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Burton et al. - 2006 - The design of simulation studies in medical statistics

STATISTICS IN MEDICINE

Statist.Med.2006;25:4279–4292

Published online31August2006in Wiley InterScience

(https://www.wendangku.net/doc/f015613361.html,)DOI:10.1002/sim.2673

The design of simulation studies in medical statistics

Andrea Burton1,2,?,?,Douglas G.Altman1,Patrick Royston1,3and Roger L.Holder4 1Cancer Research UK/NHS Centre for Statistics in Medicine,Oxford,U.K.

2Cancer Research UK Clinical Trials Unit,University of Birmingham,Birmingham,U.K.

3MRC Clinical Trials Unit,London,U.K.

4Department of Primary Care and General Practice,University of Birmingham,Birmingham,U.K.

SUMMARY

Simulation studies use computer intensive procedures to assess the performance of a variety of statistical methods in relation to a known truth.Such evaluation cannot be achieved with studies of real data alone. Designing high-quality simulations that re?ect the complex situations seen in practice,such as in prognostic factors studies,is not a simple process.Unfortunately,very few published simulation studies provide suf?cient details to allow readers to understand fully all the processes required to design a simulation study.When planning a simulation study,it is recommended that a detailed protocol be produced,giving full details of how the study will be performed,analysed and reported.This paper details the important considerations necessary when designing any simulation study,including de?ning speci?c objectives of the study,determining the procedures for generating the data sets and the number of simulations to perform.

A checklist highlighting the important considerations when designing a simulation study is provided.

KEY WORDS:simulation study;design;protocol;bias;mean square error;coverage

1.INTRODUCTION

Simulation studies use computer intensive procedures to test particular hypotheses and assess the appropriateness and accuracy of a variety of statistical methods in relation to the known truth.These techniques provide empirical estimation of the sampling distribution of the parameters of interest that could not be achieved from a single study and enable the estimation of accuracy measures, such as the bias in the estimates of interest,as the truth is known[1].Simulation studies are increasingly being used in the medical literature for a wide variety of situations,(e.g.References

?Correspondence to:Andrea Burton,Cancer Research UK/NHS Centre for Statistics in Medicine,Wolfson College Annexe,Linton Road,Oxford OX26UD,U.K.

?E-mail:andrea.burton@https://www.wendangku.net/doc/f015613361.html,

Contract/grant sponsor:Cancer Research U.K.

4280 A.BURTON ET AL.

[2–4]).In addition,simulations can be used as instructional tools to help with the understanding of many statistical concepts[5,6].

Designing high-quality simulations that re?ect the complex situations seen in practice,such as in randomized controlled trials or prognostic factor studies,is not a simple process.Simulation studies should be designed with similar rigour to any real data study,since the results are expected to represent the results of simultaneously performing many real studies.Unfortunately,in very few published simulation studies are suf?cient details provided to assess the integrity of the study design or allow readers to understand fully all the processes required when designing their own simulation study.Performing any simulation study should involve careful consideration of all design aspects of the study prior to commencement of the study from establishing the aims of the study,the procedures for performing and analysing the simulation study through to the presentation of any results obtained.These are generic issues that should be considered irrespective of the type of simulation study but there may also be further criteria speci?c to the area of interest,for example survival data. It is important for researchers to know the criteria for designing a good quality simulation study. The aim of this paper is to provide a comprehensive evaluation of the generic issues to consider when performing any simulation study,together with a simple checklist for researchers to follow to help improve the design,conduct and reporting of future simulation studies.The basic concepts underpinning the important considerations will be discussed,but full technical details are not pro-vided and the readers are directed towards the literature(e.g.References[7,8]).General considera-tions are addressed rather than the speci?c considerations for particular situations where simulations are extremely useful,such as in Bayesian clinical trials designs(e.g.Reference[9]),sample size determination(e.g.References[3,10]),or in studies of missing data(e.g.Reference[4]).A small formal review of the current practice within published simulation studies is also presented.

2.ISSUES TO CONSIDER WHEN DESIGNING A SIMULATION STUDY

When planning any simulation study,as with randomized controlled trials,a detailed protocol should be produced giving full details of how the study is to be performed,analysed and reported. The protocol should document the speci?c objectives for the simulation study and the procedures for generating multivariate data sets and,if relevant,with censored survival times.The choices for the different scenarios to be considered,for example different sample sizes,and the methods that will be evaluated should also be included in the protocol together with the number of simulations that will be performed.It is also important to give careful consideration to which data and results will be stored from each run,and which summary measures of performance will be used.If an aim of the study is to judge which is the best of two or more methods,then the criteria should be pre-speci?ed in the protocol,where possible.The rationale behind all the decisions made throughout the design stage should be included in the protocol.

Each of the preceding considerations will be discussed in more detail in the following sections.

A checklist of the important issues that require consideration when designing a simulation study is provided in Figure1.

2.1.Clearly de?ned aims and objectives

Establishing clearly de?ned aims for the simulation study prior to its commencement is an essential part of any research.This focuses the study and avoids unnecessary repetition and time wasting from having to repeat simulations when new aims are conceptualized.

DESIGNING SIMULATION STUDIES4281

0. Detailed protocol of all aspects of the simulation study

a. Justifications for all the decisions made

1. Clearly defined aims and objectives

2. Simulation procedures

a. Level of dependence between simulated datasets

b. Allowance for failures

c . Software to perform simulations

d. Random number generator to use

e . Specification o

f the startin

g seeds

3. Methods for generating the datasets

4. Scenarios to be investigated

5. Statistical methods to be evaluated

6. Estimates to be stored for each simulation and summary

measures to be calculated over all simulations

7. Number of simulations to be performed

8. Criteria to evaluate the performance of statistical methods for different

scenarios

a. Assessment of bias

b. Assessment of accuracy

c . Assessment of coverage

9. Presentation of the simulation results

Figure1.Important considerations when designing any simulation study.

2.2.Simulation procedures

Once the aims and objectives have been formalized,the procedures for performing the simulations can be considered including the level of dependence between simulations,the allowance for failures, the choice of random number generator,starting seeds and the software package to be used.The statistical software package must be able to handle the complexities involved in the proposed simulation study and have a reliable random number generator.

