Effects of Staircase on the Seismic Behavior of RC Moment Frame Buildings

In order to evaluate the impact of staircase on reduction of period and drift ratio of structure, southern building with a width of 6 meter is selected and analyzed under three conditions including bare frame, frame with staircase slabs (Fig. 5) and frame with stringer beams. Period of first three modes is presented in Table 2 and drift ratio of two direction of X and Y in Fig. 6, 7. The results show that in the model with staircase slabs, period is reduced by about 16% in X direction and by 20% in Y direction. But, in the model with stringer beams, the reduction of period is negligible in X direction and it is about 16% in Y direction. In the model with staircase slabs, depending on the stories the drift ratio is reduced from 37% to 57% in both directions. But, in the model with stringer beams, the reduction of drift ratio is not considerable in X direction and it is about 23% to 43% in Y direction. In fact, reinforced concrete slab performs as a K-shaped bracing in longitudinal direction and as an inclined shear wall in transverse direction of staircase and therefore makes fundamental changes in stiffness, period and lateral displacement in both directions. But, stringer beam performs as bracing only in longitudinal direction and its effects in transverse direction is negligible.

Figure 5.

Figure 6.

Figure 7.

An important part of staircase effects on seismic behavior of structures is related to the geometry of architectural and structural plan. Three main variables including the number of structural bays, dimension of staircase frame and location of staircase in building can be investigated. Since the span of staircase frame is usually shorter than other frames, so stiffness concentration is formed because of shorter length of this beam. This is very important in one-bay structures and especially in moment frames.

When the staircase is connected to the structure, it performs as a structural element. Due to inclined form, it acts as a diagonal strut and half-turn landing staircase forms K-shaped bracing (Fig. 8). Because the stiffness of staircase is more than moment frames, so it could absorb more forces [4, 8] and stiffness concentration in a part of the structure will occur. Since these elements are not considered in structural modeling as usual, so their effects on the structure are not analyzed.

Figure 8.

According to fire standards, walls around the staircase should be fire stopping, so they are constructed as 20 cm thick walls. On the other hand, there are structural frames around the staircase, so these walls act as infill walls against lateral forces and lead to concentrating the stiffness in a part of the structure. The stairs, which were usually placed in an eccentric position, often emphasizes the irregular stiffness distribution provided by the position of the infill panels [16].

As it has been attempted to study detrimental effects of staircase on the structure, so simulation is based on macro model. In this method infill walls are modeled with an equivalent compression diagonal strut. Accordingly, the assumptions of a research paper about infill walls with similar material are used. Modulus of elasticity has been assumed 800 times of compressive strength of wall, the effective width of equivalent strut, 0.2 times of wall diameter and thickness of strut, the same as wall. The compressive strength of wall with hollow clay blocks and cement mortar is assumed 3.8 MPa [17]. For considering the effect of opening on the strength and stiffness of walls, Equation 1 recommended by the New Zealand Society for Earthquake Engineering, has been used [18]. (1) λ Opening = 1 - 1.5 L Opening L inf

According to Iranian seismic code of practice [19], lateral load resisting elements shall be configured in a manner that the torsion resulting from earthquake loading is minimized. For this purpose it is recommended that the eccentricity between the center of mass and center of stiffness, at each floor level, be less than 5% of the building dimension in that level and in the direction under consideration [19]. In this standard there is no mandatory for the maximum allowable amount of eccentricity. In Australian standard 1170.4–1993, when the distance between center of mass and center of rigidity is in excess of 10% of the structure dimension perpendicular to the direction of earthquake force, torsional irregularity shall be considered but this provision is omitted in new version of 2007 [20, 21]. In Nepal National Building Code about mandatory rules for reinforced concrete buildings with masonry infill walls, the distance between center of mass and center of rigidity including the effects of infill wall shall be less than 10% of building dimension at the same direction [22].

There is another provision in seismic codes to control torsion. According to this, in each story the maximum drift, including accidental torsion, at one end of the structure shall not exceed 20% of the average of the story drift of the two ends of the structure [19, 23, 24, 25].

