Theoretical Basis of Steel Design
This article summarises which code rules the StatiCAD steel beam and column design module applies and which assumptions it makes. The rule set used is ÇYTHYE (Turkish Design, Calculation and Construction Principles for Steel Structures) and the provisions of TBDY 2018 relating to steel. For usage see Steel Beam and Column Design.
Design approach: LRFD (Load and Resistance Factor Design). Resistance factors: flexure φb = 0.90; shear φv = 0.90 (1.00 for a compact rolled I web); compression φc = 0.90.
1. Material and section properties
The steel grade is assigned to the member together with the section; Fy and E are read from the material library. Section properties (A, Ix, Iy, Wel = Sx, Wpl = Zx, Zy, Sy, J, Cw, h, b, tf, tw, r) are taken from the prefabricated section database. The modulus of elasticity in the material library may be defined in kg/cm² or MPa; the program recognises the unit automatically.
2. Beam flexural strength (Chapter 9)
For doubly symmetric rolled I/H beams bending about their strong axis, three regions are defined (Lb: laterally unbraced length):
| Region | Condition | Nominal strength Mn |
|---|---|---|
| Yielding (plastic) | Lb ≤ Lp | Mp = Fy·Zx |
| Inelastic lateral–torsional buckling | Lp < Lb ≤ Lr | Mn = Cb·[Mp − (Mp − 0.7·Fy·Sx)·(Lb − Lp)/(Lr − Lp)] ≤ Mp |
| Elastic lateral–torsional buckling | Lb > Lr | Mn = Fcr·Sx ≤ Mp |
- Lp = 1.76·ry·√(E/Fy)
- Fcr = Cb·π²·E/(Lb/rts)² · √(1 + 0.078·(J·c/(Sx·h0))·(Lb/rts)²)
- Lr = 1.95·rts·(E/(0.7·Fy))·√(J·c/(Sx·h0) + √((J·c/(Sx·h0))² + 6.76·(0.7·Fy/E)²))
- rts² = √(Iy·Cw)/Sx, h0 = h − tf, c = 1 (doubly symmetric I)
The report shows Lp, Lr and the name of the governing limit state, so that the consistency of Lb with the limit state can be checked by the engineer. The flexural compactness of the cross-section is classified; for a non-compact flange the local flange buckling limit state comes into play.
2.1 Unbraced length Lb and moment modification factor Cb
Lb is the length of the compression flange that is laterally unbraced (ÇYTHYE 9.2.2). If the lateral support (Le) fields in the member properties are left automatic, Lb = member length L (between its end nodes) is used and the ends are assumed to be laterally braced (ÇYTHYE 9.1(a)). If there is intermediate lateral bracing, the largest unbraced length is entered manually in the Le field.
- Cb = 12.5·Mmax / (2.5·Mmax + 3·MA + 4·MB + 3·MC) (ÇYTHYE Eq. 9.1), 1.0 ≤ Cb ≤ 3.0
- Mmax, MA, MB, MC: absolute values of the largest moment along the unbraced length and of the moments at the 1/4, 1/2 and 3/4 points
The moments are taken from the beam stations (segment ends) of the finite element solution; the values at the quarter points are found by linear interpolation between adjacent stations.
- Each strength combination is checked with its own moment diagram. Because Cb varies with the combination, the governing combination is not the one giving the largest moment but the one with the largest demand/capacity ratio. In the report the design moment, the combination name and Cb belong to this combination; shear and axial force are taken from the envelope.
- Cantilever (a member with one end not connected to a column, wall or another beam): Lb = L and Cb = 1.0 (ÇYTHYE 9.1(e)).
- The end of a secondary beam framing into another beam is treated as a support, like a column or wall support; Cb is calculated from the diagram.
- If a Lb shorter than L is entered in the Le field (intermediate bracing; the moment diagram of the unbraced segment is not separately known), or the beam is divided into fewer than 4 finite element segments, Cb = 1.0 is used (safe side).
3. Beam shear strength (Chapter 10)
Vn = 0.6·Fy·Aw·Cv1, Aw = d·tw
If h/tw ≤ 2.24·√(E/Fy) is satisfied for a rolled I web, Cv1 = 1.0 and φv = 1.00 are used; otherwise the general web rules are applied.
4. Combined effect in beams (Chapter 11)
The interaction of flexure, shear and axial force is checked and written to the report as a single ratio; if the ratio is ≤ 1.0, UYGUN (OK) is marked.
5. Beam deflection
f = 5/48 · Mmax · L² / (E·Ix) (simple beam, uniformly distributed load equivalent); condition f ≤ L/K.
Mmax is the service moment: the largest of the unfactored G+Q and G+S combinations. For a cantilever beam the coefficient 1/3 is used instead of 5/48 (tip-load equivalent; conservative for a distributed load). K comes from the deflection K selection in the member properties (default 300; 200 / 150 / manual).
6. Column compressive strength (Chapter 8)
For a doubly symmetric I/H compression member, three elastic buckling stresses are calculated and the smallest governs:
- Flexural: Fex = π²·E/(KLx/rx)², Fey = π²·E/(KLy/ry)²
- Torsional: Fez = [π²·E·Cw/(KLz)² + G·J] / (Ix + Iy), G = E/2.6
- Fe = min(Fex, Fey, Fez)
- Fy/Fe ≤ 2.25 → Fcr = 0.658^(Fy/Fe) · Fy (inelastic); Fy/Fe > 2.25 → Fcr = 0.877·Fe (elastic)
- Pn = Fcr·Ag, φc = 0.90
The governing mode is stated in the report. The weak-axis length is used for the torsional buckling length.
