Models for structural design
11.1 Models for structural analysis
Discussion
Although a unified treatment has been developed for both the verification of models for the structural analysis of frameworks and models for the structural design of components, it is convenient to report on each as a separate entity. Verifying the correctness of structural engineering calculations means establishing the truth of software models by examination or demonstration. Verification is defined as the satisfactory completion of a batch of 996 runs of engineered data from the parameter table. To obtain satisfactory completion of all 996 runs at the first attempt is rare, generally half a dozen, or more, batch runs are necessary to engineer out the blips, each time learning something about the nature of the data and modifying the parameter table accordingly, typically adding extra dependencies to prevent unpractical sets of data being generated.
11.1 Models for structural analysis
Discussions are given in order of verified model number as listed in section 7.8. The discussions are distilled versions of those given with each model. For reason of space, only the kernel of each model is included in appendix A but the unabridged set of verified models with notes on both theory and practical matters will be found in Appendix A. Before embarking on a thousand runs for full verification, it is prudent to vary just one parameter at a time keeping all others constant. The easiest way to do this, is to assign the required parameter to be varied, on a new line following the line commencing #cc924.stk, which imports sets of data for verification, leaving all the default values as originally set.
vm110 Deflection of beams including shear cf. Chebyshev polynomials
Verification is by comparison with both classical theory & the Chebyshev polynomials theory (Rolfe, 2004) for the shear force, bending moment, rotation & deflection at the centre & quarter points on a simply supported beam. Deflection computed using Chebyshev polynomials ignored shear deformation as shear deformation was not taken into account in the derivation of the theory.
vm112 Cantilevered beam cf. equilibrium, compatibility & energy
Verification for the cantilevered/propped cantilevered beam model is by comparison
cantilever results with those for the unpropped cantilever, it is clear that, for accuracy, a much higher number of segments is needed for the propped case than the unpropped case. The significant percentage differences are due to the audit of internal strain energy and external work done. For the unpropped cantilever the bending moment diagram does not have a point of contraflexure, for the propped case it does; the much curvier bending moment diagram needs a closer spacing of nodes to give the same accuracy for the energy audit as for the unpropped case. Following the first set of runs for verification, the default number of segments was increased from 16 to 32 to give more accurate results.
vm113 Cantilevered beam with many loads cf. unit load method
Grassie (1957) derives the unit load method from first principles and subsequently notes that the working formula for the determination of the deflection at any section of a straight beam is the same form as that derived by Castigliano's First Theorem Method.
From preliminary runs of the model it was found that for a reinforced concrete beam, if the span:depth ratio for the cantilever is not less than 7.5, then shear deformation will not exceed 2% of bending deformation and may be ignored as 2% is small in comparison to the percentage variability of the concrete.
Most cantilevered beams are made from reinforced concrete. The deflection of a reinforced concrete beam depends on the amount of reinforcement contained in the beam and the creep factor of the concrete. When, as is usually the case for a cantilever, it is important that the deflection be controlled, steel-beam theory is the traditional method of design. In this method, the tension and compression reinforcement are made equal and the moment of inertia is computed from the reinforcement acting alone.
When the amount of tension and compression reinforcement are equal, and the area of concrete is ignored, then the creep deformation of the concrete may be ignored.
┌───────┐ EcIc concrete, for creep factor =2,
│ o o o │ ─┬─ =(28E6/(2+1))*bd^3/12 =0.7777E6*bd^3
│ │ d EsIs steel, for 1% reinforcement t&b
│ o o o │ ─┴─ =205E6*(bd/100)*d^2 =2.05E6*bd^3.
└───────┘ Then EcIc/EsIs =0.7777/2.05 =0.379,
├───b───┤ thus EcIc ≈0.38% reinforcement t&b.
vm114 Tapered cantilevered beam cf. unit load method
See vm113 above, for notes on the deflection of a reinforced concrete beam. The cantilever is tapered, this means it has to be segmented. The beam of span s, has nsg segments; thus for 4 segments the joint and member numbers will be as shown, members in brackets. The cantilever is rigidly supported at joint 1, this implies that a tie-down span is provided.
Y ├─────────a────────┤p
\│ ▼
\╪═══════════╪═══════════╪═══════════╪════════════
\│1 (1) 2 (2) 3 (3) 4 (4) 5
\├───────────────────────s───────────────────────┤
Many loads may be applied at various distances from the left support, thus which member they come within and the distance from the start of that member is required.
Simple logic will suffice, for a point load p at distance a from joint 1:
Load contained within member mn=INT(nsg*a/s)+1 and distance from the start of that member l=a-(mn-1)*s/nsg.