As shown in Fig. 1, three groups have been defined to study torsional effects of staircase on building structure. In the following, characteristics of each group, outputs of analysis and results are presented. To evaluate torsional effects of each model, mode shapes and two above criteria are used as the main tools. Due to the large amount of outputs of the second criterion, only the results of one story will be presented.

In moment frame, due to shorter span of staircase frame, its stiffness is more than other spans, so it absorbs more seismic forces [7]. In this group, to study architectural and structural geometry, two variables including dimension of staircase frame and its location are investigated. To study the location of staircase, two buildings with a width of 10 meter and two-bay structure are selected. In southern building, the staircase is located at the corner and in the northern building it is located at the middle of length. In each of these two plans, three types of structural geometry are selected. In the first type, stairs, elevator and entrance space of each floor are designed in a structural bay, so the entire structure has a regular geometry. In the second type, only the frame dimension of staircase landing is determined based on the staircase dimension. In the third type, dimension of both frames of staircase is determined based on the staircase dimension. The entire structure of three types is in the form of two-bay (Fig. 9, 10). Models of this group are bare frame without inclined slab of staircase. In this group, six models are analyzed.

Figure 9.

Figure 10.

According to the results, mode shapes of three types of structure in southern building with staircase at the corner are the same and torsion occurs at the third mode of modal analysis. But, in two cases of one shorter frame and two shorter frames, eccentricity along Y axis is greater than 10% of building dimension and the ratios of the maximum relative story drift to the average relative story drift are greater than 1.2 (Table 3 to 5). In the northern building with staircase at the middle of the length, due to proximity of shorter frames to the center of mass, in all three cases torsion occurs at the third mode of modal analysis and eccentricity is less than 10% of building dimension. Only in a few cases the ratios of the maximum relative story drift to the average relative story drift are greater than 1.2. Therefore, torsional irregularity of this building in all structural geometry is very low (Table 6 to 8). Based on the results, when the staircase is located at the corner, even in the absent of staircase inclined slabs and infill walls, structural geometry is prone to torsion alone. Therefore, in this case, the most appropriate geometry of the structure is to design stair, elevator and entrance space in one structural bay and use a uniform geometry in the entire structure. When the staircase is located at the middle of the length, structural geometry has no significant impact and all three types are acceptable. Although using uniform geometry in the structure has its advantages.

Since the staircase has a significant effect on the onebay structure [4], in this group to study architectural and structural geometry, worst case of two variable including number of structural bays and location of staircase has been investigated. Accordingly, southern building with a width of 6 meter and staircase at the corner which has one structural bay with the exception of the staircase location is selected. Models of this group are bare frame without inclined slabs of staircase. This group includes four types of structure; main moment frame, hinge connections on the side of staircase, extra column on the opposite side of the staircase, columns with larger cross section on the opposite side of the staircase (Fig. 11).

Figure 11.

The results show that torsion in all four types of structure is at the third mode. The distance between center of mass and center of rigidity, only in main structure is greater than 10% of building dimension and the ratios of the maximum relative story drift to the average relative story drift are greater than 1.2 in the main structure and in some cases in the fourth one (Table 9 to 11). Therefore, in one-bay structure with the staircase at the corner, using rigid connection for all structural elements is not allowed. To prevent torsional effects, one of two solutions including hinge connections on the side of staircase or extra column on the opposite side of the staircase will be effective. Using columns with larger cross section on the opposite side of the staircase is not as effective as two other solutions and economically cost effective.

This is the main group regarding effective variables on stiffness distribution. In this group number of structural bays and location of staircase as the subset of the main variable of architectural and structural geometry and two other variables including inclined slab of staircase and infills around the staircase are studied. All three selected building in the form of one-bay and two-bay structure with staircase at the corner and middle of the length are analyzed. In the buildings with a width of 10 meter, the case in which all frames have the same dimension and in the southern building with a width of 6 meter, the case in which the structure has hinge connection on the side of staircase are selected. To study inclined slab of staircase and infills around the staircase, four cases including bare frame, bare frame with inclined slabs of staircase, infilled frame and infilled frame with inclined slabs of staircase are modeled. There are 12 models in this group.