All simulation studies involve generating several independent simulated data sets.These gener-ated data sets should also be completely independent for the different scenarios considered,such as different sample sizes.However,when more than one statistical methodology is being inves-tigated,there is an added complication of determining the level of dependence of the simulated data sets for the different methods,although still retaining independent data sets for each scenario studied.Two feasible simulation strategies are possible.Firstly,fully independent simulated data sets involve generating a completely different set of independent data sets for each method and scenario considered.Secondly,moderately independent simulations use the same set of simulated Copyright2006John Wiley&Sons,Ltd.Statist.Med.2006;25:4279–4292

4282 A.BURTON ET AL.

independent data sets to compare a variety of statistical methods for the same scenario,but a different set of data sets is generated for each scenario investigated.These moderately dependent samples are like a matched pair design where the within sample variability is eliminated and therefore are sensitive to detecting any differences between methods.The relationship between the generated samples should form an important consideration when designing the study.

The simulation procedures should have some allowance for failing to estimate the outcome or parameter of interest,e.g.due to rare events or lack of convergence of models,to avoid premature stopping of the study.The simulations can be set up so that a failed sample is discarded and the whole process is repeated.The number of failures that occur should be recorded to gauge how likely this could happen in practice in order to judge whether the applied statistical procedure can reliably be used in the situation being investigated.If many failures occur for a particular scenario causing the early termination of the simulation study,researchers must consider whether in their situation the failures would lead to bias,and hence unacceptable results,or unbiased but imprecise results in order to determine the usefulness of the results from the partial set of completed simulations.Failures for some simulations may result in a post hoc change of the protocol to omit scenarios,which cannot be simulated reliably.

2.2.1.Random number generation.A fundamental part of any simulation study is the ability to generate random numbers.The many different types of random number generator have been detailed elsewhere[11,12].Any random number generator should be long in sequence before repetition and subsets of the random number sequence should be independent of each other[13].

A variety of statistical tests for randomness exist,including Marsaglia’s Diehard battery of tests for randomness[14],which each random number generator must pass before it can be reliably adopted as a means of generating random numbers.

A random number generator must be able to reproduce the identical set of random numbers when the same starting value,known as a seed,is speci?ed[13].This is also essential when performing simulation studies to enable the generated data sets and hence results to be reproduced, if necessary,for monitoring purposes.The speci?cation of the starting seed also facilitates the choice of simulation strategy.The simulations will be fully independent if completely different starting seeds are used to generate the data sets for each scenario and method combination con-sidered or moderately independent if the same starting seeds are used to compare various methods for the same scenario but different seeds are employed for alternative scenarios.Any simulation strategy involves running several independent simulations for the same scenario,known as par-allel simulations,which require independent sequences of random numbers.Random numbers can be generated for parallel simulations by setting different starting values for the individual simulations that are greater than the number of random numbers required for each simulation, which reduces the possibility of correlations between samples[13].For example,if each sim-ulated data set had a sample size of500,then each of the250,say,simulations would require 500random numbers,therefore the starting seed for each simulation should be separated by at least500.

2.3.Methods for generating the data sets

The methods for obtaining simulated data sets should be carefully considered and a thorough description provided in both the protocol and any subsequent articles published.Simulating data sets requires an assumed distribution for the data and full speci?cation of the required parameters. Copyright2006John Wiley&Sons,Ltd.Statist.Med.2006;25:4279–4292

DESIGNING SIMULATION STUDIES4283 The simulated data sets should have some resemblance to reality for the results to be generalizable to real situations and have any credibility.A good approach is to use a real data set as the motivating example and hence the data can be simulated to closely represent the structure of this real data set.The actual observed covariate data could be used and only the outcome data generated or just certain aspects,such as the covariate correlation structure,could be borrowed.Alternatively,the speci?cations could be arbitrary,but the generated data set may be criticized for not resembling realistic situations.The rationale for any choices made regarding the distributions of the data, parameters of any statistical models and the covariate correlation structure used to generate the data set should accompany their speci?cations.The generated data should be veri?ed to ensure they resemble what is being simulated,for example using summary measures for the covariate distributions,Kaplan–Meier survival curves for survival data or?tting appropriate regression models.

2.3.1.Univariate data.Simple situations may involve generating a vector of random numbers sampled from a known distribution.Demirtas[15]provides procedures for obtaining a variety of univariate distributions from initial values generated from the uniform distribution,if the required distribution is unavailable within the statistical package.

2.3.2.Multivariate data.Generating multivariate data involves the additional speci?cation of cor-relations between covariates unless the covariates are assumed fully independent,which is unlikely in practice.The speci?cation of the means and associated covariance matrix is more straightforward if based on real data,especially with a large number of covariates,and the generated data will re?ect reality.Conversely,the choice of the correlations between covariates can be arbitrary but it is often problematic to determine what are valid relationships.The simplest approach to generate multivariate covariate data with a speci?ed mean and correlation structure is to assume a multi-variate normal distribution.Any continuous but non-normally distributed variables in the real data should be transformed to make the assumption of normality more appropriate.Binary variables can be generated as latent normal,i.e.generated as continuous variables and then dichotomized,but the covariate correlation structure used to generate the continuous variable needs to be adjusted to provide the correct correlation with the resulting binary variable[16].For example,the correction factor for a continuous variable that is to be dichotomized with a50:50split is0.80,suggesting that the correlation between a continuous variable and a binary variable is20per cent less than the correlation between two continuous variables[16].

2.3.3.Time to event data.When the outcome is time to an event,such as in prognostic modelling, several additional considerations must be addressed.The simulations require the speci?cation of a model for the multivariate covariate data and a distribution for the survival data,which may be censored.In order to simulate censored survival data,two survival distributions are required,one for the uncensored survival times that would be observed if the follow-up had been suf?ciently long to reach the event and another representing the censoring mechanism.

The empirical survival distribution from a similar real data set would provide a reasonable choice for the survival distribution.The uncensored survival distribution could be generated to depend on a set of covariates with a speci?ed relationship with survival,which represents the true prognostic importance of each covariate.Time-dependent covariates could also be simulated and incorporated following the procedures described by Mackenzie and Abrahamowicz[17].Bender et al.[18]discuss the generation of survival times from a variety of survival distributions including Copyright2006John Wiley&Sons,Ltd.Statist.Med.2006;25:4279–4292

4284 A.BURTON ET AL.

the exponential for constant hazards,Weibull for monotone increasing or decreasing hazards and Gompertz for modelling human mortality,in particular for use with the Cox proportional hazards model.

