Structural Design Calculators

Load parameters

• ⚠ Loads are nominal (service-level) values — factor them with the ASCE 7-22 §2.3 combinations before member design.

1. 1. Influence areaTable 4.7-1OK
   $$K_{LL}{A}_T$$
   $$K_{LL}{A}_T=4.00\times 400.00=1600.00{\text{ft}}^2$$

   ≥ 400 ft² — reduction permitted.

2. 2. Reduction multiplierEq. 4.7-1
   $$L=L_o\left(0.25+\frac{15}{\sqrt{K_{LL}{A}_T}}\right)$$
   $$0.25+\frac{15}{\sqrt{1600.00}}=0.625$$
3. 3. Lower-bound limit§4.7.2
   $$L\ge 0.50L_o\text{(one floor)}$$
   $$\text{multiplier used}=0.625$$
4. 4. Design live load§4.7
   $$L=L_o\cdot m$$
   $$L=50.00\times 0.625=31.25\text{psf}$$
5. 5. Inference & recommended next iterationEngineering judgement

   Carry this nominal load (31.25 psf) into the load-combination calculator; the factored effect, not this value, sizes the member.

Element, tributary geometry & surface loads

• ⚠ Lr and S are alternatives in the §2.3 combinations — the larger governs; they are never added.

1. 1. Element and load pathASCE 7-22 §4.7

   Interior columns — KLL = 4. Collects 3 floors plus the roof — loads accumulate down the load path slab → beam → girder → column → foundation.

2. 2. Unreduced live load from the occupancy tableASCE 7-22 Table 4.3-1
   $$L_o$$
   $$L_o=50.00\text{psf}(\text{Office — offices})$$
3. 3. Tributary area served§4.7.2
   $$A_T=A_{T,floor}\times n_{floors}$$
   $$A_T=900.00\times 3=2700.00{\text{ft}}^2$$
4. 4. Influence areaTable 4.7-1
   $$K_{LL}{A}_T$$
   $$K_{LL}{A}_T=4.00\times 2700.00=10800.00{\text{ft}}^2$$
5. 5. Live-load reduction multiplierEq. 4.7-1, §4.7.2–4.7.5OK
   $$L=L_o\left(0.25+\frac{15}{\sqrt{K_{LL}{A}_T}}\right)\ge 0.40L_o$$
   $$m=0.400$$

   Eq. 4.7-1 gives 0.394; the 0.40Lo floor for members supporting two or more floors controls the lower bound.

6. 6. Reduced live load delivered to the element§4.7
   $$L=m{L}_o$$
   $$L=0.400\times 50.00=20.00\text{psf}$$
7. 7. Roof live load reduced for tributary area§4.8.2, Eq. 4.8-1
   $$L_r=L_{r,o}{R}_1{R}_2(12\le L_r\le 20\text{psf})$$
   $$A_t=900.00{\text{ft}}^2\Rightarrow R_1=0.600,L_r=12.00\text{psf}$$
8. 8. D transferred to the interior columnASCE 7-22 §3.1
   $$P_D=q_D{A}_T/1000$$
   $$P_D=85.00\text{psf}\times 2700.00{\text{ft}}^2=270.00\text{kips}$$
9. 9. L transferred to the interior columnASCE 7-22 §4.7
   $$P_L=q_L{A}_T/1000$$
   $$P_L=20.00\text{psf}\times 2700.00{\text{ft}}^2=54.00\text{kips}$$
10. 10. Lr transferred to the interior columnASCE 7-22 §4.8
    $$P_{Lr}=q_{Lr}{A}_T/1000$$
    $$P_{Lr}=12.00\text{psf}\times 900.00{\text{ft}}^2=10.80\text{kips}$$
11. 11. S transferred to the interior columnASCE 7-22 Ch. 7
    $$P_S=q_S{A}_T/1000$$
    $$P_S=25.00\text{psf}\times 900.00{\text{ft}}^2=22.50\text{kips}$$
12. 12. Governing LRFD combination on this element§2.3.1
    $$R_u=\max (\text{§2.3 combinations})$$
    $$=421.65\text{kips}(LRFD2)$$
13. 13. Conclusions & next stepASCE 7-22 Ch. 2 & 4
    • Lo = 50.00 psf from Table 4.3-1 reduces to L = 20.00 psf on this element (60% reduction) — the reduction belongs to the element, so the slab, beam, girder and column each get a different L.
    • Carry Pu = 421.65 kips into the column / brace calculator; that is ≈ 140.55 kips per supported level.
    • Only D, L and Lr/S/R change with tributary area — W and E arrive from the lateral analysis and are not area-reduced (§4.7 applies to live load only).
    • Re-run this takedown whenever the framing layout changes — the tributary area, KLL and therefore L all move with it.