Effective buckling length: By default KL = storey height + the bottom/top level differences of the member, K = 1.0. A separate length can be defined for each axis by entering K·L manually in the Le fields of the vertical bond beam window. If KL/r > 200, a red warning is written to the report (the calculation is not stopped).
Cross-section slenderness (Table 5.1A, axial compression): Flange b/(2·tf) ≤ 0.56·√(E/Fy); web hw/tw ≤ 1.49·√(E/Fy). For a cross-section exceeding the limit (slender), the member is marked “narin enkesit” (slender cross-section) in the report and is considered inadequate.
7. Column weak-axis flexural capacity
Mny = Fy·Zy ≤ 1.6·Fy·Sy (lateral–torsional buckling does not occur about the weak axis).
8. Combined effect in columns (Chapter 11.3)
- Pr/Pc ≥ 0.2: Pr/Pc + 8/9·(Mrx/Mcx + Mry/Mcy) ≤ 1.0 (11.3a)
- Pr/Pc < 0.2: Pr/(2·Pc) + (Mrx/Mcx + Mry/Mcy) ≤ 1.0 (11.3b)
Pc = φc·Pn; Mcx is the strong-axis capacity including lateral–torsional buckling (Lb = KLy, Cb = 1.0), and Mcy is calculated with the weak-axis rule. The equation used is stated in the report.
9. Internal forces and load combinations
For the beam M, V (and N), and for the column N, Mx, My and V are taken from the program's finite element solution as the envelope of the following combinations:
- The program's combinations, as converted for steel design: in combinations with earthquake, 1.0G → 1.2G and Q at least 1.0Q (TBDY 4.4.4.2). The names of these combinations appear in the report with the suffix “[1.2G+1.0Q]”.
- ÇYTHYE LRFD combinations (from the plain G, Q, S, W load cases): 1.4G; 1.2G + 1.6Q (+0.5S); 1.2G + 1.6S (+1.0Q); if wind is defined, for each wind case 1.2G + 1.6S + 0.8W, 1.2G + 1.0Q (+0.5S) + 1.6W and 0.9G + 1.6W.
If no wind load case is defined, an analysis warning is given. The name of the combination that gives the envelope is shown next to the demand in the report.
9.1 Artificial axial force caused by the offset
If Take Offset1 and Offset2 values into account in the system stiffness matrix or, with a defined axis, Take eccentricities into account in the system stiffness matrix is ticked in Project General Settings, the beam axis is modelled lower than the joint axis by e2 = offset2 − h/2. Because the rigid diaphragm or the supports restrain the axial displacement of the joints, an axial force that does not exist in reality arises in the beam (the effect of a composite section with an infinitely rigid flange). This force is constant in each chain between diaphragm joints:
- Nk = ck · M̄3, ck = A·e2 / (I3 + A·e2²)
- M̄3: length-weighted average moment along the chain; M3 is the total moment about the joint axis.
For IPE 300 (A = 53.81 cm², I3 = 8356 cm⁴, e2 = −15 cm), ck = −3.944 m⁻¹; the finite element segment forces reproduce this value to four-digit accuracy. Without this subtraction, the force reaches 318–721 kN in the test models and increases the M–N–V ratio by about 30%. As with cold-formed steel beams, the program subtracts this share from the segment axial force of each load case (see Theoretical Basis of Cold-Formed Steel Design, Section 12): the moment does not change; the subtracted share does not exceed |N| and does not reverse its sign; chains are recognised from consecutive segments with equal N (if the “Do not connect beam intermediate joints to the rigid diaphragm” option is ticked, the whole beam is a single chain). The remaining axial force is the trapezoidal-rule remainder arising from the average of the end moments ((2/3)/n² of the peak value under a distributed load; 0.5% with 12 segments, on the safe side). The effect of the offset on the beam stiffness is a model result and is retained: in a rigid-ended beam the end moments may come out larger with the offset on (84.8 kNm against 74.5 kNm in one example).
10. Scope limits
- The load-bearing capacity check is limited to hot-rolled doubly symmetric I/H sections; U, L, T, box and pipe sections can be assigned but are marked in the report as “kesit tipi desteklenmiyor” (section type not supported).
- The design of cold-formed sections is performed in the light-gauge steel module.
- In beams Cb is calculated from the moment diagram of the combination (Section 2.1); Cb = 1.0 is used for strong-axis bending of columns. For columns K = 1.0 is assumed; different support conditions are provided by the user through the Le fields.
- There is no effective area method for slender cross-sections, and no torsional–flexural buckling of singly symmetric sections.
- The seismic ductility cross-section requirements of TBDY Chapter 9, the column–beam capacity ratio and bracing members are outside the scope of this edition.
- Lateral support in a beam is taken as a single length; there is no definition of an intermediate support list (Cb = 1.0 for a beam with intermediate bracing). The automatic Lb is the length between end nodes: for a beam divided into segments without intermediate supports in the same direction, the real unbraced length must be entered in the Le field.