This article discusses two reinforced concrete slab design examples, including one-way and two-way bending. The main goal is to compare the results obtained between hand calculations and SkyCiv Plate Design Module. We will use Eurocode 2 用于钢筋混凝土结构.

建筑规范在定义板的典型案例时有类似的方法. 如果您想了解更多有关此主题的信息, 我们建议您阅读以下有关楼板设计的文章 ACI 平板设计示例和与 SkyCiv 的比较 和 澳大利亚标准 AS3600 楼板设计实例及与 SkyCiv 的比较

单向板设计示例

The first case to analyse is a small one-floor building (数字 1, 数字 2) which has a slab behaviour described as in one-direction.

数字 1. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

数字 2. 小型建筑中的单向板示例 (plan dimensions). (结构3D, SkyCiv 云工程).

对于平板示例, 总之, 材料, 元素属性, 和需要考虑的负载 :

• 板型分类: 一 – 方式行为 \(\压裂{L_2}{L_1} > 2 ; \压裂{14米}{6米}=2.33 > 2.00 \) 好的!
• 建筑占用: 住宅用途
• 板坯厚度 \(t_{平板}=0.25m\)
• Reinforced concrete density \(\罗_w = 25 \压裂{千牛}{米^3}\)
• 混凝土特性抗压强度 28 天 (C25\30) \(fck = 25 兆帕 \)
• 板坯自重 \(Dead = \rho_w \times t_{平板} = 25 \压裂{千牛}{米^3} \乘以 0.25m = 6.25 \压裂 {千牛}{米^2}\)
• 超载恒载 \(标准差= 3.0 \压裂 {千牛}{米^2}\)
• 活荷载 \(L = 2.0 \压裂 {千牛}{米^2}\)

Hand calculations according to EN-2

在这个部分, we will calculate the required reinforced steel rebar using the reference of the Eurocode Standard. 我们首先获得由板的单一宽度条带执行的总因子弯矩.

• 恒载, \(g = (3.0 + 6.25) \压裂{千牛}{米^2} \次 1 m = 9.25 \压裂{千牛}{米}\)
• 活荷载, \(q = (2.0) \压裂{千牛}{米^2} \次 1 m = 2.0 \压裂{千牛}{米}\)
• 极限载荷, \(Fd = 1.35\times g + 1.5\乘以 q = (1.35\次 9.25 + 1.5\次 2.0)\压裂{千牛}{米} =15.5 \frac{千牛}{米} \)

Before obtaining the steel reinforcement area, we have to check the span-effective depth ratios. Two main cases:

Structural System		Basic span-effective depth ratio
	Factor for structural sistem K	Concrete highly stressed %(\(\rho = 1.5 )\)	Concrete lightly stressed %(\(\rho = 0.5 )\)
1. End span of continuous beam or one-way continuous slab or two-way slab continuous over one long side	1.3	18	26
2. Interior span of continuous beam or one-way or two-way spanning slab	1.5	20	30

The most critical case is for number one, so, we select a ratio of 26.

• \(t_{分}= 分数{大号}{我知道}+cover+0.5\dot bar_{直径}= 分数{6米}{26}+0.025m+0.5\times 12mm=0.26m \) ~ \(0.25米). The overall thickness is still adequate, 好的!

现在, it is time to use the table for one-way continuous slabs:

	End support condition				At first interior support	At middle of interior spans	At interior supports
	固定		连续
	Outer support	Near middle of end span	End support	End span
时刻	0	0.086FL	–	0.075FL	–	0.063FL	–
			0.04FL		0.086FL		0.063FL
剪力	0.4F	–		–		–
			0.46F		0.6F		0.5F

在哪里:

• L is the effective span
• F is the total ultimate load in the span (1.35Gk + 1.5Qk; Gk is the dead load and Qk the live load, 分别)

It will be explained only one case (continuous end support) and the rest will show in the following table.

