Calculation of the bent elements of the T-section.

In the first period of the use of reinforced concrete, the overlapping of structures was carried out in the form of solid plates. However, such constructions are not rational, since the height of the compressed zone is 1/10 ... 1/15 of their height, and the stretched zone of the cross-section of concrete is not taken into account in the calculation of strength and serves to accommodate the reinforcement.

If part of the concrete is removed from the stretched zone, leaving it only near the reinforcing bars, then a ribbed plate will be obtained. The plate carrying capacity will not change, and the consumption of concrete and the weight of the structure will be significantly reduced. Such elements, called Tauri, are widely used in the form of beams, decking, as a part of monolithic ribbed floors.

Experience shows that in areas remote from the edge of the voltage will be less. This is taken into account by the conditional decrease in the width of the overhangs entered into the calculation.

1) b'f = b + 12h'f npu h'f Calculation of the bent elements of the T-section. 0, lh

2) b'f = b + 6h'f at 0.05 h Calculation of the bent elements of the T-section. h'f <0.1h

3) b'f = b with h'f <0.05h

When calculating T-shaped beams, two cases are distinguished: 1. A compressed section area is within a shelf, or below the shelf

CASE 1 (x h f ')

Calculation of the bent elements of the T-section.

Case 1 occurs in sections with a developed shelf, when the external bending moment is less than or equal to the internal moment perceived by the compressed section shelf relative to the center of gravity of the reinforcement.

T-shaped section of this type is calculated as rectangular with dimensions b'f and h, since the area of the stretched concrete does not affect the bearing capacity. For the calculation, the formulas obtained for rectangular section with single reinforcement are used, in which "b" is replaced with "b'f".

l. Rb • b'f • x = Rs • As (1)

2. M Calculation of the bent elements of the T-section. Rb • b'f • x • (ho - x / 2)

3. M Calculation of the bent elements of the T-section. Rs • As • (ho - x / 2)

Selection of a T-section can be made, as for a rectangular section according to tabular data using the formulas:

4. A0 = M / Rb • b'f • ho2 Calculation of the bent elements of the T-section. Aomax.

5. As = M / Calculation of the bent elements of the T-section. • ho • Rs

CASE 2 (x> h'f)

The neutral axis extends beyond the shelf and intersects the edge, (x> h'f), i.e. the external design moment will be greater than the internal moment perceived only by the compressed shelf. Tavrovye sections of this type are found in the calculation of beam structures with a small width of the overhangs of the shelf.

To obtain the calculated formulas, the bending moment perceived by the section is divided into two points:

but). Mfl - perceived by the shelf overhangs and the corresponding Asfl reinforcement.

b). Mrib - perceived by the compressed concrete edges and the corresponding reinforcement As.rib.

Calculation of the bent elements of the T-section.

The concrete of the overhangs of the flange works for compression, and the tension is the corresponding part of the entire reinforcement As.rib. This section takes the moment:

6). Mfl = Nb, fl • z = Rb • (b'f-b) • h'f • (ho - h'f / 2) (6)

From the condition of equality "0" the sum of the projections of all forces on the element axis:

Nb, fl = Nsfl

Rb • (b'f-b) • h'f = Rs • As fl, from where

7). As fl = Rb • (b'f-b) • h'f / Rs (7)

This reinforcement constitutes only a part of the complete reinforcement As, now we will find the rest of the reinforcement corresponding to the reinforcement of the rib As.rib.

Calculation of the bent elements of the T-section.

The rib concrete is working in compression at the height of the compressed zone x, and in tension the rest of the entire tensile reinforcement As.rib.

The stress state of this scheme is fully consistent with the stress state of a rectangular section with a height of "h" and a width of "b" with a single reinforcement, which perceives the moment:

eight). Mrib = M- Mfl (8)

The area of the As.rib reinforcement is defined as for a rectangular section with a single reinforcement width “b”:

9). A0 = Mrib / Rb • b • ho2 Calculation of the bent elements of the T-section. Aomax. (9)

10. As rib = Mrib / Calculation of the bent elements of the T-section. • ho • Rs (100)

The total cross section of tensile reinforcement is determined by the formula:

eleven). As = As fl + As.rib (11)

Further, according to table 7 of the application, we find the diameter and the number of longitudinal bars required for the reinforcement of the beam.

When calculating T-sections, determine which design case (x Calculation of the bent elements of the T-section. h f ') or (x> h'f) we have, can be as follows:

find the boundary moment MX = h'f = Rb • b'f • h'f • (ho - h'f / 2) (12)

If the current moment from external forces M Calculation of the bent elements of the T-section. Mh = h'f, then we have the 1st case, if M> Mx = h'f, we have the 2nd case.

