概要

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 鉄筋コンクリート構造用.

スラブの典型的なケースを定義する場合、建築基準法にも同様のアプローチがあります。. このトピックについてもう少し詳しく知りたい場合, スラブ設計に関する次の記事を読むことをお勧めします ACI スラブの設計例と SkyCiv との比較 そして オーストラリア規格 AS3600 スラブの設計例と SkyCiv との比較

One-Way Slab Design Example

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

図 1. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

図 2. One-way slabs in a small building example (plan dimensions). (構造3D, SkyCiv Cloud Engineering).

For the slab example, in summary, the material, elements properties, and loads to consider :

• Slab type classification: 1 – way behaviour \(\フラク{L_2}{L_1} > 2 ; \フラク{14メートル}{6メートル}=2.33 > 2.00 \) OK!
• Building occupation: Residential use
• Slab thickness \(t_{スラブ}=0.25m\)
• Reinforced concrete density \(\rho_w = 25 \フラク{kN}{m^3}\)
• Concrete characteristic compressive strength at 28 日々 (C25\30) \(fck = 25 MPa \)
• Slab Self-Weight \(Dead = \rho_w \times t_{スラブ} = 25 \フラク{kN}{m^3} \times 0.25m = 6.25 \フラク {kN}{m^2}\)
• Super-imposed dead load \(SD = 3.0 \フラク {kN}{m^2}\)
• 活荷重 \(L = 2.0 \フラク {kN}{m^2}\)

Hand calculations according to EN-2

このセクションで, we will calculate the required reinforced steel rebar using the reference of the Eurocode Standard. We first obtain the total factored bending moment to be carried out by the slab’s unitary width strip.

• Dead load, \(g = (3.0 + 6.25) \フラク{kN}{m^2} \回 1 m = 9.25 \フラク{kN}{メートル}\)
• 活荷重, \(q = (2.0) \フラク{kN}{m^2} \回 1 m = 2.0 \フラク{kN}{メートル}\)
• Ultimate load, \(Fd = 1.35\times g + 1.5\times q = (1.35\回 9.25 + 1.5\回 2.0)\フラク{kN}{メートル} =15.5 \frac{kN}{メートル} \)

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, それで, we select a ratio of 26.

• \(t_{分}= frac{L}{知っている}+cover+0.5\dot bar_{直径}= frac{6メートル}{26}+0.025m+0.5\times 12mm=0.26m \) ~ \(0.25m). The overall thickness is still adequate, OK!

今, 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
	ピン留め		継続的
	Outer support	Near middle of end span	End support	End span
瞬間	0	0.086FL	–	0.075FL	–	0.063FL	–
			0.04FL		0.086FL		0.063FL
剪断	0.4F	–		–		–
			0.46F		0.6F		0.5F

どこ:

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

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

• \(F=Fd\times L = 15.5 \フラク{kN}{メートル} \times 6m = 93.0 kN \)
• \(M=0.04FL=0.04 \times 93.0 kN \times 6m= -22.32{kN}{メートル}\)
• \(d =230 mm \)
• \(K=\frac{M}{{b}{d^2}{f_{ck}}}= frac{22.32\times 10^6 {N}{んん}}{{1000んん}\回{(230 んん)^ 2}\回 {25 \フラク{N}{mm^2}}}=0.016877\)
• \(l_a = 0.95 \)
• \(z=l_a \times d = 0.95\times 230mm = 218.50 mm\)
• \(A_s = frac{M}{{0.87}{f_{yk}}{と}}= frac{22.32\times 10^6 {N}{んん}}{0.87\回 500 {N}{mm^2} \回 {218.50んん} = 234.83 mm^2 }\)
• \(A_{s,分}=0.0013{b}{d}=0.0013\times 1000mm \times 230 mm =299 mm^2\)
• \(A_{st}=max(として, A_{s,分}) = max(234.83, 299) mm^2 = 299 mm^2 \)

瞬間	Exterior Negative Left	Exterior Positive	Exterior Negative Right	Interior Negative Left	Interior Positive	Interior Negative Right
M value, kN-m	22.32	35.15	41.85	48.00	35.15	35.15
K	0.0168	0.0266	0.03164	0.0362	0.0266	0.0266
と, んん	218.50	218.50	218.50	218.50	218.50	218.50
\(として, mm^2\)	234.83	369.815	440.31	505.011	369.815	369.815
\(A_{s,分},mm^2\)	299.00	299.00	299.00	299.00	299.00	299.00
\(A_{st} {mm^2}\)	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. お願いします, keep reading the following section!.