• \(F=Fd\times L = 15.5 \压裂{千牛}{米} \times 6m = 93.0 千牛 \)
• \(M=0.04FL=0.04 \times 93.0 kN \times 6m= -22.32{千牛}{米}\)
• \(d =230 mm \)
• \(K=\frac{中号}{{b}{d^2}{F_{钢底板设计欧洲规范}}}= 分数{22.32\times 10^6 {ñ}{毫米}}{{1000毫米}\次{(230 毫米)^ 2}\次 {25 \压裂{ñ}{毫米^2}}}=0.016877\)
• \(l_a = 0.95 \)
• \(z=l_a \times d = 0.95\times 230mm = 218.50 mm\)
• \(A_s = \frac{中号}{{0.87}{F_{yk}}{与}}= 分数{22.32\times 10^6 {ñ}{毫米}}{0.87\次 500 {ñ}{毫米^2} \次 {218.50毫米} = 234.83 毫米^2 }\)
• \(一个_{s,分}=0.0013{b}{d}=0.0013\times 1000mm \times 230 mm =299 mm^2\)
• \(一个_{圣}=最大(作为, 一个_{s,分}) = 最大值(234.83, 299) mm^2 = 299 毫米^2 \)

片刻	外部负左	外部积极	外部负权	内部负左	内部积极	内部负权
M值, 千牛·米	22.32	35.15	41.85	48.00	35.15	35.15
ķ	0.0168	0.0266	0.03164	0.0362	0.0266	0.0266
与, 毫米	218.50	218.50	218.50	218.50	218.50	218.50
\(作为, mm^2\)	234.83	369.815	440.31	505.011	369.815	369.815
\(一个_{s,分},mm^2\)	299.00	299.00	299.00	299.00	299.00	299.00
\(一个_{圣} {毫米^2}\)	299.00	369.815	440.31	505.011	369.815	369.815

The next move is to calculate the reinforcement rebar steel using the Plate Design Module in SkyCiv. 请, keep reading the following section!.

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SkyCiv S3D 板设计模块结果

This section deals with obtaining the steel reinforcement area but just using the software, 的 板材设计模块. In a concise way, we will only show the results or important information through images.

在分析模型之前, 我们必须定义板网格尺寸. 一些参考资料 (2) 推荐的壳单元尺寸为 1/6 短跨度或 1/8 长跨度的, 其中较短的一个. 遵循这个值, 我们有 \(\压裂{L2}{6}= 分数{6米}{6} = 1 米 \) 要么 \(\压裂{L1}{8}= 分数{14米}{8}=1.75m \); 我们以 1m 作为最大推荐尺寸，并采用 0.50m 的网格尺寸.

数字 3. Plate meshed. (结构3D, SkyCiv 云工程).

一旦我们改进了分析结构模型, 我们进行线性弹性分析. 设计楼板时, 我们必须检查垂直位移是否小于代码允许的最大值. 欧洲规范 2 stablished a maximum serviciability vertical displacement of \(\压裂{大号}{250}= 分数{6000毫米}{250}=24.0 mm\).

数字 4. Vertical displacement, maximum values at center of spans. (结构3D, SkyCiv 云工程).

Comparing the maximum vertical displacement against the code-referenced value, 板的刚度足够. \(4.822 毫米 < 24.00mm\).

板跨度中的最大力矩位于中心，负力矩位于外部和内部支撑处. 让我们在下图中查看这些时刻值.

数字 5. Bending moments in X direction. (结构3D, SkyCiv 云工程).

数字 6. Bending moments in Y direction. (结构3D, SkyCiv 云工程).

数字 7. Steel Reinforcement for direction X at top. (结构3D, SkyCiv 云工程).

数字 8. Steel Reinforcement for direction X at bottom. (结构3D, SkyCiv 云工程).

数字 9. Steel Reinforcement for direction Y at top. (结构3D, SkyCiv 云工程).

数字 10. Steel Reinforcement for direction Y at bottom. (结构3D, SkyCiv 云工程).

结果比较

此单向板设计示例的最后一步是比较通过 S3D 分析获得的钢筋面积 (局部轴 “2”) 和手工计算.

力矩和钢面积	外部负左	外部积极	外部负权	内部负左	内部积极	内部负权
\(一个_{圣, 手算} {毫米^2}\)	299.00	369.82	440.31	505.011	369.82	369.82
\(一个_{圣, S3D} {毫米^2}\)	308.41	337.82	462.61	462.61	262.75	308.41
\(\三角洲_{差异}\) (%)	3.051	8.653	4.820	8.400	28.95	16.610

我们可以看到数值的结果非常接近. 这意味着计算是正确的!