When calculating T-sections, it is most often necessary to solve the problem of determining the cross-sectional area of As reinforcement for given section sizes, material classes and the calculated bending moment "M"

Decision plan.

1. Set the estimated width of the shelf b'f.

2. According to the formula (12), calculate Мh = h'f and determine which calculation case the section relates to;

3. Find the cross-sectional area of the armature As:

3.1 at (x Calculation of the bent elements of the T-section. h f ') - as for a rectangular section of width b'f according to formulas (4) and (5)

3.2 when (x> h'f) - first determine Mfl by the formula (6) and As fl

according to the formula (7). Then Mrib is determined by the formula (8) and As rib by the formula (10), as for a rectangular beam with the width "b". The total cross section of tensile reinforcement is found by the formula (11).

Type 2 Task

If you want to establish the carrying capacity of a given section of the formula (1) determine the height of the compressed zone:

X = Rs • As / Rb • b'f

If (x Calculation of the bent elements of the T-section. hf '), my moment perceived by the section, is determined

according to formula 2:

M Calculation of the bent elements of the T-section. Rb • b'f • x • (ho - x / 2)

If x> h'f, then the calculation procedure is as follows:

1. Determine As, fl no by the formula (7) and Mfl by the formula 6.

2. Find As, rib from formula 11

3. Count Mrib, as for a rectangular section with

single valve:

first determine:

x = Rs • As, rib / Rb • b, and then:

4. Mrib = Rs • As, rib (ho - x / 2)

5. Determine the total moment perceived by the section (its carrying capacity):

M = Mfl + Mrib

EXAMPLE N1

Determine the area of As reinforcement in the element of the T-section according to the following data:

M = 42 kNm;

Armature class A111

Concrete B15;

b'f = 80cm; Yi = 1

h = 40 cm; h'f = 4 cm;

b = 14 cm

a = 3cm

Decision.

According to the tables of the application, we determine the calculated characteristics:

Rb = 8.5 • 1 = 8.5mPa; Rs = 365mPa

To determine the estimated width of the flange, we find the ratio h'f / h = 4/40 = 0.1– taken b'f = b + 12h'f = 14 + 12 • 4 = 62cm

1). Find the useful (working) section height ho = h-a = 40-3 = 37cm

2). We calculate the boundary moment Мх = h'f = Rb • b'f • h'f • (ho - h'f / 2) (100) =

8.5 • 62 • 4 • (37-4 / 2) • (100) = 7378000nsm = 73.8kNm> 42 kNm, therefore, we have the 1st calculation case.

We define As as for a rectangular section of width b'f

3) .A0 = M / Rb • b'f • ho2 Calculation of the bent elements of the T-section. Aomax

A0 = 42 • 105 / 8.5 • 62 • 372 • (100) = 0.058 <Aomax = 0.42 (tab.5 of the appendix)

By A0 we define Calculation of the bent elements of the T-section. = 0.976 (tab.6 of the application)

4). Find As = M / Calculation of the bent elements of the T-section. • ho • Rs (100) = 42 • 105 / 0.976 • 37 • 365 (100) = 3.26 cm2

According to table 7 of the application we take 3¯12АIII with As, fact = 3.39 cm2

EXAMPLE N2

According to the previous example, determine As if M = 86 kNm.

Decision.

Set the settlement case:

Since M> MX = h'f (86> 73.8), we have 2 cases (x> h'f)

1). Determine:

Mfl = Rb • (b'f-b) • h'f • (ho-h'f / 2) = 8.5 • (62-14) • 4 • (37-4 / 2) (100) = 5712000Nsm = 57.1kNm

2). We determine As fl = Rb • (b'f-b) • h'f / Rs = 57.1 • 105/365 • (37-4 / 2) (100) = 4.46 cm2

3). We calculate Mrib = M- Mfl = 86-57.1 = 29 kNm

4). We calculate A0 = Mrib / Rb • b • ho2 Calculation of the bent elements of the T-section. Aomax

A0 = 29 • 105 / 8.5 • 14 • 372 (100) = 0.172 <Aomax; Calculation of the bent elements of the T-section. = 0.9

5) .As rib = Mrib / Calculation of the bent elements of the T-section. • ho • Rs (100) = 29 • 105 / 0.9 • 37 • 365 (100) = 2.39 cm2

6). We calculate As = As fl + As.rib = 4.46 + 2.39 = 6.85 cm2

According to table 7 of the application we take 4¯16АIII with As, fact = 8.04 cm2