Random non-informative right censoring with a speci?ed proportion of censored observations can be generated in a similar manner to the uncensored survival times by assuming a particular distribution for the censoring times,such as an exponential,Weibull or uniform distribution but without including any covariates.Determining the parameters of the censoring distribution given the censoring probability is often achieved by iteration.However,Halabi and Singh [10]pro-vide formulas for achieving this for standard survival and censoring distributions.The censoring mechanism can also be extended to incorporate dependent,informative censoring [19].

The survival times incorporating both events and censored observations are calculated for each case by combining the uncensored survival times and the censoring times.If the uncensored survival time for a case is less than or equal to the censored time,then the event is considered to be observed and the survival time equals the uncensored survival time,otherwise the event is considered censored and the survival time equals the censored time.

2.4.Scenarios to be investigated and methods for evaluation

Simulation studies usually examine the properties of one or more statistical methods in several scenarios de?ned by values of various factors such as sample size and proportion of censoring.These factors are generally examined in a fully factorial arrangement.The number of scenarios to be investigated and the methods for evaluation must be determined and justi?cations for these choices provided in the protocol.The scenarios investigated should aim to re?ect the most common circumstances and if possible cover the range of plausible parameter values.The number of scenarios and statistical methods to investigate will depend on the study objectives but may be constrained by the amount of time available,the ef?ciency of the programming language and the speed and availability of several computers to run simulations simultaneously [20].

2.5.Estimates obtained from each simulation

It is essential to plan how the estimates will be stored after each simulation.Storing estimates enables consistency checks to be performed and allows for the identi?cation of any errors or outlying values and the exploration of any trends and patterns within the individual simulations that may not be observed from the summary measure alone.Storing estimates also allows different ways of summarizing the estimates to be calculated retrospectively,if necessary,without the need to repeat all the simulations.A thorough consideration at the design stage of the possible estimates that may be of interest can ensure that all the required estimates are included,analysed and the results stored,and will avoid the risk of needing to repeat simulations.The estimate of interest,? i ,could include the mean value of a variable,the parameter estimate after ?tting a regression model,the log hazard ratio for survival models or the log odds ratios for logistic regression models.An associated within simulation standard error (SE)for the estimate of interest,SE (? i ),is generally required.

It is also important to establish how to summarize these estimates once all simulations have been performed.Many published simulation studies report the average estimate of interest over

the B simulations performed,e.g.? = B i =1? i /B as a measure of the true estimate of interest.Simulations are generally designed to mimic the results that could have been obtained from a

DESIGNING SIMULATION STUDIES 4285

single study and therefore an assessment of the uncertainty in the estimate of interest between simulations,denoted SE (? ),is usually the empirical SE,calculated as the standard deviation of the estimates of interest from all simulations, [1/(B ?1)] B i =1(? i ?? )2.Alternatively,the average of the estimated within simulation SE for the estimate of interest B i =1SE (? i )/B could be used.The empirical SE should be close to the average of the estimated within simulation SE if the estimates are unbiased [21]and therefore,it may be sensible to consider both estimates of uncertainty.Alternatively,if using the mean and SE of the estimates over all simulations is not considered appropriate then non-parametric summary measures using quantiles of the distribution could be obtained.

2.6.Number of simulations required

The number of simulations to perform can be based on the accuracy of an estimate of interest,e.g.a regression coef?cient,as with determining the sample size for any study [22,23].The number of simulations (B )can be calculated using the following equation:

B =

Z 1?( /2)

2(1)

where is the speci?ed level of accuracy of the estimate of interest you are willing to accept,i.e.the permissible difference from the true value ,Z 1?( /2)is the 1?( /2)quantile of the standard normal distribution and 2is the variance for the parameter of interest [22,23].A realistic estimate of the variance may be obtained from real data if the simulations are based on a real data set and are trying to maintain the same amount of variability.If the variance is unknown or cannot be estimated reliably then it may be possible to perform an identical simulation study to obtain realistic estimates for the variance or consider the measure of accuracy as a percentage of the SE.For example,if the variance from ?tting a single covariate in a Cox regression model was 0.0166,then the number of simulations required to produce an estimate to within 5per cent accuracy of the true coef?cient of 0.349with a 5per cent signi?cance level would be only 209.To estimate the regression coef?cient to within 1per cent of the true value would require 5236simulations.Alternatively,the number of simulations could be determined based on the power (1? )to detect a speci?c difference from the true value as signi?cant [22],such that

B =

(Z 1?( /2)+Z 1? )

In fact,this formula is equivalent to equation (1)if the power to detect a speci?ed difference is assumed to be 50per cent.

The number of simulations to perform is thus dependent on the true value of the estimate of interest,the variability of the estimate of interest,and the required accuracy.For example,more simulations are needed if the regression coef?cient is small or the estimate has little variability.Increasing the number of simulations will reduce the SE of the simulation process,i.e.SE (? )/√B ,but this can be computational expensive and therefore variance reduction techniques could be employed [24].The rationale for the number of simulations to perform should be included in the protocol.

4286 A.BURTON ET AL.

2.7.Evaluating the performance of statistical methods for different scenarios

After the simulations have been performed,the required estimates stored after each replication and summary measures calculated,it is necessary to consider the criteria for evaluating the performance of the obtained results from the different scenarios or statistical approaches being studied.The comparison of the simulated results with the true values used to simulate the data provides a measure of the performance and associated precision of the simulation process.Performance measures that are often used include an assessment of bias,accuracy and coverage.Collins et al.[4]emphasized the importance of examining more than one performance criterion such as mean square error (MSE),coverage and width of the con?dence intervals in addition to bias,as results may vary across criteria.In general,the expectation of the simulated estimates is the main interest and hence the average of the estimates over all simulations is used to calculate accuracy measures,such as the bias.When judging the performance of different methods,there is a trade-off between the amount of bias and the variability.Some argue that having less bias is more crucial than producing a valid estimate of sampling variance [25].However,methods that result in an unbiased estimate with large variability or conversely a biased estimate with little variability may be considered of little practical use.The most commonly used performance measures are considered in turn.Table I provides a summary of the most applicable performance measures and formulas.

Table I.Performance measures for evaluating different methods.

Evaluation criteria Formula BIAS

Bias

=? ? Percentage bias ??? ?

