叠加的通用计算方法抗弯强度的钢骨混凝土组合梁

FOR CALCULATING BENDING STRENGTH

OF CONCRETE ENCASED COMPOSITE BEAM

Li Shaoquan and Sha Zhenping

Department of Civil Engineering, Xiamen University, Xiamen, P. R. China

ABSTRACT

Based on the lower limit principle of plastic theory, the axial force can be arbitrarily given to the steel and the reinforced concrete according to balance condition, and the bending strength of relative parts can be calculated. The maximum sum value of bending strength on the steel and the reinforced concrete is the concrete encased composite beam bending strength. In this paper, the formulas of the universal superposition method are derived. As a result, the universal superposition is no longer in need of pilot calculation. It is shown that the results are in good agreement with the accuracy method. The accuracy method is based on the plane section assumption. Therefore, the bending strength of the concrete encased composite beam can be calculated as reinforced concrete beam. Besides, the suggested method is easier than the accuracy method.

Keywords: concrete encased composite beam, universal superposition method, bending strength

INTRODUCTION

In structural engineering, common sections of concrete encased composite beams are shown in Figure 1. Among the bending design approaches developed for concrete encased composite beams, the design provision suggested by the American building specifications and by the Japanese building codes are the two widely employed ones. Based on the plane section assumption, the bending strength of concrete encased composite beam can be calculated as reinforced concrete. This method, referred as the accuracy method in this communication, is used in the American Concrete Institute (ACI) (1999) building code. However, the accuracy method is quite complex to calculate. Another method, based onthe lower limit principle of plastic theory, is the universal superposition method. The universal superposition method is precise and in good agreement with the accuracy method, but it can be too complex for design purposes. The simple superposition method is adopted by the Architectural

N rc+N ss=0 (1a)

M=M rc+M ss + N rc e0(1b) rc ss

INTERACTION CURVES OF AXIAL FORCE VERSUS MOMENT

Interaction curves of axial force versus moment for reinforcement concrete are plotted in Figure 3. The eccentric compression of reinforcement concrete can be divided into compression failure and tension failure. The eccentric tension of reinforced concrete can be divided into whole section in tension and partial section in tension. Within the limits of economic ratio, the compression failure or the whole section in tension does not occur for reinforced concrete. Hence, only the tension failure of eccentric compression and the partial section in tension of eccentric tension are discussed for reinforced concrete.

Interaction curves of axial force versus moment for steel are plotted in Figure 4, they are determined as follows ：

1ss y0

ss

ss

p0ss

=+N N M M

(2a) ss ss ss y0f A N = (2b) ss pss ss p0f W M = (2c)

where ss y0

N = axial load-bearing capacity of steel; ss

p0M = pure bending capacity of steel; A ss = sectional area of steel; W pss = plastic section modulus of steel; f ss = specified yield strength of steel.

Thus,

ss

pss ss

ss pss ss A W N f W M −= (2d)

when N ss ≤ 0 (steel in tension), Equation 2d can be written as

ss

pss rc

ss pss ss A W N f W M −= (3a)

when N ss ＞ 0 (steel in compression), Equation 2d can be written as

ss

pss rc

ss pss ss A W N f W M += (3b)

Steel in Tension (N ss ≤ 0)

According to the balance Equation 1a, the reinforced concrete means eccentric compression (N rc ≥ 0) as shown in Fig. 5. Taking moments about the sectional center,

(

⎞

⎛+′′−−+⎟⎠

⎞

⎜⎝⎛′−′′+⎟⎠⎞⎜⎝⎛−=f A f A N N a h f A a h f A M

rc cr y s y s rc

22

where A s and A s and compression reinforcement respectively; f y and f y respectively; f c = specified concrete strength; h and b moment M becomes:

ss pss

rc

ss pss A W N f W M −=⎟⎠

⎞⎜⎝⎛′−′′+⎟⎠⎞⎜⎝⎛−+a h f A a h f A 22y s y s ()

0rc c y s y s rc

y s y s rc 22e N b f f A f A N h f A f A N +⎟⎟⎠

⎞

⎜⎜⎝

⎛+′′−−+′′−+ (5)

In order to determine the maximum bending capacity of the concrete encased composite beam M , we put dM / dN rc = 0 to find axial force of the reinforced concrete N rc .

y s y s c ss pss 0rc 2f A f A b f A W e h

N ′′+−⎟⎟

⎠⎞⎜⎜⎝

⎛−+= (6a)

Putting α = A s f y / f c bh 0 which is reinforcement density; β = W pss / h 0A ss which is relative depth of steel; λ = A s 'f y ' / A s f y which is strength ratio of compression reinforcement and tensile reinforcement. And approximately taking (h 0-a )/h 0 = 0.9, Equation 6a can be written as

()⎥⎦

⎤⎢⎣⎡−−⎟⎟⎠⎞⎜⎜⎝⎛−+=λαβ155.000

0c rc h e bh f N (6b)

Steel in Compression (N ss > 0)

According to the balance Equation 1a, the reinforced concrete is under eccentric tension (N rc < 0). The stress distribution of eccentric tension with partial section in tension is the same as that of eccentric compression of tension failure. Hence, using the same method above, the axial force of the reinforced concrete N rc can be written as

()⎥⎦

⎤⎢⎣⎡−−⎟⎟⎠⎞⎜⎜⎝⎛++=λαβ155.000

0c rc h e bh f N (7)

T-BEAM

()⎥⎦⎤⎢⎣⎡−−′

⎟⎠⎞⎜⎝⎛−′+⎟⎟⎠⎞⎜⎜⎝⎛−+=λαβ1155.00f f 000c rc h h b b h e bh f N (8) ()⎥⎦

⎤⎢⎣⎡−−′

⎟⎠⎞⎜⎝⎛−′+⎟⎟

⎠⎞⎜⎜⎝⎛++=λαβ1155.00f f 000c rc

h h b b h e bh f N

(9)

where b f ' and h f ' = width and depth of flange.

For steel in tension (N ss ≤0), N rc can be calculated using Equation 8, hence, N rc ≥0. Conversely, for steel in compression (N ss > 0), N rc can be calculated using Equation 9, hence, N rc < 0.

DISCUSSIONS AND CONCLUSIONS

The comparisons of the accuracy method and the universal superposition method are shown in Fig.7 by varying the relative depth of steel β, the reinforcement density α, the strength ratio of compression

need of pilot calculation. The practical design method described in the paper is easier to use than the accuracy method.

References

American Concrete Institute (ACI). (1999). “Buildings code requirements for structural concrete.” ACI 318-99, Detroit.

Architectural Institute of Japan (AIJ). (1987). Standards for structural calculation of steel reinforced concrete structures, Tokyo.

Hodge P. G. (1959). Plastic analysis of structures. McGraw-Hill, New York. Sherif

El-Tawil, Gregory G. Deierlein (1999). “Strength and ductility of concrete encased composite columns.” J. Struct. Engrg., ASCE, 125(9), 1009 – 1019.

7