Nominal (service) load effects

• ⚠ Net uplift of 9.00 kips — connections, anchorage and foundation stability must be designed for load reversal.

1. 1. LRFD 1§2.3.1 (1)
   $$1.4D$$
   $$LRFD1=56.00\text{kips}$$
2. 2. LRFD 2§2.3.1 (2)OK
   $$1.2D+1.6L+0.5(L_r\text{or}S\text{or}R)$$
   $$LRFD2=103.50\text{kips}$$

   Governing combination.

3. 3. LRFD 3§2.3.1 (3)
   $$1.2D+1.6(L_r\text{or}S\text{or}R)+(0.5L\text{or}0.5W)$$
   $$LRFD3=87.00\text{kips}$$
4. 4. LRFD 4§2.3.1 (4)
   $$1.2D+1.0W+0.5L+0.5(L_r\text{or}S\text{or}R)$$
   $$LRFD4=95.50\text{kips}$$
5. 5. LRFD 5§2.3.6 (5)
   $$1.2D+1.0E+0.5L+0.2S$$
   $$LRFD5=86.00\text{kips}$$
6. 6. LRFD 6§2.3.1 (6)
   $$0.9D+1.0W$$
   $$LRFD6=61.00\text{kips}$$
7. 7. LRFD 7§2.3.6 (7)
   $$0.9D+1.0E$$
   $$LRFD7=56.00\text{kips}$$
8. 8. Required strengthASCE 7-22 §2.3
   $$R_u=\max (\text{combinations})$$
   $$R_u=103.50\text{kips}(\text{LRFD 2})$$
9. 9. LRFD 6 (uplift)§2.3.1 (6)NG
   $$0.9D-1.0W_{up}$$
   $$-9.00\text{kips}$$

   Net reversal — the effect changes sign (uplift / net tension).

10. 10. LRFD 7 (uplift)§2.3.6 (7)OK
    $$0.9D-1.0E_{up}$$
    $$16.00\text{kips}$$

    No reversal from this case.

11. 11. Conclusions & recommended next iterationASCE 7-22 Ch. 2
    • Governing case: LRFD 2 → Ru = 103.50 kips, 7.7% above the next case (LRFD 4 = 95.50 kips).
    • A gravity combination governs — the lateral cases are not critical for strength here, but drift and stability (App. 8 B1/B2) still must be checked.
    • Net UPLIFT of 9.00 kips occurs in LRFD 6 (uplift). Next: design the roof-to-wall / column-to-base connection, anchor rods and hold-downs for this net tension, check overturning and sliding of the footing with only 0.9D resisting, and add ballast or a larger footing if the resisting moment margin is under 1.5.
    • The governing case is clearly separated, so a single design iteration on this combination is sufficient.
    • Recommended next step: carry Ru = 103.50 kips into the member calculator (φRn ≥ Ru), then re-run this sheet if any nominal load changes by more than 10%.

Material & trial shape

AISC Manual Part 1 shape families. A992 governs W-shapes; A572 Gr. 50 plates; A500 Gr. C for HSS and pipe (AISC 360-22 Table A3.1).