T-section reinforcement

Given: 2¯ 22 ASH L = 7.8m

Product brand	Item Position	Name	Qty	Mass 1det. (kg.)	Product weight
CR2	one	¯10АIII L = 1940	four	1.2	5.88
CR2	2	¯3ВР1 L = 980	20	0.054	5.88

Product brand	Item Position		Name		Qty	Mass 1det. (kg.)			Product weight
KP1	one		0 22 AIII / = 7780		one	23.2			34.15
	2		0 10A1 / = 7780		one	4.8
	3		08A1 / = 38O		41	0.15
Position		Designation		Name			Count	Note
				Assembly units
one		KPSK 2902 C-32 OSI KR-1		skeleton flat KR-1			2
2		KR-2		skeleton flat KR-2			2
				Details
3				0UA1 / = 18O			25	0.111
				Materials
four				Concrete B20			m3	1.17

Steel consumption statement

Brand of elements.	Reinforcement products								Total
	Armature class			Armature class			Armature class		82,84
B1	AIII			AI			BP1
	GOST 5781-82			GOST 5781-82			GOST 5781-82
	022	010	Total	010	08	Total	03	Total
	46.4	9.6	56	12.38	12.3	24.68	2.16	2.16

Questions for self-monitoring on the topic :

"The calculation of the bent elements of the T-section"

1. What is the advantage of T-shaped sections over rectangular?

2. How many cases are there for calculating T-sections?

3. How to determine the case of calculating the T-section?

4. What is the boundary moment?

5. What is the difference between reinforcing a T-section and a rectangular section?

6. What are the KR-2 frameworks for in beams of T-section?

7. How to determine the diameter and spacing of transverse reinforcement in the frame of the beam?

8. Name the types of reinforcement according to the nature of work in the frame of the beam.

Table 1

Design concrete resistance for limiting states of the first group of Rb and Rbt MPa depending on the class of concrete in compressive strength

Kind of resistance	Concrete					Concrete compressive strength class
						B12.5			B15					IN 20		B 25			VZO				B 35			B40
Compression oseeoe (prismatic strength) Rb	Heavy had a grained					7.5			8.5					11.5		14.5			17				19.5			22
Axial stretch Rbt	Heavy					0.6			0.75					0.9		1.05			1.2				1.3			1.4
table 2

			Calculated resistance fittings for group I of limit states Rs, MPa
			Stretched																		Compressed rsc
Type and grade of steel;			longitudinal, transverse (clamps and limb) for bending on an inclined section. Rs								transverse (clamps and bends) in the calculation of the transverse force Rsw

Hot rolled round grade A- I			225								175										225							: '
Hot rolled periodic
class profile: A-ll			280								225										280
A-III d-8 mm			355								285										355
A-III d- 10. ..40 mm \|			365								290										365
Wire reinforcing periodic profile class bp-i											.
d, mm:
3			375								270										375
four			370								265										365
five			360								260										360
table 3 The ratio between the diameters of the welded rods
The diameters of the rods in one direction d1 (mm)		3		four	five		6	eight		ten		12	14		sixteen		18	20		22		25		28	32		36		40
The smallest allowable diameters of rods in a different direction d2 (mm)		3		3	3		3	3		3		four	five		five		6	6		eight		eight		ten	ten		12		12

table 4

The modulus of elasticity of reinforcing steel Es

Type and grade of steel	The modulus of elasticity of reinforcement Es.MPa
Core steel AI and A-II	210,000
A-IIIb	200,000
A-IV, A-VI, AT-IIIC	190,000
Reinforcing Wire B-II, BP-II, BP-I	200,000 170,000
Reinforcement ropes K-7, K-19	180,000

table 5

Table: Values Calculation of the bent elements of the T-section. iAoMAh

Reinforcement class	Coefficient
		IN 20	VZO	B40	B50	B60
AI	at	0.65	0.59	0.55	-	-
AI	AOmax	0.48	0.42	0.4	-	-
AP	at	0.62	0.57	0.52	0.47	0.44
AP	AOmax	0.43	0.41	0.38	0.36	0.34
A-III with ¯ 6-8 mm	at	0.59	0.54	0.5	0.44	0.41
A-III with ¯ 6-8 mm	AOmax	0.42	0.39	0.37	0.34	0.33