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SkyCiv S3D Plate Design Module Results

This section deals with obtaining the steel reinforcement area but just using the software, の プレート設計モジュール. In a concise way, we will only show the results or important information through images.

Before analyzing the model, we must define a plate mesh size. Some references (2) recommend a size for the shell element of 1/6 of the short span or 1/8 of the long span, the shorter of them. Following this value, 我々は持っています \(\フラク{L2}{6}= frac{6メートル}{6} = 1m \) または \(\フラク{L1}{8}= frac{14メートル}{8}=1.75m \); we take 1m as a maximum recommended size and 0.50m applied mesh size.

図 3. Plate meshed. (構造3D, SkyCiv Cloud Engineering).

Once we improved our analytical structural model, we run a linear elastic analysis. When designing slabs, we have to check if the vertical displacement are less than the maximum allowed by code. ユーロコード 2 stablished a maximum serviciability vertical displacement of \(\フラク{L}{250}= frac{6000んん}{250}=24.0 mm\).

図 4. Vertical displacement, maximum values at center of spans. (構造3D, SkyCiv Cloud Engineering).

Comparing the maximum vertical displacement against the code-referenced value, the slab’s stiffness is adequate. \(4.822 んん < 24.00mm\).

The maximum moments in the slab’s spans are located for positive in the center and for negative at the exterior and interior supports. Let’s see these moments values in the following images.

図 5. Bending moments in X direction. (構造3D, SkyCiv Cloud Engineering).

図 6. Bending moments in Y direction. (構造3D, SkyCiv Cloud Engineering).

図 7. Steel Reinforcement for direction X at top. (構造3D, SkyCiv Cloud Engineering).

図 8. Steel Reinforcement for direction X at bottom. (構造3D, SkyCiv Cloud Engineering).

図 9. Steel Reinforcement for direction Y at top. (構造3D, SkyCiv Cloud Engineering).

図 10. Steel Reinforcement for direction Y at bottom. (構造3D, SkyCiv Cloud Engineering).

結果比較

The last step in this one-way slab design example is compare the steel rebar area obtained by S3D analysis (local axes “2”) and handcalculations.

Moments and steel area	Exterior Negative Left	Exterior Positive	Exterior Negative Right	Interior Negative Left	Interior Positive	Interior Negative Right
\(A_{st, HandCalcs} {mm^2}\)	299.00	369.82	440.31	505.011	369.82	369.82
\(A_{st, S3D} {mm^2}\)	308.41	337.82	462.61	462.61	262.75	308.41
\(\デルタ_{dif}\) (%)	3.051	8.653	4.820	8.400	28.95	16.610

We can see that the results of the values are very close to each other. This means the calculations are correct!

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Two-way Slab Design Example

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 (図 11).

図 11. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

For the slab example, in summary, the material, elements properties, and loads to consider :

• Slab type classification: Two – way behaviour \(\フラク{L_2}{L_1} \インクルード 2 ; \フラク{7メートル}{6メートル}=1.17 \le 2.00 \) OK!
• Building occupation: Residential use
• Slab thickness \(t_{スラブ}=0.30m\)
• Reinforced concrete density \(\rho_w = 25 \フラク{kN}{m^3}\)
• Concrete characteristic compressive strength at 28 日々 (C25\30) \(fck = 25 MPa \)
• Slab Self-Weight \(Dead = \rho_w \times t_{スラブ} = 25 \フラク{kN}{m^3} \times 0.30m = 7.5 \フラク {kN}{m^2}\)
• Super-imposed dead load \(SD = 3.0 \フラク {kN}{m^2}\)
• 活荷重 \(L = 2.0 \フラク {kN}{m^2}\)

Hand calculations according EN-2

The first step is define the total ultimate load:

• Dead load, \(g = (3.0 + 7.5) \フラク{kN}{m^2} \回 7 m = 73.50 \フラク{kN}{メートル}\)
• 活荷重, \(q = (2.0) \フラク{kN}{m^2} \回 7 m = 14.00 \フラク{kN}{メートル}\)
• Ultimate load, \(Fd = 1.35\times g + 1.5\times q = (1.35\回 73.50 + 1.5\回 14.00)\フラク{kN}{メートル} =120.225 \frac{kN}{メートル} \)

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
• 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 (んん)
	Slab thickness, hs	Axis distance, a
REI 60	180	15
REI 90	200	25
REI 120	200	35
REI 240	200	50

このセクションで, 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).

