Results.
| Example | Reference | Records | Largest deviation |
|---|---|---|---|
| Prismatic beam | Independent calculation, Published source | 2 | 0% |
| Cantilever, self weight | Closed form | 2 | 0% |
| Mixed model of shell, frame and bar elements | Closed form | 1 | 0% |
| Tapered beam, fixed at both ends | Independent calculation | 1 | < 0.001% |
Problem. Self-weight comes from the consistent mass matrix times the acceleration. The results for frames, tapered members, shells and bars are compared with closed forms. For beams the fixed-end moments are compared as well.
Reference document. For the cantilever the tip deflection and support moment, and for the prismatic beam the fixed-end moment, come from closed forms. For the mixed model the reference is the total weight computed by hand. For the tapered member the reference is the force method.
Self-weight is the mass times the gravitational acceleration and produces this fixed-end moment in a prismatic beam.
The tip deflection and the support moment of a cantilever under self-weight are given by these relations.
Numerical model. A prismatic beam, a cantilever, a mixed model and a fixed-fixed tapered beam are used. The mixed model contains shell, frame and axial bar elements together. For the tapered member the weight is computed with the flexibility matrix method.
Prismatic beam
Independent calculation. The same load computed along two independent paths, from the mass matrix and from the Hermite integral
| Quantity | Condition | Reference | Computed | Deviation |
|---|---|---|---|---|
| Relative deviation between the mass matrix path and the distributed load path | ρ = 2.5 · A = 0.18 · g = 9.81 | 0 | 0% | |
| Fixed-end moment at end i | w = 4.4145 · L = 8 | 23.544 | 23.544 | 0% |
Cantilever, self weight
Parameters. L = 8 · ρg = 24.525 · element count = 8
Closed form. The closed form for a cantilever
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Tip deflection | -0.013952 | -0.013952 | 0% |
| Moment at the fixed support | 141.264 | 141.264 | 0% |
Mixed model of shell, frame and bar elements
Parameters. shells = 4×3×0.2 · columns = 2 · bars = 1
Closed form. A hand calculation, the volumes of the slab and the two columns times density and gravity
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Total vertical support reaction | 72.4115 | 72.4115 | 0% |
Tapered beam, fixed at both ends
Parameters. hᵢ = 1.2 · hⱼ = 0.4
Independent calculation. Force method
| Quantity | Reference | Computed | Deviation |
|---|---|---|---|
| Fixed-end moment at the deep end | 66.7373 | 66.7373 | < 0.001% |