table 6

Coefficient values Calculation of the bent elements of the T-section. AO,

		AO			AO
0.41	0.795	0.326	0.57	0.715	0.408
0.42	0.79	0.332	0.58	0.71	0.412
-	-	-	0.59	0.705	0.416
0.43	0.785	0.337	0.6	0.7	0.42
0.44	0.78	0.343
0.45	0.775	0.349	0.61	0.695	0.424
0.46	0.77	0.354	0.62	0,69	0.428
0.47	0.765	0.359	0.63	0.685	0.432
0.48	0.76	0.365	0.64	0.68	0.435
			0.65	0.675	0.439
0.49	0.755	0.37	0.66	0.672	0.442
0.5	0.75	0.375
0.51	0.745	0.38	0.67	0.665	0.446
0.52	0.74	0.385	0.68	0.66	0.449
0.53	0.735	0.39	0,69	0.655	0.452
0.54	0.73	0.394	0.7	0.65	0.455
0.55	0.725	0.399	_	-	-
0.56	0.72	0.403	-	-	-
0.01	0.995	0.01	0.21	0.895	0.188
0.02	0.99	0.02	0.22	0.89	0.196
0.03	0.985	0.03	0.23	0.885	0.203
0.04	0.98	0.039	0.24	0.88	0.211
0.05	0.975	0.048
0.06	0.97	0.058	0.25	0.875	0.219
			0.26	0.87	0,226
0.07	0.965	0.067	0.27	0.865	0.236
0.08	0.96	0.077	0.28	0.86	0.241
0.09	0.955	0.085	0.29	0.855	0.248
0.1	0.95	0.095	0.3	0.85	0.255
0.11	0.945	0,104	-	-	-
0.12	0.94	0.113	0.31	0.845	0,262
			0.32	0.84	0.269
0.13	0.935	0.121	0.33	0.835	0.275
0.14	0.93	0.13	0.34	0.83	0.282
0.15	0.925	0.13	0.35	0.825	0,289
0.16	0.92	0.147	0.36	0.82	0,295
0.17	0.915	0.155	-	-	-
0.18	0.91	0.164	0.37	0.815	0.301
0.19	0.905	0.172	0.39	0.805	0.314
0.2	0.9	0.18	0.4	0.8	0.32

table 7

Cross-sectional areas and mass of reinforcement bars

d	cross-sectional area (cm.kv.) when the number of rods									mass1 m	d
mm	one	2	3	four	five	6	7	eight	9	kg	mm
3	0.071	0.14	0.21	0.28	0.35	0.42	0.49	0.57	0.64	0.055	3
four	0.126	0.25	0.38	0.50	0.63	0.76	0.88	1.01	1.13	0.098	four
five	0.196	0.39	0.59	0.79	0.98	1.18	1.37	1.57	1.77	0.154	five
6	0.283	0.57	0.85	1.13	1.42	1.70	1.98	2.26	2.55	0.222	6
7	0.385	0.77	1.15	1.54	1.92	2.31	2.69	3.08	3.46	0.302	7
eight	0.503	1.01	1.51	2.01	2.51	3.02	3.52	4.02	4.53	0.395	eight
9	0.636	1.27	1.91	2.54	3.18	3.82	4.45	5.09	5.72	0.499	9
ten	0.785	1.57	2.36	3.14	3.93	4.71	5.50	6.28	7.07	0.617	ten
12	1,510	2.26	3.39	4.52	5.65	6.79	7.92	9.05	10.18	0.888	12
14	1.539	3.08	4.62	6.16	7,69	9.23	10.77	12.31	13.85	1,208	14
sixteen	2,011	4.02	6.03	8.04	10.05	12.06	14.07	16.08	18.10	1.578	sixteen
18	2.545	5.09	7.63	10.18	12.72	15.27	17.81	20.36	22.90	1,993	18
20	3.142	6.28	9.41	12.56	15.71	18.85	21.99	25.14	28.28	2.466	20
22	3,801	7.60	11.40	15.20	19.00	22.81	26.61	30.41	34.21	2,994	22
25	4,909	9.82	14.73	19.63	24.54	29.45	34.36	39.27	44.18	3,853	25
28	6.158	12.32	18.47	24.63	30.79	36.95	43.10	49.26	55.42	4.83	28
32	8,043	16.08	24.13	32.17	40.21	48.25	56.30	64.34	72.38	6.313	32
36	10.18	20.36	30,54	40.72	50.9	61.08	71.26	81.44	91.62	7.99	36
40	12.56	25.12	37.68	50.24	62,8	75.36	87.92	100.48	113.04	9.87	40

CALCULATION OF TARPSETS (TYPE 2) A s =?