1. 1. Material specificationASTM A992/A992M — AISC 360-22 Table A3.1
   $$\text{Grade}$$
   $$\text{A992 W-shapes }:F_y=50\text{ksi},F_u=65\text{ksi}$$

   Rolled W-shapes — default for buildings

2. 2. Modulus of elasticityAISC 360-22 §A3 / Table 1-1
   $$E=29,000\text{ksi},G=11,200\text{ksi}$$
   $$E=29000\text{ksi}$$
3. 3. Required plastic modulusAISC 360-22 Eq. F2-1
   $$Z_{x,req}=\frac{M_u}{{\varphi }_b{F}_y}$$
   $$Z_{x,req}=\frac{250.00(12)}{0.90(50)}=66.67{\text{in}}^3$$
4. 4. Trial shape capacity§F2.1OK
   $${\varphi }_b{M}_p={\varphi }_b{F}_y{Z}_x$$
   $${\varphi }_b{M}_p=\frac{0.90(50)(101.00)}{12}=378.75\text{kip-ft}$$
5. 5. Selection guidanceAISC Manual Tables 3-2 / 1-1
   $$\text{Choose the lightest shape with }{Z}_x\text{or}{A}_g\ge \text{required}$$
   $$\text{required}=66.67\text{in³}$$

   Use the AISC shape picker above — Manual Table 3-2 values are listed for Fy = 50 ksi.

6. 6. Inference & recommended next iterationAISC Manual Part 3 / Part 1
   • The trial shape is lightly stressed (D/C = 0.66). Re-iterate downward: the lightest shape with Zx ≥ 70.00 in³ will save weight and cost.
   • Next: confirm Lb versus Lp and Lr (Chapter F calculator) — LTB can reduce φbMn well below φbMp — then check shear (Ch. G) and deflection.
   • Grade check: 50 ksi is assumed throughout; if a different grade is substituted, the required Zx scales by 50/Fy,new.

Strength & load effects

• ⚠ The β estimate uses generic coefficients of variation (VR = VQ = 0.35) and is illustrative only.

1. 1. Limit-state format§B3.1
   $$\varphi R_n\ge R_u=\sum {\gamma }_i{Q}_i$$
   $$\varphi =0.90,R_n=400.00\text{kips}$$
2. 2. Factored (required) strengthASCE 7-22 §2.3.1 (2)
   $$R_u=1.2D+1.6L$$
   $$R_u=1.2(80.00)+1.6(120.00)=288.00\text{kips}$$
3. 3. Design strength§B3.1OK
   $$\varphi R_n$$
   $$\varphi R_n=0.90(400.00)=360.00\text{kips}$$
4. 4. LRFD demand / capacity§B3.1OK
   $$\frac{R_u}{\varphi R_n}\le 1.0$$
   $$\frac{288.00}{360.00}=0.800$$
5. 5. Equivalent ASD safety factor§B3.2, Commentary B3
   $$\Omega =\frac{1.5}{\varphi }$$
   $$\Omega =\frac{1.5}{0.90}=1.67$$
6. 6. ASD check§B3.2OK
   $$\frac{R_n}{\Omega }\ge R_a=D+L$$
   $$\frac{400.00}{1.67}=240.00\text{vs}{R}_a=200.00\Rightarrow \text{D/C}=0.833$$
7. 7. Live-to-dead ratio effectCommentary B3.1
   $$\frac{L}{D}$$
   $$\frac{L}{D}=1.50$$

   LRFD and ASD give identical members at L/D ≈ 3; LRFD is more economical for larger L/D and more conservative for small L/D.

8. 8. Nominal factor of safetyReliability basis
   $$FS=\frac{R_n}{D+L}$$
   $$FS=\frac{400.00}{200.00}=2.00$$
9. 9. Approximate reliability indexAISC Commentary B3.1
   $$\beta \approx \frac{\ln (R_n/Q_n)}{\sqrt{V_R^2+V_Q^2}}$$
   $$\beta \approx \frac{\ln (2.00)}{\sqrt{0.35^2+0.35^2}}=1.40$$

   Target β ≈ 2.6 (members, gravity) and 4.0 (connections).