Beam Deflection by The Unit-load Method:
Grassie (1957) derives the Unit-load Method from first principles, and subsequently notes that the working formula for the determination of the deflection at any section of a straight beam i.e.
⌠L M.m.dx del = │ ──────
⌡0 EI
is the same form as that derived by Castigliano's first theorem method. For the application of the unit load method for a tapered cantilever, see vm114.ndf in Appendix A.
vm115 Cantilevered beam with tie down span cf. Roark
══════════════════════
▲ ▲ Span l is the tie-down span
├───────l───────┼──c─┤ for the cantilever of span c.
Point load │p ┌─────────────────┐ ┌───┐
▼ │w load/unit leng.│ │ cw│
═══════════════════ ═══════════════════ ══════════════════
▲ (a) ▲ ▲ (b) ▲ ▲ (c) ▲ Three types of load are considered (a-c) as above. To ignore any type, set its magnitude to zero i.e. p=0, w=0, cw=0. Both w & cw are given as load per unit length. Downward loads are negative. By combining (b) & (c) with a sign change, loading on the span but not on the cantilever may be modelled. See vm115.ndf in Appendix A for formulae used.
vm117 Subframe, continuous beam + columns cf. equilibrium, compatibility &
energy
Subframes (as with continuous beams) are subjected to loading patterns of live load; BS 8110 specifies adjacent bays and alternate bays. Forty years ago great emphasis was placed on saving material at the expense of design office time. Today the emphasis is on simplicity in the design with generous imposed loads sufficient to accommodate change of use of the building over its working life; thus several point loads on a span are often lumped into the distributed load.
The first load case is for serviceability (unfactored dead plus imposed); the second for factored dead+imposed on odd spans (left to right); the third for factored dead+imposed
thereafter for factored dead+imposed on adjacent spans, thus for a general continuous beam, diagrams follow where . . . denotes unfactored dead load only.
▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ Case 1
╦═══════════╦═══════════╦═══════════╦═══════════╦═► unfactored
▼▼▼▼▼▼▼▼▼▼▼▼▼...▼▼▼▼▼▼▼▼▼▼▼▼▼... Case 2
╦═══════════╦═══════════╦═══════════╦═══════════╦═► odd spans ...▼▼▼▼▼▼▼▼▼▼▼▼▼...▼▼▼▼▼▼▼▼▼▼▼▼▼ Case 3
╦═══════════╦═══════════╦═══════════╦═══════════╦═► even spans ▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ Case 4
╦═══════════╦═══════════╦═══════════╦═══════════╦═► all spans
▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼...▼▼▼▼▼▼▼▼▼▼▼▼▼ Case 5
╦═══════════╦═══════════╦═══════════╦═══════════╦═►
...▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼... Case 6
╦═══════════╦═══════════╦═══════════╦═══════════╦═►
▼▼▼▼▼▼▼▼▼▼▼▼▼...▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼▼ Case 7
╦═══════════╦═══════════╦═══════════╦═══════════╦═►
For compatibility, energy, local & overall equilibrium check see chapter 8. For a full discussion see the notes in vm117.ndf in Appendix A.
vm120 Continuous beam cf. Hardy Cross
Verification is by comparison of NL-STRESS with the Moment Distribution method of Prof Hardy Cross (Cross, 1929 & 1932), for a continuous beam subjected to a mix of UDL's, point loads & linearly varying loads, with moment enveloping for the various combinations of DL & LL, with factoring in accordance with BS 8110. The moment distribution was simple to program. To envelope the various loading combinations specified in BS 8110, the number of loadings (load cases) is four for one or two span beams, else seven. Although a continuous beam of more than one span is expected, a single span may be analysed. Forty years ago great emphasis was placed on saving material at the expense of design office time. Today the emphasis is on simplicity in the design with generous imposed loads, sufficient to accommodate change of use of the building over its working life; thus several concentrated or partial distributed loads on a span are often lumped into the distributed load rather than being treated separately as was done forty years ago.
vm122 Two member lean-to or Mansard beam cf. equilibrium, compatibility &
energy
For compatibility, energy, local & overall equilibrium, and Clerk Maxwell, Betti, Southwell check see the notes in vm112.ndf. A rectangular section is assumed, so that the model may be used with steel, concrete & timber sections. When the section thickness is given as zero, a solid section is assumed. To allow for the considerable shear deformation associated with timber, BS 5268 states that the modulus of rigidity
should be taken as Young's Modulus divided by 16. The modulus of rigidity g=e/(2*(1+nu)), where nu=Poisson's ratio. For g=e/16 then e/16=e/(2*(1+nu)), equating the denominators 8=1+nu, thus nu=7. This may seem strange, but it is a BS 5268 requirement.