No. Var	M (kNm)	in (cm)	′ f	h′f	concrete	armature	Twi	a (cm)	h (cm)	l (m)
one	120	nineteen	60	ten	B15	AII	0.8	3.0	40	6.0
2	123	18	61	eleven	IN 20	AIII	0.85	3.2	41	6.5
3	136	17	62	12	B25	AII	0.9	3.3	42	6.6
four	130	22	63	14	B30	AIII	0.85	3.4	43	6.7
five	124	24	64	15	B15	AII	0.9	3.5	44	6.8
6	132	25	65	ten	IN 20	AIII	0.8	3.6	45	6.9
7	118	26	66	eleven	B25	AII	0.85	3.7	46	7.0
eight	129	20	67	12	B30	AIII	0.9	3.8	47	7.1
9	137	27	68	13	B15	AII	0.8	3.9	48	7.2
ten	119	28	69	14	IN 20	AIII	0.85	4.0	49	7.3
eleven	121	29	70	15	B25	AII	0.9	3.0	50	7.4
12	127	thirty	71	ten	B30	AIII	0.8	3.2	40	7.5
13	135	15	72	eleven	B15	AII	0.85	3.3	41	7.6
14	140	sixteen	73	12	IN 20	AIII	0.9	3.4	42	7.7
15	141	17	60	13	B25	AII	0.8	3.5	43	7.8
sixteen	142	18	61	14	B30	AIII	1.0	3.6	44	7.9
17	143	nineteen	62	15	B15	AII	1.0	3.7	45	8.0
18	144	20	63	ten	IN 20	AIII	1.0	3.8	46	8.1
nineteen	145	21	64	eleven	B25	AII	1.0	3.9	47	8.2
20	149	22	65	12	B30	AIII	1.0	4.0	48	8.3
21	125	23	66	13	B15	AII	1.0	3.0	49	8.4
22	126	24	67	14	B25	AIII	1.0	3.2	50	8.5
23	129	25	68	15	B30	AII	1.0	3.3	51	8.6
24	138	26	69	12	B15	AIII	1.0	3.4	40	8.7
25	140	27	70	14	IN 20	AII	1.0	3.5	52	8.8

CALCULATION OF TARED SECTIONS. Check M <Msech

No. Var	M (kNm)	in (cm)	′ f	h′f	concrete	armature	Twi	a (cm)	h (cm)	l (m)	As
one	120	nineteen	60	ten	B15	AII	0.8	3.0	40	6.0	14.3
2	123	18	61	eleven	IN 20	AIII	0.85	3.2	41	6.5	11.5
3	136	17	62	12	B25	AII	0.9	3.3	42	6.6	13.1
four	130	22	63	14	B30	AIII	0.85	3.4	43	6.7	15.4
five	124	24	64	15	B15	AII	0.9	3.5	44	6.8	15.2
6	132	25	65	ten	IN 20	AIII	0.8	3.6	45	6.9	12.9
7	118	26	66	eleven	B25	AII	0.85	3.7	46	7.0	11.6
eight	129	20	67	12	B30	AIII	0.9	3.8	47	7.1	16,1
9	137	27	68	13	B15	AII	0.8	3.9	48	7.2	15.3
ten	119	28	69	14	IN 20	AIII	0.85	4.0	49	7.3	17.2
eleven	121	29	70	15	B25	AII	0.9	3.0	50	7.4	14.6
12	127	thirty	71	ten	B30	AIII	0.8	3.2	40	7.5	15.3
13	135	15	72	eleven	B15	AII	0.85	3.3	41	7.6	16,1
14	140	sixteen	73	12	IN 20	AIII	0.9	3.4	42	7.7	12.8
15	141	17	60	13	B25	AII	0.8	3.5	43	7.8	13.0
sixteen	142	18	61	14	B30	AIII	1.0	3.6	44	7.9	17.0
17	143	nineteen	62	15	B15	AII	1.0	3.7	45	8.0	10.9
18	144	20	63	ten	IN 20	AIII	1.0	3.8	46	8.1	15.1
nineteen	145	21	64	eleven	B25	AII	1.0	3.9	47	8.2	16.3
20	149	22	65	12	B30	AIII	1.0	4.0	48	8.3	14.6
21	125	23	66	13	B15	AII	1.0	3.0	49	8.4	15.7
22	126	24	67	14	B25	AIII	1.0	3.2	50	8.5	18.2
23	129	25	68	15	B30	AII	1.0	3.3	51	8.6	18.6
24	138	26	69	12	B15	AIII	1.0	3.4	40	8.7	11.3
25	140	27	70	14	IN 20	AII	1.0	3.5	52	8.8	15.0

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Calculation of the bent elements of the T-section.

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