According to the results of the southern building with a width of 6 meter, in the cases with infills, torsion occurs at the second mode. The distance between center of mass and center of rigidity, in the cases with inclined slabs of staircase is greater than 10% and also the ratios of the maximum relative story drift to the average relative story drift are greater than 1.2. In these models due to plan geometry, infills balance the concentration of stiffness caused by inclined slabs of staircase (Table 12 to 14). In the southern building with a width of 10 meter, torsion is at the second mode in cases with infilled frames. The distance between center of mass and center of rigidity, in all cases with the exception of bare frame is greater than 10% and the ratios of the maximum relative story drift to the average relative story drift are greater than 1.2 in the models with inclined slabs of staircase (Table 15 to 17). In the northern building with a width of 10 meter, for all models torsion occurs at the third mode and eccentricity is less than 10%. Only in the models with infills the ratios of the maximum relative story drift to the average relative story drift for seismic force along X direction with negative eccentricity are greater than 1.2. In this case where the staircase is located at the middle of length, torsional effects are less than others (Table 18 to 20).

To study impact of staircase on lateral force distribution, southern building with a width of 6 meter is selected. Diagrams of shear forces and bending moments under seismic force along Y direction are drawn in both cases with and without inclined slab of staircase (Fig. 12, 13). It has been observed that with incorporation of staircase slab, in columns touching landing beam, shear forces increase about 1.2–3 times and bending moments about 1.1–1.6 times than the model without staircase slab. This is over the capacity of the columns in many parts of structure and will lead to failure. In other columns shear forces decrease about 0.03–0.7 times and bending moments about 0.1–0.7 times than the model without staircase slab. Diagrams of shear forces and bending moments under seismic force along X direction are drawn in both cases with and without inclined slab of staircase (Fig. 14, 15). Because an inclined slab of staircase performs as shear wall in transverse direction, internal forces and moments in all structural elements will reduce. It should be noted that if the staircase is constructed with stringer beams, its effects in longitudinal direction is similar to the staircase constructed with inclined slab and in transverse direction is similar to bare frame and has no significant effects.

Figure 12.

Figure 13.

Figure 14.

Figure 15.

The results of models in group 1 show that RC slab of staircase performs as a K-shaped bracing in longitudinal direction and as an inclined shear wall in transverse direction, so in both directions structural stiffness increases, period and lateral displacement of structure decrease, but staircase constructed by stringer beam only acts as a bracing in longitudinal direction and its effect in transverse direction can be neglected.

By isolating the staircase from master structure, only location and dimension of staircase and arrangement of infill walls shall be studied. Based on the analyses of group 2, when the staircase is located at the corner, even the stair and infill walls are isolated from the structure, structural geometry is prone to torsion alone. Therefore, in this case the most appropriate geometry of the structure is to design stair, elevator and entrance space in one structural bay and use a uniform geometry in the entire structure. When the staircase is located at the middle of the length, structural geometry has no significant impact and all three structural geometries including uniform frames in the entire structure, shorter frame on the landing and shorter frames on both sides of staircase are acceptable. Although using uniform geometry in the structure has its advantages.

Based on the results of group 3, to prevent torsional effects in one-bay structure with the staircase at the corner, hinge connections on the side of staircase or extra column on the opposite side of the staircase will be effective.

The results of models in group 4 show that stiffness caused by small span, inclined RC slabs and perimeter infill walls of staircase, based on the location of staircase and the number of structural bays could change mode shape and lead to torsion due to changing the center of stiffness. Based on the geometry of plan and arrangements of infill walls, sometimes infills balance the concentration of stiffness caused by inclined slabs of staircase and sometimes intensify it.

Based on the analyses of group 5, staircase changes force distribution fundamentally. In longitudinal direction shear force and bending moment of columns adjacent to landing beams increase enormously and short column occurs. Instead, forces of other members decrease. But in transverse direction, internal forces and moments in all structural elements reduce because an inclined slab of staircase performs as shear wall.