Materials · Composites · Frames and Trusses · Loads and Dynamics · Joints · Stress and Strain · Stability · Plates and Shells · Thin-Walled Structures · Contact · Stress Concentrations · Fatigue · Damage Tolerance