???100Standardized bias

??? ? SE (? )???100ACCURACY

Mean square error (? ? )2+(SE (? ))2COVERAGE Proportion of times the 100(1? )%con?dence interval ? i

±Z 1? /2SE (? i )include ,for i =1,...,B .Average 100(1? )%

B i =12Z 1? /2SE (? i )

B con?dence interval length

Key : is the true value for estimate of interest,? = B i =1? i /B ,B is the number of simulations performed,? i is the estimate of interest within each of the i =1,...,B simulations,SE (? )is the empirical SE of the estimate of interest over all simulations,SE (? i )is the SE of the estimate of interest within each simulation and Z 1?( /2)is the 1?( /2)quantile of the standard normal distribution.

DESIGNING SIMULATION STUDIES 4287

2.7.1.Assessment of bias.The bias is the deviation in an estimate from the true quantity,which can indicate the performance of the methods being assessed.One assessment of bias is the difference

between the average estimate and the true value,i.e. =? ? (Table I).The amount of bias that is considered troublesome has varied from 12SE (? )[21]to 2SE (? )[26].Another approach is to calculate the bias as a percentage of the true value (Table I),providing the true value does not equal

to zero.The percentage bias could have a detrimental effect on the results if the bias is greater than the amount speci?ed when determining the number of simulations required.Alternatively,the bias as a percentage of the SE (? )(Table I)can be more informative,as the consequence of the bias depends on the size of the uncertainty in the parameter estimate [4].A standardized bias of greater than 40per cent in either direction has been shown to have noticeable adverse impact on the ef?ciency,coverage and error rates [4].

Testing the signi?cance of the amount of bias in the estimates [27]or obtaining a 95per cent

con?dence interval using the average parameter estimate,? ,seem counterintuitive,since these statistics are based on the number of simulations through the SE (? )=SE (? )/√and hence these statistics can be improved or penalized by changing the number of simulations performed (see

Section 2.6).Collins et al.[4]warned that with a large number of simulations,the bias may be deemed statistically signi?cant but not be practically signi?cant.Therefore do not rely solely on the p -value but consider the amount of bias as well.

2.7.2.Assessment of accuracy.The MSE provides a useful measure of the overall accuracy (Table I),as it incorporates both measures of bias and variability [4].The square root of the MSE transforms the MSE back onto the same scale as the parameter [4].

2.7.

3.Power,type I and II errors.The empirical power of a test,where relevant,can be determined as the proportion of simulation samples in which the null hypothesis of no effect is rejected at the nominal,usually 5per cent,signi?cance level,when the null hypothesis is false (e.g.References [3,28]).Hence the empirical type II error rate is 1-power.The empirical type I error can be calculated as the proportion of p -values from testing the null hypothesis of no difference on each simulated sample that are less than the nominal 5per cent signi?cance level,when the null hypothesis is true (e.g.Reference [29]).

2.7.4.Assessment of coverage.The coverage of a con?dence interval is the proportion of times that the obtained con?dence interval contains the true speci?ed parameter value (Table I).The coverage should be approximately equal to the nominal coverage rate,e.g.95per cent of samples for 95per cent con?dence intervals,to properly control the type I error rate for testing a null hypothesis of no effect [4].Over-coverage,where the coverage rates are above 95per cent,suggests that the results are too conservative as more simulations will not ?nd a signi?cant result when there is a true effect thus leading to a loss of statistical power with too many type II errors.In contrast,under-coverage,where the coverage rates are lower than 95per cent,is unacceptable as it indicates over-con?dence in the estimates since more simulations will incorrectly detect a signi?cant result,which leads to higher than expected type I errors.A possible criterion for acceptability of the coverage is that the coverage should not fall outside of approximately two SEs of the nominal coverage probability (p ),SE (p )=√p (1?p )/B [27].For example,if 95per cent con?dence intervals are calculated using 1000independent simulations then SE (?p )is Copyright 2006John Wiley &Sons,Ltd.Statist.Med.2006;25:4279–4292

4288 A.BURTON ET AL.

0.006892and hence between936and964of the con?dence intervals should include the true value.

The average length of the95per cent con?dence interval for the parameter estimate? (Table I) is often considered as an evaluation tool in simulation studies(e.g.References[4,30]).If the parameter estimates are relatively unbiased then narrower con?dence intervals imply more precise estimates,suggesting gains in ef?ciency and power[30].

2.8.Presentation of the simulation results

Simulation studies can generate a substantial amount of results that need to be summarized and displayed in a clear and concise manner for the conclusions to be understood.The appropriate format is highly dependent on the nature of the information presented and hence there is a lack of a consistency in the literature.Structuring a report of any simulation study using separate subheadings for the objectives,methods,results and discussion provides clarity and can aid interpretation.

3.REVIEW OF CURRENT PRACTICE

A small formal review of articles published during2004in the Statistics in Medicine journal that included‘simulation’in the title,abstract or as a keyword was carried out to identify the current practices within published simulation studies.Of all270articles published in2004,58(21per cent) were identi?ed as reporting a simulation study;their characteristics are summarized in Table II. The speci?cs of the random number generator and the choice of starting seeds were generally omitted from the publications.Only one of the58articles explicitly stated the random number generator that was used;drand48on the Unix/LINUX system[31].Twenty-two articles gave some indication of the software package that was being used to generate the data or for the analysis, but it was unclear in the remaining36articles what statistical package was used to conduct the simulations.The relationship between generated samples was rarely stated within published simulation studies.Only one article stated that the simulations started with different seeds[32], whilst two other articles reported that independent samples were generated but did not explicitly mention anything about the starting seeds.

The number of simulations performed varied from100to100000replications,with the most common choices being1000(19articles)and10000(12articles)replications.It was unclear in four articles how many simulations were performed.Only six of these58articles provided any justi?cation for the number of simulations performed.Three articles based their justi?cations on the expected SE given the number of simulations[33–35].Two articles provided a justi?cation in terms of the power to detect differences of a speci?ed level from the true value as statistically signi?cant[36,37].The last considered the chosen number of simulations to be suf?cient,as they were not aiming to estimate any quantities with high accuracy[38].

The distributions and parameter speci?cations for generating the data were based on a real data set in eight of the simulation studies.In a further16articles,the simulated data intended to be typical of real data,although not explicitly based on a particular data set.The remaining34articles had no clear justi?cation for the choices of parameters for the speci?ed models used to generate the simulated data sets.

Generally the results from only a small proportion of the scenarios investigated were reported in an article,probably due to the limited space available.The choice of results to publish is fairly

DESIGNING SIMULATION STUDIES4289

Table II.Summary of results from review of58articles.