vm123 Three member lean-to/Mansard beam cf. equilibrium, compatibility &
energy
vm124 Three member cranked beam cf. equilibrium, compatibility & energy Both models listed above i.e. vm123 & vm124 have similar discussions to that for vm122, so are omitted for reason of space.
vm130 Ground beam on an elastic foundation cf. Hetényi
Verification is by comparison of NL-STRESS with the classical solution (Hetényi, 1948). An engineers' arithmetic check is also included, in which it is assumed that the beam is infinitely stiff; thus from the centroid of the loads a linear pressure beneath the beam is given by P/A±MY/I and from this linear pressure the bending moment at each load position may be calculated.
From Terzaghi (1955), the engineer assesses k', the modulus of subgrade reaction (units kN/m3 i.e. pressure to give the soil unit deflection) by means of charts and tables, taking due account of the foundation size and the distribution of loads. The coefficient of subgrade reaction is then multiplied by the area (assumed lumped at a spring support) and the resulting spring stiffness used in the data. If the soil is of poor quality, the value of k' can be increased by: compacting soil; stabilizing the soil with cement or lime; applying a well compacted subbase of sufficient thickness; removing the poor quality layer and replacing it with well compacted sand or crushed stone, stabilised sand or lean concrete. The k' value cannot be used as a measure of settlement. The settlement must be calculated on the basis of the results of a geotechnical study.
A.A.Alexandrou, formerly of the University of Greenwich has provided a table of moduli of subgrade reactions (k') which is contained in Appendix A. Soils do not behave in a linear elastic manner in the long term, they settle due to pore water dissipation and other effects which compact the soil, such as vibrations. Engineers measure the void ratio of the soil to estimate the amount of consolidation expected in the long term. It is normal to assume that the self weight of the ground beam is supported directly by the supporting soil, therefore the self weight is omitted from the model.
From preliminary runs, varying the centres of joints i.e. centres of springs, a close comparison between the two methods was obtained when the beam depth was greater than twice the joint centres. From an engineering viewpoint, a point load may be assumed to be spread at 45° from the top of the beam to the neutral axis. The vertical distance from the top of the beam to the neutral axis is half the beam depth, i.e. equal to the suggested centres for the joints. From preliminary runs varying the modulus of
with sand, the average percentage difference between NL-STRESS & Hetényi (1948), increases from 0.21% to 3.19%.
Following the comparison between Hetényi (1948) and the stiffness method, comes the traditional method of analysis, entitled Engineers' Arithmetic. For this method, the centre of loading is first found, then pressures at each end of the ground beam are computed from P/A±M.y/I, assuming a linear pressure distribution beneath the ground beam, then moments & shears at load positions are calculated. The bending moments and shears computed by engineers' arithmetic do not agree with those computed by NL-STRESS or Hetényi. NL-NL-STRESS/Hetényi take the soil stiffness into account, engineers' arithmetic does not; however, engineers' arithmetic and NL-STRESS/Hetényi can be reconciled by reducing the modulus of subgrade reaction to a very low value, thereby making the beam so stiff by comparison with the soil, that the pressure distribution beneath the beam is linear. Exact agreement between NL-STRESS
& engineers' arithmetic can be seen from the table below, when the modulus of subgrade reaction =1E-3 kN/m3, i.e. 1 kN/m² (weight of a 16 stone man) spread over an area of 1 m² and resulting in a deflection of 1000 m =1 km.
Modulus of subgrade Bending moment at the first load, Case 1.
reaction kN/m3 Nl-STRESS Engineers' Arithmetic 10000 158.346 153.409
100 154.039 153.409 1 153.416 153.409 1E-3 153.409 153.409
Further runs, again varying just one parameter at a time, showed that the number of springs (nj) modelling the subgrade reaction, was insufficient. The dependency conditions were adjusted in the light of the above, such that the number of joints
=INT(10*l/d), where l is the length of the beam and d is the depth. Using this new dependency condition, a study was made of the results of 996 runs with shear deformation suppressed, for which the average percentage difference was found to be 0.399%. Inspection of the averages for each run showed that most of the runs had an average percentage difference near zero, whereas one or two runs had appreciable differences, the largest being in run 832 which had a difference of 34.55%, accordingly this set of data was studied. The first obvious item of data to be considered in run 832, was the high number of joints =183, the previous dependency condition was
=INT(2*l/d) which had been changed to INT(10*l/d), for the beam length=5.5758, breadth=4.9758, depth=0.30424, thus: number of joints =INT(10*5.5758/0.30424)
=183.