• 列ストリップ幅: \(6m/4 = 1.50m\)
• 中間ストリップ幅: \(7メートル – 2\times 1.50m = 4.0m\)

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 \(0.17{b_e}{d^2}{f_{ck}}\)	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
	ピン留め		継続的
	Outer support	Near middle of end span	End support	End span
瞬間	0	0.086FL	–	0.075FL	–	0.063FL	–
			0.04FL		0.086FL		0.063FL
剪断	0.4F	–		–		–
			0.46F		0.6F		0.5F

どこ:

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

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

• \(F=Fd\times L = 120.225 \フラク{kN}{メートル} \times 6m = 721.35 kN \)
• \(M=0.04FL=0.04 \times 721.35 kN \times 6m= -173.124 {kN}{メートル}\)
• \(d =280 mm \)
• \(K=\frac{M}{{b}{d^2}{f_{ck}}}= frac{173.124\times 10^6 {N}{んん}}{{1500んん}\回{(280 んん)^ 2}\回 {25 \フラク{N}{mm^2}}}=0.012637\)
• \(l_a = 0.95 \)
• \(z=l_a \times d = 0.95\times 280mm = 266.0 mm\)
• \(A_s = frac{M}{{0.87}{f_{yk}}{と}}= frac{173.124\times 10^6 {N}{んん}}{0.87\回 500 {N}{mm^2} \回 {266.0んん} = 214.0523 mm^2 }\)
• \(A_{s,分}=0.0013{b}{d}=0.0013\times 1500mm \times 280 mm =546 mm^2\)
• \(A_{st}=max(として, A_{s,分}) = max(234.83, 546) mm^2 = 299 mm^2 \)

瞬間	Exterior Negative Left	Exterior Positive	Exterior Negative Right	Interior Negative Left	Interior Positive	Interior Negative Right
M value, kN-m	173.124	191.125	260.064	298.281	191.125	218.429
K	0.05897	0.06500	0.0884	0.101	0.06500	0.0743
と, んん	266.00	266.00	266.00	266.00	266.00	266.00
\(として, mm^2\)	1498.366	1651.761	2247.55	2577.835	1651.761	1887.727
\(A_{s,分},mm^2\)	546.00	546.00	546.00	546.00	546.00	546.00
\(A_{st} {mm^2}\)	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. お願いします, keep reading the following section!

SkyCiv S3D Plate Design Module Results

図 12. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

図 13. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

When designing slabs, we have to check if the vertical displacement are less than the maximum allowed by code. Eurocode stablished a maximum serviciability vertical displacement of \(\フラク{L}{250}= frac{6000んん}{250}=24.0 mm\).

図 14. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

The image above gaves to us the vertical displacement. The maximum value is -4.148mm being less than the maximum allowed of -24mm. したがって, the slab’s stiffeness is adequate.

図 15. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

Images 15 そして 16 consist of the bending moment in each main direction. Taking the moment distribution and values, the software, SkyCiv, can obtain then the total steel reinforcement area.

図 16. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

Steel reinforcement areas:

図 17. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

図 18. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

図 19. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

図 20. One-way slabs in a small building example. (構造3D, SkyCiv Cloud Engineering).

結果比較

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

Moments and steel area	Exterior Negative Left	Exterior Positive	Exterior Negative Right	Interior Negative Left	Interior Positive	Interior Negative Right
\(A_{st, HandCalcs} {mm^2}\)	1498.366	1651.761	2247.55	2577.835	1651.761	1887.727
\(A_{st, S3D} {mm^2}\)	3889.375	1040.00	4196.145	4196.145	520.00	3175.00
\(\デルタ_{dif}\) (%)	61.475	37.04	46.44	38.566	68.52	40.544

 

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参考文献

1. B. Mosley, R. Hulse, J.H. Bungey , “Reinforced Concrete Design to Eurocode 2”, Seventh edition, Palgrave MacMillan.
2. Bazan Enrique & Meli Piralla, “Diseño Sísmico de Estructuras”, 1ed, LIMUSA.
3. ユーロコード 2: Design of concrete structures.