Criteria Frequency

Random number generator

drand48on the Unix/LINUX system1

Not stated57

Statistical Software used for analysis

Splus6

SAS6

STATA1

Mathematica1

BUGS1

MLWIN1

MATLAB1

Standalone package2

Not stated36

Dependence of samples/starting seed

Samples independent2

Different seeds used1

Not stated55

Number of simulations

1006

2003

4001

5008

100019

50002

1000013

500001

1000001

Unclear4

Any justi?cation for number of simulations

Yes6

No52

Justi?cation for data generation

Based on a real data set8

Typical of real data16

Not stated34

arbitrary and can depend on the important conclusions to be portrayed.However,one article has made available the full set of simulation results,which can be downloaded from a website speci?ed in the article[3].

4.DISCUSSION

The advances in computer technology have allowed simulation studies to be more accessible. However,performing simulations is not simple.In any simulation study,many decision are required Copyright2006John Wiley&Sons,Ltd.Statist.Med.2006;25:4279–4292

4290 A.BURTON ET AL.

prior to the commencement of simulations,but there is generally no single correct answer.The choices made at each stage of the simulation process are open to criticism if not supplemented with thorough justi?cations.

Monte Carlo methods encompass any technique of statistical sampling employed to give approximate solutions to quantitative problems.They include,in addition to simulations,the Monte Carlo Markov chain methods such as Gibbs sampling,which are explicitly used for solving com-plicated integrals[39,40].This paper discusses simulation studies where data sets are formulated to imitate real data.Resampling studies[41,42],where multiple data sets are sampled from a large real data set,require the same rigorous planning as simulation studies,differing from simulation studies only in terms of the generation of the data sets.Hence,similar considerations as discussed throughout this manuscript are relevant.Simulations are also useful in decision-making and engi-neering systems,where computer experiments are used to model dynamic processes in order to assess the effects over time and of varying any inputs(e.g.Reference[43]).Speci?c considerations for designing these studies in terms of formulating the problem,de?ning and designing the model and the choice of inputs and outputs have been discussed elsewhere(e.g.References[43,44]). This paper has discussed the important considerations when designing a simulation study.They include the choice of data to simulate and the procedures for generating the required data.Choices of distributions,parameters of any models,and covariate correlation structures used to generate the data set should be justi?ed.Before commencing simulations,careful consideration should be given to the identi?cation of the estimates of interest,the appropriate analysis,the methods for comparison,the criteria for evaluating these methods,the number of situations to consider,and the reporting of the results.In addition,every simulation study should have a detailed protocol, documenting the speci?c objectives and providing full details of how the study will be performed, analysed and reported.Modi?cations of the simulation processes,such as altering the number of simulations or collecting additional parameters or choices of scenarios,as a consequence of emerging data are possible,but can be time-consuming if they require all simulations to be rerun. Therefore,thorough planning at the start of any simulation study can ensure that the simulations are performed ef?ciently and only the necessary criteria and scenarios assessed.This paper has provided a concise reference(Figure1)for researchers to follow when designing simulation studies.

A small review of published articles in one journal has suggested that the majority of simulation studies reported in the literature are not providing suf?cient details of the simulation process to allow exact replication or clear justi?cations for the choices made.Future published simulation studies should include details of all the simulation procedures to enable the results to be https://www.wendangku.net/doc/f015613361.html,ing separate subheadings for the objectives,methods,results and discussion,irrespective of whether it is the main focus of the article,as in Reference[33],provides clarity and can aid interpretation. In addition,encouraging researcher to consider the suggested criteria(Figure1)might encourage more sound and reliable simulation studies to be performed and reported with credible results.

ACKNOWLEDGEMENT

Andrea Burton was supported by a Cancer Research U.K.project grant.

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2019版新人如何选择滑雪板精品文档23页

滑雪新人如何挑选滑雪双板 2015修正版主要是给新人普及大众滑雪所用器材的相关知识，随着雪具技术的更新和滑雪方式的变更，2010版帖子中的很多观点现在看有点过时了，这里更新一下，欢迎一起探讨。目录 1、滑雪运动和个人滑雪器材的概述滑雪被称为是绅士运动之一（滑雪、网球、马术、高尔夫），是冬季最为贴近大自然的一种运动休闲方式。集速度、力量、勇气、技巧融于一体，但最重要的精神始终是自信与参与。冬季周末，离开烟尘（PM2.5）污染严重的都市，和饭局、酒局、车流、麻将桌暂时分别一下，在群山环绕的银色世界中飘逸的滑行是很多时尚人士选择。滑雪者在滑雪运动中将感受强烈的、其它户外运动项目所无法比拟的成功感、征服感和刺激感，洗洗肺顺便想静静，也是一种都市生活的重启方式。国内的冬季滑雪运动开展时间也不短了，但是受天气和地域的限制颇多，滑雪场的建设对比欧美、日韩等要落后一些。国内玩粉状雪（野雪）、自由式的场地很少，大多都是机压雪道，很多雪场不能保证每天都进行雪道平整。个别的U槽道、越野道只是对比赛和专业运动队开放，很多项目开展受限，很多类型的雪板在国内没有用武之地，如FREESKI自由式、Cross-country skiing越野滑雪、Ski jumping跳台滑雪、Nordic combined 北欧两项、Biathlon现代冬季2项等根本无法在国内普及。那么在国内比较适合的就是高山Alpine ski 滑雪板。在国内销售和出租

的滑雪板的也大都是这个类型的板子。首先要确定自己滑雪的目的，这就解决了方向性问题。简单的说就是我想要玩什么？玩竞技追求最快的速度？还是顺山而下、征服最极限的坡度？或者就是随心而欲的回转，享受在银色世界滑行的感受。老王始终推崇的就是：“慢滑高级道，充分去体会在银白世界中滑行的感受。”另外高级道的初级滑雪者少，安全系数也要高很多，大众滑雪是娱乐享受，时刻要牢记安全第一。国内雪场的出租雪板这几年档次有所提升，长白万达，北大湖万科等新开的雪场出租雪具投资的规模都还可以，雪具对得起缆车票的价格了。租用雪具还是有两个需要解决的问题：一、公用的出租雪鞋，周末和节假日滑雪者多的时候，不能及时的消毒和烘干，非常不利于个人卫生，甚至会导致某些疾病的传染，还有雪鞋是否完全合脚的问题。二、很难保证滑雪者每次去都能用自己专用的滑雪板，不同的雪板的磨损和性能都不同，每次都要重新去适应雪板，不利于个人基本滑雪动作的规范养成。随着大众滑雪的普及和滑雪爱好者的增加，一些滑雪者想到了应该购买自己的滑雪器材。什么情况下到了该买自己专用雪具的时候呢？有雪友计算过，如果每个雪季能保证滑雪次数大于8天/次，就到了拥有自己的雪具的时候了。如果每年就是凑热闹的去几次雪场，还是算了，租用雪具比较适合。选购滑雪板并不是一件难事，最最关键的就是要选择适合自己的，这个是