Changing the:
Number of joints 366 183 91 45 Average %age diff. 0.001 34.55 0.049 0.50 The mystery deepens, closing in:
Number of joints 181 183 185 187 189 Average %age diff. 0.022 34.55 34.54 34.55 0.040 Inspection of the structure with the loading superimposed, gives a clue to what is
difference is very low both sets come on the beam, when the percentage difference is high, only 4 out of the five loads in the second set come on the beam; thus the problem is due to roundoff. Placing load on the member was controlled by the following which is copied from the data.
IF x>som AND x<=som+crs THEN m FORCE Y CONCENTR. P p(lc) L x-som The logic for this is robust, for all spans except the last one. Any load which occurs almost at a joint position will either be considered on the current span, or left for the next one, but for the last span there is a possibility that sometimes a load will be fluked onto the end of the beam, and sometimes will not. The problem could be solved by a special case for the last span.
IF x>som AND x<=l THEN m FORCE Y CONCENTRATED P p(lc) L l-som More elegantly both cases are covered by:
eom=som+crs l'=x-som ;IF m=nm THEN eom=l l'=l-som
IF x>som AND x<=eom THEN m FORCE Y CONCENTRATED P p(lc) L l' The dependency conditions were adjusted in the light of the above to be nj=INT(4*l/d).
vm131 Ground beam on elastic piles cf. flexibility
Piles supporting a ground beam rarely behave as rigid supports. Loads applied to the beam cause the piles beneath to shorten and the beam to spread the load to adjacent piles. The amount of spreading is a function of the ratio of beam flexibility to pile flexibility. Verification of the NL-STRESS analysis is by the flexibility method to compute the shears, forces and settlements for a ground beam subjected to a train of loads such as those from a crane. The load train position may be stepped automatically.
For a continuous beam on elastic supports, if the internal support reactions are chosen as releases, expressions may be derived for:
a) settlement due to loading
b) flexure due to loading on released structure
c) settlement (-ve) due to unit bi-actions at and corresponding to the releases d) flexure due to unit bi-actions at and corresponding to the releases.
Expressions a,b,c and d are continuous functions between end supports. Separate expressions are required when the loading comes on any cantilever. This choice of releases has the advantage that it is unnecessary to check in which internal span any particular load occurs and is therefore particularly convenient as the program is designed to step the loading any chosen increment to the right. The resulting flexibility matrix is not very well conditioned, but the degree of ill-conditioning has been checked and has been found to be quite acceptable when using double-precision arithmetic (15+
decimal digits). The terms of the flexibility matrix are evaluated using Simpson's rule.
The matrix is inverted using Fox's method. In 1968, the writer, then a Civil design engineer with George Wimpey, developed the necessary theory and used it for the design of the ground beams supporting the Goliath crane at the Harland & Woolf shipyard. Since then, whenever there is an outside broadcast from Belfast, the Goliath
crane is used as a backdrop, alas its days and the shipyard's days are numbered. The theory is contained in the file vm131.NDF in Appendix A.
It is normal to connect the piles to the ground beam: for concrete piles by taking the pile reinforcement into the ground beam; for steel bearing piles, casting the pile heads into the ground beam. Thus, although lift off can occur for ground beams supported by soil, lift off will not normally occur for ground beams supported by piles as the friction between the pile shaft and the soil will normally prevent lift off. The connections between pile heads and the ground beam are normally assumed to be pinned, thus the rotational stiffnesses are set to zero in the data but they may be changed if required. As the flexibility theory ignores rotational stiffness, the structural effect of any moment connection between pile heads and ground beam can be seen by inspection of the percentage differences given in the summary.
Before launching a thousand runs - to simulate the mixture of parameter variation likely in general usage - it is prudent to carry out several single runs varying just one parameter at a time. These preliminary runs are reported in the file vm131.NDF in Appendix A. As in all numerical studies, odd values crop up which are not as expected;
the maximum percentage differences can be greatly influenced by one or two low values, of course one way to avoid this is to relate the difference between the NL-STRESS values and the flexibility approach to the average bending moment, but such a device complicates the matter.
vm140 Influence lines cf. Müller-Breslau
Verification is by comparison with the classical methods of Müller-Breslau for creating influence lines for: reaction, shear, and moment. The procedure used was originally derived by Donald Alcock in the seventies, but never published. The values for shear are those due to a unit load applied just to the right of the position to which the value is
Verification is by comparison with the classical methods of Müller-Breslau for creating influence lines for: reaction, shear, and moment. The procedure used was originally derived by Donald Alcock in the seventies, but never published. The values for shear are those due to a unit load applied just to the right of the position to which the value is