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双向板设计示例

SkyCiv 3D Plate Design Module is a powerful software that can analyze and design any type of building you can imaging. For the second design slab example, we’ve decided to run a flat slab system (数字 11).

数字 11. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

对于平板示例, 总之, 材料, 元素属性, 和需要考虑的负载 :

• 板型分类: 二 – 方式行为 \(\压裂{L_2}{L_1} \这 2 ; \压裂{7米}{6米}=1.17 \le 2.00 \) 好的!
• 建筑占用: 住宅用途
• 板坯厚度 \(t_{平板}=0.30m\)
• Reinforced concrete density \(\罗_w = 25 \压裂{千牛}{米^3}\)
• 混凝土特性抗压强度 28 天 (C25\30) \(fck = 25 兆帕 \)
• 板坯自重 \(Dead = \rho_w \times t_{平板} = 25 \压裂{千牛}{米^3} \times 0.30m = 7.5 \压裂 {千牛}{米^2}\)
• 超载恒载 \(标准差= 3.0 \压裂 {千牛}{米^2}\)
• 活荷载 \(L = 2.0 \压裂 {千牛}{米^2}\)

Hand calculations according EN-2

The first step is define the total ultimate load:

• 恒载, \(g = (3.0 + 7.5) \压裂{千牛}{米^2} \次 7 m = 73.50 \压裂{千牛}{米}\)
• 活荷载, \(q = (2.0) \压裂{千牛}{米^2} \次 7 m = 14.00 \压裂{千牛}{米}\)
• 极限载荷, \(Fd = 1.35\times g + 1.5\乘以 q = (1.35\次 73.50 + 1.5\次 14.00)\压裂{千牛}{米} =120.225 \frac{千牛}{米} \)

For hand calculation, the structure has to be divided into a series of equivalent frames. We can use the following methods to reach up this goal:

• Moment distribution (Hardy Cross Method) for frame analysis.
• Stiffness method for frame analysis on computer. 试试我们的 刚度矩阵计算器.
• A simplified method using the moments coefficients for one-way direction adjusted to the following requirements (We selected this method due the simplicity of the model analyzed):
  • The lateral stability is not dependent on the slab-column connections (We don’t analyze the building for lateral loads);
  • There are at least three rows of panels of approximately equal span in the direction being considered (We have four and three rows of panels in both main directions);
  • The bay size exceeds \(30m^2\) (Our model area is \(42m^2\)

The thickness selected for the slab example is greater than the maximum minimum value for fire resistance indicated in the table below.

Standard fire resistance	Minimum dimensions (毫米)
	板坯厚度, hs	Axis distance, 一个
REI 60	180	15
REI 90	200	25
REI 120	200	35
REI 240	200	50

在这个部分, we will develop only the calcs for the longitudinal direction and column strip (feel free to calculate for another direction, the transverse, and for middle strips). Before going deep in numbers, first we have to divide in strips: middle and column. (For more details about design strips, check this SkyCiv article: Design slabs with ACI-318).

• 柱条宽度: \(6m/4 = 1.50m\)
• 中带宽度: \(7米 – 2\times 1.50m = 4.0m\)

EC2 allows assigning moments in each design strip according to the following table

	Column strip	Middle strip
Negative moment at edge column	100% but no more than \(0.17{b_e}{d^2}{F_{钢底板设计欧洲规范}}\)	0
Negative moment at internal column	60-80%	40-20%
Positive moment in span	50-70%	50-30%

We selected the percentages of moments for the column strip being analyzed:

• Negative moment at edge column: 100%.
• Negative moment at internal column: 80%
• Positive moment in span: 70%

Total design strips moments calculation:

	End support condition				At first interior support	At middle of interior spans	At interior supports
	固定		连续
	Outer support	Near middle of end span	End support	End span
时刻	0	0.086FL	–	0.075FL	–	0.063FL	–
			0.04FL		0.086FL		0.063FL
剪力	0.4F	–		–		–
			0.46F		0.6F		0.5F

在哪里:

• L is the effective span
• F is the total ultimate load in the span (1.35Gk + 1.5Qk; Gk is the dead load and Qk the live load, 分别)

It will be explained only one case (continuos end support) and the rest will show in the following table.