板式楼梯计算实例

板式楼梯计算实例 "OU 1OT 用U ----------------------------------------- ------------------------------------- r

58C 11X300=3500 1800 - 240 1------------ ：——：——：------------- 7 5800 B2J有承就戕碱板式儀粕桝f 【例题2.1《楼梯、阳台和雨篷设计》37页，PDF版47页】图 2.1为某实验楼楼梯的平面图和剖面图。采用现浇板式楼梯，混凝土强度等级为 C25, f c -11.9N/mm2, f t -1.27N/mm2钢筋直径d> 12mm9寸采用HRB40（级钢筋，f y =360N/mm2; d< 10mrtJ寸采用HPB300级钢筋，f y =270N / mm2,楼梯活荷载为 3.5KN/m2。楼梯的结构布置如图 2.8所示。斜板两端与平台梁和楼梯梁整结，平台板一端与平台梁整结，平台板一端与平台梁整结，另一端则与窗过梁整结，平台梁两端都搁置在楼梯间的侧墙上。

580 11X3003300 1800 120 d 1——11 ---------------------------------------------------------- p *--------------------------------- 屮 5800 02.8 试对此现浇板式楼梯进行结构设计。解： 1)斜板TB1设计除底层第一跑楼梯的斜板外，其余斜板均相同，而第一跑楼梯斜板的下端为混凝土基础，可按净跨计算。这里只对标准段斜板TB1进行设计。对斜板TB1取1m宽作为其计算单元。 (1) 确定斜板厚度t 斜板的水平投影净长为I in=3300mm 斜板的斜向净长为 -= ------------ = 3691mm cosa 300 / J150+002

板式楼梯设计典型例题

3.4.5 楼梯设计例题设计资料 ?某公共建筑标准层层高为3.6m，采用现浇板式楼梯，其平面布置见图3.53。

?楼梯活荷载标准值为q =2.5KN/m2，踏 k 步面层采用30mm厚水磨石面层(自重为0.65 KN/m2)，底面为20mm 厚混合砂浆(自重为17 KN/m3)抹灰。 ?采用C25混凝土，梁纵筋采用HRB335级钢筋，其余钢筋均采用HPB235级钢筋。梯段板设计估算斜板厚h=lo/30=3500/30=117（mm），取=120mm。板倾斜角为tanα=150/300=0.5 (由踏步倾斜得来)

取1m 宽板带进行计算。（1）荷载计算恒荷载标准值水磨石面层: (0.3+0.15)×0.65×3.01 =0.98(KN/m) 三角形踏步: 2 1×0.3×0.15×25×3.01 =1.88(KN/m) 混凝土斜板: 0.12×25×1/0.894=3.36(KN/m) 板底抹灰: 0.02×17×1/0.894=0.38(KN/m) 恒荷载标准值 g k =6.60 KN/m 恒荷载设计值g =1.2×6.60=7.92 KN/m 活荷载设计值q =1.4×2.5=3.5 KN/m 合计 p =g+q =11.42 KN/m (2)截面设计水平投影计算跨度为 lo=ln+b =3.3+0.2=3.5m

弯矩设计值 2 0)(101l q g M +==25.342.1110 1?? =13.99(KN ·m) 斜板有效高度: ho=120-20=100(mm) 2 01bh f M c s αα= =26 10010009.110.110 99.13???? =0.188， 937.0=s γ 0h f M A s y s γ= =100937.02101099.136 ???=711(mm 2) 选配φ10@110，As=714mm 2 ，梯段板的配筋见图3.54。配筋要求见P89。 ?受力钢筋：沿斜向布置。 ?构造负筋：在支座处板的上部设置一定数量，以承受实际存在的负弯矩和防止产生过宽的裂缝。一般取φ8@200，长度为l n /4。本题取φ8@200，3300/4=825mm ，取850mm 。 ?分布钢筋：在垂直于受力钢筋方向按构造配置，每个踏步板内至少放置一根分布钢筋。放置在受力钢筋

各种滑雪装备选择使用详解

各种滑雪装备选择使用（详解）滑雪装备滑雪板滑雪板一般分为高山板、越野冬季两项板、跳台板、自由式板、单板等。高山板由多层结构组成，主要包括弹性反材、搞扭较往年盒形结构、板芯、玻璃纤维复合材料、高分子材料底板、金属边刃等。在先择滑雪板的长度时，最长应以不超过本人手臂上举手腕部高度为限，最短不应短于胯部。选择长的浮雕雪板，使用起来速度快，稳定性好，短的滑雪板速度慢，易颤动，稳定性差。对于初学者来说，太长的滑雪板不容易控制，转弯较困难不利于提高自己的技术水平，初学者最好以自己的身高再加5厘米左右即可。初学者还应选用弹性较大的滑雪板。因为这种滑雪板遇到不平的雪面时不易颠簸，制动效果也较好，操作起来比较容易。技术好的滑雪者可以选择长一点。弹性小一点，稍微重一些的滑雪板，它可以增加滑行中的稳定性，使滑雪板的金履带边刃紧紧地卡在雪面上，有利于滑雪者充分地操纵滑雪板，滑出漂亮的弧形，滑雪板底板的材料主要由塑料或高分子尼龙材料制成，高分子材料的底板摩擦系数小，比塑料底坂要好，滑雪板的边刃要随时保持锋利，这样在你对它施加重力时，不会产生侧滑，据说专业滑雪运动员使用的滑雪板其边刃可以刮胡子。滑雪装备滑雪鞋滑雪鞋一般分为高山鞋、越野鞋、跳台鞋和单板鞋等。高山鞋一般由内外两部分构成，外壳是由塑料或ABS注塑而成，较硬不易变形，内层由化纤织物和保温材料组成，鞋的踝关节角度和鞋的肥瘦等可根据需要进行调节，具有保护功能。越野鞋一般分为尼龙和皮革制品，鞋腰矮软且轻便。跳台鞋一般是用皮革制成，鞋腰较高且前倾大，有利于运动员跳跃和空中习行前倾姿势。滑雪鞋的选择要使人感到即舒适又很合脚，脚趾在鞋中能活动自如，但脚掌、脚背、脚弓、脚跟应能紧紧的被裹住，外壳上的卡子要卡得恰到好处，使踝关节可以向前屈膝，只有这样才能控制滑雪板和滑雪速度。初学者应选择轻便、灵活、富有弹性的滑雪鞋，它的可操纵余地较大。而技术好的滑雪者，可选择能将脚与滑雪鞋紧紧连为一体的滑雪鞋，从而使滑雪者任何一点微