10. 10. Inference & recommended next iterationAISC 360-22 §B3.1 · ASCE 7-22 §2.3
    • ADEQUATE and efficient — D/C = 0.80, governed by LRFD satisfied. This is the target band (0.75–0.95) for an economical design.
    • Utilisation is 0.80. A lighter section with about 0.89× less capacity would still satisfy this limit state.
    • Target for the next trial: capacity ≈ 320.00 kips (D/C ≈ 0.90) against the required 288.00 kips.
    • Next: carry this result into the governing load combination and confirm the remaining limit states for this member.

Material response parameters

1. 1. Yield strain§A3.1
   $${\epsilon }_y=\frac{F_y}{E}$$
   $${\epsilon }_y=\frac{50}{29000}=0.00172$$
2. 2. Elastic branchHooke's law
   $$f=E\epsilon (\epsilon \le {\epsilon }_y)$$
   $$E=29000\text{ksi}$$
3. 3. Applied strain regime§A3.1OK
   $$\epsilon =0.01000$$
   $$\epsilon /{\epsilon }_y=5.80\Rightarrow \text{Yield plateau plastic flow}$$
4. 4. Stress at applied strainIdealized tri-linear model
   $$f(\epsilon )$$
   $$f=50.00\text{ksi}$$
5. 5. Plastic (permanent) strainUnload elastically along slope E
   $${\epsilon }_p=\epsilon -\frac{f}{E}$$
   $${\epsilon }_p=0.01000-\frac{50.00}{29000}=0.00828$$
6. 6. Ductility ratioSeismic detailing basis
   $$\mu =\frac{\epsilon }{{\epsilon }_y}$$
   $$\mu =5.80$$

   Compact seismically-detailed members are expected to reach μ ≈ 8–12.

7. 7. Elongation of the memberCompatibility
   $$\Delta =\epsilon L_0$$
   $$\Delta =0.01000\times 120.00=1.200\text{in}$$
8. 8. Axial force at this strainEquilibrium
   $$P=fA$$
   $$P=50.00\times 10.00=500.00\text{kips}$$
9. 9. Strain-hardening modulusTypical 2–4% of E
   $$E_{sh}\approx \frac{F_u-F_y}{{\epsilon }_u-{\epsilon }_{sh}}$$
   $$E_{sh}=800\text{ksi}\approx 2.8\%\text{of}E$$
10. 10. Yield ratioAISC Seismic §A3.2 limits Ry, RtOK
    $$\frac{F_y}{F_u}$$
    $$\frac{50}{65}=0.769$$

    Yield ratio ≤ 0.85 preserves inelastic reserve for seismic systems.

11. 11. Section shape factorAISC 360-22 §F2
    $$SF=\frac{M_p}{M_y}=\frac{Z}{S}$$
    $$SF=1.120$$

    ≈1.12 for W-shapes about the strong axis; 1.5 for a rectangle.

12. 12. Strain-energy densitiesResilience & toughness
    $$u_e=\frac{1}{2}{F}_y{\epsilon }_y,u_t\approx \frac{1}{2}(F_y+F_u){\epsilon }_u$$
    $$u_e=0.0431\text{ksi},u_t\approx 8.63\text{ksi}$$
13. 13. Inference & recommended next iterationAISC 360-22 §A3 · Table A3.1
    • ADEQUATE but essentially fully utilised — D/C = 1.00. Only 0% reserve remains against Yield plateau (plastic flow); any load increase during design development will fail this check.
    • Utilisation is 1.00. A lighter section with about 1.11× less capacity would still satisfy this limit state.
    • Target for the next trial: capacity ≈ 55.56 ksi (D/C ≈ 0.90) against the required 50.00 ksi.
    • Next: carry this result into the governing load combination and confirm the remaining limit states for this member.