• \(F=Fd\times L = 120.225 \压裂{千牛}{米} \times 6m = 721.35 千牛 \)
• \(M=0.04FL=0.04 \times 721.35 kN \times 6m= -173.124 {千牛}{米}\)
• \(d =280 mm \)
• \(K=\frac{中号}{{b}{d^2}{F_{钢底板设计欧洲规范}}}= 分数{173.124\times 10^6 {ñ}{毫米}}{{1500毫米}\次{(280 毫米)^ 2}\次 {25 \压裂{ñ}{毫米^2}}}=0.012637\)
• \(l_a = 0.95 \)
• \(z=l_a \times d = 0.95\times 280mm = 266.0 mm\)
• \(A_s = \frac{中号}{{0.87}{F_{yk}}{与}}= 分数{173.124\times 10^6 {ñ}{毫米}}{0.87\次 500 {ñ}{毫米^2} \次 {266.0毫米} = 214.0523 毫米^2 }\)
• \(一个_{s,分}=0.0013{b}{d}=0.0013\times 1500mm \times 280 mm =546 mm^2\)
• \(一个_{圣}=最大(作为, 一个_{s,分}) = 最大值(234.83, 546) mm^2 = 299 毫米^2 \)

片刻	外部负左	外部积极	外部负权	内部负左	内部积极	内部负权
M值, 千牛·米	173.124	191.125	260.064	298.281	191.125	218.429
ķ	0.05897	0.06500	0.0884	0.101	0.06500	0.0743
与, 毫米	266.00	266.00	266.00	266.00	266.00	266.00
\(作为, mm^2\)	1498.366	1651.761	2247.55	2577.835	1651.761	1887.727
\(一个_{s,分},mm^2\)	546.00	546.00	546.00	546.00	546.00	546.00
\(一个_{圣} {毫米^2}\)	1498.366	1651.761	2247.55	2577.835	1651.761	1887.727

The next move is to calculate the reinforcement rebar steel using the Plate Design Module in SkyCiv. 请, keep reading the following section!

SkyCiv S3D 板设计模块结果

数字 12. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

数字 13. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

设计楼板时, 我们必须检查垂直位移是否小于代码允许的最大值. Eurocode stablished a maximum serviciability vertical displacement of \(\压裂{大号}{250}= 分数{6000毫米}{250}=24.0 mm\).

数字 14. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

上图为我们提供了垂直位移. The maximum value is -4.148mm being less than the maximum allowed of -24mm. 因此, 板坯的刚度足够.

数字 15. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

图片 15 和 16 由每个主方向的弯矩组成. 获取矩分布和值, 软件, SkyCiv, 即可求得总钢筋面积.

数字 16. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

钢筋加固区域:

数字 17. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

数字 18. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

数字 19. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

数字 20. 小型建筑中的单向板示例. (结构3D, SkyCiv 云工程).

结果比较

The last step in this two-way slab design example is to compare the steel rebar area obtained by S3D analysis and hand calculations.

Rebar steel for X direction and Column Strip

力矩和钢面积	外部负左	外部积极	外部负权	内部负左	内部积极	内部负权
\(一个_{圣, 手算} {毫米^2}\)	1498.366	1651.761	2247.55	2577.835	1651.761	1887.727
\(一个_{圣, S3D} {毫米^2}\)	3889.375	1040.00	4196.145	4196.145	520.00	3175.00
\(\三角洲_{差异}\) (%)	61.475	37.04	46.44	38.566	68.52	40.544

 

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参考资料

1. 乙. 莫斯利, [R. Hulse, J.H. 蹦极 , “Reinforced Concrete Design to Eurocode 2”, Seventh edition, Palgrave MacMillan.
2. 巴赞·恩里克 & 梅利·皮拉拉, “结构抗震设计”, 1编辑, 清除.
3. 欧洲规范 2: 混凝土结构设计.