板式和梁式楼梯手算及实例

1. 板式楼梯例8-1 某公共建筑现浇板式楼梯，楼梯结构平面布置见图(8-6)。层高3.6m ，踏步尺寸150× 300mm 。采用混凝土强度等级C25，钢筋为HPB235 和 HRB335。楼梯上均布活荷载标准值=3.5kN ／m 2，试设计此楼梯。 1. 楼梯板计算板倾斜度 ,5.000150==αtg 894.0cos =α 设板厚h=120mm ；约为板斜长的1／30。取lm 宽板带计算 (1) 荷载计算图8-6 例8-1的楼梯结构平面荷载分项系数 2.1=G γ 4.1=Q γ 基本组合的总荷载设计值 m kN p /82.124.15.32.16.6=?+?= 表8-1 梯段板的荷载 (2) 截面设计

板水平计算跨度m l n 3.3= 弯矩设计值 m kN pl M n ?=??== 96.133.382.1210110122 mm h 100201200=-= 117.010010009.111096.132 62 01=???== bh f M c s αα 614.0124.0117.0211211=<=?--=--=b s ξαξ 2 01703210124 .010010009.11mm f bh f A y c s =???= = ξ α %27.021027 .145.045.0%59.01201000703min 1===>=?== y t s f f bh A ρρ 选配?10@110mm, A s =714mm 2 分布筋?8，每级踏步下一根，梯段板配筋见图(8-7)。表8-2 平台板的荷载 2. 平台板计算设平台板厚h=70mm, 取lm 宽板带计算。 (1) 荷载计算总荷载设计值 m kN p /19.85.34.174.22.1=?+?= (2) 截面设计板的计算跨度 m l 76.12/12.02/2.08.10=+-= 弯矩设计值 mm h 5020700=-= m kN pl M ? = ? ? = = 54 . 2 76 . 1 19 . 8 10 1 10 1 2 2 0

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如何挑选滑雪板——双板购买一套滑雪板真个是头痛的问题。进入滑雪器材商店，看到琳琅满目的产品，感觉哪一个都很吸引人。可是面对诸如长度、转弯半径、三围尺寸、弹性、硬度等等各种数据，以及不同的滑雪板分类，相信多数人的头都会大上一号了。实选购滑雪板并不是一件难事，最最关键的就是要选择适合自己。首先明确了自己的目的就解决了方向性问题。简单的说就是我想要玩什么？是追求速度？还是最求灵活性？或者就是想简单的回转，体验滑行的乐趣。们先来确定长度：现代的滑雪板都是经过改良的“大头板”，比较灵活也很好控制。所以，我们现在选购滑雪板不必按照传统直板时代身高加10厘米的方法来确定长度。一般情况下，我们应该选择身高减去5到15厘米这个长度的滑雪板。比如说你的身高是170厘米，那么你适合的滑雪板长度就是在155到165厘米之间。如果你想追求灵活的操控性，那就选155厘米的板长。如果就是想简单的体验滑雪乐趣，那160厘米的板长是你最好选择。另外，你想追求刺激的速度，那就选择165厘米，甚至与身高相等的都可以。在速度的大回转比赛中，有些运动员使用比自己身高长很多的滑雪板，不过劝各位玩家还是小心谨慎的好，他们都是经过严格训练的，我们可不能和他们相比。长度确定了，再来看看转弯半径和三围：这几个数据是标在滑雪板尾部的，例如106－72－97＝14m。这是说

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《混凝土结构设计原理》实验报告实验三楼梯设计土木工程专业10 级 3 班姓名学号指导老师二零一三年一月仲恺农业工程学院城市建设学院

目录一、主体介绍 (4) 二、现浇板式楼梯设计 (5) 1. 梯段板TB2设计 (5) 1) 荷载计算 (5) 2) 截面设计 (6) 2. 平台板PTB2设计 (6) 1）荷载计算 (7) 2）截面设计 (7) 3. 平台梁TL4设计 (8) 1）荷载计算 (8) 2)截面设计 (9) 3)斜截面受剪承载力计算 (9) 4）配筋图 (10) 4. 平台梁TL5设计 (10) 1）荷载计算 (10) 2)截面设计 (10) 3)斜截面受剪承载力计算 (12) 4) 配筋图 (12) 5.构造柱GZ设计 (12) 三、现浇梁式楼梯设计 (13)

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结构图实物图上视图

2，将后固定器和雪耙组合在一起，形成一个平面的连接。（关键点：正确的连接是形成一个平面，雪耙和后固定器有连接的锚点） 3，打开前固定器的锁止按钮，从滑轨前部分划入前固定器并锁定。（关键点：锁止按钮每个厂家不一样，但是大同小异，是一个小拇指家大的扳手，能抬起来的。）（对于金鸡雪板是从中间往两边滑动安装） 4，打开后固定器的锁止按钮，从滑轨后部分划入雪耙和后固定器的组合体，并且锁定。（关键点：从后部滑入）（对于金鸡雪板是从中间往两边滑动安装）5，找到雪鞋外壳尺寸，并按外壳尺寸调节前后固定器位置。（关键点：调节到包含雪鞋外壳尺寸的档位，例如雪鞋外壳尺寸是305mm，那么档位应该打到例如300-310mm这个包含305mm数字的档位，前后固定器都是如此，切记，是雪鞋的外壳尺寸，不是您的脚长）

6，把雪鞋踩入固定器，试试是否合适，合适的标准是能正好掐住雪鞋，雪鞋不晃。 7，DIN值调整。刚开始适应一个雪板不要滑的太快，头2次使用固定器也不要调节的太高，摸清楚雪板特性再调节到国际DIN值比较合适，刚开始用最好降低1-1.5个din标准。（金鸡的雪板调节螺丝要在锁定雪鞋状态调节）

祝您享受滑雪的愉快，安全，健康滑雪！

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试对此现浇板式楼梯进行结构设计。解： 1）斜板TB1设计除底层第一跑楼梯的斜板外，其余斜板均相同，而第一跑楼梯斜板的下端为混凝土基础，可按净跨计算。这里只对标准段斜板TB1进行设计。对斜板TB1取1m宽作为其计算单元。 (1)确定斜板厚度t 斜板的水平投影净长为l1n=3300mm

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FLOW单板的技术特点图列(快穿固定器)

Reflex木芯 Flow最轻，最灵敏的木芯，带给人类已知的最高强度-重量比的木芯结构。在你所需要的区域侧重响应性，特定区域使用更多中低密度木芯，在增加强度的同时减轻重量。皇家威士忌使用玻璃纤维的全新剪裁，好处就是消除了固定器脚跟和脚尖的区域的抖动。增加了边刃控制力并提供流畅平顺的滑行。枪手威士忌一种反V字型设计，使用4条炭纤维实现疯狂的弹性，同时让板头、板尾不受约束，拥有轻快和灵活的滑行感觉。双轴纤维（0/90度）大部份纤维在0度轴上，用于增加额外的弹性和使雪板拥有更自然的扭力弹性。 OPTIX 2000 快速并少维护。 EZDT（E-DT）边刃为EZ-Rock和另一个版本的EZ Flat-Rock设计，EZDT在固定器和板头之间是一条小弧度的边刃，在固定器区域是更加积极抓地的边刃。这样更加容易上手滑行，同时混合的大弧度刃也可以做出快速的换刃。纯双向板头板尾的距离相等，使滑行在任何方向滑行都有相同的感觉。单向板头比板尾略长，固定器安装孔后置，板尾跳跃会获得更多的弹性，更好的野雪漂浮。

3DT（3D-过渡）边刃平顺并拥有的宽泛的抓地能力，3D 过渡边刃的混合方式很与众不同，两脚之间弧度小的是为I-Rock，弧度超大的是为了采用Pop-Cam的雪板。小弧度板头和固定器区域较大的弧度提供一个无限的抓地性，具综合性的超平顺和轻快的滑行感觉。摇椅传送带一条炭纤维带子水平的放置在木芯下面，在每款摇椅雪板上创造出一个轻微凸起的板底。从上铁杆儿到呲铁杆滑行都变得更平顺了。摇滚威士忌位于脚跟和脚尖下面的4片玄武岩薄片直接将滑手的能量传递到雪板上的关键控制区域。同时也使强度和全地形的边刃控制得到加倍，并且不会影响到滑行的平顺。 I-ROCK 摇椅凸起板底的摇椅拱形在铁杆上更自然的感觉，在多边的情况下也能保持稳定。在固定器区域的外侧结合了反弓，I-ROCK提供了稳定和反弓的弹性，使之拥有了从高山到公园的性能。 EZ-ROCK 固定器之间超微小的凸起摇椅板底和固定器区域外侧反弓提供了独一无二的稳定。无论你将固定器安装在哪，压板都非常容易和稳定。 POP-CAM 三段式的反弓使之成为市场上最有趣的反弓结构。脚下还是传统反弓，固定器之间是平面，这种特有的设计增大冰面的抓地性，在野雪也有更好的漂浮性呲杆儿更稳定。 4000烧结最快，超硬，雪蜡高吸收性和高耐久性。四向纤维（0/30）有着最多的纤维部署，从板头到板尾每30度方向，这种矩阵在提供了大量的弹力和自由扭力同时将固定器的力量分散到了一个更大片

教你如何滑雪分解

板式楼梯配筋计算实例.

9 板式楼梯设计 9.1 楼梯建筑设计 (1)楼梯甲（净面积3350×7150）、楼梯乙（净面积3325×7150）、楼梯丙（净面积3350×7150）计算（梯段宽取1600mm ）：设h =150mm ，N=1503900 =26, 2h +b =600～620mm 且h +b ≈450mm ，取b =300mm 。平台宽度≥900＋300＝1200，取1700mm 。第一梯段：7150－1700－900＝4550mm 4550/300=15.2，取15步。平台实际长度＝300×15＝4500mm 首层平台高度＝150×16＝2400mm 平台梁下与室内地面净高差2400－400＝2000mm , 满足平台下过人要求净高≥2000mm ，故无需降低平台梁下地面标高。第二梯段：踏步数26－16＝10，踏面数9。水平长度＝300×9＝2700mm 。二层以上每跑梯段踏步数相等26/2=13,水平面长度＝300×12＝3600mm 。核算首层中间平台到二层平台底的净高(3.9+1.95)-2.4=3.45m > 2.2m ，满足要求。 (2)楼梯丁（净面积6950×7150）计算（中间梯段宽取3000mm ，两边梯段宽取1800 mm ）：设h =150mm ，N=1503900 =26, 2h +b =600～620mm 且h +b ≈450mm ，取b =300mm 。平台宽度≥900＋300＝1200，取1900mm 。第一梯段：7150－1900－600＝4650mm 4650/300=15.5，取15步。

平台实际长度＝300×15＝4500mm 首层平台高度＝150×16＝2400mm 平台梁下与室内地面净高差2400－400＝2000mm , 满足平台下过人要求净高≥2000mm ，故无需降低平台梁下地面标高。第二梯段：踏步数26－16＝10，踏面数9。水平长度＝300×9＝2700mm 。二层以上每跑梯段踏步数相等26/2=13,水平面长度＝300×12＝3600mm 。核算首层中间平台到二层平台底的净高(3.9+1.95)-2.4=3.45m > 2.2m ，满足要求。 9.2 楼梯斜板设计以下均以楼梯甲的设计为例。考虑到第一跑楼梯梯段斜板两端与混凝土楼梯梁的固结作用，斜板跨度可按净跨计算，对斜板取1m 宽作为其计算单元。 (1)确定斜板厚度t ，斜板的水平投影净长l 1n =300×12＝3600mm 斜板的斜向净长l′1n ＝mm l n 4027894 .03600 300150/3003600cos 2 21== += α 斜板厚度t 1 =n l 1 )30 1 ~251( '= (134～161)mm,取板厚t 1 ＝140mm (2)荷载计算，楼梯梯段斜板的荷载计算列于下表(取1m 宽的板带作为计算单元)：表9－1 楼梯梯段斜板的荷载计算表